estimation of the eigenvalues and the smallest singular value of matrices abstract the purpose of this paper is to discuss the estimation of eigenvalues and the smallest singular value of matrices firstly we prove that all the eigenvalues of arbitrarily complex matrix are located in one closed disk around trm n of radius n 1 n m f 2 trm 2 n max 1 k n 1 b k n k f c n k k f 2 secondly we present a lower bound for the smallest singular value of matrices finally we give some numerical examples which will show the effectiveness of our results

the determinants of gcd matrices abstract let s x1 x2 xn be an ordered set of distinct positive integers and s the gcd matrix defined on s the value of the determinant of s is obtained it is shown that det s x1 x2 xn iff s is factor closed

the role of network topology on extremism propagation with the relative agreement opinion dynamics abstract in deffuant et al j artif soc soc simulation 5 2002 4 we proposed a simple model of opinion dynamics which we used to simulate the influence of extremists in a population simulations were run without any specific interaction structure and varying the simulation parameters we observed different attractors such as predominance of centrism or of extremism we even observed in certain conditions that the whole population drifts to one extreme of the opinion even if initially there are an equal number of extremists at each extreme of the opinion axis in the present paper we study the influence of the social networks on the presence of such a dynamical behavior in particular we use small world networks with variable connectivity and randomness of the connections we find that the drift to a single extreme appears only beyond a critical level of connectivity which decreases when the randomness increases

spectral properties of the laplacian of multiplex networks omez et al phys rev lett 110 028701 2013 some of the authors proposed a framework for the study of diffusion processes in such networks here we extend the previous framework to deal with general configurations in several layers of networks and analyze the behavior of the spectrum of the laplacian of the full multiplex we derive an interesting decoupling of the problem that allow us to unravel the role played by the interconnections of the multiplex in the dynamical processes on top of them capitalizing on this decoupling we perform an asymptotic analysis that allow us to derive analytical expressions for the full spectrum of eigenvalues this spectrum is used to gain insight into physical phenomena on top of multiplex specifically diffusion processes and synchronizability

higher dimensional gcd matrices abstract we evaluate the higher dimensional determinants of the greatest common divisor matrix defined on a gcd closed set and on an arbitrary set of distinct positive integers

a lower bound for the smallest singular value abstract a lower bound for the smallest singular value of a n x n is given in terms of the determinant and the 2 norm of the columns and the rows of a examples are given to show that this bound performs well for a large class of matrices

divisibility of determinants of power gcd matrices and power lcm matrices on finitely many quasi coprime divisor chains abstract let a n 1 be integers and s x1 xn be a set of n distinct positive integers the matrix having the ath power xi xj a of the greatest common divisor of xi and xj as its i j entry is called ath power greatest common divisor gcd matrix defined on s denoted by sa similarly we can define the ath power lcm matrix sa we say that the set s consists of finitely many quasi coprime divisor chains if we can partition s as s s1 sk where k 1 is an integer and all si 1 i k are divisor chains such that max si max sj gcd s for 1 i j k in this paper we first obtain formulae of determinants of power gcd matrices sa and power lcm matrices sa on the set s consisting of finitely many quasi coprime divisor chains with gcd s s using these results we then show that det sa det sb det sa det sb and det sa det sb if a b and s consists of finitely many quasi coprime divisor chains with gcd s s but such factorizations fail to be true if such divisor chains are not quasi coprime

an operator theoretic formulation of asynchronous exponential growth a strongly continuous semigroup of bounded linear operators t t t o in the banach space x has asynchronous exponential growth with intrinsic growth constant a0 provided that there is a nonzero finite rank operator p0 in x such that limr ooe x tt t po necessary and sufficient cvonditions are established for t t t o to have asynchronous exponential growth applications are made to a maturity time model of cell population growth and a transition probability model of cell population growth

rational group decision making a random field ising model at t 0 abstract a modified version of a finite random field ising ferromagnetic model in an external magnetic field at zero temperature is presented to describe group decision making fields may have a non zero average a postulate of minimum inter individual conflicts is assumed interactions then produce a group polarization along one very choice which is however randomly selected a small external social pressure is shown to have a drastic effect on the polarization individual bias related to personal backgrounds cultural values and past experiences are introduced via quenched local competing fields they are shown to be instrumental in generating a larger spectrum of collective new choices beyond initial ones in particular compromise is found to results from the existence of individual competing bias conflict is shown to weaken group polarization the model yields new psychosociological insights about consensus and compromise in groups

the bifurcating autoregression model in cell lineage studies a model for cell lineage data is presented and analysed the model is an extension of the classical first order autoregression used in time series studies to bifurcating data trees of general size and shape maximum likelihood theory is developed and compared with an extensive simulation study some properties of moment estimators are also presented

continuous opinion model in small world directed networks abstract in the compromise model of continuous opinions proposed by deffuant et al the states of two agents in a network can start to converge if they are neighbors and if their opinions are sufficiently close to each other below a given threshold of tolerance in directed networks if agent i is a neighbor of agent j j need not be a neighbor of i in watts strogatz networks we performed simulations to find the averaged number of final opinions f and their distribution as a function of and of the network structural disorder in directed networks f exhibits a rich structure being larger than in undirected networks for higher values of and smaller for lower values of

structural measures for multiplex networks many real world complex systems consist of a set of elementary units connected by relationships of different kinds all such systems are better described in terms of multiplex networks where the links at each layer represent a different type of interaction between the same set of nodes rather than in terms of single layer networks in this paper we present a general framework to describe and study multiplex networks whose links are either unweighted or weighted in particular we propose a series of measures to characterize the multiplexicity of the systems in terms of i basic node and link properties such as the node degree and the edge overlap and reinforcement ii local properties such as the clustering coefficient and the transitivity and iii global properties related to the navigability of the multiplex across the different layers the measures we introduce are validated on a genuinely multiplex data set of indonesian terrorists where information among 78 individuals are recorded with respect to mutual trust common operations exchanged communications and business relationships

robust estimation in generalized linear mixed models generalized linear mixed models glmms are widely used to analyse non normal response data with extra variation but non robust estimators are still routinely used we propose robust methods for maximum quasi likelihood and residual maximum quasi likelihood estimation to limit the influence of outlying observations in glmms the estimation procedure parallels the development of robust estimation methods in linear mixed models but with adjustments in the dependent variable and the variance component the methods proposed are applied to three data sets and a comparison is made with the nonparametric maximum likelihood approach when applied to a set of epileptic seizure data the methods proposed have the desired effect of limiting the influence of outlying observations on the parameter estimates simulation shows that one of the residual maximum quasi likelihood proposals has a smaller bias than those of the other estimation methods we further discuss the equivalence of two glmm formulations when the response variable follows an exponential family their extensions to robust glmms and their comparative advantages in modelling are described some possible modifications of the robust glmm estimation methods are given to provide further flexibility for applying the method

a note on a lower bound for the smallest singular value abstract we give a short proof of a lower bound for the smallest singular value mentioned by hong and pan one of our results improves their theorem we obtain another lower bound that is sharp an example shows that these results can be quite accurate

dynamics of bounded confidence opinion in heterogeneous social networks concord against partial antagonism abstract bounded confidence models of opinion dynamics in social networks have been actively studied in recent years in particular opinion formation and extremism propagation along with other aspects of social dynamics in this work after an analysis of limitations of the deffuant weisbuch dw bounded confidence relative agreement model we propose the mixed model that takes into account two psychological types of individuals concord agents c agents are friendly people they interact in a way that their opinions always get closer agents of the other psychological type show partial antagonism in their interaction pa agents opinion dynamics in heterogeneous social groups consisting of agents of the two types was studied on different social networks erdos renyi random graphs small world networks and complete graphs limit cases of the mixed model pure c and pa societies were also studied we found that group opinion formation is qualitatively almost independent of the topology of networks used in this work opinion fragmentation polarization and consensus are observed in the mixed model at different proportions of pa and c agents depending on the value of initial opinion tolerance of agents as for the opinion formation and arising of dissidents the opinion dynamics of the c agents society was found to be similar to that of the dw model except for the rate of opinion convergence nevertheless mixed societies showed dynamics and bifurcation patterns notably different to those of the dw model the influence of biased initial conditions over opinion formation in heterogeneous social groups was also studied versus the initial value of opinion uncertainty varying the proportion of the pa to c agents bifurcation diagrams showed an impressive evolution of collective opinion in particular radical changes of left to right consensus or vice versa at an opinion uncertainty value equal to 0 7 in the model with the pa c mixture of population near 50 50

nonsingularity of matrices associated with classes of arithmetical functions let s x1 xn be a set of n distinct positive integers let f be an arithmetical function let f xi xj denote the n n matrix having f evaluated at the greatest common divisor xi xj of xi and xj as its i j entry and f xi xj denote the n n matrix having f evaluated at the least common multiple xi xj of xi and xj as its i j entry the set s is said to be gcd closed if xi xj s for all 1 i j n for an integer x let x denote the number of distinct prime factors of x in this paper by using the concept of greatest type divisor introduced by s hong in adv math china 25 1996 566 568 j algebra 218 1999 216 228 we obtain a new reduced formula for detf xi xj if s is gcd closed then we show that if s x1 xn is a gcd closed set satisfying maxx s x 2 and if f is a strictly increasing respectively decreasing completely multiplicative function or if f is a strictly decreasing respectively increasing completely multiplicative function satisfying 0 f p 1p respectively f p p for any prime p then the matrix f xi xj respectively f xi xj defined on s is nonsingular as a corollary we show the following interesting result the lcm matrix xi xj defined on a gcd closed set is nonsingular if maxx s x 2

an approach to the analysis of repeated measurements summary a linear model for repeated measurements is proposed in which the correlation structure within each time sequence of measurements includes parameters for measurement error variation between experimental units and serial correlation within units an approach to data analysis is presented which involves preliminary analysis by ordinary least squares use of the empirical semi variogram of residuals to suggest a suitable correlation structure and formal inference using likelihood based methods applications to two biological data sets are described

on a conjecture of ilmonen haukkanen and merikoski concerning the smallest eigenvalues of certain gcd related matrices abstract let k n be the set of all n n lower triangular 0 1 matrices with each diagonal element equal to 1 l n y y t y k n and let c n be the minimum of the smallest eigenvalue of y y t as y goes through k n the ilmonen haukkanen merikoski conjecture the ihm conjecture states that c n is equal to the smallest eigenvalue of y 0 y 0 t where y 0 k n with y 0 i j 1 1 i j 2 for i j in this paper we present a proof of this conjecture in our proof we use an inequality for spectral radii of nonnegative matrices

on meet and join matrices associated with incidence functions we study recently meet matrices on meet semilattices as an abstract generalization of greatest common divisor gcd matrices analogously in this paper we consider join matrices on lattices as an abstract generalization of least common multiple lcm matrices a formula for the determinant of join matrices on join closed sets bounds for the determinant of join matrices on all sets and a formula for the inverse of join matrices on join closed sets are given the concept of a semi multiplicative function gives us formulae for meet matrices on join closed sets and join matrices on meet closed sets finally we show what new the study of meet and join matrices contributes to the usual gcd and lcm matrices

divisibility properties of power gcd matrices and power lcm matrices abstract let a b and n be positive integers and the set s x 1 x n of n distinct positive integers be a divisor chain i e there exists a permutation on 1 n such that x 1 x n in this paper we show that if a b then the a th power gcd matrix s a having the a th power x i x j a of the greatest common divisor of x i and x j as its i j entry divides the b th power gcd matrix s b in the ring m n z of n n matrices over integers we show also that if a b and n 2 then the a th power gcd matrix s a does not divide the b th power gcd matrix s b in the ring m n z similar results are also established for the power lcm matrices

on greatest common divisor matrices and their applications abstract let s x 1 x 2 x n be a set of distinct positive integers the n n matrix s s ij where s ij x i x j the greatest common divisor of x i and x j is called the greatest common divisor gcd matrix on s we study the structure of a gcd matrix and obtain interesting relations between its determinant euler s totient function and moebius function we also determine some arithmetic progressions related to gcd matrices then we generalize the results to general partially ordered sets and show a variety of applications

total nonnegativity of gcd matrices and kernels abstract let x x 1 x n be a vector of distinct positive integers the n n matrix s s x gcd x i x j i j 1 n where gcd x i x j denotes the greatest common divisor of x i and x j is called the greatest common divisor gcd matrix on x by a surprising result of beslin and ligh 1989 7 all gcd matrices are positive definite in this paper we completely characterize the gcd matrices satisfying the stronger property of being totally nonnegative tn or totally positive tp as we show a gcd matrix is never tp when n 3 and is tn if and only if it is t n 2 i e all its 2 2 minors are nonnegative we next demonstrate that a gcd matrix is tn 2 if and only if the exponents of each prime divisor in the prime factorization of the x i s form a monotonic sequence reformulated in the language of kernels our results characterize the subsets of integers over which the kernel k x y gcd x y is totally nonnegative the proofs of our characterizations depend on gantmacher and krein s notion of a green s matrix we conclude by showing that a gcd matrix is tn if and only if it is a green s matrix as a consequence we obtain explicit formulas for all the minors and for the inverse of totally nonnegative gcd matrices

effective masses for laplacians on periodic graphs abstract we consider laplacians on both periodic discrete and periodic metric equilateral graphs their spectrum consists of an absolutely continuous part which is a union of non degenerate spectral bands and flat bands i e eigenvalues of infinite multiplicity we estimate effective masses associated with the ends of each spectral band in terms of geometric parameters of the graphs moreover in the case of the bottom of the spectrum we determine two sided estimates on the effective mass in terms of geometric parameters of the graphs the proof is based on floquet theory factorization of fiber operators perturbation theory and the relation between effective masses for laplacians on discrete and metric graphs obtained in our paper

pairwise stochastic bounded confidence opinion dynamics heavy tails and stability unlike traditional graph based linear dynamics where agents exchange opinions with their neighbors in a static social graph regardless of their differences in opinions the bounded confidence opinion dynamics models exchange between agents with similar opinions we generalize the bounded confidence opinion dynamics model by incorporating pairwise stochastic interactions probabilistic influencing based on opinion differences and the self or endogenous evolutions of the agent opinions which are represented by random processes the opinion exchanges resulting from influencing have pairwise contraction effects whereas endogenous motions have an expansive effect for instance of a diffusive nature we analytically characterize the conditions under which this stochastic dynamics is stable in an appropriate sense in the diffusive case the presence of heavy tailed influence functions with a pareto exponent of 2 is critical for stability for a pair of agents an influence function maps opinion differences to probabilities of influence moreover this model sheds light on dynamics that combine aspects of graph based and bounded confidence dynamics

steady state size distributions in probabilistic models of the cell division cycle a model for the steady state size distribution in an exponentially growing population of single cells is derived and studied the model incorporates a general growth law for individual cells and a probability density for interdivision times a uniqueness theorem is proved and it is shown that no solution exists when individual cells grow exponentially for linear growth infinite series solutions are found in two specific cases statistical data are obtained for these solutions and comparisons are made with the results of some numerical simulations and some experiments

on the factorization of lcm matrices on gcd closed sets abstract let s x1 xn be a set of n distinct positive integers the matrix having the greatest common divisor gcd xi xj of xi and xj as its i j entry is called the greatest common divisor matrix denoted by s n the matrix having the least common multiple lcm xi xj of xi and xj as its i j entry is called the least common multiple matrix denoted by s n the set is said to be gcd closed if xi xj s for all 1 i j n in this paper we show that if n 3 then for any gcd closed set s x1 xn the gcd matrix on s divides the lcm matrix on s in the ring m n z of n n matrices over the integers for n 4 there exists a gcd closed set s x1 xn such that the gcd matrix on s does not divide the lcm matrix on s in the ring m n z this solves a conjecture raised by the author in 1998

opinion dynamics of random walking agents on a lattice the opinion dynamics of random walking agents on finite two dimensional lattices is studied in the model the opinion is continuous and both the lattice and the opinion can be either periodic or nonperiodic at each time step all agents move randomly on the lattice and update their opinions based on those of neighbors with whom the differences of opinion are not greater than a given threshold due to the effect of repeated averaging opinions first converge locally and eventually reach steady states as in other models with bounded confidence steady states in general are those with one or more opinion groups in which all agents have the same opinion when both the lattice and the opinion are periodic however metastable states can emerge in which the whole spectrum of location dependent opinions can coexist this result shows that when a set of continuous opinions forms a structure like a circle unlike the typically used linear opinions rich dynamic behavior can arise when there are geographical restrictions in real situations a complete consensus is rarely reached and metastable states here might be one of the explanations for these situations especially when opinions are not linear

generalized eigenvalue problems for meet and join matrices on semilattices abstract we study generalized eigenvalue problems for meet and join matrices with respect to incidence functions on semilattices we provide new bounds for generalized eigenvalues of meet matrices with respect to join matrices under very general assumptions the applied methodology is flexible and it is shown in the case of gcd and lcm matrices that even sharper bounds can be obtained by applying the known properties of the divisor lattice these results can also be easily modified for the dual problem of eigenvalues of join matrices with respect to meet matrices which we briefly consider as well we investigate the effectiveness of the obtained bounds for select examples involving number theoretical lattices

upper bounds for the condition numbers of the gcd and the reciprocal gcd matrices in spectral norm let s x 1 x n be a set of n distinct positive integers the nxn matrix having the greatest common divisor x i x j of x i and x j as its i j entry is called the greatest common divisor gcd matrix defined on s denoted by x i x j or abbreviated as s the nxn matrix s 1 g i j where g i j 1 x i x j is called the reciprocal greatest common divisor gcd matrix on s in this paper we present upper bounds for the spectral condition numbers of the reciprocal gcd matrix s 1 and the gcd matrix s defined on s 1 2 n with n 2 as a function of euler s f function and n

a survey of condition number estimation for triangular matrices we survey and compare a wide variety of techniques for estimating the condition number of a triangular matrix and make recommendations concerning the use of the estimates in applications each of the methods is shown to bound the condition number the bounds can broadly be categorised as upper bounds from matrix theory and lower bounds from heuristic or probabilistic algorithms for each bound we examine by how much at worst it can overestimate or underestimate the condition number numerical experiments are presented in order to illustrate and compare the practical performance of the condition estimators

efficient implementation of second order stochastic approximation algorithms in high dimensional problems stochastic approximation sa algorithms have been widely applied in minimization problems when the loss functions and or the gradient information are only accessible through noisy evaluations stochastic gradient sg descent a first order algorithm and a workhorse of much machine learning is perhaps the most famous form of sa among all sa algorithms the second order simultaneous perturbation stochastic approximation 2spsa and the second order stochastic gradient 2sg are particularly efficient in handling high dimensional problems covering both gradient free and gradient based scenarios however due to the necessary matrix operations the per iteration floating point operations flops cost of the standard 2spsa 2sg is o p 3 where p is the dimension of the underlying parameter note that the o p 3 flops cost is distinct from the classical spsa based per iteration o 1 cost in terms of the number of noisy function evaluations in this work we propose a technique to efficiently implement the 2spsa 2sg algorithms via the symmetric indefinite matrix factorization and show that the flops cost is reduced from o p 3 to o p 2 the formal almost sure convergence and rate of convergence for the newly proposed approach are directly inherited from the standard 2spsa 2sg the improvement in efficiency and numerical stability is demonstrated in two numerical studies

on the divisibility of meet and join matrices abstract let p p be a lattice let s x 1 x 2 x n be a meet closed subset of p and let f p z be a function we characterize the matrix divisibility of the join matrix s f f x i x j by the meet matrix s f f x i x j in the ring z n n in terms of the usual divisibility in z and we present two algorithms for constructing certain classes of meet closed sets s such that s f divides s f as an example we present the lattice theoretic structure of all meet closed sets with at most five elements possessing the matrix divisibility property finally we show that our methods solve some open problems in the divisor lattice concerning the divisibility of gcd and lcm matrices

non divisibility of lcm matrices by gcd matrices on gcd closed sets abstract in this paper we consider the divisibility problem of lcm matrices by gcd matrices in the ring m n z proposed by shaofang hong in 2002 and in particular a conjecture concerning the divisibility problem raised by jianrong zhao in 2014 we present some certain gcd closed sets on which the lcm matrix is not divisible by the gcd matrix in the ring m n z this could be the first theoretical evidence that zhao s conjecture might be true furthermore we give the necessary and sufficient conditions on the gcd closed set s with s 8 such that the gcd matrix divides the lcm matrix in the ring m n z and hence we partially solve hong s problem finally we conclude with a new conjecture that can be thought as a generalization of zhao s conjecture

asymptotic analysis of a cell cycle model based on unequal division we provide an analysis of the asymptotic behavior of a novel cell cycle model offering a uniform description of the processes of rna production and division and explaining the cell generation time variability at least in certain cell lines we prove that the distribution m t x of the rna level x in the dividing cells at a given time t tends to exp lambda t mu x as t tends to infinity i e that there exists the so called exponential steady state for our model and that perturbed cell populations reach this state in the limit the result is obtained by applying the semigroup theory to a functional integral equation describing the evolution of m t x in time for a variety of reasons the analysis requires explicit characterization of the semigroup spectrum our asymptotic result may be viewed as a generalization of similar results for the generalized branching processes with continuous time under less restrictive assumptions also we discuss related models and asymp

bounds for determinants of matrices associated with classes of arithmetical functions abstract let be an arithmetical function and s x 1 x n a set of distinct positive integers let x i x j denote the n n matrix having evaluated at the greatest common divisor of and as its entry and denote the matrix having evaluated at the least common multiple x i x j of x i and x j as its i j entry in this paper we show for a certain class of arithmetical functions new bounds for det x i x j which improve the results obtained by bourque and ligh in 1993 as a corollary we get new lower bounds for det x i x j which improve the results obtained by rajarama bhat in 1991 we also show for a certain class of semi multiplicative function new bounds for det x i x j which improve the results obtained by bourque and ligh in 1995

robust statistical modeling using the t distribution abstract the t distribution provides a useful extension of the normal for statistical modeling of data sets involving errors with longer than normal tails an analytical strategy based on maximum likelihood for a general model with multivariate t errors is suggested and applied to a variety of problems including linear and nonlinear regression robust estimation of the mean and covariance matrix with missing data unbalanced multivariate repeated measures data multivariate modeling of pedigree data and multivariate nonlinear regression the degrees of freedom parameter of the t distribution provides a convenient dimension for achieving robust statistical inference with moderate increases in computational complexity for many models estimation of precision from asymptotic theory and the bootstrap is discussed and graphical methods for checking the appropriateness of the t distribution are presented

divisibility among power gcd matrices and among power lcm matrices on finitely many coprime divisor chains let a b and h be positive integers and s x1 xh be a set of h distinct positive integers the set s is called a divisor chain if there is a permutation on 1 h such that x 1 x h we say that the set s consists of finitely many coprime divisor chains if there is a positive integer k such that we can partition s as s s1 sk where all the si are divisor chains and each element of si is coprime to each element of sj for 1 i j k the matrix having the ath power xi xj a of the greatest common divisor of xi and xj as its i j entry is called ath power greatest common divisor gcd matrix defined on s denoted by sa similarly we can define the ath power lcm matrix sa in this paper we show that if a b and s consists of finitely many coprime divisor chains with 1 s then in the ring mh z of h h matrices over integers we have sa sb sa sb and sa sb but such results fail to be true if a b these results confirm partially hong s conjectures raised in 2008

on inverses of gcd matrices associated with multiplicative functions and a proof of the hong loewy conjecture abstract let c 1 2 m and f be a multiplicative function such that f k 0 for every positive integer k and the euler product f 1 1 f 1 converges let c f f i j be the m m matrix defined on the set c having f evaluated at the greatest common divisor i j of i and j as its ij entry in the present paper we first obtain the least upper bounds for the ij entry and the absolute row sum of any row of c f 1 the inverse of c f in terms of f specializing these bounds for the arithmetical functions f n e j e and e we examine the asymptotic behavior the smallest eigenvalue of each of matrices c n e c j e and c e depending on e when m tends to infinity we conclude our paper with a proof of a conjecture posed by hong and loewy s hong r loewy asymptotic behavior of eigenvalues of greatest common divisor matrices glasg math j 46 2004 551 569

variance components models for dependent cell populations cells grown in culture can be tracked for several generations and measurements taken on size or age at division and other cell characteristics the observations for the offspring of each cell form a family tree of dependent data such cell lineage data are here modeled as repeated measurements on different family trees arising from individual ancestor cells selected at random from a population of cultured cells the bifurcating autoregression model is embedded in a process that allows for measurement error and variation from tree to tree robust methods are presented that accommodate outliers in this time dependent and branching environment while allowing the statistician to interactively build a variance components model for the process the methodology is illustrated on a substantial data set of 41 trees of emt6 cells with the surprising conclusion that after removing measurement error sister cell lifetimes are nearly identical

consensus formation on coevolving networks groups formation and structure we study the effect of adaptivity on a social model of opinion dynamics and consensus formation we analyse how the adaptivity of the network of contacts between agents to the underlying social dynamics affects the size and topological properties of groups and the convergence time to the stable final state we find that while on static networks these properties are determined by percolation phenomena on adaptive networks the rewiring process leads to different behaviors adaptive rewiring fosters group formation by enhancing communication between agents of similar opinion though it also makes possible the division of clusters we show how the convergence time is determined by the characteristic time of link rearrangement we finally investigate how the adaptivity yields nontrivial correlations between the internal topology and the size of the groups of agreeing agents

understanding nonparametric estimation for clustered data summary in this note we give an alternative formulation of the nonparametric estimators of wang 2003 with the identity link this results in a closed form of the estimator that has computational advantages and gives insight into the rationale behind the estimator

on smith s determinant abstract we give a brief review of papers relating to smith s determinant and point out a common structure that can be found in many extensions and analogues of smith s determinant we present the common structure in the language of posets we also make an investigation on a conjecture of beslin and ligh on greatest common divisor gcd matrices in the sense of meet matrices and give characterizations of the posets satisfying the conjecture further we give a counterexample for the conjecture of bourque and ligh that the least common multiple matrix on any gcd closed set is invertible

mass media and heterogeneous bounds of confidence in continuous opinion dynamics abstract this work focuses on the effects of an external mass media on continuous opinion dynamics with heterogeneous bounds of confidence we modified the original deffuant et al and hegselmann and krause models to incorporate both an external mass media and a heterogeneous distribution of confidence levels we analysed two cases one where only two bounds of confidence are taken into account and other where each individual of the system has her his own characteristic level of confidence we found that in the absence of mass media diversity of bounds of confidence can improve the capacity of the systems to reach consensus we show that the persuasion capacity of the external message is optimal for intermediate levels of heterogeneity our simulations also show the existence for certain parameter values of a counter intuitive effect in which the persuasion capacity of the mass media decreases if the mass media intensity is too large we discuss similarities and differences between the two heterogeneous versions of these continuous opinion dynamic models under the influence of mass media

high breakdown inference for mixed linear models mixed linear models are used to analyze data in many settings these models have a multivariate normal formulation in most cases the maximum likelihood estimator mle or the residual mle reml is usually chosen to estimate the parameters however the latter are based on the strong assumption of exact multivariate normality welsh and richardson have shown that these estimators are not robust to small deviations from multivariate normality this means that in practice a small proportion of data even only one can drive the value of the estimates on their own because the model is multivariate we propose a high breakdown robust estimator for very general mixed linear models that include for example covariates this robust estimator belongs to the class of s estimators from which we can derive asymptotic properties for inference we also use it as a diagnostic tool to detect outlying subjects we discuss the advantages of this estimator compared with other robust estimators proposed previously and i

on meet matrices on posets abstract we consider meet matrices on posets as an abstract generalization of greatest common divisor gcd matrices some of the most important properties of gcd matrices are presented in terms of meet matrices

determinants and divisibility of power gcd and power lcm matrices on finitely many coprime divisor chains let a b and h be positive integers and s x 1 x h be a set of h distinct positive integers the hxh matrix s a x i x j a having the ath power x i x j a of the greatest common divisor of x i and x j as its i j entry is called the ath power gcd matrix on s the ath power lcm matrix on s can be defined similarly in this paper we first obtain the formulae for determinants of power gcd and power lcm matrices on the set s consisting of finitely many coprime divisor chains i e there is a positive integer k such that we can partition s as s s 1 s k where s i and s j are divisor chains and each element of s i is coprime to each element of s j for any 1 j

controllability and observability of network of networks via cartesian products the paper presents a system theoretic analysis framework for a network of networks formed from smaller factor networks via graph cartesian products we provide a compositional framework for extending the controllability and observability of the factor networks to that of the composite network of networks we then delve into the effectiveness of designing control and estimation algorithms for the composite network via its symmetry and gramian structure an example demonstrating the usefulness of our results in the context of social networks with a cartesian product structure is then presented

robust statistical analysis of interlaboratory studies summary a common procedure in testing analytical methods is to send a portion of each of a number of samples to each of several laboratories the results of such a study are submitted to statistical analysis to determine the two important variance components in the problem replication error and laboratory bias outliers are relatively common in these data both among laboratory effects and among the residuals this paper presents a method of analysis for interlaboratory studies that is robust to the existence of outliers and long tailed distributions of random effects theoretical considerations as well as a monte carlo study are adduced as support for this new technique

greatest common divisor matrices abstract let s x 1 x 2 x n be a set of distinct positive integers then n n matrix s s ij where s ij x i x j the greatest common divisor of x i and x j is call the greatest common divisor gcd matrix on s we initiate the study of gcd matrices in the direction of their structure determinant and arithmetic in z n several open problems are posed

on the bourque ligh conjecture of least common multiple matrices abstract let s x1 xn be a set of n distinct positive integers the matrix s n having the least common multiple xi xj of xi and xj as its i j entry is called the least common multiple lcm matrix on s a set s is gcd closed if xi xj s for 1 i j n bourque and ligh conjectured that the lcm matrix s n defined on a gcd closed set s is nonsingular in this paper we prove that the conjecture is true for n 7 and is not true for n 8 so the conjecture is solved completely

on eigenvalues of meet and join matrices associated with incidence functions abstract let p be a locally finite meet semilattice let s x 1 x 2 x n x i x j i j be a finite subset of p and let f be a complex valued function on p then the n n matrix s f where s f ij f x i x j is called the meet matrix on s with respect to f the join matrix on s with respect to f is defined dually on a locally finite join semilattice in this paper we give lower bounds for the smallest eigenvalues of certain positive definite meet matrices with respect to f on any set s we also estimate eigenvalues of meet matrices respect to any f on meet closed set s and with respect to semimultiplicative f on join closed set s the same is carried out dually for join matrices

a proof of a conjecture on monotonic behavior of the smallest and the largest eigenvalues of a number theoretic matrix abstract in this study we investigate the monotonic behavior of the smallest eigenvalue t n and the largest eigenvalue t n of the n n matrix e n t e n where the ij entry of e n is 1 if j i and 0 otherwise we present a proof of the mattila haukkanen conjecture which states that for every n z t n 1 t n and t n t n 1 also we prove that lim n t n 0 and lim n t n and we give a lower bound for t n

α and β curves sister sister and mother daughter correlations in cell population dynamics abstract the and curves and sister sister and mother daughter cycle length correlations are analyzed for a general model of age and size structured cell population dynamics some basic facts concerning these concepts are established a criterion is given for log a and log a to have asymptotically equal slopes the significance of size dependence upon sister sister and mother daughter correlations is demonstrated some numerical examples are given to illustrate the concepts

notes on hong s conjectures of real number power lcm matrices abstract let e be a real number and s x 1 x n be a set of n distinct positive integers the set s is said to be gcd closed respectively lcm closed if x i x j s respectively x i x j s for all 1 i j n the matrix having eth power x i x j e of the least common multiple of x i and x j as its i j entry is called the eth power least common multiple lcm matrix denoted by x i x j e or abbreviated by s e in this paper we show that for any real number e 1 and n 7 the power lcm matrix x i x j e defined on any gcd closed respectively lcm closed set s x 1 x n is nonsingular this confirms partially two conjectures raised by hong in s hong nonsingularity of matrices associated with classes of arithmetical functions j algebra 281 2004 1 14 similar results are established for reciprocal real number power gcd matrices

on the modelling and simulation of the competition for a secession under media influence by active particles methods and functional subsystems decomposition this paper deals with the development of a mathematical model for complex socio economical systems where external actions play a key role the aim of the paper is to show the emergence of collective behaviors or macroscopic trends from individual based interactions where agents are identified by functional subsystems the approach is based on the methods of the mathematical kinetic theory for active particles which describes the evolution of large systems of interacting entities which are carriers of specific functions in our specific application socio political activities subjected to the influence of the media

effects of heterogeneous convergence rate on consensus in opinion dynamics abstract the deffuant model has attracted much attention in the study of opinion dynamics here we propose a modified version by introducing into the model a heterogeneous convergence rate which is dependent on the opinion difference between interacting agents and a tunable parameter we study the effects of heterogeneous convergence rate on consensus by investigating the probability of complete consensus the size of the largest opinion cluster the number of opinion clusters and the relaxation time we find that the decrease of the convergence rate is favorable to decreasing the confidence threshold for the population to always reach complete consensus and there exists optimal resulting in the minimal bounded confidence threshold moreover we find that there exists a window before the threshold of confidence in which complete consensus may be reached with a nonzero probability when is not too large we also find that within a certain confidence range decreasing the convergence rate will reduce the relaxation time which is somewhat counterintuitive

boundary effects in a three state modified voter model for languages abstract the standard three state voter model is extended by including the outside pressure favouring one of the three language choices and by adding some biased internal noise the monte carlo simulations are motivated by states with the population divided into three groups of various affinities to each other we show the crucial influence of the boundaries for moderate lattice sizes like 500 500 by removing the fixed boundary at one side we demonstrate that this can lead to the victory of one single choice noise in contrast stabilizes the choices of all three populations in addition we compute the persistence probability i e the number of sites who have never changed their opinion during the simulation and we consider the case of rigid minded decision makers

cell lineage analysis variance components models for dependent cell populations abstract cells grown in culture can be tracked for several generations and measurements taken on size or age at division and other cell characteristics the observations for the offspring of each cell form a family tree of dependent data such cell lineage data are here modeled as repeated measurements on different family trees arising from individual ancestor cells selected at random from a population of cultured cells the bifurcating autoregression model is embedded in a process that allows for measurement error and variation from tree to tree robust methods are presented that accommodate outliers in this time dependent and branching environment while allowing the statistician to interactively build a variance components model for the process the methodology is illustrated on a substantial data set of 41 trees of emt6 cells with the surprising conclusion that after removing measurement error sister cell lifetimes are nearly identical

opinion clusters in a modified hegselmann krause model with heterogeneous bounded confidences and stubbornness abstract in opinion dynamics with continuous opinion bounded confidence is a critical parameter agents can interact with each other only when the opinion difference between them is less than the bounded confidence larger bounded confidence always leads to fewer opinion clusters stubbornness characterizing the insistence of an agent on her own opinion is thought to only affect the transition time in this work a modified hegselmann krause model with heterogeneous population is investigated where agents in different same subpopulation have different same bounded confidence and stubbornness we find that due to the interaction among subpopulations increasing the stubbornness of agents in the subpopulation with the largest bounded confidence favors fewer opinion clusters and the expansion of the largest cluster we also find that increasing the bounded confidence of a subpopulation leads to fewer clusters and a larger largest cluster provided that all the others have large bounded confidence while one subpopulation is with a small bounded confidence there exist an optimal bounded confidence of another subpopulation for the smallest number of opinion clusters and that for the largest size of the largest cluster

a nonlinear model of population dynamics containing an arbitrary number of continuous structure variables a nonlinear model is presented for the dynamics of a population in which each individual is characterized by its chronological age and by an arbitrary finite number of additional structure variables the nonlinearities are introduced by assuming that the birth and loss processes as well as the maturation rates of individuals are controlled by a functional of the population density the model is a generalization of the classical sharpe lotka mckendrick model of age structured population growth the nonlinear age structured model of gurtin and maccamy and the age size structured cell population model of bell and anderson based on a reformulation of the model in terms of a coupled system of equations the existence for all positive time of unique solutions to the model is proved using a contraction mapping argument the existence of equilibrium solutions is discussed and sufficient conditions are proved for the local asymptotic stability of equilibria using results from the theory of strongly continuous

singularity wielandt s lemma and singular values in this study some upper and lower bounds for singular values of a general complex matrix are investigated according to singularity and wielandt s lemma of matrices especially some relationships between the singular values of the matrix a and its block norm matrix are established based on these relationships one may obtain the effective estimates for the singular values of large matrices by using the lower dimension norm matrices in addition a small error in piazza 2002 g piazza t politi an upper bound for the condition number of a matrix in spectral norm j comput appl math 143 1 2002 141 144 is also corrected some numerical experiments on saddle point problems show that these results are simple and sharp under suitable conditions

on gcd and lcm matrices let s x1 x2 xn be a set of distinct positive integers the matrix s having the greatest common divisor xi xj of xi and xj as its i j entry is called the greatest common divisor gcd matrix on s the matrix s having the least common multiple of xi and xj as its i j entry is called the least common multiple lcm matrix on s the set s is factor closed if it contains every divisor of each of its elements if s is factor closed we calculate the inverses of the gcd and lcm matrices on s and show that s s 1 is an integral matrix we also extend a result of h j s smith by calculating the determinant of s when xi xj s for 1 i j n

consensus formation on adaptive networks the structure of a network can significantly influence the properties of the dynamical processes that take place on them while many studies have been paid to this influence much less attention has been devoted to the interplay and feedback mechanisms between dynamical processes and network topology on adaptive networks adaptive rewiring of links can happen in real life systems such as acquaintance networks where people are more likely to maintain a social connection if their views and values are similar in our study we consider different variants of a model for consensus formation our investigations reveal that the adaptation of the network topology fosters cluster formation by enhancing communication between agents of similar opinion although it also promotes the division of these clusters the temporal behavior is also strongly affected by adaptivity while on static networks it is influenced by percolation properties on adaptive networks both the early and late time evolutions of the system are determined by the rewiring process the investigation of a variant of the model reveals that the scenarios of transitions between consensus and polarized states are more robust on adaptive networks

asymptotic properties of maximum likelihood estimates in the mixed poisson model this paper considers the asymptotic behavior of the maximum likelihood estimators mle s of the probabilities of a mixed poisson distribution with a nonparametric mixing distribution the vector of estimated probabilities is shown to converge in probability to the vector of mixed probabilities at rate n 1 2 varepsilon for any varepsilon 0 under a generalized chi 2 distance function it is then shown that any finite set of the mle s has the same joint limiting distribution as does the corresponding set of sample proportions when the support of the mixing distribution g 0 is an infinite set with a known upper bound and g 0 satisfies a certain condition at zero

nonsingularity of matrices associated with classes of arithmetical functions on lcm closed sets abstract let s x1 xn be a set of n distinct positive integers and f be an arithmetical function let f xi xj denote the n n matrix having f evaluated at the greatest common divisor xi xj of xi and xj as its i j entry and f xi xj denote the n n matrix having f evaluated at the least common multiple xi xj of xi and xj as its i j entry the set s is said to be lcm closed if xi xj s for all 1 i j n for an integer x 1 let x denote the number of distinct prime factors of x define 1 0 in this paper we show that if s x1 xn is an lcm closed set satisfying max x s lcm s x 2 and if f is a strictly increasing resp decreasing completely multiplicative function or if f is a strictly decreasing resp increasing completely multiplicative function satisfying 0 f p 1 p resp f p p for any prime p then the matrix f xi xj resp f xi xj defined on s is nonsingular by using the concept of least type multiple introduced in s hong j algebra 281 2004 1 14 we also obtain reduced formulas for det f xi xj and det f xi xj when f is completely multiplicative and s is lcm closed we also establish several results about the nonsingularity of lcm matrices and reciprocal gcd matrices

a model of the effects of authority on consensus formation in adaptive networks impact on network topology and robustness abstract opinions of individuals in real social networks are arguably strongly influenced by external determinants such as the opinions of those perceived to have the highest levels of authority in order to model this we have extended an existing model of consensus formation in an adaptive network by the introduction of a parameter representing each agent s level of authority based on their opinion relative to the overall opinion distribution we found that introducing this model along with a randomly varying opinion convergence factor significantly impacts the final state of converged opinions and the number of interactions required to reach that state we also determined the relationship between initial and final network topologies for this model and whether the final topology is robust to node removals our results indicate firstly that the process of consensus formation with a model of authority consistently transforms the network from an arbitrary initial topology to one with distinct measurements in mean shortest path clustering coefficient and degree distribution secondly we found that subsequent to the consensus formation process the mean shortest path and clustering coefficient are less affected by both random and targeted node disconnection speculation on the relevance of these results to real world applications is provided

robust inference for variance components models for single trees of cell lineage data previously huggins and staudte examined robust estimators for a variance components formulation of the bifurcating autoregressive model for cell lineage data they gave asymptotic properties of the estimators if a large number of trees were observed however for single trees the derivation of these asymptotic properties is more complex here the asymptotic distributions of robust estimators of parameters associated with the stationary bifurcating autoregressive process as a single tree becomes large are obtained these results follow from the formulation of the estimating functions as the product of a nonrandom matrix and the sum of vectors of functions of an infinite sequence of exchangeable random variables

deffuant model of opinion formation in one dimensional multiplex networks complex systems in the real world often operate through multiple kinds of links connecting their constituents in this paper we propose an opinion formation model under bounded confidence over multiplex networks consisting of edges at different topological and temporal scales we determine rigorously the critical confidence threshold by exploiting probability theory and network science when the nodes are arranged on the integers evolving in continuous time it is found that the existence of multiplexity impedes the convergence and that working with the aggregated or summarized simplex network is inaccurate since it misses vital information analytical calculations are confirmed by extensive numerical simulations

matrices associated with multiplicative functions abstract let f be a multiplicative function and s x 1 x 2 x n a set of distinct positive integers denote by f x i x j the n n matrix having f evaluated at the least common multiple x i x j of x i and x j as its i j entry if s is factor closed we calculate the determinant of this matrix and if it is invertible its inverse and show that for a certain class of functions the n n matrix f x i x j having f evaluated at the greatest common divisor of x i and x j as its i j entry is a factor of the matrix f x i x j in the ring of n n matrices over the integers we also determine conditions on f that will guarantee the matrix f x i x j is positive definite

further bounds for the smallest singular value and the spectral condition number abstract we derive monotonic sequences of bounds for the extreme singular values in particular we find further lower bounds for the smallest singular value which improve the bounds of yu and gun also we give new upper bounds for the spectral condition number

analysis of complex contagions in random multiplex networks we study the diffusion of influence in random multiplex networks where links can be of r different types and for a given content e g rumor product or political view each link type is associated with a content dependent parameter ci in 0 that measures the relative bias type i links have in spreading this content in this setting we propose a linear threshold model of contagion where nodes switch state if their perceived proportion of active neighbors exceeds a threshold namely a node connected to mi active neighbors and ki mi inactive neighbors via type i links will turn active if cimi ciki exceeds its threshold under this model we obtain the condition probability and expected size of global spreading events our results extend the existing work on complex contagions in several directions by i providing solutions for coupled random networks whose vertices are neither identical nor disjoint ii highlighting the effect of content on the dynamics of complex contagions and iii showing that content dependent propagation over a multiplex network leads to a subtle relation between the giant vulnerable component of the graph and the global cascade condition that is not seen in the existing models in the literature

determinants of matrices associated with arithmetic functions on finitely many quasi coprime divisor chains let s x 1 x n be a set of n distinct positive integers and f be an arithmetic function we use f s f x i x j resp f s f x i x j to denote the n i n matrix having f evaluated at the greatest common divisor resp the least common multiple of x i and x j as its i j entry the set s is called a divisor chain if there is a permutation of 1 n such that x 1 x n if s can be partitioned as s i 1 k s i with all s i 1 i k being divisor chains and max s i max s j gcd s for 1 i j k then we say that s consists of finitely many quasi coprime divisor chains in this paper we introduce a new method to give the formulas for the determinants of the matrices f s and f s on finitely many quasi coprime divisor chains s we show also that det f s det f s holds under some natural conditions these extend the results obtained by tan and lin 2010 and tan et al 2013 respectively

an eigenvalue problem related to cell growth abstract in this paper the eigenvalues of the operator corresponding to the partial differential equation which describes the evolution of a population reproducing by simple fission are investigated this is done by transforming the eigenvalue problem to an integral equation the theory concerning positive operators on a banach space appears to be very useful

determinant and inverse of join matrices on two sets abstract let p be a lattice and f a complex valued function on p we define meet and join matrices on two arbitrary subsets x and y of p by x y f f x i y j and x y f f x i y j respectively here we present expressions for the determinant and the inverse of x y f our main goal is to cover the case when f is not semimultiplicative since the formulas presented earlier for x y f cannot be applied in this situation in cases when f is semimultiplicative we obtain several new and known formulas for the determinant and inverse of x y f and the usual meet and join matrices s f and s f we also apply these formulas to lcm max gcd and min matrices which are special cases of join and meet matrices