a characterization of decentralized fixed modes in terms of transmission zeros a simple development is given to show that a linear time invariant plant has a decentralized fixed mode at if and only if is a transmission zero of certain subsystems of the plant

transport in simple networks described by an integrable discrete nonlinear schrodinger equation we elucidate the case in which the ablowitz ladik al type discrete nonlinear schrodinger equation nlse on simple networks e g star graphs and tree graphs becomes completely integrable just as in the case of a simple one dimensional 1d discrete chain the strength of cubic nonlinearity is different from bond to bond and networks are assumed to have at least two semi infinite bonds with one of them working as an incoming bond the present work is a nontrivial extension of our preceding one sobirov et al phys rev e 81 066602 2010 on the continuum nlse to the discrete case we find 1 the solution on each bond is a part of the universal bond independent al soliton solution on the 1d discrete chain but it is multiplied by the inverse of the square root of bond dependent nonlinearity 2 nonlinearities at individual bonds around each vertex must satisfy a sum rule and 3 under findings 1 and 2 there exist an infinite number of constants of motion as a practical issue with the use of an al soliton injected through the incoming bond we obtain transmission probabilities inversely proportional to the strength of nonlinearity on the outgoing bonds

oscillations of networks the role of soft nodes to describe the flow of a miscible quantity on a network we consider the graph wave equation where the standard continuous laplacian is replaced by the graph laplacian the structure of the graph influences strongly the dynamics assuming the graph is forced and damped at specific nodes we derive the amplitude equations using a basis of eigenvectors of the graph laplacian these lead us to introduce the notion of soft nodes we give sufficient conditions for their existence in general graphs they can cause several effects as we show on small graphs for example the ineffectiveness of damping applied to them soft nodes may be of critical importance for complex physical networks and engineering networks like power grids

the berry keating operator on l 2 mathbb r _ gt rm d x and on compact quantum graphs with general self adjoint realizations the berry keating operator berry and keating 1999 siam rev 41 236 governing the schrodinger dynamics is discussed in the hilbert space and on compact quantum graphs it has been proved that the spectrum of hbk defined on is purely continuous and thus this quantization of hbk cannot yield the hypothetical hilbert polya operator possessing as eigenvalues the nontrivial zeros of the riemann zeta function a complete classification of all self adjoint extensions of hbk acting on compact quantum graphs is given together with the corresponding secular equation in form of a determinant whose zeros determine the discrete spectrum of hbk in addition an exact trace formula and the weyl asymptotics of the eigenvalue counting function are derived furthermore we introduce the squared berry keating operator which is a special case of the black scholes operator used in financial theory of option pricing again all self adjoint extensions the corresponding secular equation the trace formula and the weyl asymptotics are derived for h2bk on compact quantum graphs while the spectra of both hbk and h2bk on any compact quantum graph are discrete their weyl asymptotics demonstrate that neither hbk nor h2bk can yield as eigenvalues the nontrivial riemann zeros some simple examples are worked out in detail

inverse spectral problem for quantum graphs the inverse spectral problem for the laplace operator on a finite metric graph is investigated it is shown that this problem has a unique solution for graphs with rationally independent edges and without vertices having valence 2 to prove the result a trace formula connecting the spectrum of the laplace operator with the set of periodic orbits for the metric graph is established

time complexity of decentralized fixed mode verification given an interconnected system this note is concerned with the time complexity of verifying whether an unrepeated mode of the system is a decentralized fixed mode dfm it is shown that checking the decentralized fixedness of any distinct mode is tantamount to testing the strong connectivity of a digraph formed based on the system it is subsequently proved that the time complexity of this decision problem using the proposed approach is the same as the complexity of matrix multiplication this work concludes that the identification of distinct dfms by means of a deterministic algorithm rather than a randomized one is computationally very easy although the existing algorithms for solving this problem would wrongly imply that it is cumbersome this note provides not only a complexity analysis but also an efficient algorithm for tackling the underlying problem

imaging geometric graphs using internal measurements abstract this article presents two topologically motivated algorithms for reconstructing a metric graph from solutions to a wave equation on it our algorithms rely on narrowband and visibility measurements and are therefore of considerable value to urban sensing applications this is an advancement over the wideband methods traditionally used to study metric graphs our methodology exposes and separates the impact of the graph topology and geometry and is driven by a complete cohomological characterization of the space of wave equation solutions

quantum motion on a half line connected to a plane in this paper the free motion of a particle on a manifold that consists of a one dimensional and a two dimensional part connected in one point is discussed the class of admissible hamiltonians is found using the theory of self adjoint extensions particular attention is paid to those hamiltonians that allow the particle to pass through the point singularity the reflection coefficient and other quantities characterizing scattering on the connection point are calculated a possible application is also discussed

on the characterization of fixed modes in decentralized control characterizations of decentralized fixed modes in terms of remnant zeros are presented a novel criterion for testing the fixed modes is obtained the result can be used to derive an existing algebraic criterion through a simple rank evaluation of certain matrices

high energy analysis and levinson s theorem for the selfadjoint matrix schrodinger operator on the half line the matrix schrodinger equation with a selfadjoint matrix potential is considered on the half line with the general selfadjoint boundary condition at the origin when the matrix potential is integrable the high energy asymptotics are established for the related jost matrix the inverse of the jost matrix and the scattering matrix under the additional assumption that the matrix potential has a first moment levinson s theorem is derived relating the number of bound states to the change in the argument of the determinant of the scattering matrix

decentralized stabilization and pole assignment for general proper systems considered is the problem of finding existence conditions and a controller synthesis procedure using decentralized control for assigning the poles of a linear time invariant proper system described by a state space model c a b d where no assumption is made regarding the structure of d this problem has direct application to the decentralized stabilization problem decentralized robust servomechanism problem etc and is a nontrivial extension to the standard decentralized problem where it is assumed that the direct feedthrough terms either are absent or have a block diagonal structure

index theorems for quantum graphs in geometric analysis an index theorem relates the difference of the numbers of solutions of two differential equations to the topological structure of the manifold or bundle concerned sometimes using the heat kernels of two higher order differential operators as an intermediary in this paper the case of quantum graphs is addressed a quantum graph is a graph considered as a singular one dimensional variety and equipped with a second order differential hamiltonian h a laplacian with suitable conditions at vertices for the case of scale invariant vertex conditions i e conditions that do not mix the values of functions and of their derivatives the constant term of the heat kernel expansion is shown to be proportional to the trace of the internal scattering matrix of the graph this observation is placed into the index theory context by factoring the laplacian into two first order operators h a a and relating the constant term to the index of a an independent consideration provides an index formula for any differential operator on a finite quantum graph in terms of the vertex conditions it is also found that the algebraic multiplicity of 0 as a root of the secular determinant of h is the sum of the nullities of a and a

can one hear the shape of a graph we show that the spectrum of the schroperator on a finite metric graph determines uniquely the connectivity matrix and the bond lengths provided that the lengths are non commensurate and the connectivity is simple no parallel bonds between vertices and no loops connecting a vertex to itself that is one can hear the shape of the graph we also consider a related inversion problem a compact graph can be converted into a scattering system by attaching to its vertices leads to infinity we show that the scattering phase determines uniquely the compact part of the graph under similar conditions as above

inverse spectral problems for the sturm liouville operator on a d star graph abstract in this work we consider inverse spectral problems for the sturm liouville differential operator on a d star type graph with standard matching conditions in the internal vertex where the integer d 2 by using the yurko s method yurko 2008 27 yurko 2009 28 we show that 1 if the potential function q j x on a fixed edge e j is prescribed on the interval 2 then the reciprocal of d of the spectrum suffices to determine q j x on the whole interval 0 2 the 2 over d of the spectrum suffices to determine q j x on a fixed edge e j

inverse eigenvalue problems on directed graphs the differential operators id and d2 p are constructed on certain finite directed weighted graphs two types of inverse spectral problems are considered first information about the graph weights and boundary conditions is extracted from the spectrum of d2 second the compactness of isospectral sets for d2 p is established by computation of the residues of the zeta function

technical communique a graph theoretic method to find decentralized fixed modes of lti systems this paper deals with the decentralized pole assignability of interconnected systems by means of linear time invariant lti controllers a simple graph theoretic approach is proposed to identify the distinct decentralized fixed modes dfms of the system i e the unrepeated modes which cannot be moved by means of a lti decentralized controller the state space representation of the system is transformed to the decoupled form using a proper change of coordinates for any unrepeated mode a matrix is then computed which resembles the transfer function matrix of the system at some point in the complex plane a bipartite graph is constructed accordingly in terms of the computed matrix now the problem of verifying if this mode is a dfm of the system reduces to checking if the constructed graph has a complete bipartite subgraph with a certain property the sole restriction of this work is that it is only capable of identifying the distinct dfms of a system however it is axiomatic that most of the modes of the real world systems are normally distinct the primary advantage of the present paper is its simplicity compared to the existing ones which often require evaluating the rank of several matrices

a new proof of the voronoi summation formula we present a short alternative proof of the voronoi summation formula which plays an important role in dirichlet s divisor problem and has recently found an application in physics as a trace formula for a schrodinger operator on a noncompact quantum graph

complex geodesics and iterates of holomorphic maps on convex domains in ⁿ we study complex geodesics f where is the unit disk in c and belongs to a class of bounded convex domains in c n with no boundary regularity assumption along with continuity up to the boundary existence of such complex geodesics with two prescribed values z w is established as a consequence we obtain some new results from iteration theory of holomorphic self maps of bounded convex domains in c n

iteration of holomorphic maps in hilbert spaces abstract previous work gong ning chen j math anal appl 98 1984 305 313 on iteration of holomorphic maps of cn is continued the purpose of this note is to extend results given in the above mentioned reference to the case of complex hilbert spaces other comments are appended

soliton propagation on chains with simple nonlocal defects we study the propagation of solitons on complex chains built by inserting finite graphs at two sites of an unbranched chain we compare numerical findings with the results of an analytical linear approximation scheme describing the interaction of large and fast solitons with nonlocal topological defects on a chain we show that the transmission properties of the solitons strongly depend on the structure of the inserted graphs giving a tool for controlling the soliton propagation through the choice of pertinent graphs to attach to the chain

the number of eigenvalues of the matrix schrodinger operator on the half line with general boundary conditions we prove a bound of the bargmann birman schwinger type on the number of eigenvalues of the matrix schrodinger operator on the half line with the most general self adjoint boundary condition at the origin and with selfadjoint matrix potentials that are integrable and have a finite first moment

the spectrum of the hilbert space valued second derivative with general self adjoint boundary conditions abstract we consider a large class of self adjoint elliptic problems associated with the second derivative acting on a space of vector valued functions we present and survey several results that can be obtained by means of two different approaches to the study of the associated eigenvalues problems the first more general one allows to replace a secular equation which is well known in some special cases by an abstract rank condition the second one though available in general seems to apply particularly well to a specific boundary condition the sometimes dubbed anti kirchhoff condition in the literature that arises in the theory of differential operators on graphs it also permits to discuss interesting and more direct connections between the spectrum of the differential operator and some graph theoretical quantities in particular some results on the symmetry of the spectrum in either case

recovering a quantum graph spectrum from vertex data we study the question to what extent spectral information on a schrodinger operator on a finite compact metric graph subject to standard or type matching conditions can be recovered from a corresponding titchmarsh weyl function on the boundary of the graph in contrast to the case of ordinary or partial differential operators the knowledge of the titchmarsh weyl function is in general not sufficient for recovering the complete spectrum of the operator or the potentials on the edges however it is shown that those eigenvalues with sufficiently high depending on the cyclomatic number of the graph multiplicities can be recovered moreover we prove that under certain additional conditions the titchmarsh weyl function even contains the full spectral information

probabilistic methods for discrete nonlinear schrodinger equations we show that the thermodynamics of the focusing cubic discrete nonlinear schrodinger equation are exactly solvable in dimension 3 and higher a number of explicit formulas are derived

convergence of iterates of nonexpansive mappings and orbits of nonexpansive semigroups abstract let x be a uniformly convex banach space and let c be a bounded closed and convex subset of x assume that c has nonempty interior and is locally uniformly rotund let t be a nonexpansive self mapping of c if t has no fixed point in the interior of c then there exists a unique point x on the boundary of c such that each sequence of iterates of t converges in norm to x we also establish an analogous result for nonexpansive semigroups

reducible boundary conditions in coupled channels we study hamiltonians with point interactions in spaces of vector valued functions using some information from the theory of quantum graphs we describe a class of the operators which can be reduced to the direct sum of several one dimensional problems it shown that such a reduction is closely connected with the invariance under channel permutations examples are provided by some model interactions in particular the so called and the kirchhoff couplings

on the inverse scattering problem on branching graphs the inverse scattering problem on branching graphs is studied the definition of the schrodinger operator on such graphs is discussed the operator is defined with real potentials with finite first momentum and using special boundary conditions connecting values of the functions at the vertices it is shown that in general the scattering matrix does not determine the topology of the graph the potentials on the edges and the boundary conditions uniquely

some inverse scattering problems on star shaped graphs having in mind applications to the fault detection diagnosis of lossless electrical networks here we consider some inverse scattering problems for schrodinger operators over star shaped graphs we restrict ourselves to the case of minimal experimental setup consisting in measuring at most two reflection coefficients when an infinite homogeneous potential less branch is added to the central node first by studying the asymptotic behavior of only one reflection coefficient in the high frequency limit we prove the identiability of the geometry of this star shaped graph the number of edges and their lengths next we study the potential identification problem by inverse scattering noting that the potentials represent the inhomogeneities due to the soft faults in the network wirings potentials with bounded h1 norms the main result states that under some assumptions on the geometry of the graph the measurement of two reflection coefficients associated to two different sets of boundary conditions at the external vertices of the tree determines uniquely the potentials it can be seen as a generalization of the theorem of the two boundary spectra on an interval

kirchhoff s rule for quantum wires we formulate and discuss one particle quantum scattering theory on an arbitrary finite graph with n open ends and where we define the hamiltonian to be minus the laplace operator with general boundary conditions at the vertices this results in a scattering theory with n channels the corresponding on shell s matrix formed by the reflection and transmission amplitudes for incoming plane waves of energy e 0 is given explicitly in terms of the boundary conditions and the lengths of the internal lines it is shown to be unitary which may be viewed as the quantum version of kirchhoff s law we exhibit covariance and symmetry properties it is symmetric if the boundary conditions are real also there is a duality transformation on the set of boundary conditions and the lengths of the internal lines such that the low energy behaviour of one theory gives the high energy behaviour of the transformed theory finally we provide a composition rule by which the on shell s matrix of a graph is factorizable in terms of the s matrices of its subgraphs all proofs use only known facts from the theory of self adjoint extensions standard linear algebra complex function theory and elementary arguments from the theory of hermitian symplectic forms

random iterations fixed points and invariant crf horospheres in complex banach spaces abstract for holomorphic noncontractive maps on not necessarily bounded domains in complex banach spaces we establish the conditions guaranteeing locally uniform convergence of random iterations and study the existence of fixed points and boundary behaviour of iterations in particular we show that the problem concerning the existence of the horospheres determined by caratheodory reiffen finsler pseudometrics defined on unbounded domains has the solution and we prove new results of type of julia s lemma and wolff s theorem

asymptotically isospectral quantum graphs and generalised trigonometric polynomials abstract the theory of almost periodic functions is used to investigate spectral properties of schrodinger operators on metric graphs also known as quantum graphs in particular we prove that two schrodinger operators may have asymptotically close spectra if and only if the corresponding reference laplacians are isospectral our result implies that a schrodinger operator is isospectral to the standard laplacian on a may be different metric graph only if the potential is identically equal to zero

horoballs and iteration of holomorphic maps on bounded symmetric domains abstract given a fixed point free compact holomorphic self map f on a bounded symmetric domain d which may be infinite dimensional we establish the existence of a family h 0 of convex f invariant domains at a point in the boundary d of d which generalises completely wolff s theorem for the open unit disc in c further we construct horoballs at and show that they are exactly the f invariant domains when d is of finite rank consequently we show in the latter case that the limit functions of the iterates f n with weakly closed range all accumulate in one single boundary component of d

overlapping control systems with optimal information exchange this paper deals with the stabilizability of interconnected systems via linear time invariant lti decentralized controllers given a controllable and observable system with some distinct decentralized fixed modes dfms it is desired to find a suitable control structure in terms of information flow for it since a decentralized controller consists of a number of non interacting local controllers the objective here is to establish certain interactions between the local controllers in order to eliminate the undesirable dfms this objective is achieved by translating the knowledge of the system into some bipartite graphs then the notions of minimal sets and maximal subgraphs are introduced which lead to a simple combinatorial algorithm for solving the underlying problem the efficacy of the results obtained is demonstrated by an illustrative example

schrodinger operators on graphs and geometry iii general vertex conditions and counterexamples schrodinger operators on metric graphs with general vertex conditions are studied explicit spectral asymptotics is derived in terms of the spectrum of reference laplacians a geometric version of

random iterations of holomorphic contractions in locally convex spaces and of weaker contractions in uniform spaces conditions which guarantee the uniform convergence of random iterations of holomorphic contractions on unbounded domains in complex hausdorff locally convex and sequentially complete topological vector spaces are established also conditions concerning the convergence of random iterations of weaker contractions in hausdorff uniform spaces are given

an exact trace formula and zeta functions for an infinite quantum graph with a non standard weyl asymptotics we study a quantum hamiltonian that is given by the negative laplacian and an infinite chain of like potentials with strength 0 on the half line and which is equivalent to a one parameter family of laplacians on an infinite metric graph this graph consists of an infinite chain of edges with the metric structure defined by assigning an interval in 0 ln to each edge with length we show that the one parameter family of quantum graphs possesses a purely discrete and strictly positive spectrum for each 0 and prove that the dirichlet laplacian is the limit of the one parameter family in the strong resolvent sense the spectrum of the resulting dirichlet quantum graph is also purely discrete the eigenvalues are given by n n2 with multiplicities d n where d n denotes the divisor function we can thus relate the spectral problem of this infinite quantum graph to dirichlet s famous divisor problem and infer the non standard weyl asymptotics for the eigenvalue counting function based on an exact trace formula the voronoi summation formula we derive explicit formulae for the trace of the wave group the heat kernel the resolvent and for various spectral zeta functions these results enable us to establish a well defined renormalized secular equation and a selberg like zeta function defined in terms of the classical periodic orbits of the graph for which we derive an exact functional equation and prove that the analogue of the riemann hypothesis is true

concrete method for recovering the euler characteristic of quantum graphs trace formulas play a central role in the study of spectral geometry and in particular of quantum graphs the basis of our work is the result by kurasov which links the euler characteristic of metric graphs to the spectrum of their standard laplacian these ideas were shown to be applicable even in an experimental context where only a finite number of eigenvalues from a physical realization of quantum graph can be measured in the present work we analyse sufficient hypotheses which guarantee the successful recovery of we also study how to improve the efficiency of the method and in particular how to minimise the number of eigenvalues required finally we compare our findings with numerical examples surprisingly just a few dozens of eigenvalues can be enough

brief paper robust controllability and observability degrees of polynomially uncertain systems this paper deals with the class of polynomially uncertain continuous time linear time invariant lti systems whose uncertainties belong to a semi algebraic set the objective is to determine the minimum of the smallest singular value of the controllability or observability gramian over the uncertainty region this provides a quantitative measure for the robust controllability or observability degree of the system to this end it is shown that the problem can be recast as a sum of squares sos problem in the special case when the uncertainty region is polytopic the corresponding sos formulation can be simplified significantly one can apply the proposed method to any large scale interconnected system in order to identify those inputs and outputs that are more effective in controlling the system in a robust manner this enables the designer to simplify the control structure by ignoring those inputs and outputs whose contribution to the overall control operation is relatively weak a numerical example is presented to demonstrate the efficacy of the results

discrete time control of continuous systems with approximate decentralized fixed modes in this paper the discrete time control of decentralized continuous time systems which have approximate decentralized fixed modes is studied it is shown that under certain conditions discrete time controllers can improve the overall performance of the decentralized control system when a linear time invariant continuous time controller is ineffective in order to obtain these conditions a quantitative measure for different types of approximate fixed modes in a decentralized system is given in this case it is shown that discrete time zero order hold zoh controllers and in particular that generalized sampled data hold functions gshf can significantly improve the overall performance of the resultant closed loop system the proposed sampled data controller is in fact a linear time varying controller for the continuous time system