nonzero bound on fiedler eigenvalue causes exponential growth of h infinity norm of vehicular platoon we consider platoons composed of identical vehicles and controlled in a distributed way that is each vehicle has its own onboard controller the regulation errors in spacing to the immediately preceeding and following vehicles are weighted differently by the onboard controller which thus implements an asymmetric bidirectional control scheme the weights can vary along the platoon we prove that such platoons have a nonzero uniform bound on the second smallest eigenvalue of the graph laplacian matrix the fiedler eigenvalue furthermore it is shown that existence of this bound always signals undesirable scaling properties of the platoon namely the h infinity norm of the transfer function of the platoon grows exponentially with the number of vehicles regardless of the controllers used hence the benefits of a uniform gap in the spectrum of a laplacian with an asymetric distributed controller are paid for by poor scaling as the number of vehicles grows

reduced difference polynomials and self intersection computations abstract a reduced difference polynomial f u v p u p v u v may be associated with any given univariate polynomial p t t 0 1 such that the locus f u v 0 identifies the pairs of distinct values u and v satisfying p u p v the bernstein coefficients of f u v on 0 1 2 can be determined from those of p t through a simple algorithm and can be restricted to any subdomain of 0 1 2 by standard subdivision methods by constructing the reduced difference polynomials f u v and g u v associated with the coordinate components of a polynomial curve r t x t y t a quadtree decomposition of 0 1 2 guided by the variation diminishing property yields a numerically stable scheme for isolating real solutions of the system f u v g u v 0 which identify self intersections of the curve r t through the kantorovich theorem for guaranteed convergence of newton raphson iterations to a unique solution the self intersections can be efficiently computed to machine precision by generalizing the reduced difference polynomial to encompass products of univariate polynomials the method can be readily extended to compute the self intersections of rational curves and of the rational offsets to pythagorean hodograph curves

distributed control for identical dynamically coupled systems a decomposition approach we consider the problem of designing distributed controllers for a class of systems which can be obtained from the interconnection of a number of identical subsystems if the state space matrices of these systems satisfy a certain structural property then it is possible to derive a procedure for designing a distributed controller which has the same interconnection pattern as the plant this procedure is basically a multiobjective optimization under linear matrix inequality constraints with system norms as performance indices the explicit expressions for computing these controllers are given for both h infin or h 2 performance and both for static state feedback and dynamic output feedback in discrete time at the end of the paper two application examples illustrate the effectiveness of the approach

mistuning based control design to improve closed loop stability margin of vehicular platoons we consider a decentralized bidirectional control of a platoon of n identical vehicles moving in a straight line the control objective is for each vehicle to maintain a constant velocity and inter vehicular separation using only the local information from itself and its two nearest neighbors each vehicle is modeled as a double integrator to aid the analysis we use continuous approximation to derive a partial differential equation pde approximation of the discrete platoon dynamics the pde model is used to explain the progressive loss of closed loop stability with increasing number of vehicles and to devise ways to combat this loss of stability if every vehicle uses the same controller we show that the least stable closed loop eigenvalue approaches zero as o 1 n2 in the limit of a large number n of vehicles we then show how to ameliorate this loss of stability by small amounts of mistuning i e changing the controller gains from their nominal values we prove that with arbitrary small amounts of mistuning the asymptotic behavior of the least stable closed loop eigenvalue can be improved to o 1 n all the conclusions drawn from analysis of the pde model are corroborated via numerical calculations of the state space platoon model

distributed control of spatially invariant systems we consider distributed parameter systems where the underlying dynamics are spatially invariant and where the controls and measurements are spatially distributed these systems arise in many applications such as the control of vehicular platoons flow control microelectromechanical systems mems smart structures and systems described by partial differential equations with constant coefficients and distributed controls and measurements for fully actuated distributed control problems involving quadratic criteria such as linear quadratic regulator lqr h sub 2 and h sub spl infin optimal controllers can be obtained by solving a parameterized family of standard finite dimensional problems we show that optimal controllers have an inherent degree of decentralization and this provides a practical distributed controller architecture we also prove a general result that applies to partially distributed control and a variety of performance criteria stating that optimal controllers inherit the spatial invariance structure of the plant connections of this work to that on systems over rings and systems with dynamical symmetries are discussed

approximating smooth planar curves by arc splines when a smooth curve is used to describe the path of a computer controlled cutting machine the path is usually approximated by many straight line segments it is preferable to describe the cutting path as an arc spline a tangent continuous piecewise curve made of circular arcs and straight line segments this paper presents an algorithm for finding an arbitrarily close arc spline approximation of a smooth 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

optimal design for synchronization of cooperative systems state feedback observer and output feedback this technical note studies synchronization of identical general linear systems on a digraph containing a spanning tree a leader node or command generator is considered which generates the desired tracking trajectory a framework for cooperative tracking control is proposed including full state feedback control observer design and dynamic output feedback control the classical system theory notion of duality is extended to networked systems it is shown that unbounded synchronization regions that achieve synchronization on arbitrary digraphs containing a spanning tree can be guaranteed by using linear quadratic regulator based optimal control and observer design methods at each node

sensitivity results for a general class of generalized vector quasi equilibrium problems with set valued maps abstract in this paper we give sufficient conditions for the local openness local closedness properties and the lower upper semicontinuity properties of the solution sets of a general model which includes as special cases many generalized vector quasi equilibrium problems with set valued maps the obtained results generalize and improve several known results an application is also given for a model which can be interpreted as a system of generalized vector quasi equilibrium problems

a new method to compute the singularities of offsets to rational plane curves given a planar curve defined by means of a real rational parametrization we prove that the affine values of the parameter generating the real singularities of the offset are real roots of a univariate polynomial that can be derived from the parametrization of the original curve without computing or making use of the implicit equation of the offset by using this result a finite set containing all the real singularities of the offset and in particular all the real self intersections of the offset can be computed we also report on experiments carried out in the computer algebra system maple showing the efficiency of the algorithm for moderate degrees

on computing of arbitrary positive integer powers for one type of tridiagonal matrices of even order abstract this paper is an extension of the work on computing of arbitrary positive integer powers for one type of tridiagonal matrices applied mathematics and computation to appear in which the general expression of the l th power l n for one type of tridiagonal matrices of order n 2 p 1 p n is presented in this new paper we derive the general expression of the l th power l n for the same type of tridiagonal matrices of order n 2 p p n expressions of eigenvectors and jordan s form of the matrix and of transforming matrix and its inverse are given too

sensitivity in mixed generalized vector quasiequilibrium problems with moving cones in this paper we consider a parametric generalized vector quasiequilibrium problem which is mixed in the sense that several different relations can simultaneously appear in this problem the moving cones and other data of the problem are assumed to be set valued maps defined in topological spaces and taking values in topological spaces or topological vector spaces the main result of this paper gives general verifiable conditions for the solution mapping of this problem to be semicontinuous with respect to a parameter varying in a topological space the result is proven with the help of notions of cone semicontinuity of set valued maps weaker than the usual concepts of semicontinuity and an assumption imposed on the set valued map whose values are the dual cones of the corresponding values of the moving cones

bounds for the extremal eigenvalues of a class of symmetric tridiagonal matrices with applications abstract we consider a class of symmetric tridiagonal matrices which may be viewed as perturbations of toeplitz matrices the toeplitz structure is destroyed since two elements on each off diagonal are perturbed based on a careful analysis we derive sharp bounds for the extremal eigenvalues of this class of matrices in terms of the original data of the given matrix in this way we also obtain a lower bound for the smallest singular value of certain matrices some numerical results indicate that our bounds are extremely good

on the eigenvalues of some tridiagonal matrices a solution is given for a problem on eigenvalues of some symmetric tridiagonal matrices suggested by william trench the method presented can be generalizable to other problems

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

eigenvalues of tridiagonal pseudo toeplitz matrices abstract in this article we determine the eigenvalues of sequences of tridiagonal matrices that contain a toeplitz matrix in the upper left block

generalized vector quasi equilibrium problems with applications in this paper we consider the generalized vector quasi equilibrium problem with or without involving condensing maps and prove the existence of its solution by using known fixed point and maximal element theorems as applications of our results we derive some existence results for a solution to the vector quasi optimization problem for nondifferentiable functions and vector quasi saddle point problem

coherence in large scale networks dimension dependent limitations of local feedback we consider distributed consensus and vehicular formation control problems specifically we address the question of whether local feedback is sufficient to maintain coherence in large scale networks subject to stochastic disturbances we define macroscopic performance measures which are global quantities that capture the notion of coherence a notion of global order that quantifies how closely the formation resembles a solid object we consider how these measures scale asymptotically with network size in the topologies of regular lattices in 1 2 and higher dimensions with vehicular platoons corresponding to the 1 d case a common phenomenon appears where a higher spatial dimension implies a more favorable scaling of coherence measures with a dimensions of 3 being necessary to achieve coherence in consensus and vehicular formations under certain conditions in particular we show that it is impossible to have large coherent 1 d vehicular platoons with only local feedback we analyze these effects in terms of the underlying energetic modes of motion showing that they take the form of large temporal and spatial scales resulting in an accordion like motion of formations a conclusion can be drawn that in low spatial dimensions local feedback is unable to regulate large scale disturbances but it can in higher spatial dimensions this phenomenon is distinct from and unrelated to string instability issues which are commonly encountered in control problems for automated highways

on computing of arbitrary positive integer powers for one type of tridiagonal matrices in this paper we derive the general expression of the l th power email protected n for one type of tridiagonal matrices of order n 2p 1 email protected n

brief paper an internal model principle is necessary and sufficient for linear output synchronization output synchronization of a network of heterogeneous linear state space models under time varying and directed interconnection structures is investigated it is shown that assuming stabilizability and detectability of the individual systems and imposing very mild connectedness assumptions on the interconnection structure an internal model requirement is necessary and sufficient for synchronizability of the network to polynomially bounded trajectories the resulting dynamic feedback couplings can be interpreted as a generalization of existing methods for identical linear systems

spectra of certain large tridiagonal matrices abstract we characterize the eigenvalues and eigenvectors of a class of complex valued tridiagonal n by n matrices subject to arbitrary boundary conditions i e with arbitrary elements on the first and last rows of the matrix for large n we show there are up to 4 eigenvalues the so called special eigenvalues whose behavior depends sensitively on the boundary conditions the other eigenvalues the so called regular eigenvalues vary very little as function of the boundary conditions for large n we determine the regular eigenvalues up to o n 2 and the special eigenvalues up to o n for some 0 1 the components of the eigenvectors are determined up to o n 1 the matrices we study have important applications throughout the sciences among the most common ones are arrays of linear dynamical systems with nearest neighbor coupling and discretizations of second order linear partial differential equations in both cases we give examples where specific choices of boundary conditions substantially influence leading eigenvalues and therefore the global dynamics of the system

distributed control of networked dynamical systems static feedback integral action and consensus this paper analyzes distributed control protocols for first and second order networked dynamical systems we propose a class of nonlinear consensus controllers where the input of each agent can be written as a product of a nonlinear gain and a sum of nonlinear interaction functions by using integral lyapunov functions we prove the stability of the proposed control protocols and explicitly characterize the equilibrium set we also propose a distributed proportional integral pi controller for networked dynamical systems the pi controllers successfully attenuate constant disturbances in the network we prove that agents with single integrator dynamics are stable for any integral gain and give an explicit tight upper bound on the integral gain for when the system is stable for agents with double integrator dynamics throughout the paper we highlight some possible applications of the proposed controllers by realistic simulations of autonomous satellites power systems and building temperature control

semicontinuity of the solution set of parametric multivalued vector quasiequilibrium problems abstract in this paper we establish sufficient conditions for the solution set of parametric multivalued vector quasiequilibrium problems to be semicontinuous all kinds of semicontinuity are considered lower semicontinuity upper semicontinuity hausdorff upper semicontinuity and closedness moreover we investigate both the weak and strong solutions of quasiequilibrium problems

rigidity of h type groups on h type groups n the left invariant horizontal vector fields span a subbundle of the tangent bundle called the horizontal bundle hn generalized contact mappings f on n are smooth mappings which preserve hn the question is how many such mappings exist in the case of the heisenberg group these are the contact mappings in the classical sense and they exist in abundance in this paper it is shown that if the dimension of the center of n is at least three then the generalized contact mappings are in the automorphism group of a finite dimensional lie algebra g the elements in vec g are the infinitesimal generators of local one parameter subgroups of generalized contact transformations rigidity is defined as the property that vec g is finite dimensional for the case of the complexified heisenberg group i e the case when the dimension of the center of n is two it has been shown rr that g is infinite dimensional

geometric hermite interpolation with tschirnhausen cubics explicit formulae are found that give the unique tschirnhausen cubic that solves a geometric hermite interpolation problem that solution is used to create a planar g1 spline by joining segments of tschirnhausen cubics if the geometric hermite data is from a smooth function the tschirnhausen cubic approximates the smooth function the error in the approximation of a short segment of length h can be expressed as a power series in h the error is o h4 and the coefficient of the leading term is found

c 2 hermite interpolation by pythagorean hodograph space curves we solve the problem of c 2 hermite interpolation by pythagorean hodograph ph space curves more precisely for any set of c 2 space boundary data two points with associated first and second derivatives we construct a four dimensional family of ph interpolants of degree 9 and introduce a geometrically invariant parameterization of this family this parameterization is used to identify a particular solution which has the following properties first it preserves planarity i e the interpolant to planar data is a planar ph curve second it has the best possible approximation order 6 third it is symmetric in the sense that the interpolant of the reversed set of boundary data is simply the reversed original interpolant this particular ph interpolant is exploited for designing algorithms for converting possibly piecewise analytical curves into a piecewise ph curve of degree 9 which is globally c 2 and for simple rational approximation of pipe surfaces with a piecewise analytical spine curve the algorithms are presented along with an analysis of their error and approximation order

passivity as a design tool for group coordination we pursue a group coordination problem where the objective is to steer the differences between output variables of the group members to a prescribed compact set to stabilize this set we study a class of feedback rules that are implementable with local information available to each member when the information flow between neighboring members is bidirectional we show that the closed loop system exhibits a special interconnection structure which inherits the passivity properties of its components by exploiting this structure we develop a passivity based design framework which results in a broad class of feedback rules that encompass as special cases some of the existing formation stabilization and group agreement designs in the literature the passivity approach offers additional design flexibility compared to these special cases and systematically constructs a lurie type lyapunov function for the closed loop system we further study the robustness of these feedback laws in the presence of a time varying communication topology and present a persistency of excitation condition which allows the interconnection graph to lose connectivity pointwise in time as long as it is established in an integral sense

powers of tridiagonal matrices with constant diagonals abstract in this paper we derive a general expression for the entries of the qth power q n of the n n complex tridiagonal matrix tridiag n a 1 a 0 a 1 for all n n in terms of the chebyshev polynomials of the second kind

on computing of arbitrary positive integer powers for one type of even order skew symmetric tridiagonal matrices with eigenvalues on imaginary axis ii this paper is an extension of the work on computing of arbitrary positive integer powers for one type of even order skew symmetric tridiagonal matrices with eigenvalues on imaginary axis i appl math comput in press in which the general expression of the lth power l n for one type of tridiagonal matrices of order n 2p p n is given in this new paper we present the complete derivation of this general expression expressions of eigenvectors of the matrix and of the transforming matrix and its inverse are given too

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

transients of platoons with asymmetric and different laplacians abstract we consider an asymmetric control of platoons of identical vehicles with nearest neighbor interaction recent results show that if the vehicle uses different asymmetries for position and velocity errors the platoon has a short transient and low overshoots in this paper we investigate the properties of vehicles with friction to achieve consensus an integral part is added to the controller making the vehicle a third order system we show that the parameters can be chosen so that the platoon behaves as a wave equation with different wave velocities simulations suggest that our system has a better performance than other nearest neighbor scenarios moreover an optimization based procedure is used to find the controller properties

a unified pythagorean hodograph approach to the medial axis transform and offset approximation algorithms based on pythagorean hodographs ph in the euclidean plane and in minkowski space share common goals the main one being rationality of offsets of planar domains however only separate interpolation techniques based on these curves can be found in the literature it was recently revealed that rational ph curves in the euclidean plane and in minkowski space are very closely related in this paper we continue the discussion of the interplay between spatial mph curves and their associated planar ph curves from the point of view of hermite interpolation on the basis of this approach we design a new simple interpolation algorithm the main advantage of the unifying method presented lies in the fact that it uses after only some simple additional computations an arbitrary algorithm for interpolation using planar ph curves also for interpolation using spatial mph curves we present the functionality of our method for g 1 hermite data however one could also obtain higher order algorithms

positive integer powers of complex skew symmetric circulant matrices abstract in j rimas on computing of arbitrary positive integer powers for one type of odd order symmetric circulant matrices i applied mathematics and computation 165 2005 137 141 j rimas on computing of arbitrary positive integer powers for one type of even order symmetric circulant matrices i applied mathematics and computation 172 2006 86 90 rimas derived a general expression for the entries of the qth power q n of the n n real symmetric circulant matrix circ n 0 1 0 0 0 0 1 for all n n in this paper we present an extension of that interesting work deriving a similar expression for the entries of the positive integer powers of any complex symmetric circulant matrix

stepwise stability for the heat equation with a nonlocal constraint the authors prove the stepwise stability for a finite difference scheme for the heat equation with an integral constraint the resulting matrix is nonsymmetric and does not have the usual band structure the proof is based on the method of matrix analysis the eigenvalues of several matrices are found explicitly or their location described precisely this method relies upon the relationship of the characteristic polynomials of these matrices with orthogonal polynomials

positive integer powers of certain complex tridiagonal matrices in this paper we firstly present a general expression for the entries of the rth email protected n power of a certain n square complex tridiagonal matrix in terms of the chebyshev polynomials of the first kind secondly we obtain a complex factorization formula for generalized fibonacci pell numbers obeying also classical fibonacci and pell numbers

approximate duality for vector quasi equilibrium problems and applications abstract in this paper we introduce new versions of dual problems of a vector quasi equilibrium problem with set valued maps and we give an duality result between approximate solutions of the primal and dual problems as the first application of the main result we obtain an duality for a vector quasi equilibrium problem whose solutions are understood in the sense of proper efficiency the second application is devoted to an duality for a vector optimization problem with set valued maps

new nonlinear scalarization functions and applications abstract in this paper new nonlinear scalarization functions which are different from the gerstewitz function are introduced some properties of these functions are discussed and are used to prove new results on the existence of solutions of generalized vector quasi equilibrium problems with moving cones and the lower semicontinuity of solution mappings of parametric vector quasi equilibrium problems detailed comparisons between our results and those obtained by using the gerstewitz function for existence theorems and by other approaches for the case of solution stability are given illustrating examples are provided

algebraic analysis of offsets to hypersurfaces in this paper we present a complete algebraic analysis of degeneration and existence of simple and special components of generalized offsets to irreducible hypersurfaces over algebraically closed fields of characteristic zero more precisely we analyze the degeneration situations when offseting and we state that there exist at most a finite set of distances for which the offset of a hypersurface may degenerate as a consequence of this analysis an algorithmic method to determine such distances is derived furthermore as an application of these results a complete degeneration analysis of the generalized offset to the sphere is developed in addition we study the existence of simple and special components of the offset in this context we prove that in the case of classical offsets there always exists at least one simple component and in the case of generalized offsets we prove that for almost every distance and for almost every isometry all components of the offset are simple

powers of real persymmetric anti tridiagonal matrices with constant anti diagonals in this paper we present an eigenvalue decomposition with an orthogonal eigenvector matrix for any real persymmetric anti tridiagonal matrix with constant anti diagonals furthermore we derive a general expression for the entries of the powers of these anti tridiagonal matrices in terms of the chebyshev polynomials of the second kind

distributed control design for spatially interconnected systems this paper deals with analysis synthesis and implementation of distributed controllers designed for spatially interconnected systems we develop a state space framework for posing problems of this type and focus on systems whose model is spatially discrete in this paper analysis and synthesis results are developed for this class of systems using the l sub 2 induced norm as the performance criterion the results are stated in terms of linear matrix inequalities and are thus readily amenable to computation a special implementation of the resulting controllers is presented which is particularly attractive for distributed operation of the controller several examples are provided to further illustrate the application of the results

bezout matrices subresultant polynomials and parameters the main purpose of this paper is to analyze new determinantal expressions which define the subresultant sequence of two polynomials when the coefficients of such polynomials depend on parameters

upper semicontinuity in a parametric general variational problem and application in this paper we give sufficient conditions for the upper semicontinuity property of the solution mapping of a general model which includes many generalized vector quasi equilibrium problems with set valued maps as special cases the main result generalizes and improves several recent results an example is given to illustrate such generalization and improvement the main result is also applied to a model which can be interpreted as a system of generalized vector quasi equilibrium problems with moving cones the main tools of the paper are some new notions of cone semicontinuity properties and openness closedness properties of families of set valued maps

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

a multi temperature kinetic ising model and the eigenvalues of some perturbed jacobi matrices in this paper we analyze the eigenvalues of some perturbed jacobi matrices the results contain as particular cases the known spectra of several classes of tridiagonal matrices studied recently as a motivation we discuss a three and a four temperature kinetic ising model that can be analyzed using some perturbed jacobi matrices the analytical results can also be used for the associated reaction diffusion systems to solve for the particle density

on computing of arbitrary positive integer powers for tridiagonal matrices with elements 1 0 0 0 1 in principal and 1 1 1 1 in neighbouring diagonals ii this paper is an extension of the work j rimas on computing of arbitrary positive integer powers for tridiagonal matrices with elements 1 0 0 0 1 in principal and 1 1 1 1 in neighbouring diagonals i applied mathematics and computation in press doi 10 1016 j amc 2006 07 145 in which the general expression of the lth power l n for one type of tridiagonal matrices of arbitrary order is given in this new paper we present the complete derivation of this general expression expressions of eigenvectors of the matrix and of the transforming matrix and its inverse are given too