infinitely divisible nonnegative matrices m matrices and the embedding problem for finite state stationary markov chains abstract this paper explicitly details the relation between m matrices nonnegative roots of nonnegative matrices and the embedding problem for finite state stationary markov chains the set of nonsingular nonnegative matrices with arbitrary nonnegative roots is shown to be the closure of the set of matrices with matrix roots in im the methods presented here employ nothing beyond basic matrix analysis however it answers a question regarding m matrices posed over 30 years ago and as an application a new characterization of the set of all embeddable stochastic matrices is obtained as a corollary

the complexity of divisibility we address two sets of long standing open questions in linear algebra and probability theory from a computational complexity perspective stochastic matrix divisibility and divisibility and decomposability of probability distributions we prove that finite divisibility of stochastic matrices is an np complete problem and extend this result to nonnegative matrices and completely positive trace preserving maps i e the quantum analogue of stochastic matrices we further prove a complexity hierarchy for the divisibility and decomposability of probability distributions showing that finite distribution divisibility is in p but decomposability is np hard for the former we give an explicit polynomial time algorithm all results on distributions extend to weak membership formulations proving that the complexity of these problems is robust to perturbations

the complexity of relating quantum channels to master equations completely positive trace preserving cpt maps and lindblad master equations are both widely used to describe the dynamics of open quantum systems the connection between these two descriptions is a classic topic in mathematical physics one direction was solved by the now famous result due to lindblad kossakowski gorini and sudarshan who gave a complete characterisation of the master equations that generate completely positive semi groups however the other direction has remained open given a cpt map is there a lindblad master equation that generates it and if so can we find its form this is sometimes known as the markovianity problem physically it is asking how one can deduce underlying physical processes from experimental observations

the generation and evolution of lump solitary waves in surface tension dominated flows three dimensional solitary waves or lump solitons are known to be solutions to the kadomtsev petviashvili i equation which models small amplitude shallow water waves when the bond number is greater than 1 recently pego and quintero presented a proof of the existence of such waves for the benney luke equation with surface tension here we establish an explicit connection between the lump solitons of these two equations and numerically compute the benney luke lump solitons and their speed amplitude relation furthermore we numerically collide two benney luke lump solitons to illustrate their soliton wave character finally we study the flow over an obstacle near the linear shallow water speed and show that three dimensional lump solitons are periodically generated

fully localized solitary waves for the forced kadomtsevpetviashvili equation the kadomtsevpetviashvili kp equation is a generalized form of the kortewegde vries equation for a three dimensional channel flow three dimensional fully localized solitary wave solutions can be obtained when the effect of surface tension is significant we refer to this as the kp i equation an exact solution of a three dimensional fully localized stationary solution for the kp i equation has been found in the absence of any obstacle on the rigid channel bottom however three dimensional fully localized stationary solutions for the kp i equation have not been found in the presence of a forcing neither analytically nor numerically this forcing term comes from some obstacles in the rigid channel bottom in this work we focus on the fully localized solitary wave solutions when a positive bump or a negative hole is given as the rigid bottom configuration the forced kp i equation is defined in an infinite domain and it is reduced to a finite computational domain by introducing artificial boundary conditions interestingly there are at least two distinct stationary lump type solutions for a positive bump while there are at least three distinct stationary lump type solutions for a negative hole furthermore we investigate their numerical stability by solving the forced time dependent kp i equation using them as initial conditions our numerical results confirm that there exists a stable stationary lump type solitary wave solution for both a positive bump and a negative hole when they evolve in time

diffusion and consensus on weakly connected directed graphs abstract let g be a weakly connected directed graph with asymmetric graph laplacian l consensus and diffusion are dual dynamical processes defined on g by x l x for consensus and p p l for diffusion in this survey we consider both these processes as well their discrete time analogues we define a basis of row vectors i i 1 k of the left null space of l and a basis of column vectors i i 1 k of the right null space of l in terms of the partition of g into strongly connected components this allows for complete characterization of the asymptotic behavior of both diffusion and consensus discrete and continuous in terms of these eigenvectors as an application of these ideas we present a treatment of the pagerank algorithm that is dual to the usual one we further show that the teleporting see below feature usually included in the algorithm is not strictly necessary together with 1 this is a complete and self contained overview of the asymptotics of consensus and diffusion on digraphs many of the ideas presented here can be found scattered in the literature though mostly outside mainstream mathematics and not always with complete proofs this paper seeks to remedy this by providing a compact and accessible survey

computing short interval transition matrices of a discrete time markov chain from partially observed data markov chains constitute a common way of modelling the progression of a chronic disease through various severity states for these models a transition matrix with the probabilities of moving from one state to another for a specific time interval is usually estimated from cohort data quite often however the cohort is observed at specific times with intervals that may be greater than the interval of interest the transition matrix computed then needs to be decomposed in order to estimate the desired interval transition matrix suited to the model although simple to implement this method of matrix decomposition can yet result in an invalid short interval transition matrix with negative or complex entries in this paper we present a method for computing short interval transition matrices that is based on regularization techniques our method operates separately on each row of the invalid short interval transition matrix aiming to minimize an appropriate distance measure we test our method on various matrix structures and sizes and evaluate its performance on a real life transition model for hiv infected individuals

growth motion and 1 parameter families of symmetry sets associated to every plane curve there is the locus of centres of circles bitangent to that curve the so called symmetry set of the curve we can view this set as the spine of our curve which can be recovered by taking the envelope of circles of varying radii along this spine varying the symmetry set in some isotopy while keeping the radius function fixed may be viewed as crudely modelling motion of the original curve viewed as a biological object fixing the symmetry set and varying the radius function can be considered to model growth crudely in this paper we shall describe the generic changes in the curves which take place in the process of growth and motion and outline the corresponding results for spheres centred on a space curve we also use the idea of a stratified morse function to describe the generic changes which occur in one parameter families of full bifurcation sets in the plane applying this to the bifurcation set of distance squared functions we find all the transitions of a symmetry set and evolute which occur in a generic isotopy of a plane curve

decentralized control of vehicle formations this paper investigates a method for decentralized stabilization of vehicle formations using techniques from algebraic graph theory the vehicles exchange information according to a pre specified communication digraph g a feedback control is designed using relative information between a vehicle and its in neighbors in g we prove that a necessary and sufficient condition for an appropriate decentralized linear stabilizing feedback to exist is that g has a rooted directed spanning tree we show the direct relationship between the rate of convergence to formation and the eigenvalues of the directed laplacian of g various special situations are discussed including symmetric communication graphs and formations with leaders several numerical simulations are used to illustrate the results

modelling nonlinear electrohydrodynamic surface waves over three dimensional conducting fluids the evolution of the free surface of a three dimensional conducting fluid in the presence of gravity surface tension and vertical electric field due to parallel electrodes is considered based on the analysis of the dirichlet neumann operators a series of fully nonlinear models is derived systematically from the euler equations in the hamiltonian framework without assumptions on competing length scales can therefore be applied to systems of arbitrary fluid depth and to disturbances with arbitrary wavelength for special cases well known weakly nonlinear models in shallow and deep fluids can be generalized via introducing extra electric terms it is shown that the electric field has a great impact on the physical system and can change the qualitative nature of the free surface i when the separation distance between two electrodes is small compared with typical wavelength the boussinesq benney luke bl and kadomtsev petviashvili kp equations with modified coefficients are obtained and electric forces can turn kp i to kp ii and vice versa ii as the parallel electrodes are of large separation distance but the thickness of the liquid is much smaller than typical wavelength we generalize the bl and kp equations by adding pseudo differential operators resulting from the electric field iii for a quasi monochromatic plane wave in deep fluid we derive the cubic nonlinear schrodinger nls equation but its type focusing or defocusing is strongly influenced by the value of the electric parameter for sufficient surface tension numerical studies reveal that lump type solutions exist in the aforementioned three regimes particularly even when the associated nls equation is defocusing for a wave train lumps can exist in fully nonlinear models

on achieving size independent stability margin of vehicular lattice formations with distributed control we study the stability margin of a vehicular formation with distributed control in which the control at each vehicle only depends on the information from its neighbors in an information graph we consider a d dimensional lattice as information graph of which the 1 d platoon is a special case the stability margin is measured by the real part of the least stable eigenvalue of the closed loop state matrix which quantifies the rate of decay of initial errors in 1 it was shown that with symmetric control in which two neighbors put equal weight on information received from each other the stability margin of a 1 d vehicular platoon decays to 0 as 0 1 n2 where n is the number of vehicles moreover a perturbation analysis was used to show that with vanishingly small amount of asymmetry in the control gains the stability margin scaling can be improved to 0 1 n in this technical note we show that with judicious choice of nonvanishing asymmetry in control the stability margin of the closed loop can be bounded away from zero uniformly in n asymmetry in control gains thus makes the control architecture highly scalable the results are also generalized to d dimensional lattice information graphs that were studied in 2 and the correspondingly stronger conclusions than those derived in 2 are obtained in addition we show that the size independent stability margin can be achieved with relative position and relative velocity rprv feedback as well as relative position and absolute velocity rpav feedback while the analysis in 1 2 was only for the rpav case

inverse m matrices ii this is an update of the 1981 survey by the first author in the meantime a considerable amount has been learned about the very special structure of the important class of inverse m matrices developments since the earlier survey are emphasized but we have tried to be somewhat complete and some results have not previously been published some proofs are given where appropriate and references are given for others after some elementary preliminaries results are grouped by certain natural categories

modeling complex systems by generalized factor analysis we propose a new modeling paradigm for large dimensional aggregates of stochastic systems by generalized factor analysis gfa models these models describe the data as the sum of a flocking plus an uncorrelated idiosyncratic component the flocking component describes a sort of collective orderly motion which admits a much simpler mathematical description than the whole ensemble while the idiosyncratic component describes weakly correlated noise we first discuss static gfa representations and characterize in a rigorous way the properties of the two components the extraction of the dynamic flocking component is discussed for time stationary linear systems and for a simple classes of separable random fields

two dimensional solitary waves for a benny luke equation abstract we prove the existence of finite energy solitary waves for isotropic benney luke equations that arise in the study of the evolution of small amplitude three dimensional water waves when the horizontal length scale is long compared with the depth the family of benney luke equations discussed in this paper includes the effect of surface tension and a variety of equivalent forms of dispersion these equations reduce formally to the korteweg de vries kdv equation and to the kadomtsev petviashvili kp i or kp ii equation in the appropriate limits existence of finite energy solitary waves or lumps is proved via the concentration compactness method when surface tension is sufficiently strong bond number larger than 1 3 we prove that a suitable family of benney luke lump solutions converges to a nontrivial lump solution for the kp i equation

asymptotic expansions for solitary gravity capillary waves in two and three dimensions high order asymptotic series are obtained for two and three dimensional gravity capillary solitary waves in two dimensions the first term in the asymptotic series is the well known sech 2 solution of the korteweg de vries equation in three dimensions the first term is the rational lump solution of the kadomtsev petviashvili equation i the two dimensional series is used with nine terms included to investigate how small surface tension affects the height and energy of large amplitude waves and waves close to the solitary version of stokes extreme wave in particular for small surface tension the solitary wave with the maximum energy is obtained for large surface tension the two dimensional series is also used to study the energy of depression solitary waves energy considerations suggest that for large enough surface tension there are solitary waves that can get close to the fluid bottom in three dimensions analytic solutions for the high order perturbation terms are computed numerically and the resulting asymptotic series to three terms is used to obtain the speed versus maximum amplitude curve for solitary waves subject to sufficiently large surface tension finally the above asymptotic method is applied to the benney luke bl equation and the resulting asymptotic series to three terms is verified to agree with the solitary wave solution of the bl equation

computing real square roots of a real matrix bjorck and hammarling 1 describe a fast stable schur method for computing a square root x of a matrix a x2 a we present an extension of their method which enables real arithmetic to be used throughout when computing a real square root of a real matrix for a nonsingular real matrix a conditions are given for the existence of a real square root and for the existence of a real square root which is a polynomial in a the number of square roots of the latter type is determined the conditioning of matrix square roots is investigated and an algorithm is given for the computation of a well conditioned square root

dividing quantum channels we investigate the possibility of dividing quantum channels into concatenations of other channels thereby studying the semigroup structure of the set of completely positive trace preserving maps we show the existence of indivisible channels which can not be written as non trivial products of other channels and study the set of infinitesimal divisible channels which are elements of continuous completely positive evolutions for qubit channels we obtain a complete characterization of the sets of indivisible and infinitesimal divisible channels moreover we identify those channels which are solutions of time dependent master equations for both positive and completely positive evolutions for arbitrary finite dimension we prove a representation theorem for elements of continuous completely positive evolutions based on new results on determinants of quantum channels and markovian approximations

forest matrices around the laplacian matrix abstract we study the matrices q k of in forests of a weighted digraph and their connections with the laplacian matrix l of the i j entry of q k is the total weight of spanning converging forests in forests with k arcs such that i belongs to a tree rooted at j the forest matrices q k can be calculated recursively and expressed by polynomials in the laplacian matrix they provide representations for the generalized inverses the powers and some eigenvectors of l the normalized in forest matrices are row stochastic the normalized matrix of maximum in forests is the eigenprojection of the laplacian matrix which provides an immediate proof of the markov chain tree theorem a source of these results is the fact that matrices q k are the matrix coefficients in the polynomial expansion of adj i l thereby they are precisely faddeev s matrices for l

the nonnegative inverse eigenvalue problem abstract let 1 n be a list of complex numbers the nonnegative inverse eigenvalue problem niep is the problem of determining necessary and sufficient conditions in order that be the spectrum of an entrywise nonnegative n n matrix our purpose is to give an overview of the niep its history modern results and subproblems with particular emphasis on symmetric realizing matrices a substantial bibliography is included

on pth roots of stochastic matrices in markov chain models in finance and healthcare a transition matrix over a certain time interval is needed but only a transition matrix over a longer time interval may be available the problem arises of determining a stochastic pth root of a stochastic matrix the given transition matrix by exploiting the theory of functions of matrices we develop results on the existence and characterization of matrix pth roots and in particular on the existence of stochastic pth roots of stochastic matrices our contributions include characterization of when a real matrix has a real pth root a classification of pth roots of a possibly singular matrix a sufficient condition for a pth root of a stochastic matrix to have unit row sums and the identification of two classes of stochastic matrices that have stochastic pth roots for all p we also delineate a wide variety of possible configurations as regards existence nature primary or nonprimary and number of stochastic roots and develop a necessary condition for existence of a stochastic root in terms of the spectrum of the given matrix

automated traffic and the finite size resonance we investigate in detail what one might call the canonical automated traffic problem a long string of n 1 cars numbered from 0 to n moves along a one lane road in formation at a constant velocity and with a unit distance between successive cars each car monitors the relative velocity and position of only its neighboring cars this information is then fed back to its own engine which decelerates brakes or accelerates according to the information it receives the question is what happens when due to an external influence a traffic light turning green the zero th car the leader accelerates

inverse m matrices abstract this is an attempt at a comprehensive expository study of those nonnegative matrices which happen to be inverses of m matrices and is aimed at an audience conversant with basic ideas of matrix theory a theme is the parallels anddifferences between the class of m matrices and the class of inverse m matrices among the primary tools used are diagonal multiplications the neumann expansion and the form of the inverse of a partitioned matrix some of the results seem not to have appeared before and several unsolved problems are mentioned

roots of completely positive maps abstract we introduce the concept of completely positive roots of completely positive maps on operator algebras we do this in different forms as asymptotic roots proper discrete roots and as continuous one parameter semigroups of roots we present structural and general existence and non existence results some special examples in settings where we understand the situation better and several challenging open problems our study is closely related to elfving s embedding problem in classical probability and the divisibility problem of quantum channels

a characterization of the drazin inverse abstract if a is a nonsingular matrix of order n the inverse of a is the unique matrix x such that rank a i i x rank a in this paper we generalize this fact to any matrix a of order n over the complex field to obtain an analogous result for the drazin inverse of a we then give an algorithm for computing the drazin inverse of a

global theorems for symmetry sets of smooth curves and polygons in the plane etablissement des formules globales de structure des ensembles de symetrie a l aide d un theoreme d ozawa etude des transitions sur de tels ensembles cas de polygones plans

operational characterization of divisibility of dynamical maps in this work we show the operational characterization to the divisibility of dynamical maps in terms of the distinguishability of quantum channels it is proven that the distinguishability of any pair of quantum channels does not increase under divisible maps in which the full hierarchy of divisibility is isomorphic to the structure of entanglement between system and environment this shows that i channel distinguishability is the operational quantity signifying detecting divisibility indivisibility of dynamical maps and ii the decision problem for the divisibility of maps is as hard as the separability problem in entanglement theory we also provide the information theoretic characterization to the divisibility of maps with conditional min entropy

a pseudospectral procedure for the solution of nonlinear wave equations with examples from free surface flows an algorithm for the solution of general isotropic nonlinear wave equations is presented the algorithm is based on a symmetric factorization of the linear part of the wave operator followed by its exact integration through an integrating factor in spectral space the remaining nonlinear and forcing terms can be handled with any standard pseudospectral procedure solving the linear part of the wave operator exactly effectively eliminates the stiffness of the original problem characterized by a wide range of temporal scales the algorithm is tested and applied to several problems of three dimensional long surface waves solitary wave propagation interaction diffraction and the generation of waves by flow over slowly varying bottom topography other potential applications include waves in rotating and stratified flows and wave interaction with more pronounced topographic features

flocks and formations given a large number the flock of moving physical objects we investigate physically reasonable mechanisms of influencing their orbits in such a way that they move along a prescribed course and in a prescribed and fixed configuration or in formation each agent is programmed to see the position and velocity of a certain number of others this flow of information from one agent to another defines a fixed directed loopless graph in which the agents are represented by the vertices this graph is called the communication graph to be able to fly in formation an agent tries to match the mean position and velocity of his neighbors his direct antecedents on the communication graph to his own this operation defines a directed laplacian on the communication graph a linear feedback is used to ensure stability of the coherent flight patterns we analyze in detail how the connectedness of the communication graph affects the coherence of the stable flight patterns and give a characterization of these stable flight patterns we do the same if in addition the flight of the flock is guided by one or more leaders finally we use this theory to develop some applications examples of these are flight guided by external controls flocks of flocks and some results about flocks whose formation is always oriented along the line of flight such as geese

existence and conditional energetic stability of three dimensional fully localised solitary gravity capillary water waves in this paper we show that the hydrodynamic problem for three dimensional water waves with strong surface tension effects admits a fully localised solitary wave which decays to the undisturbed state of the water in every horizontal direction the proof is based upon the classical variational principle that a solitary wave of this type is a critical point of the energy which is given in dimensionless coordinates by epsilon eta phi integral r2 1 2 integral 1 eta 0 phi 2 x phi 2 y phi 2 z dy 1 2 eta 2 beta root 1 eta 2 x eta 2 z 1 dxdz subject to the constraint that the momentum i eta phi integral r2 eta x phi vertical bar y 1 eta dzdz is fixed here x y z x z is an element of r y is an element of 0 1 eta x z is the fluid domain phi is the velocity potential and beta 1 3 is the bond number these functionals are studied locally for eta in a neighbourhood of the origin in h 3 r 2 we prove the existence of a minimiser of epsilon subject to the constraint i 2 mu where 0 mu 1 the existence of a small amplitude solitary wave is thus assured and since epsilon and i are both conserved quantities a standard argument may be used to establish the stability of the set d mu of minimisers as a whole stability is however understood in a qualified sense due to the lack of a global well posedness theory for three dimensional water waves we show that solutions to the evolutionary problem starting near d mu remain close to d mu in a suitably defined energy space over their interval of existence they may however explode in finite time due to higher order derivatives becoming unbounded c 2012 elsevier inc all rights reserved