complexity of critical functions for hamiltonian systems transition to predominantly chaotic motion in hamiltonian systems with two degrees of freedom is described by a complicated function of frequency which is called the critical function a graph of this function is a fractal set with the local structure which is believed to depend only on the arithmetic nature of the frequency we calculated numerically fractal dimensions of this function for a few typical systems using the method of modular smoothing and an efficient algorithm for computation of the fractal dimensions the dimensions which measure the complexity of the fractal are indeed the same within the error bounds and are equal to the dimension of the exponent of the brjuno function which is a purely arithmetic function

on optimal finite state digital transmission systems the digital transmission of signals by transmitters and receivers that are time invariant finite state machines were investigated in general form an average distortion criterion is used some general structure theorems are proved and some examples given that show improvement in performance as the memories of the transmitter and receiver are increased many conjectures and unsolved problems are mentioned and areas for further study are indicated

critical functions for complex analytic maps critical functions measure the width of the domain of stability around a given fixed point or an invariant circle for complex analytic and area preserving maps the author discusses their dependence on the rotation number of the invariant curves and proposes some new methods to determine them based on the existence of critical points and on some properties of quasiconformal maps by means of the majorant series method some rigorous estimates are given for complex area preserving maps like the semistandard map and the modulated singular map in particular the author makes use of the brjuno function to interpolate critical maps and proves that the convergence of the brjuno function is a necessary and sufficient condition for the existence of an analytic invariant curve of a given rotation number the author also discusses the optimality of the rigorous bounds obtained

source coding and graph entropies a sender wants to accurately convey information to a receiver who has some possibly related data we study the expected number of bits the sender must transmit for one and for multiple instances in two communication scenarios and relate this number to the chromatic and korner 1973 entropies of a naturally defined graph

fundamental mode shape and its derivatives in structural damage localization abstract various techniques for structural damage detection involving modal parameters have widely been used over the past few decades this is because the modal parameters of a structure can easily be obtained from forced free or ambient vibration measurements in many of these techniques mode shape curvature has been used for localization of damage in this paper a mathematical basis is provided to show the correlation between a structural damage and a change in the fundamental mode shape and its derivatives this has been achieved by deriving the expression of a damaged mode shape utilizing a perturbation approach for a cantilever shear beam discretized into a large number of elements this approach demonstrates that the change in the fundamental mode shape due to any damage is an excellent indicator of damage localization as it is found to be discontinuous at the location of damage further the change in higher derivatives i e slope and curvature of the fundamental mode shape is shown to be sensitive enough in damage localization a numerical study involving a shear building and a steel moment resisting frame is conducted to show the effectiveness of the proposed approach in damage localization it has been found from this study that spline fitting of mode shapes in case of a limited number of degrees of freedom which is generally adopted for plotting mode shapes and their derivatives may lead to false detection of damage

modular smoothing and kam tori abstract transformation properties of kam tori under the action of a modular group on frequencies are analysed and used for calculating the kam tori of a hamiltonian system with one and a half degrees of freedom first we derive a relation between the kam torus with a frequency v and the kam torus with the frequency v 1 v 1 v where 1 v is the integer part of 1 v given by a continuous function l1 v in the double limit v m n and k 0 where m n is a rational frequency and k is a perturbation parameter the function l1 v was introduced by buric percival and vivaldi and also gives the transformation properties of the critical parameter as a function of the frequency this function provides a zeroth order modular smoothing approximation to the invariant tori which is better than a perturbation expansion the first order corrections to this approximation can be calculated from a function r1 v that appears to be smooth

damage localization in plate structures from uniform load surface curvature although a large number of methods exist for detecting damage in a structure using measured modal parameters many of them require a correlated finite element model or at least modal data of the structure for the intact state as baseline for one dimensional beam like structures curvature techniques e g mode shape curvature and flexibility curvature have been applied to localize damage in this paper a damage localization method based on changes in uniform load surface uls curvature is developed for two dimensional plate structures a new approach to compute the uls curvature is proposed based on the chebyshev polynomial approximation instead of the central difference method the proposed method requires only the frequencies and mode shapes of the first few modes of the plate before and after damage or only the eigenpairs for the damaged state if a gapped smoothing technique is applied numerical simulations considering different supported conditions measurement noise mode truncation and sensor sparsity are studied to evaluate the effectiveness of the proposed method it is found that the uls curvature is sensitive to the presence of local damages even with truncated incomplete and noisy measurements

single beam analysis of damaged beams verified using a strain energy based damage measure analytical expression of a new damage measure which relates the strain energy to the damage location and magnitude is presented in this paper the strain energy expression is calculated using modes and natural frequencies of damaged beams that are derived based on single beam analysis considering both decrease in mass and stiffness decrease in mass and stiffness are a fallout of geometric discontinuity and no assumptions regarding the physical behavior of damage are made the method is applicable to beams with notch like non propagating cracks with arbitrary boundary conditions the analytical expressions derived for mode shapes curvature shapes natural frequencies and an improved strain energy based damage measure are verified using experiments the improvement in the damage measure is that it is not assumed that the bending stiffness of the damaged beam is constant and equal to that of undamaged beam when calculating the strain energy of the entire beam it is also not assumed that the bending stiffness of the element in which the damage is located is constant

optimal causal coding decoding problems the symbols produced by a finite markov source are causally encoded so as to be transmitted through a noisy memoryless channel the encoder is assumed to have channel feedback information and the decoder to be causal the feedback information is shown to be useful in general separation results are derived and used to prove that encoding is useless for a class of symmetric channels

exact results for an approximate renormalisation scheme and some predictions for the breakup of invariant ori abstract an approximate renormalisation scheme is derived for the breakup of invariant tori of arbitrary winding ratio in hamiltonian systems of one and a half degrees of freedom similar to that of escande and doveil it is a free semi group with two generators this scheme is solved exactly for its orbits stable manifolds unstable manifolds and critical set various results are found including a cantor set of universal fractal diagrams the robustness of noble tori and a scaling law for areas near critical circles

damage visualization based on local dynamic perturbation theory and application to characterization of multi damage in a plane structure abstract previously an inverse damage characterization framework was proposed by quantifying the perturbation to local dynamic equilibrium of a beam like structure showing advantages in some aspects over the traditional global vibration based and local guided wave based methods residing on the plate theory this framework was expanded to a two dimensional domain inheriting the attributes of localized canvassing using high order spatial derivatives this approach has proven effectiveness in quantitatively characterizing damage of small dimension regardless of its number and type in addition the approach requires no benchmarks baseline signals global models additional excitation sources pre modal analysis nor prior knowledge on structural boundary a damage imaging algorithm using the quantified dynamic perturbation was further established enabling presentation of damage characterization results in an intuitive and prompt manner integrating the detection capacities in one and two dimensional domains a hybrid damage visualization strategy was developed for systems comprising structural components of different types various geometries and diverse boundary conditions two independent de noising techniques low pass wavenumber filtering and adjustment of measurement density together with a hybrid data fusion algorithm were proposed as auxiliary means to enhance the robustness of the strategy in noisy measurement conditions the strategy was applied experimentally to the evaluation of multi damage in a plane structure comprising beam and plate components showing satisfactory results

identification of the stiffness distribution in statically indeterminate beams an identification procedure is presented for the stiffness distribution in statically indeterminate structures the proposed procedure analogous to direct calculation of the dynamic stiffness the inverse problem algorithm used in it is based on a finite element fe model of the damaged structure with unknown stiffness distribution and on a subset of measured vibration frequencies and vibration modes the bending stiffness is updated for each element in the fe model by an iterative procedure in order to improve its practicability the effect of random measurement noise is taken into consideration a numerical study of statically indeterminate beams shows that it is able to locate both localized and multiple damage and determine its severity with a high level of reliability

closed form solutions of a multi cracked circular arch under static loads abstract generalised functions have been widely adopted in structural mechanics to treat singularities of beam like structures however due to the curved geometry that couples axial and transversal displacements their use has never been explored for curved beams in this paper the capability of distributions of leading to closed form exact solutions for multi cracked circular arch is shown the exact closed form solution of a circular euler arch in presence of any number of discontinuities due to concentrated damage and subjected to an arbitrary distribution of static loads is obtained damage under the form of concentrated cracks has been modeled through the widely adopted and validated equivalent elastic hinge concept and has been introduced in the governing differential equations by making use of dirac s delta functions the resulting nontrivial generalised six order differential equations have been derived and solved in closed form independently of the number of along arch concentrated cracks the solution is expressed as a function of six integration constants only in which the damage positions and intensities are given data appearing explicitly in the solution expression this latter aspect constitutes a fundamental aid towards the resolution of the static damage inverse identification problem the results have been validated through some comparisons with finite element numerical simulations examples referred to multi cracked euler arches with different boundary conditions damage and load scenarios are presented

damage detection using the lipschitz exponent estimated by the wavelet transform applications to vibration modes of a beam damage detection by the wavelet transform of the fundamental vibration mode receives much attention recently many investigations report successful applications of the wavelet transform in damage detection but most of them appear to lack theoretical justifications the objective of this contribution is to show the effectiveness of the wavelet transform by means of its capability to estimate the lipschitz exponent it is also addressed that the magnitude of the lipschitz exponent can be used as a useful indicator of the damage extent as a specific example damaged beams are investigated both numerically and experimentally the continuous wavelet transform cwt by a mexican hat wavelet having two vanishing moments is utilized for the estimation of the lipschitz exponent the analysis by the cwt also gives a guideline to choose appropriate discrete wavelet transforms

renormalization method for computing the threshold of the large scale stochastic instability in two degrees of freedom hamiltonian systems an approximate renormalization procedure is derived for the hamiltonianh v x t v2 2 m cosx p cosk x t it gives an estimate of the large scale stochastic instability threshold which agrees within 5 10 with the results obtained from direct numerical integration of the canonical equations it shows that this instability is related to the destruction of kam tori between the two resonances and makes the connection with kam theory possible improvements of the method are proposed the results obtained forh allow us to estimate the threshold for a large class of hamiltonian systems with two degrees of freedom

a numerical study of the eigenparameters of a damaged cantilever abstract when a structure is subjected to damage its dynamic response changes characterized by shifts in the eigenvalues and modifications of the eigenvectors considerable effort has been put into investigating the relationship between the damage location the damage size and the corresponding changes in the eigenparameters in most cases emphasis has been on using the shift in eigenvalues as a means of determining the damage location and the information derived from the change in eigenvectors has largely remained obscure in this paper a systematic study is presented of the relationship between damage location damage size and the changes in the eigenvalues and eigenvectors of a cantilever when subjected to damage a finite element model of a uniform cross sectioned cantilever was chosen to provide data for analysis the changes in the eigenvalues and eigenvectors are shown to follow a definite trend in relation to the location and extent of damage

structural damage detection based on proper orthogonal decomposition experimental verification the present paper describes a new structural damage detection method based on the monitoring of vibrational properties of the structure sensors record the accelerations of several points of the structure the recorded data are then used to compute the proper orthogonal modes the comparison between the proper orthogonal modes of the undamaged structure and those of the damaged structure provides information on damage location experimental results are presented

vibration analysis of edge cracked beam on elastic foundation with axial loading using the differential quadrature method abstract the eigenvalue problems of clamped free and hinged hinged bernoulli euler beams on elastic foundation with a single edge crack axial loading and excitation force were numerically formulated using the differential quadrature method dqm appropriate boundary conditions accompanied the dqm to transform the partial differential equation of a bernoulli euler beam with a single edge crack into a discrete eigenvalue problem the dqm results for the natural frequencies of cracked beams agree well with other literature values the sampling point number effect the location of the crack effect and the depth of the crack effect on the accuracy variation of calculated natural frequencies are presented by using two elements in this work the effects of axial loading foundation stiffness opening crack and closing crack are also studied

on causal source codes with side information we study the effect of the introduction of side information into the causal source coding setting of neuhoff and gilbert we find that the spirit of their result namely the sufficiency of time sharing scalar quantizers followed by appropriate lossless coding for attaining optimum performance within the family of causal source codes extends to many scenarios involving availability of side information at both encoder and decoder or only on one side for example in the case where side information is available at both encoder and decoder we find that time sharing side information dependent scalar quantizers at most two for each side information symbol attains optimum performance this remains true even when the reproduction sequence is allowed noncausal dependence on the side information and even for the case where the source and the side information rather than consisting of independent and identically distributed i i d pairs form respectively the output of a memoryless channel and its stationary ergodic input

damage estimation on beam like structures using the multi resolution analysis abstract this work proposes a vibration based damage evaluation method that can detect locate and size damage utilizing only a few of the lower mode shapes the proposed method is particularly advantageous for beam like structures with uncertain applied axial load mass density and foundation stiffness based on a small damage assumption a linear relationship between damaged and undamaged curvatures is revealed in the context of elasticity it turns out that the resulting damage index equation inherently suffers from singularities near inflection nodes the transformation of the problem into the multi resolution wavelet domain provides a set of coupled linear equations with the aid of the singular value decomposition technique the solution to the damage index equation is achieved in the wavelet space next the desired physical solution to the damage index equation is reconstructed from the one in the wavelet space the performance of the proposed method is compared with two existing damage detection methods using a set of numerical simulations the proposed method attempts to resolve the mode selection problem the singularity problem the axial force problem and the absolute severity estimation problem all of which remained unsolved by earlier attempts

on optimal zero delay coding of vector markov sources optimal zero delay coding quantization of a vector valued markov source driven by a noise process is consid ered using a stochastic control problem formulation the exis tence and structure of optimal quantization policies are studied for a finite horizon problem with bounded per stage distortion measure the existence of an optimal zero delay quantization policy is shown provided that the quantizers allowed are ones with convex codecells the bounded distortion assumption is relaxed to cover cases that include the linear quadratic gaussian problem for the infinite horizon problem and a stationary markov source the optimality of deterministic markov coding policies is shown the existence of optimal stationary markov quantization policies is also shown provided randomization that is shared by the encoder and the decoder is allowed

converse kam theory and practice we unify extend reinterpret and apply criteria of birkhoff 1 herman 9 mather 2 3 aubry et al 4 5 and newman and percival 6 for the nonexistence of invariant circles for area preserving twist maps the criteria enable one to establish regions of phase space through which no rotational invariant circles pass for families of maps the same can be done for regions of the combined space of phase points and parameters the criteria can be implemented rigorously on a computer and give a practical method of proving quite strong results as an example we present a computer program which proved that the standard map has no rotational invariant circles for any parameter value k 63 64

damage assessment of structures using modal test data abstract system health monitoring of structures is important not only for conducting safe operation but also maintaining system performance in this paper two identification algorithms for assessing structural damages using the modal test data have been developed an important characteristic in the present approaches is that the employment of the global numerical models e g fem model and some important information e g young s modulus of structures are avoided to a great extent as the first step of the damage identification two algorithms for the detection of damage location are proposed which are similar in concept to the subspace rotation algorithm or best achievable eigenvector technique furthermore a quadratic programming model is set up for the two approaches to predict the damage extent to demonstrate the capability of the proposed approaches an example of a 10 bay planar truss structure is employed for checking the present approaches numerically furthermore the experimental data from the vibration test of a beam with two fixed ends are used directly in the present approaches the final results show that the present techniques perform quite well in spite of the little structural information and measurement inaccuracies

modular smoothing of action the central quantity in the theory of transport for hamiltonian systems and in particular the area preserving twist maps is the action of rotational periodic orbits usually this is a complicated discontinuous function of two arguments some perturbation parameter k and a rational rotation number denoted by we applied the idea of modular smoothing to this complicated fractal our main result is that all the information contained in the fractal can be retrieved from a set of continuous or smooth functions of one variable

refinements of damage detection methods based on wavelet analysis of dynamical shapes this manuscript aims at illustrating significant refinements concerning the use of wavelets when these latter are used in the guise of continuous wavelet transforms cwt for identifying damage on transversally vibrating structural components e g beams plates and shells the refinements regard the presentation of wavelet algorithms which are aimed at significantly reducing those border distortions normally arising during a wavelet damage detection procedure the main advantage of the algorithms is that they are self contained namely i the wavelet transforms do not undergo any own variation and their application follows the convolution laws established in the past ii it is not necessary to design a specific boundary wavelet iii no significant analytical treatment neither of the wavelets nor of the signal is required and finally iv the algorithms can be adapted to different boundary conditions and different physical situations besides all the specified advantages the wavelet damage detection procedure is still carried out by excluding historical data the effectiveness of the algorithms is shown through numerical and experimental examples these latter are illustrated along with reduced outliers of experimental estimation through a related consistent statistical procedure

a pseudo excitation approach for structural damage identification from strong to weak modality a damage characterization framework based on the pseudo excitation pe approach has recently been established aimed at quantitatively identifying damage in beam plate and shell like structural components however it is envisaged that the effectiveness of the pe approach can be restricted in practical implementation due to the involvement of high order derivatives of structural dynamic deflections in which measurement noise and uncertainties can overwhelm the damage associated signal features upon mathematical differentiation in this study the pe approach was revamped by introducing the weighted integration whereby the prerequisite of satisfying the local equilibrium conditions was relaxed from point by point to region by region the revamped modality was thus colloquially referred to as weak formulation of the pe approach as opposed to its original version which is contrastively termed as strong formulation by properly configuring a weight function noise immunity of the pe approach was enhanced giving rise to improved detection accuracy and precision even under noisy measurement conditions furthermore the weak formulation was extended to a series of coherent variants through partial integration rendering a multitude of detection strategies by selecting measurement parameters and configurations this endowed the pe approach with flexibility in experimental manipulability so as to accommodate various detection requirements as an application of the weak formulation a continuous gauss smoothing cgs based detection scheme was developed and validated by localizing multiple cracks in a beam structure showing fairly improved noise tolerance

on the structure of optimal real time encoders and decoders in noisy communication the output of a discrete time markov source must be encoded into a sequence of discrete variables the encoded sequence is transmitted through a noisy channel to a receiver that must attempt to reproduce reliably the source sequence encoding and decoding must be done in real time and the distortion measure does not tolerate delays the structure of real time encoding and decoding strategies that jointly minimize an average distortion measure over a finite horizon is determined the results are extended to the real time broadcast problem and a real time variation of the wyner ziv problem

transformation properties of perturbation expansions around elliptic fixed points transformation properties of perturbation expansions of vibrational quasi periodic orbits upon the modular transformations of frequencies are studied using simple but nontrivial examples the siegel complex quadratic map and special solutions of the real area preserving quadratic map it is shown that the transformation properties are similar to those in the previously studied case of the rotational invariant tori except for some special features of the real map related to the atypical nature of 1 3 resonance in this case

chaotic boundary of a hamiltonial map abstract the half plane maps are a class of complex hamiltonial maps whose invariant curves at most fixed irrational frequencies can be obtained as a convergent taylor series expansion for these maps the boundary between regular ordered motion on invariant curves and irregular chaotic motion is given by the radius of convergence of the series the successive terms of the series oscillate wildly due to the presence of small divisors methods are presented for taming the series based on the conversion of the convergenceexponentc ln c into the integral of a continuous but nondifferentiable lambda function whose graph whows a similarity structure on small scales self similarity properties are illustrated for the chaotic boundary function c where v is the frequency

on the wyner ziv problem for individual sequences we consider a variation of the wyner ziv w z problem pertaining to lossy compression of individual sequences using finite state encoders and decoders there are two main results in this paper the first characterizes the relationship between the performance of the best m state encoder decoder pair to that of the best block code of size lscr for every input sequence and shows that the loss of the latter relative to the former in terms of both rate and distortion never exceeds the order of logm lscr independently of the input sequence thus in the limit of large m the best rate distortion performance of every infinite source sequence can be approached universally by a sequence of block codes which are also implementable by finite state machines while this result assumes an asymptotic regime where the number of states is fixed and only the length n of the input sequence grows without bound we then consider the case where the number of states m mn is allowed to grow concurrently with n our second result is then about the critical growth rate of mn such that the rate distortion performance of mn state encoder decoder pairs can still be matched by a universal code we show that this critical growth rate of mn is linear in n

optimal design of sequential real time communication systems optimal design of sequential real time communication of a markov source over a noisy channel is investigated in such a system the delay between the source output and its reconstruction at the receiver should equal a fixed prespecified amount an optimal communication strategy must minimize the total expected symbol by symbol distortion between the source output and its reconstruction design techniques or performance bounds for such real time communication systems are unknown in this paper a systematic methodology based on the concepts of information structures and information states to search for an optimal real time communication strategy is presented this methodology trades off complexity in communication length linear in contrast to doubly exponential with complexity in alphabet sizes doubly exponential in contrast to exponential as the communication length is usually order of magnitudes bigger than the alphabet sizes the proposed methodology simplifies the search for an optimal communication strategy in spite of this simplification the resultant optimality equations cannot be solved efficiently using existing algorithmic techniques the main idea is to formulate a zero delay communication problem as a dynamic team with nonclassical information structure then an appropriate choice of information states converts the dynamic team problem into a centralized stochastic control problem in function space thereafter markov decision theory is used to derive nested optimality equations for choosing an optimal design for infinite horizon problems these optimality equations give rise to a fixed point functional equation communication systems with fixed finite delay constraint a higher order markov source and channels with memory are treated in the same manner after an appropriate expansion of the state space thus this paper presents a comprehensive methodology to study different variations of real time communication

damage detection from changes in curvature mode shapes abstract a damage in a structure alters its dynamic characteristics the change is characterized by changes in the eigenparameters i e natural frequency damping values and the mode shapes associated with each natural frequency considerable effort has been spent in obtaining a relationship between the changes in eigenparameters the damage location and the damage size most of the emphasis has been on using the changes in the natural frequencies and the damping values to determine the location and the size of the damage in this paper a new parameter called curvature mode shape is investigated as a possible candidate for identifying and locating damage in a structure by using a cantilever and a simply supported analytical beam model it is shown here that the absolute changes in the curvature mode shapes are localized in the region of damage and hence can be used to detect damage in a structure the changes in the curvature mode shapes increase with increasing size of damage this information can be used to obtain the amount of damage in the structure finite element analysis was used to obtain the displacement mode shapes of the two models by using a central difference approximation curvature mode shapes were then calculated from the displacement mode shapes

structure theorems for real time variable rate coding with and without side information the output of a discrete markov source is to be encoded instantaneously by a variable rate encoder and decoded by a finite state decoder our performance measure is a linear combination of the distortion and the instantaneous rate structure theorems pertaining to the encoder and next state functions are derived for every given finite state decoder which can have access to side information

universal small scale structure near the boundary of siegel disks of arbitrary rotation number abstract we present numerical evidence that for analytic maps of the complex plane with a siegel disk of any irrational rotation number and a quadratic critical point on its boundary the small scale structure near the critical point can be described asymptotically by a universal two parameter family we propose an explanation in terms of an ergodic attractor for a renormalisation operator