on the method of tangent hyperbolas on banach spaces abstract as an application of the technique employed by the author in a series of papers 15 18 some results are established concerning convergence of the method of tangent hyperbolas for solving nonlinear equations in banach spaces as well as existence and uniqueness of solution the results are compared with those obtained by doring 5

convergence for a class of improved sixth order chebyshev halley type methods in this paper we consider the semilocal convergence on a class of improved chebyshev halley type methods for solving f x 0 where f x y is a nonlinear operator x and y are two banach spaces is a non empty open convex subset in x to solve the problems that f x is unbounded in and it can not satisfy the whole lipschitz or h o lder continuity f x n is replaced by f x 0 n for all x where n n 0 x0 is an initial point moreover f x is assumed to be local h o lder continuous so the convergence conditions are relaxed we prove an existence uniqueness theorem for the solution which shows that the r order of these methods is at least 5 q where q 0 1 especially when f x is local lipschitz continuous the r order will become six

a lower bound for nodal count on discrete and metric graphs we study the number of nodal domains maximal connected regions on which a function has constant sign of the eigenfunctions of schrodinger operators on graphs under a certain genericity condition we show that the number of nodal domains of the nth eigenfunction is bounded below by n l where l is the number of links that distinguish the graph from a tree

stability of nodal structures in graph eigenfunctions and its relation to the nodal domain count the nodal domains of eigenvectors of the discrete schr dinger operator on simple finite and connected graphs are considered courant s well known nodal domain theorem applies in the present case and sets an upper bound to the number of nodal domains of eigenvectors arranging the spectrum as a non decreasing sequence and denoting by n the number of nodal domains of the nth eigenvector courant s theorem guarantees that the nodal deficiency n n is non negative the above applies for generic eigenvectors special care should be exercised for eigenvectors with vanishing components the main result of this work is that the nodal deficiency for generic eigenvectors is equal to a morse index of an energy functional whose value at its relevant critical points coincides with the eigenvalue the association of the nodal deficiency to the stability of an energy functional at its critical points was recently discussed in the context of quantum graphs band et al 2011 commun math phys at press doi 10 1007 s00220 011 1384 9 and dirichlet laplacian in bounded domains in berkolaiko et al 2011 arxiv 1107 3489 this work adapts this result to the discrete case the definition of the energy functional in the discrete case requires a special setting substantially different from the one used in band et al and berkolaiko et al and it is presented here in detail

inverse nodal problems it is shown that nodal sequences determine the underlying manifold up to scaling within classes of rectangles with dirichlet boundary conditions separable two dimensional tori two dimensional flat klein bottles and flat tori in two and three dimensions

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

a note on the halley method in banach spaces abstract we provide a convergence theorem under the usual ostrowski kantorovich conditions for an equivalent form of the halley method in banach spaces the best possible upper error bound for the method is obtained some applications to the solution of quadratic operator equations are provided

on the r order of the halley method an r order bound for the halley method is obtained in this work where an analysis of the convergence of the method is also presented under mild differentiability conditions to do this a new technique is developed where the involved operator must satisfy some recurrence relations

stabilization of infinte dimensional systems with periodic feedback gains and sampled output abstract a novel control technique periodic output feedback is investigated in the context of infinite dimensional linear dynamic systems with unbounded input and output operators this control scheme differs from classical control schemes in that the output of the system in question need not be monitored continuously but only monitored at sampling instants it is shown that for systems satisfying an analog of detectability and stabilizability periodic output feedback provides a stabilization scheme which is robust under perturbations of the sampling time sequence and is easily implementable

elimination of fixed modes in decentralized systems by means of sampling abstract in this contribution it is shown that fixed modes in continuous time decentralized control systems can be eliminated by the use of sampling criteria are derived which distinguish the fixed modes which are still present in the sampled data system from those which are eliminated

the number of nodal domains on quantum graphs as a stability index of graph partitions the courant theorem provides an upper bound for the number of nodal domains of eigenfunctions of a wide class of laplacian type operators in particular it holds for generic eigenfunctions of a quantum graph the theorem stipulates that after ordering the eigenvalues as a non decreasing sequence the number of nodal domains n of the n th eigenfunction satisfies n n here we provide a new interpretation for the courant nodal deficiency d n n n in the case of quantum graphs it equals the morse index at a critical point of an energy functional on a suitably defined space of graph partitions thus the nodal deficiency assumes a previously unknown and profound meaning it is the number of unstable directions in the vicinity of the critical point corresponding to the n th eigenfunction to demonstrate this connection the space of graph partitions and the energy functional are defined and the corresponding critical partitions are studied in detail

pole placement for linear discrete time systems by periodic output feedbacks abstract necessary and sufficient conditions for the existence of a solution to the pole placement problem by periodic output feedbacks for single input single output linear discrete time systems are established an algorithm for finding a sequence of output feedback gains is presented and illustrated by a simple numerical example

feedback stabilization of linear diffusion systems with periodically scanning sensors the feedback stabilization of linear diffusion systems with periodically scanning sensors is considered the control inputs to the system are given by the feedback of the sakawa type observer output in this case since the output operator becomes time periodic the characteristic multiplier assignment is used to design the observer and the bounded perturbation theorem of the evolution operators is used to prove the closed loop stability

recurrence relations for semilocal convergence of a newton like method in banach spaces abstract the aim of this paper is to establish the semilocal convergence of a multipoint third order newton like method for solving f x 0 in banach spaces by using recurrence relations the convergence of this method is studied under the assumption that the second frechet derivative of f satisfies holder continuity condition this continuity condition is milder than the usual lipschitz continuity condition a new family of recurrence relations are defined based on the two new constants which depend on the operator f these recurrence relations give a priori error bounds for the method two numerical examples are worked out to demonstrate the applicability of the method in cases where the lipschitz continuity condition over second derivative of f fails but holder continuity condition holds

a note on halley s method we introduce the degree of logarithmic convexity which provides a measure of the convexity of a function at each point making use of this concept we obtain a new theorem of global convergence for halley s method

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

nodal domains statistics a criterion for quantum chaos we consider the distribution of the properly normalized numbers of nodal domains of wave functions in 2d quantum billiards we show that these distributions distinguish clearly between systems with integrable separable or chaotic underlying classical dynamics and for each case the limiting distribution is universal system independent thus a new criterion for quantum chaos is provided by the statistics of the wave functions which complements the well established criterion based on spectral statistics

periodically varying compensation of time invariant systems linear time invariant discrete time scalar systems under periodically varying output feedback and the associated problems of stabilization and arbitrary eigenvalue assignment are investigated the general form of the characteristic polynomial of the closed loop periodically varying system is obtained and some detailed results are provided for the second order case interesting research questions are pointed out

a homotopy method for eigenvalue assignment using decentralized state feedback this paper addresses the following problem given an interconnected system m composed of n subsystems of the form a i b i k i i 1 n a i b i a controllable pair and where the off diagonal blocks of m lie in the image of the appropriate b i then is it possible to arbitrarily assign the characteristic polynomial of m by a suitable selection of the characteristic polynomials of a i b i k i moreover is it possible to compute the appropriate characteristic polynomials of the a i b i k i or equivalently construct the k i needed to do so the first question is answered by constructing a mapping f r n rightarrow r n which maps a prescribed set of n of the feedback gains elements of k i i 1 n to the n coefficients of the characteristic polynomial of m the question then becomes given a p in r n does f x p have a solution the answer is found by constructing a homotopy h r n x o 1 rightarrow r n where h x 1 f x and h x 0 is some trivial function degree theory is then applied to guarantee that there exists an x t such that h x t t p for all t in 0 1 the parameterized sard s theorem is then utilized to prove that with probability 1 x t is a smooth curve and hence can be followed numerically from x 0 to x 1 by the solution of a differential equation davidenko s method

monodromy eigenvalue assignment in linear periodic systems this note addresses the problem of assigning the eigenvalues of the monodromy matrix of a linear continuous time periodic system by periodic state feedback a controllability criterion is given which slightly strengthens known results an explicit expression for a periodic feedback gain which performs the pole assignment is given when the system is controllable over one period the whole monodromy matrix is assignable by periodic feedback an example is given

robust tracking and regulation of linear periodic discrete time systems in this paper the robust asymptotic tracking and disturbance rejection problem is studied for linear periodic discrete time systems and for general classes of disturbance functions and reference signals consisting in free responses of periodic processes the necessary and sufficient conditions for the existence of a solution which involve the periodic notion of invariant zero are found a constructive design procedure of the periodic compensator is given it is based on the internal model principle

indices of convexity and concavity application to halley method we define an index to measure the convexity of a convex function f at each point we use this index to establish conditions on the convergence of the halley method in the complex plane and in banach spaces

simultaneous pole assignment in linear periodic systems by constrained structure feedback the authors present methods for pole assignment by feedback with constrained structure in linear periodic systems using generalized sampled data hold functions gshf constrained structure is taken to mean that the input of each channel is restricted to depend only on the measurements of some specific output channels the basic idea of gshf control is to use sample and hold but to consider the hold function as a design parameter four strategies are proposed for closing a control loop using gshf the key to the results obtained is that when gshf control with constrained structure is applied to a periodic system a discrete time time invariant decentralized system is obtained for which control design methods are available

a note on the kantorovich theorem for newton iteration abstract in this paper a new theorem for the newton method convergence is obtained its condition is different from that of the kantorovich theorem and therefore it has theoretical and practical value

semilocal convergence for halley s method under weak lipschitz condition the semilocal convergence properties of halley s method for nonlinear operator equations are studied under the hypothesis that the second derivative satisfies some weak lipschitz condition the method employed in the present paper is based on a family of recurrence relations which will be satisfied by the involved operator an application to a nonlinear hammerstein integral equation of the second kind is provided

resolving isospectral drums by counting nodal domains several types of systems have been put forward during the past few decades to show that there exist isospectral systems which are metrically different one important class consists of laplace beltrami operators for pairs of flat tori in r n with n 4 we propose that the spectral ambiguity can be resolved by comparing the nodal sequences the numbers of nodal domains of eigenfunctions arranged by increasing eigenvalues in the case of isospectral flat tori in four dimensions where a four parameter family of isospectral pairs is known we provide heuristic arguments supported by numerical simulations to support the conjecture that the isospectrality is resolved by the nodal count thus one can count the shape of a drum if it is designed as a flat torus in four dimensions less

nodal domains on isospectral quantum graphs the resolution of isospectrality we present and discuss isospectral quantum graphs which are not isometric these graphs are the analogues of the isospectral domains in which were introduced recently in gordon et al 1992 bull am math soc 27 134 8 chapman 1995 am math mon 102 124 buser et al 1994 int math res not 9 391 400 okada and shudo 2001 j phys a math gen 34 5911 22 jakobson et al 2006 j comput appl math 194 141 55 and levitin et al 2006 j phys a math gen 39 2073 82 all based on sunada s construction of isospectral domains sunada t 1985 ann math 121 196 86 after presenting some of the properties of these graphs we discuss a few examples which support the conjecture that by counting the nodal domains of the corresponding eigenfunctions one can resolve the isospectral ambiguity

methods with prefixed order for approximating square roots with global and general convergence a family of newton like methods is constructed to approximate the square root of a positive real number the iterative methods of the family are global or generally convergent depending on the prefixed order of convergence is even or odd we show the dynamical behaviour of some of these methods by means of several julia sets and the intricate fractal structures which arise from the order of the iterative methods are displayed

a third order method for fixed points in banach spaces this paper deals with a third order stirling like method used for finding fixed points of nonlinear operator equations in banach spaces the semilocal convergence of the method is established by using recurrence relations under the assumption that the first frechet derivative of the involved operator satisfies the holder continuity condition a theorem is given to establish the error bounds and the existence and uniqueness regions for fixed points the r order of the method is also shown to be equal to at least 2p 1 for p 0 1 the efficacy of our approach is shown by solving three nonlinear elementary scalar functions and two nonlinear integral equations by using both stirling like method and newton like method it is observed that our convergence analysis is more effective and give better results

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

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

on the convergence of a higher order family of methods and its dynamics in this paper we present the study of the local convergence of a higher order family of methods moreover the dynamical behavior of this family of iterative methods applied to quadratic polynomials is studied some anomalies are found in this family by means of studying the associated rational function parameter spaces are shown and the study of the stability of all the fixed points is presented

hermitian symplectic geometry and extension theory here we give brief account of hermitian symplectic spaces showing that they are intimately connected to symmetric as well as self adjoint extensions of a symmetric operator furthermore we find an explicit parametrization of the lagrange grassmannian in terms of the unitary matrices this allows us to explicitly describe all self adjoint boundary conditions for the schrodinger operator on the graph in terms of a unitary matrix we show that the asymptotics of the scattering matrix can be expressed simply in terms of this unitary matrix

pole assignment with improved control performance by means of periodic feedback this technical note is concerned with the pole placement of continuous time linear time invariant lti systems by means of lq suboptimal periodic feedback it is well known that there exist infinitely many generalized sampled data hold functions gshf for any controllable lti system to place the modes of its discrete time equivalent model at prescribed locations among all such gshfs this technical note aims to find the one which also minimizes a given lq performance index to this end the gshf being sought is written as the sum of a particular gshf and a homogeneous one the particular gshf can be readily obtained using the conventional pole placement techniques the homogeneous gshf on the other hand is expressed as a linear combination of a finite number of functions such as polynomials sinusoidals etc the problem of finding the optimal coefficients of this linear combination is then formulated as a linear matrix inequality lmi optimization the procedure is illustrated by a numerical example

semilocal convergence analysis on the modifications for chebyshev halley methods under generalized condition in this paper we consider the semilocal convergence for modifications of chebyshev halley methods in banach space compared with the results on super halley method studied in reference gutierrez and hernandez 1998 these modified methods need less computation of inversion the r order is improved and the lipschitz continuity of second derivative is also relaxed we prove a theorem to show existence uniqueness of solution the r order for these modified methods is analyzed under generalized condition

on the local convergence and the dynamics of chebyshev halley methods with six and eight order of convergence we study the local convergence of chebyshev halley methods with six and eight order of convergence to approximate a locally unique solution of a nonlinear equation in sharma 2015 see theorem 1 p 121 the convergence of the method was shown under hypotheses reaching up to the third derivative the convergence in this study is shown under hypotheses on the first derivative hence the applicability of the method is expanded the dynamics of these methods are also studied finally numerical examples examining dynamical planes are also provided in this study to solve equations in cases where earlier studies cannot apply

structurally fixed modes a notion of structurally fixed modes is introduced in the framework of decentralized control systems to identify the modes that cannot be shifted by decentralized feedback regardless of the numerical values of system parameters in this way a qualitative characterization of decentralized fixed modes is provided which is suitable for delineating decentrally stabilizable systems the characterization leads naturally to the notion of fixed modes with respect to arbitrary feedback structure constraints

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

newton kantorovich theorem for a family of modified halley s method under holder continuity conditions in banach space abstract in this study under p holder continuity conditions a family of new modified halley s method with 2 p order is proposed in banach space which is used to solve the nonlinear operator equations the newton kantorovich convergence theorem for the family of new modified halley s method is established under holder continuity conditions the error estimate also is gotten finally two examples are provided to show the application of our theorem

comment on resolving isospectral drums by counting nodal domains in gnutzmann et al 2005 j phys a math gen 38 8921 33 the authors studied the 4 parameter family of isospectral flat 4 tori t a b c d discovered by conway and sloane with a particular method of counting nodal domains they were able to distinguish these tori numerically by computing the corresponding nodal sequences relative to a few explicit tuples a b c d in this note we confirm the expectation expressed in gnutzmann et al 2005 j phys a math gen 38 8921 33 by proving analytically that their nodal count distinguishes any 4 tuple of distinct positive real numbers

a stirling like method with holder continuous first derivative in banach spaces in this paper the convergence of a stirling like method used for finding a solution for a nonlinear operator in a banach space is examined under the relaxed assumption that the first frechet derivative of the involved operator satisfies the holder continuity condition many results exist already in the literature to cover the stronger case when the second frechet derivative of the involved operator satisfies the lipschitz holder continuity condition our convergence analysis is done by using recurrence relations the error bounds and the existence and uniqueness regions for the solution are obtained finally two numerical examples are worked out to show that our convergence analysis leads to better error bounds and existence and uniqueness regions for the fixed points