parabolic theory of the discrete p laplace operator abstract we study the discrete version of the p laplace operator based on its variational properties we discuss some features of the associated parabolic problem we prove well posedness of the problem and obtain information about positivity and comparison principles as well as compatibility with the symmetries of the underlying graph our methods consist in an interplay of the theory of subdifferentials and of combinatorial methods we conclude briefly discussing the variational properties of a handful of nonlinear generalized laplacians appearing in different parabolic equations

relations between 2 d and multichannel linear prediction a procedure for solving 2 d normal equations is derived by relating the 2 d linear prediction problem to a multichannel linear prediction problem and applying the multichannel levinson recursion

on the laplacian eigenvalues of a graph abstract in the note we present an upper bound for the spectral radius of laplacian matrix of a graph in terms of a 2 degree of a vertex

the spectrum of unbounded operator matrices with non diagonal domain abstract systems of linear evolution equations involving non diagonal boundary conditions yield operator matrices having non diagonal domains in products of banach spaces we show how to compute the spectrum of these matrices by using virtual matrix elements and characteristic operator functions

quantum ergodicity for graphs related to interval maps we prove quantum ergodicity for a family of graphs that are obtained from ergodic one dimensional maps of an interval using a procedure introduced by pakonski et al j phys a 34 9303 9317 2001 as observables we take the l 2 functions on the interval the proof is based on the periodic orbit expansion of a majorant of the quantum variance specifically given a one dimensional lebesgue measure preserving map of an interval we consider an increasingly refined sequence of partitions of the interval to this sequence we associate a sequence of graphs whose directed edges correspond to elements of the partitions and on which the classical dynamics approximates the perron frobenius operator corresponding to the map we show that except possibly for subsequences of density 0 the eigenstates of the quantum graphs equidistribute in the limit of large graphs

limits of quantum graph operators with shrinking edges abstract we address the question of convergence of schrodinger operators on metric graphs with general self adjoint vertex conditions as lengths of some of graph s edges shrink to zero we determine the limiting operator and study convergence in a suitable norm resolvent sense it is noteworthy that as edge lengths tend to zero standard sobolev type estimates break down making convergence fail for some graphs we use a combination of functional analytic bounds on the edges of the graph and lagrangian geometry considerations for the vertex conditions to establish a sufficient condition for convergence this condition encodes an intricate balance between the topology of the graph and its vertex data in particular it does not depend on the potential on the differences in the rates of convergence of the shrinking edges or on the lengths of the unaffected edges

closed geodesics and periods of automorphic forms we study the detailed structure of the distribution of eichler shimura periods of an automorphic form on a compact hyperbolic surface we show that these periods do not cluster around the asymptotic period over a homology class discovered by zelditch

isospectral graphs with identical nodal counts according to a recent conjecture isospectral objects have different nodal count sequences gnutzmann et al 2005 j phys a math gen 38 8921 33 we study generalized laplacians on discrete graphs and use them to construct the first non trivial counterexamples to this conjecture in addition these examples demonstrate a surprising connection between isospectral discrete and quantum graphs

design of a multichannel two dimensional delta domain lattice filter for noise removal in this paper a multichannel form of the two dimensional 2 d delta domain lattice filter has been designed motivated by the success of 2 d delta domain lattice filters delta domain concepts are generalized to the multichannel case by a good choice of the transformation matrix that relates the q domain and delta domain filters the channels can be handled independently which makes the algorithm simple and attractive however it is capable of handling multichannel problems even if the noise is correlated between channels resulting in extremely low snr values the proposed algorithm provides the ability of handling multichannel image frames and colored images in the so called delta domain that is known to be always better than the traditional q domain in handling ill conditioned problems on line image frames carrying gaussian and multiplicative noise simultaneously on different channels as well as color images are taken into consideration and successful results are obtained in terms of noise removal

the eigenvalue problem for networks of beams abstract in this paper we consider the spectral analysis of different models of networks of euler bernoulli beams we first give the characteristic equation for the spectrum secondly in some particular situations we show that the spectrum depends only on the structure of the graph thirdly we investigate the asymptotic behaviour of the eigenvalues by proving the so called weyl s formula

quantum graphs ii some spectral properties of quantum and combinatorial graphs the paper deals with some spectral properties of mostly infinite quantum and combinatorial graphs quantum graphs have been intensively studied lately due to their numerous applications to mesoscopic physics nanotechnology optics and other areas a schnol type theorem is proven that allows one to detect that a point belongs to the spectrum when a generalized eigenfunction with an sub exponential growth integral estimate is available a theorem on spectral gap opening for decorated quantum graphs is established its analogue is known for the combinatorial case it is also shown that if a periodic combinatorial or quantum graph has a point spectrum it is generated by compactly supported eigenfunctions scars

spectral gap for quantum graphs and their edge connectivity the spectral gap for laplace operators on metric graphs and the relation between the graph s edge connectivity is investigated in particular what happens to the gap if an edge is added to or deleted from a graph it is shown that in contrast to discrete graphs the connection between the connectivity and the spectral gap is not one to one the size of the spectral gap depends not only on the topology of the metric graph but on its geometric properties as well it is shown that adding sufficiently large edges as well as cutting away sufficiently small edges leads to a decrease of the spectral gap corresponding explicit criteria are given

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

the structural property of a class of vector valued hyperbolic equations and applications abstract in this paper we study the structural properties of a class of vector valued hyperbolic equations with appropriate boundary conditions including the spectrum determined growth condition we prove that the equations associate with a c 0 semigroup by the structural analysis we obtained a sufficient and necessary condition for being at least an eigenvalue on the imaginary axis in particular using the asymptotic analysis technique we prove that the spectrum of the operator determined by the equations is distributed in a strip parallel to the imaginary axis and is union of finitely many separable sets furthermore we prove that the root vectors of the operator are complete and there is a sequence of root vectors that forms a riesz basis with parentheses for the hilbert state space as applications of our results we give some concrete examples in controlled complex network of euler bernoulli beams

an index theory for uniformly locally finite graphs abstract an index theory for uniformly locally finite ulf graphs is developed based on the adjacency operator a acting on the space of bounded sequences defined on the vertices it turns out that the characterization by upper and lower nonnegative eigenvectors is an appropriate tool to overcome the difficulties imposed by the l setting a distinctive property of the spectral radius r a in l is the identity r sup 0 x l x 0 a x x i whilethe l 2 spectral radius r 2 of the adjacency operator satisfies r 2 inf 0 x l x 0 a x x the index i as well as other order indices can serve in classifying ulf graphs and enables connections with various graph invariants e g the chromatic number can be estimated from above by 1 r moreover results on the index i in the periodic case the regular one and for graphs having only finitely many essential ramification nodes are presented

negative energy ground states for the l2 critical nlse on metric graphs we investigate the existence of ground states with prescribed mass for the focusing nonlinear schrodinger equation with l2 critical power nonlinearity on noncompact quantum graphs we prove that unlike the case of the real line for certain classes of graphs there exist ground states with negative energy for a whole interval of masses a key role is played by a thorough analysis of gagliardo nirenberg inequalities and by estimates of the optimal constants most of the techniques are new and suited to the investigation of variational problems on metric graphs

spectral properties and asymptotic periodicity of flows in networks we combine functional analytical and graph theoretical methods in order to study flows in networks we show that these flows can be described by a strongly continuous operator semigroup on a banach space using perron frobenius spectral theory we then prove that this semigroup behaves asymptotically periodic

control of a network of euler bernoulli beams abstract the aim is to study the boundary controllability of a system modeling the vibrations of a network of n euler bernoulli beams connected by n vibrating point masses using the classical hilbert uniqueness method the control problem is reduced to the obtention of an observability inequality the solution is then expressed in terms of fourier series so that it is also enough to show that the distance between two consecutive large eigenvalues of the spatial operator involved in this evolution problem is superior to a minimal fixed value this property called spectral gap holds as soon as the roots of a function denoted by f and giving the asymptotic behaviour of the eigenvalues are all simple for a network of n 2 different beams this assumption on the multiplicity of the roots of f denoted by a is proved to be satisfied and controllability follows for higher values of n a numerical approach allows one to prove a in many situations and no counterexample has been found but the problem of giving a general proof of controllability remains open

spectral gap of the discrete laplacian on triangulations our goal in this paper is to find an estimate for the spectral gap of the laplacian on a two simplicial complex consisting on a hypergraph of a complete graph an upper estimate is given by generalizing the cheeger constant the lower estimate is obtained from the first non zero eigenvalue of the discrete laplacian acting on the functions of certain sub graphs

variational and non variational eigenvalues of the p laplacian it is well known that all the eigenvalues of the linear eigenvalue problem u q r u in rn can under appropriate conditions on q r and be characterized by minimax principles but it has been a long standing question whether that remains true for analogous equations involving the p laplacian p it will be shown that there are corresponding nonlinear eigenvalue problems pu q r u p 1sgnu in rn with 1 0 on for which not all eigenvalues are of variational type as far as we know this is the first observation of such a phenomenon and examples will be given for one and higher dimensional equations the question of exactly which eigenvalues are variational is also discussed when n 1

a sum formula for a pair of closed geodesics on a hyperbolic surface we consider an arbitrary pair of closed geodesics and the corresponding period integrals for the eigenfunctions of the laplacian on a compact hyperbolic surface a summation formula that relates geometric information about the geodesics namely the angles of intersection and lengths of common perpendiculars between them to the period integrals is proved as a corollary an asymptotic is obtained for the second moment of the period integrals for a fixed geodesic as an average over the eigenvalue with an error term that can be interpreted in terms of the geometric data

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

anomalous nodal count and singularities in the dispersion relation of honeycomb graphs we study the nodal count of the so called bi dendral graphs and show that it exhibits an anomaly the nodal surplus is never equal to 0 or the first betti number of the graph according to the nodal magnetic theorem this means that bands of the magnetic spectrum dispersion relation of such graphs do not have maxima or minima at the usual symmetry points of the fundamental domain of the reciprocal space of magnetic parameters in search of the missing extrema we prove a necessary condition for a smooth critical point to happen inside the reciprocal fundamental domain using this condition we identify the extrema as the singularities in the dispersion relation of the maximal abelian cover of the graph the honeycomb graph being an important example in particular our results show that the anomalous nodal count is an indication of the presence of conical points in the dispersion relation of the maximal universal cover we also discover that the conical points are present in the dispersion relatio

gaussian distributions on riemannian symmetric spaces statistical learning with structured covariance matrices the riemannian geometry of covariance matrices has been essential to several successful applications in computer vision biomedical signal and image processing and radar data processing for these applications an important ongoing challenge is to develop riemannian geometric tools which are adapted to structured covariance matrices this paper proposes to meet this challenge by introducing a new class of probability distributions gaussian distributions of structured covariance matrices these are riemannian analogs of gaussian distributions which only sample from covariance matrices having a preassigned structure such as complex toeplitz or block toeplitz the usefulness of these distributions stems from three features 1 they are completely tractable analytically or numerically when dealing with large covariance matrices 2 they provide a statistical foundation to the concept of structured riemannian barycentre i e frechet or geometric mean and 3 they lead to efficient statistical learning algorithms which realise among others density estimation and classification of structured covariance matrices this paper starts from the observation that several spaces of structured covariance matrices considered from a geometric point of view are riemannian symmetric spaces accordingly it develops an original theory of gaussian distributions on riemannian symmetric spaces of their statistical inference and of their relationship to the concept of riemannian barycentre then it uses this original theory to give a detailed description of gaussian distributions of three kinds of structured covariance matrices complex toeplitz and block toeplitz finally it describes algorithms for density estimation and classification of structured covariance matrices based on gaussian distribution mixture models

stabilization of the wave equation on 1 d networks in this paper we study the stabilization of the wave equation on general 1 d networks for that we transfer known observability results in the context of control problems of conservative systems see r dager and e zuazua wave propagation observation and control in 1 d flexible multi structures math appl 50 springer verlag berlin 2006 into a weighted observability estimate for dissipative systems then we use an interpolation inequality similar to the one proved in p begout and f soria j differential equations 240 2007 pp 324 356 to obtain the explicit decay estimates of the energy for smooth initial data the obtained decay rate depends on the geometric and topological properties of the network we also give some examples of particular networks in which our results apply yielding different decay rates

mathematical aspects of vacuum energy on quantum graphs we use quantum graphs as a model to study various mathematical aspects of the vacuum energy such as convergence of periodic path expansions consistency among different methods trace formulae versus method of images and the possible connection with the underlying classical dynamics in our study we derive an expansion for the vacuum energy in terms of periodic paths on the graph and prove its convergence and smooth dependence on the bond lengths of the graph for an important special case of graphs with equal bond lengths we derive a simpler explicit formula with minor changes this formula also applies to graphs with rational up to a common factor bond lengths the main results are derived using the trace formula we also discuss an alternative approach using the method of images and prove that the results are consistent this may have important consequences for other systems since the method of images unlike the trace formula includes a sum over special bounce paths we succeed in showing that in our model bounce paths do not contribute to the vacuum energy finally we discuss the proposed possible link between the magnitude of the vacuum energy and the type chaotic versus integrable of the underlying classical dynamics within a random matrix model we calculate the variance of the vacuum energy over several ensembles and find evidence that the level repulsion leads to suppression of the vacuum energy

an algorithm for direct computation of 2 d linear prediction coefficients the authors present some simplifications to the method of computing two dimensional 2 d linear prediction coefficients lpcs directly from image data using an extension of the multichannel burg algorithm this simplification results from imposing the structure of the burg algorithm the method can also be used for computing prediction errors directly from the data results of applying the method to 2 d lp coding of images and 2 d autoregressive ar spectral estimation are presented

two dimensional orthogonal lattice structures for autoregressive modeling of random fields two dimensional orthogonal lattice filters are developed as a natural extension of the 1 d lattice parameter theory the method offers a complete solution for the levinson type algorithm to compute the prediction error filter coefficients using lattice parameters from the given 2 d augmented normal equations the proposed theory can be used for the quarter plane and asymmetric half plane models depending on the indexing scheme in the prediction region it is shown that the final order backward prediction error may correspond to different quarter plane models in addition to developing the basic theory the article includes several properties of this lattice model conditions for lattice model stability and an efficient method for factoring the 2 d correlation matrix are given it is shown that the unended forward and backward prediction errors form orthogonal bases a simple procedure for reduced complexity 2 d orthogonal lattice filters is presented the proposed 2 d lattice method is compared with other alternative structures both in terms of conceptual background and complexity examples are considered for the given covariance case

eigenvalue bracketing for discrete and metric graphs abstract we develop eigenvalue estimates for the laplacians on discrete and metric graphs using various types of boundary conditions at the vertices of the metric graph via an explicit correspondence of the equilateral metric and discrete graph spectrum also in the exceptional values of the metric graph corresponding to the dirichlet spectrum we carry over these estimates from the metric graph laplacian to the discrete case we apply the results to covering graphs and present examples where the covering graph laplacians have spectral gaps

well posedness and symmetries of strongly coupled network equations we consider a class of evolution equations taking place on the edges of a finite network and allow for feedback effects between different possibly non adjacent edges this generalizes the setting that is common in the literature where the only considered interactions take place at the boundary i e in the nodes of the network we discuss well posedness of the associated initial value problem as well as contractivity and positivity properties of its solutions finally we discuss qualitative properties that can be formulated in terms of invariance of linear subspaces of the state space i e of symmetries of the associated physical system applications to a neurobiological model as well as to a system of linear schrodinger equations on a quantum graph are discussed

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

a stabilization algorithm for multichannel multidimensional linear prediction of imagery the authors have investigated the stability problems observed in multichannel multidimensional linear predictive modeling of images it is known that based on a positive definite autocorrelation matrix singular values of the matrix h delta sub i 1 herm delta sub i 1 must lie inside the unit circle for a stable solution where delta sub i 1 is the normalized partial correlation matrix and herm denotes the hermitian operator the authors have developed a two step stabilization method to obtain stabilized linear prediction coefficients for short term analysis windows formed digitized images the authors have modified the multichannel levinson recursion algorithm to include this stability procedure they have tested the algorithm on numerous images commonly used in image coding and the results are very impressive

a characteristic equation associated to an eigenvalue problem on c2 networks abstract a characteristic equation for the spectrum of an eigenvalue problem on c 2 networks is deduced and those special cases are investigated in which the spectrum depends mainly on the structure of the abstract graph belonging to the c 2 network

on the recursive computation of interpolators with nonrectangular masks an algorithm is presented for the recursive computation of finite order interpolators and predictors for scalar random processes on multidimensional parameter sets the algorithm is able to achieve computational savings even for interpolation filters with nonrectangularly shaped support because it avoids direct exploitation of toeplitz structure in the normal equations by using the displacement invariance structure of the interpolation filter and the low displacement rank properties of the correlation matrix the paper presents the method for step by step growth of the interpolation support and shows that an interpolation filter can be constructed from the interpolator of the previous step along with certain interpolators corresponding to the boundary points of the filter support in the previous step when restricted to rectangularly shaped masks the algorithm has the same order of complexity as previous algorithms for solving toeplitz block toeplitz systems

new efficient 2 d lattice structures for general autoregressive modeling of random fields in this paper we present a new efficient general 2 d autoregressive ar lattice modeling technique of random fields our proposed approach is capable of simultaneously providing all possible types of 2 d causal quarter plane qp and asymmetric half plane ashp ar models for an arbitrary rectangular shape of the prediction support region psr it is shown that it is also possible to obtain various noncausal half plane ar models without any additional computational cost this new lattice structure introduces only one row or one column of new observation points into the existing rectangular psr when the order of the model increases in the horizontal or vertical direction at each of the recursive order incrementation stage respectively starting with a given random data field a set of auxiliary forward and backward prediction error fields pefs for the horizontal and vertical directions are generated after recursive order updates to the desired size of the rectangular psr all types of causal qp ashp and noncausal half plane 2 d lattice models are obtained by deriving the appropriate pef from the remaining ones in addition to developing the basic theory the presentation includes the derivation of the synthesis model transfer functions from the calculated lattice parameters it is shown that the proposed auxiliary horizontal and vertical backward prediction error filters inherently generates the mutually orthogonal realization vectors and thus are optimum for all stages the theory has been confirmed by computer simulations

edge connectivity and the spectral gap of combinatorial and quantum graphs we derive a number of upper and lower bounds for the first nontrivial eigenvalue of laplacians on combinatorial and quantum graph in terms of the edge connectivity i e the minimal number of edges which need to be removed to make the graph disconnected on combinatorial graphs one of the bounds corresponds to a well known inequality of fiedler of which we give a new variational proof on quantum graphs the corresponding bound generalizes a recent result of band and levy all proofs are general enough to yield corresponding estimates for the p laplacian and allow us to identify the minimizers based on the betti number of the graph we also derive upper and lower bounds on all eigenvalues which are asymptotically correct i e agree with the weyl asymptotics for the eigenvalues of the quantum graph in particular the lower bounds improve the bounds of friedlander on any given graph for all but finitely many eigenvalues while the upper bounds improve recent results of ariturk our estimates are also used to derive bounds on the eigenvalues of the normalized laplacian matrix that improve known bounds of spectral graph theory

bounds on the laplacian spectral radius of graphs the spectral radius of a graph is the largest eigenvalue of adjacency matrix of the graph and its laplacian spectral radius is the largest eigenvalue of the laplacian matrix which is the difference of the diagonal matrix of vertex degrees and the adjacency matrix some sharp bounds are obtained for the laplacian spectral radii of connected graphs as consequences some sharp upper bounds of the nordhaus gaddum type are also obtained for the sum of laplacian spectral radii of a connected graph and its connected complement

tight wavelet frames on multislice graphs we present a framework for the design of wavelet transforms tailored to data defined on multislice graphs i e multiplex or dynamic graphs graphs with multiple types of interactions are ubiquitous in real life motivating the extension of wavelets to these complex domains our framework generalizes the recently proposed spectral graph wavelet transform sgwt d hammond p vandergheynst and r gribonval wavelets on graphs via spectral graph theory appl comput harmon anal vol 30 pp 129 150 mar 2011 which is designed in the spectral frequency domain of an arbitrary finite weighted graph we extend the sgwt to form a tight frame which conserves energy in the wavelet domain and define the relationship between conventional and spectral graph wavelets we then propose a design for multislice graphs that is based on the higher order singular value decomposition hosvd a powerful tool from multilinear algebra in particular the multiple adjacency matrices are stacked to form a tensor and the hosvd decomposition provides information about its third order structure analogous to that provided by matrix factorizations we obtain a set of eigennetworks and from these graph wavelets which exploit the variability across the graphs we demonstrate the feasibility of our method 1 by capturing different orientations of a gray scale image and 2 by decomposing brain signals from functional magnetic resonance imaging we show its effectiveness to identify variability across graph edges and provide meaningful decompositions

eigenvalue estimates for the laplacian with anti kirchhoff conditions on a metric tree abstract this paper deals with the eigenvalue estimates for laplace operators with anti kirchhoff conditions on compact trees the results include lower estimates for k involving the edge lengths or diameter of when is a star and upper estimates for k involving the average edge length or the diameter of when is a general tree we get the upper eigenvalue estimates by proving a monotonicity principle for the eigenvalues with respect to the domain graph two geometric versions of the ambartsumian theorem for are also proven

multidimensional multiple order complex parametric model identification this paper presents a way to access both the multiple order and parameters of a multidimensional complex number autoregressive ar model through matrix factorization the principle of this technique consists of the transformation of the multidimensional model to a pseudo simple input simple output ar model then performing factorization of the covariance matrix of the data this factorization then leads to a recursive form of the parameter and order estimation this paper makes two principal contributions the first is a generalization of one dimensional factored form algorithm and the second is that it makes it possible to access all the possible different orders and parameters of a multidimensional complex number ar model of any dimension whereas classical approaches are limited to at most four dimensional models computer simulation results are provided to illustrate the behavior of this method

eichler shimura homology intersection numbers and rational structures on spaces of modular forms in this paper we reinterpret the main results of 8 using the intersection theory of cycles with coefficients to this end we give a functorial interpretation of eichler schimura periods introduction in this paper we reformulate the main results of s katok 8 in terms of homology theory with local coefficients in order to do so we srst give an interpretation of the period of a cusp form f of weight 2m 2 m 1 over a closed geodesic y as the kronecker index of the cohomoiogy class sh f with local coefficients associated to f by shimura 13 and a 1 cycle with dual local coefficients associated to y we will call this period the eichler shimura period there are other interpretations of eichler shimura periods but these seem to work only for arithmetic fundamental groups whereas the defining integral of the period makes sense quite generally there is a simple idea from group homology theory which is the basis of our reformulation that of a decomposable cycle we now give the definition suppose a is a representation of a group r into gl v where v is a finite dimensional vector space then we may consider the eilenberg mac lane homology group hl r we observe that if ey is an element of r and v is an element of v satisfying a y v v a ey invariant then the 1 chain oy 3 v satisfies a ty x v a ty v v o consequently ty x v is a 1 cycle and gives rise to an element of hl r a which we also denote oy q3 v definitions we will call such cycles decomposable cycles and we will say a homology class is decomposable if it has a decomposable representative remark a general 1 cycle is of the form e 1 1 aj x vj where n n 5 tyj vj vj o j l j l for more details on homology and cohomology of groups see brown 3 we can now give our formulation of eichler shimura periods let v be the space of the standard 2 dimensional complex representation of sl2 r and v be its dual the corresponding real space we will denote by vr we let smv denote received by the editors january 31 1986 1980 mathematics subject classification 1985 revision primary llf06 20h10 55n25 secondary 22e40 the second author was partially supported by an nsf grant t1987 american mathematical society 0002 9947 87 1 00 25 per page

spectrum of a network of euler bernoulli beams abstract a network of n flexible beams connected by n vibrating point masses is considered the spectrum of the spatial operator involved in this evolution problem is studied if 2 is any real number outside a discrete set of values s and if is an eigenvalue then it satisfies a characteristic equation which is given the associated eigenvectors are also characterized if 2 lies in s and if the n beams are identical same mechanical properties another characteristic equation is available it is not the case for different beams no general result can be stated some numerical examples and counterexamples are given to illustrate the impossibility of such a generalization at last the asymptotic behaviour of the eigenvalues is investigated by proving the so called weyl s formula

poincare perron s method for the dirichlet problem on stratified sets abstract we extend standard poincare perron s method to the dirichlet problem on a class of multistructures this method is based on the spherical mean theorem the construction of fundamental solutions and on harnack s inequality on such domains that we first establish

on the identification of systems from data measurements using arma lattice models methods for the autoregressive moving average arma modeling of digital systems using a two channel autoregressive ar lattice are presented one method has the advantage that one half of the lattice parameters are zero when the system s excitation signal is white noise

spectral asymptotics for quantum graphs with equal edge lengths if q is smooth on the edges of a finite metric graph with all edges of length one the eigenvalues of the schrodinger operator d2 q with standard vertex conditions have generalized asymptotic expansions to all orders

simplicity of eigenvalues and non vanishing of eigenfunctions of a quantum graph we prove that after an arbitrarily small adjustment of edge lengths the spectrum of a compact quantum graph with type vertex conditions can be simple we also show that the eigenfunctions with the exception of those living entirely on a looping edge can be made to be non vanishing on all vertices of the graph as an application of the above result we establish that the secular manifold also called determinant manifold of a large family of graphs has exactly two smooth connected components

closed geodesics periods and arithmetic of modular forms is the upper half plane and f is a discrete cocompact subgroup of sl 2 n acting by fractional linear transformations we demonstrate that in this case the periods over closed geodesics play a role somewhat similar to that of fourier coefficients of modular forms on sl 2 z and its congruence subgroups more specifically those periods uniquely determine a modular form theorem 2 this result is valid for cusp forms on any fuchsian group of the first kind with or without cusps and is closely related to the study of relative poincar6 series associated to closed geodesics for each integer k 2 and each closed geodesic 7o we define special cusp forms of weight 2k called relative poincar6 series ok t ol and prove that they generate the whole space s2k f of cusp forms theorem 1 in w 3 we give an expression for periods of a relative poincar6 series over closed geodesics in purely geometrical terms through the intersection of the corresponding geodesics theorem 3 an application of theorems 1 and 3 to arithmetic subgroups of sl 2 1r gives two natural rational structures o n s2k l n theorem 4 the relative poincar6 series have been studied for general f by petersson 18 19 and hejhal 5 g 2 wolpert 23 gives a basis of s4 f for g 2 for sl 2 tz the relative poincar6 series have been studied by zagier 25 kohnen 8 kohnen and zagier 9 and kramer 13 a related problem of constructing cusp forms of weight two associated to closed geodesics has been treated by kudla and millson in 143 in connection with the problem of

the fenchel nielsen deformation the uniformization theorem provides that a riemann surface s of negative euler characteristic has a metric of constant curvature 1 a hyperbolic structure can be understood in terms of its deformations unfortunately the variation of the hyperbolic metric arising from a deformation is not determined by local data if z is a conformal coordinate for s and a ii h ii 00 1 a beltrami differential then dw dz p dj defines a new conformal coordinate and structure si the tensor p may vanish on the open set 0 c s and yet the hyperbolic metrics of s and si do not coincide on this phenomenon presents a s erious difficulty in the deformation theory of hyperbolic structures the fenchel nielsen twist deformation does not share this defect the deformation is defined by cutting the surface along a simple closed geodesic rotating one side of the cut relative to the other and attaching the sides in their new position a geodesic intersecting the cut is deformed to a broken geodesic the hyperbolic structure in the complement of the cut extends naturally to a hyperbolic structure on the new surface the deformation is concentrated at the cut the basic idea for the deformation appears in the work of fricke klein dehn and fenchel nielsen 71 8 in the fenchel nielsen manuscript the deformation was considered extensively coordinates for teichmiller space are defined in terms of the deformation the proper definition is given by a fuchsian group construction calculations are made in terms of unimodular matrices this approach is often cumbersome our basic objective is to give a description of the deformation in terms of quasiconformal mappings the bers embedding of teichmiiller space is central to the analytic theory of deformations using our characterization we calculate the first variation of the bers embedding for the fenchel nielsen deformation the result is a poincare series 6 which already appears in the work of petersson 16

existence and uniqueness results for a nonlinear caputo fractional boundary value problem on a star graph abstract in this paper we study a nonlinear caputo fractional boundary value problem on a star graph by means of transformation an equivalent system of fractional boundary value problem is obtained then we establish the existence and uniqueness results by fixed point theory in particular using banach s contraction principle and schaefer s fixed point theorem we also present some examples to illustrate the application of our results

spectral analysis of the schrodinger operator on binary tree shaped networks and applications abstract in this paper we analyse the spectrum of the dissipative schrodinger operator on binary tree shaped networks as applications we study the stability of the schrodinger system using a riesz basis as well as the transfer function associated to the system

classical solvability of linear parabolic equations on networks on etudie la resolubilite classique des problemes aux valeurs limites et initiales pour des equations paraboliques lineaires generales sur des reseaux c 8 finis

exponential stability of a network of serially connected euler bernoulli beams the aim is to prove the exponential stability of a system modelling the vibrations of a network of n euler bernoulli beams serially connected using a result given by k ammari and m tucsnak the problem is reduced to the estimate of a transfer function and the obtention of an observability inequality the solution is then expressed in terms of fourier series so that one of the sufficient conditions for both the estimate of the transfer function and the observability inequality is that the distance between two consecutive large eigenvalues of the spatial operator involved in this evolution problem is superior to a minimal fixed value this property called spectral gap holds it is proved using the exterior matrix method due to w h paulsen two more asymptotic estimates involving the eigenfunctions are required they are established using an adequate basis

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

extension of the schur cohn stability test for 2 d ar quarter plane model the schur cohn test plays an essential role in checking the stability of one dimensional 1d random processes such as autoregressive ar models via the so called reflection coefficients partial correlations or schur szego coefficients in the context of two dimensional 2d random field modeling one of the authors recently proposed a 2d ar quarter plane model representation using 2d reflection coefficients estimated by a fast recursive adaptive algorithm based on such 2d reflection coefficients we can therefore derive two necessary stability conditions for a 2d ar quarter plane model one of these conditions can be considered as an extension of the schur cohn stability criterion based on the 2d reflection coefficients

schrodinger operators on graphs and geometry i essentially bounded potentials the inverse spectral problem for schrodinger operators on finite compact metric graphs is investigated the relations between the spectral asymptotics and geometric properties of the underlying graph are studied it is proven that the euler characteristic of the graph can be calculated from the spectrum of the schrodinger operator in the case of essentially bounded real potentials and standard boundary conditions at the vertices several generalizations of the presented results are discussed

surgery principles for the spectral analysis of quantum graphs we present a systematic collection of spectral surgery principles for the laplacian on a compact metric graph with any of the usual vertex conditions natural dirichlet or delta type which show

a direct algorithm for computing 2 d ar power spectrum estimates an algorithm for computing the parameters in a 2 d autoregressive spectral estimate without prior estimation of the correlation is described the algorithm utilizes the multichannel form of the burg algorithm and the relation between multichannel and 2 d ar modeling the procedure permits computation of the spectral matrix for several channels of 2 d data models with support in different quadrants are combined to form the spectral estimate

optimum quarter plane autoregressive modeling of 2 d fields using four field lattice approach a new orthogonal four field two dimensional 2 d quarter plane lattice structure with a complete set of reflection coefficients is developed by employing appropriately defined auxiliary prediction errors this work is the generalization of the three parameter lattice filter proposed by parker and kayran 1984 after the first stage four auxiliary forward and four auxiliary backward prediction errors are generated in order to obtain a growing number of 2 d reflection coefficients at successive stages the theory has been proven by using a geometrical formulation based on the mathematical concepts of vector space orthogonal projection and subspace decomposition it is shown that all four quarter plane filters are orthogonal and thus optimum for all stages in addition to developing the basic theory a set of orthogonal backward prediction error fields for successive lattice parameter model stages is presented

nonreal eigenvalues for second order differential operators on networks with circuits it is shown that on every finite network with at least one circuit there exist second order differential operators having an infinite number of nonreal eigenvalues the presence of nonreal eigenvalues implies that these operators cannot be selfadjoint with respect to any metric these eigenvalues reveal also the existence of oscillatory solutions for the corresponding time dependent partial differential equations

unitary dimension reduction for a class of self adjoint extensions with applications to graph like structures abstract we consider a class of self adjoint extensions using the boundary triplet technique assuming that the associated weyl function has the special form m z m z id t n z 1 with a bounded self adjoint operator t and scalar functions m n we show that there exists a class of boundary conditions such that the spectral problem for the associated self adjoint extensions in gaps of a certain reference operator admits a unitary reduction to the spectral problem for t as a motivating example we consider differential operators on equilateral metric graphs and we describe a class of boundary conditions that admit a unitary reduction to generalized discrete laplacians

decomposition spectrale de 1 formes differentielles sur une surface de riemann et series d eisenstein on etablit la formule de decomposition spectrale des 1 formes differentielles sur une surface de riemann non compacte m en introduisant le fibre en spheres de m on explicite egalement les series d eisenstein qui entrent en jeu on en donne quelques proprietes notamment dans le cas de la surface gamma 2 backslash h ou gamma 2 est le sous groupe de congruence principal d ordre 2

optimum asymmetric half plane autoregressive lattice parameter modeling of 2 d fields in this paper we present a new optimum asymmetric half plane ashp autoregressive lattice parameter modeling of two dimensional 2 d random fields this structure introduces 4n points into the prediction support region when the order of the model increases from n 1 to n starting with a given data field a set of four auxiliary prediction errors are generated in order to obtain the growing number of 2 d ashp reflection coefficients at successive stages the theory has been applied to the high resolution radar imaging problem and has also been proven using the concepts of vector space orthogonal projection and subspace decomposition it is shown that the proposed ashp structure generates the orthogonal realization subspaces for different recurse directions in addition to developing the basic theory the presentation includes a comparison between the proposed theory and other alternative structures both in terms of conceptual background and complexity while the recently developed reduced complexity ashp lattice modeling structure requires o 4n sup 3 lattice sections with n equal to the order of the error filter the proposed configuration requires only o 2n sup 2 lattice sections

relationship between scattering matrix and spectrum of quantum graphs abstractwe investigate the equivalence between spectral characteristics of the laplace op erator on a metric graph and the associated unitary scattering operator we provethat the statistics of level spacings and moments of observations in the eigenbasescoincide in the limit that all bond lengths approach a positive constant value 1 introduction quantum graphs have attracted much attention in recent years due both to their ap plicability as physical models and their interesting mathematical properties we referthe reader to 1 and the forthcoming volume 2 for a pot pourri of new results recentreviews dedicated to quantum graphs include 3 4 in this article we focus on one feature of interest which is the use of graph models toprobe the the universality of quantum systems one of the unsolved paradoxes of quantummechanics is the observation that a great many quantum systems are remarkably similarwhen one makes statistical observations in the semi classical r egime this manifests itselfboth in the energy levels and associated energy eigenfunctions despite a great deal ofe ort this universality is poorly understood mathematically generic quantum graphsexhibit this universal behavior and represent the most likely system for which a fullmathematically rigorous proof of this universality will rst be found important steps inthis direction have been taken in 5 a quantum graph can be de ned in two di erent but related ways a complete de scription appears in the following section one may consider a self adjoint realisationof the laplace operator or a scattering matrix approach mathematically the scatteringapproach appears to be more tractable and has formed the basis of most investigations 6 7 8 9 10 moreover the spectra arising from these quantizations are subtly di erent this is not as confusing as it might seem since the statistical properties of both versionsof the spectrum are believed to coincide when averaged over a large interval it is thepurpose of this article to put a concrete mathematical foundation behind this belief the plan of the article is as follows in the next section we give precise de nitions ofthe two ways to describe quantum graphs and in section 3 describe our main results insection 4 we describe the tools used and then present in section 5 the proofs of our results

the m matrix inverse problem for the sturm liouville equation on graphs abstractwe consider an inverse spectral problemfor sturm liouville boundaryvalue problemson a graph with formally self adjoint boundary conditions at the nodes where the giveninformation is the m matrix based on the results found in s currie b a watson m matrix asymptotics for sturm liouville problems on graphs j com appl math doi 10 1016 j cam 2007 11 019 using the green s function we prove that the poles ofthe m matrix are at the eigenvalues of the associated boundary value problem and aresimple located on the real axis and that the residue at a pole is a negative semi de nitematrix with rank equal to the multiplicity of the eigenvalue we de ne the so callednorming constants and relate them to the spectral measure and the m matrix thisenables us to recover from the m matrix the boundary conditions and the potential up to a unitary equivalence for co normal boundary conditions keywords inverse problem di erential operators on graphs m function sturm liouville 2000 msc 34a55 34b45 34b20 34l05 34b27

nodal statistics on quantum graphs it has been suggested that the distribution of the suitably normalized number of zeros of laplacian eigenfunctions contains information about the geometry of the underlying domain we study this distribution more precisely the distribution of the nodal surplus for laplacian eigenfunctions of a metric graph the existence of the distribution is established along with its symmetry one consequence of the symmetry is that the graph s first betti number can be recovered as twice the average nodal surplus of its eigenfunctions furthermore for graphs with disjoint cycles it is proven that the distribution has a universal form it is binomial over the allowed range of values of the surplus to prove the latter result we introduce the notion of a local nodal surplus and study its symmetry and dependence properties establishing that the local nodal surpluses of disjoint cycles behave like independent bernoulli variables

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

dirichlet series constructed from periods of automorphic forms we consider certain dirichlet series of selberg type constructed from periods of automorphic forms we study analytic properties of these dirichlet series and show that they have analytic continuation to the whole complex plane

a two dimensional fast lattice recursive least squares algorithm this paper is mainly devoted to the derivation of a new two dimensional fast lattice recursive least squares 2d flrls algorithm this algorithm updates the filter coefficients in growing order form with linear computational complexity after appropriately defining the order of 2d data and exploiting the relation with 1d multichannel order recursion relations and shift invariance property are derived the geometrical approaches of the vector space and the orthogonal projection then can be used for solving this 2d prediction problem we examine the performances of this new algorithm in comparison with other fast algorithms

multiplicity of solutions of second order linear differential equations on networks abstract an upper optimal bound is obtained for the multiplicity of solutions of second order linear differential equations on networks with general boundary and node conditions this bound depends only on the abstract topological associated graph in particular the result holds for the multiplicity of eigenvalues of the corresponding operators