characterization of oriented graphs of rank 2 abstract let d be an oriented graph i e a digraph in which all arcs are not symmetric with vertex set 1 n the adjacency matrix a a i j of d is the n n matrix defined as a i j 1 if ij is an arc of d and a i j 0 if ij is not an arc of d the rank of d is defined to be the rank of its adjacency matrix in this paper oriented graphs of rank 2 are characterized

geometric description of vakonomic and nonholonomic dynamics comparison of solutions we treat the vakonomic dynamics with general constraints within a new geometric framework which can be useful in the study of optimal control problems we compare our formulation with the one of vershik and gershkovich in the case of linear constraints we show how nonholonomic mechanics also admits a new geometrical description which allows us to develop an algorithm of comparison between the solutions of both dynamics examples illustrating the theory are treated

nonholonomic versus vakonomic dynamics on a riemann cartan manifold for the chaplygin s nonholonomic constrained systems the constraint manifold can be endowed with riemann cartan geometric structure by nonholonomic mapping into a riemann manifold the two kinds of existing dynamics nonholonomic dynamics and vakonomic dynamics are compared in the framework of riemann cartan geometry it is proved that the equations of motion for nonholonomic and vakonomic dynamics are described by the equations of autoparallel and geodesic trajectories on the riemann cartan constraint manifold respectively if the metricity condition of riemann cartan connection is satisfied the torsion contorsion of the riemann cartan manifold characterizes the difference between the autoparallel and geodesic trajectories as well as the distinction between the nonholonomic and vakonomic equations

d alembert lagrange analytical dynamics for nonholonomic systems the d alembert lagrange principle dlp is designed primarily for dynamical systems under ideal geometric constraints although it can also cover linear velocity constraints its application to nonlinear kinematic constraints has so far remained elusive mainly because there is no clear method whereby the set of linear conditions that restrict the virtual displacements can be easily extracted from the equations of constraint on recognition that the commutation rule traditionally accepted for velocity displacements in lagrangian dynamics implies displaced states that do not satisfy the kinematic constraints we show how the property of possible displaced states can be utilized ab initio so as to provide an appropriate set of linear auxiliary conditions on the displacements which can be adjoined via lagrange s multipliers to the d alembert lagrange equation to yield the equations of state and also new transpositional relations for nonholonomic systems the equations of state so obtained for systems under general nonlinear velocity and acceleration constraints are shown to be identical with those derived in appendix a from the quite different gauss principle the present advance therefore solves a long outstanding problem on the application of dlp to ideal nonholonomic systems and as an aside provides validity to axioms as the chetaev rule previously left theoretically unjustified a more general transpositional form of the boltzmann hamel equation is also obtained

skew rank of an oriented graph in terms of matching number abstract an oriented graph g is a digraph without loops and multiple arcs where g is called the underlying graph of g let s g denote the skew adjacency matrix of g the rank of s g is called the skew rank of g denoted by sr g which is even since s g is skew symmetric li and yu 2015 12 proved that the skew rank of an oriented unicyclic graph g is either 2 m g 2 or 2 m g where m g denotes the matching number of g in this paper we extend this result to general cases it is proved that the skew rank of an oriented connected graph g is an even integer satisfying 2 m g 2 g sr g 2 m g where g e g v g 1 is the number of fundamental cycles also called the first betti number besides the oriented graphs satisfying sr g 2 m g 2 g are characterized definitely

extremal values of the trace norm over oriented trees abstract the trace norm of the digraph d is defined as n d i 1 n i where 1 2 n 0 are the singular values of the adjacency matrix a of d i e the square roots of the eigenvalues of a a we find the extremal values of n over the set of oriented trees and over the set of oriented paths

on the relationship between the skew rank of an oriented graph and the rank of its underlying graph abstract an oriented graph g is a digraph without loops and multiple arcs where g is the underlying graph of g let s g denote the skew adjacency matrix of g and a g be the adjacency matrix of g the rank resp skew rank of g resp g written as r g resp s r g refers to the rank of a g resp s g it is natural and interesting to study the relationship between s r g and r g wong ma and tian european j combin 54 2016 76 86 30 determined the sharp upper bound on s r g r g as a continuance of it in this paper a sharp lower bound on s r g r g is determined as well a sharp upper bound on s r g r g is determined all the corresponding extremal oriented graphs g are characterized respectively

equivalence of the dynamics of nonholonomic and variational nonholonomic systems for certain initial data in this paper we discuss the necessary and sufficient conditions for the equivalence of the dynamics of nonholonomic mechanics and variational nonholonomic vakonomic dynamics for certain initial conditions we derive a priori results for identifying equivalence and specializing to abelian chaplygin systems prove that equivalence results if and only if the constrained nonholonomic equations are lagrangian we eliminate the need to solve the variational nonholonomic problem when checking equivalence by obtaining explicit formulae for the system s multipliers and then derive conditions under which the multiplier free lagrangian gives equivalence of the dynamics we consider nonholonomic systems possessing invariant measures showing when equivalence and hamiltonization are the same we also derive conditions under which measure preserving systems exhibit equivalence we apply the results to many of the known nonholonomic systems

the rank of a signed graph in terms of the rank of its underlying graph abstract let g be a signed graph and a be its adjacency matrix where g is the underlying graph of the rank r of is the rank of a we know that for a signed graph g is balanced if and only if g g that is when is balanced then r r g where r g is the rank of the underlying graph g of a natural problem is that how about the relations between the rank of an unbalanced signed graph and the rank of its underlying graph in this paper we first prove that r g 2 d g r r g 2 d g for an unbalanced signed graph with d g 1 where d g e g v g g is the dimension of cycle spaces of g g is the number of connected components of g as an application we also prove that 1 d g r r g 1 d g for an unbalanced signed graph with d g 1 all corresponding extremal graphs are characterized

a characterization of graphs with rank 4 the rank of a graph g is defined to be the rank of its adjacency matrix in this paper we consider the following problem what is the structure of a connected graph with rank 4 this question has not yet been fully answered in the literature and only some partial results are known in this paper we resolve this question by completely characterizing graphs g whose adjacency matrix has rank 4

newton s law and integrability of nonholonomic systems in this paper we consider nonholonomic control systems on riemannian manifolds such systems evolve on subbundles of tangent bundles defined by the nonholonomic constraints this paper promotes the view of such systems as the restriction to the nonholonomic subbundle of newton law type problems on the entire tangent bundle defined by in general non riemannian connections these connections should be related to specific geometric properties of the nonholonomic system we introduce a particular class of connections and demonstrate the richness of the class through four examples the rolling ball the constrained particle the rolling penny and the generalized rolling ball this class of connections is strongly related to questions of integrability of the original nonholonomic system this in turn provides additional insight into the relation between nonholonomic control systems formulated as kinematic equations and those that are formulated as the full dynamic equations

poisson structures for reduced non holonomic systems borisov mamaev and kilin have recently found certain poisson structures with respect to which the reduced and rescaled systems of certain non holonomic problems involving rolling bodies without slipping become hamiltonian the hamiltonian function being the reduced energy we study further the algebraic origin of these poisson structures showing that they are of rank 2 and therefore the mentioned rescaling is not necessary we show that they are determined up to a non vanishing factor function by the existence of a system of first order differential equations providing two integrals of motion we generalize the form of the poisson structures and extend their domain of definition we apply the theory to the rolling disc the routh s sphere the ball rolling on a surface of revolution and its special case of a ball rolling inside a cylinder

a characterization of long graphs of arbitrary rank abstract the rank of a graph g is defined to be the rank of its adjacency matrix a connected graph g of rank n is called long if it has maximum diameter among all connected graphs with rank n in this paper we characterize the long graphs of rank n for n an arbitrary positive integer

a relationship between sensitivity and stability of multivariable feedback systems both comparison sensitivity and robust stability analysis attempt to extend concepts which are well understood in scalar feedback systems to multivariable control system design although the two bodies of work have developed about a decade apart the basic philosophies are similar both attempt to maintain a system property sensitivity reduction versus stability in the presence of plant modeling errors this paper demonstrates how the similarity in philosophy translates into a similarity in results and interpretations

reduction of some classical non holonomic systems with symmetry two types of nonholonomic systems with symmetry are treated i the configuration space is a total space of a g principal bundle and the constraints are given by a connection ii the configuration space is g itself and the constraints are given by left invariant forms the proofs are based on the method of quasicoordinates in passing a derivation of the maurer cartan equations for lie groups is obtained simple examples are given to illustrate the algorithmical character of the main results

nonholonomic mechanical systems with symmetry this work develops the geometry and dynamics of mechanical systems with nonholonomic constraints and symmetry from the perspective of lagrangian mechanics and with a view to control theoretical applications the basic methodology is that of geometric mechanics applied to the formulation of lagrange d alembert generalizing the use of connections and momentum maps associated with a given symmetry group to this case we begin by formulating the mechanics of nonholonomic systems using an ehresmann connection to model the constraints and show how the curvature of this connection enters into lagrange s equations unlike the situation with standard configuration space constraints the presence of symmetries in the nonholonomic case may or may not lead to conservation laws however the momentum map determined by the symmetry group still satisfies a useful differential equation that decouples from the group variables this momentum equation which plays an important role in control problems involves parallel transport operators and is computed explicitly in coordinates an alternative description using a body reference frame relates part of the momentum equation to the components of the euler poincar e equations along those symmetry directions consistent with the constraints one of the purposes of this paper is to derive this evolution equation for the momentum and to distinguish geometrically and mechanically the cases where it is conserved and those where it is not an example of the former is a ball or vertical disk rolling on a flat plane and an example of the latter is the snakeboard a modified version of the skateboard which uses momentum coupling for locomotion generation we construct a synthesis of the mechanical connection and the ehresmann connection defining the constraints obtaining an important new object we call the nonholonomic connection when the nonholonomic connection is a principal connection for the given symmetry group we show how to perform lagrangian reduction in the presence of nonholonomic constraints generalizing previous results which only held in special cases several detailed examples are given to illustrate the theory september 1994 revised march 1995 revised june 1995

quantizing conditionally variational nonholonomic systems in this paper we present a quantization procedure for a class of nonholonomic systems briefly mechanical systems subject to nonintegrable constraints on the velocities whose reduced mechanics is hamiltonian we illustrate the theory developed through several examples one of which demonstrates how the classical nonholonomic constraints are sometimes manifested quantum mechanically as a shift in the ground state energy of the system

mixed graphs with h rank 3 abstract in the article mohar 2016 9 mohar determined all mixed graphs with h rank 2 and used it to classify cospectral graphs with respect to their hermitian adjacency matrix constructing a class of graphs which can not be determined by their h spectrum in the present paper we investigate the h rank of mixed graphs further determining the h ranks of those mixed graphs with trees cycles and complete bipartite graphs as underlying graphs respectively moreover we characterize all mixed graphs with h rank 3 and show that all connected mixed graphs with h rank 3 can be determined by their h spectrum

optimal control and geodesic flows in this paper we analyze and generalize from the point of view of the maximum principle a class of nonlinear optimal control problems originally introduced in brockett 1994 the optimal control problems discussed here lie on the adjoint orbits of compact lie groups or on grassmann manifolds and the structure of the equations that arise depends on the metric used the so called normal metric we consider some special cases of the general problem and their meaning from the point of view of riemannian geometry and hamiltonian mechanics

the geometrical theory of constraints applied to the dynamics of vakonomic mechanical systems the vakonomic bracket a vakonomic mechanical system can be alternatively described by an extended lagrangian using the lagrange multipliers as new variables since this extended lagrangian is singular the constraint algorithm can be applied and a dirac bracket giving the evolution of the observables can be constructed