non stationary parametric oscillations the response of a single degree of freedom system with cubic non linearity to a non stationary principal parametric excitation is investigated the method of multiple scales is used to derive two first order ordinary differential equations for the evolution of the amplitude and phase of the response the evolution equations are numerically integrated for various sweeping rates of the amplitude and frequency of the excitation the results show that the non stationary response penetrates the instability regions and the higher the sweeping rate is the deeper the penetration is

a generalized local linearization principle for non linear dynamical systems tangent spaces in non linear dynamical systems are state dependent hence it is generally not possible to exactly represent a non linear dynamical system by a linear one over finite segments of the evolving trajectories in the phase space it is known from the well known theorem of hartman and grobman that a non linear oscillator admits a linearization within a neighbourhood of any hyperbolic fixed point in the sense that the linearized flow bears a topological conjugacy to the non linear flow within the same neighbourhood this linearization is based on the construction of the tangent space to the non linear vector field at the hyperbolic fixed point even though useful in bifurcation analysis such a linearization procedure cannot be successfully applied to identically simulate the non linear flow over the phase space because the functional form of the topological conjugacy is not known with this in mind an implicit approach to local linearization is evolved in this paper such that the tangent space of the linearized equation transversally intersects the tangent space to the non linear dynamical system at that point in the state space where the solution vector is desired several numerical schemes for the implementation of this implicit and local linearization procedure are explored and illustrated with several numerical results it is shown that the locally transversal linearization ltl procedure finally reduces the given set of non linear ordinary differential equations odes to a set of transcendental algebraic equations with the desired solution vector as the unknown the variables in the state space appear as unknown quantities in this approach finally one arrives at non linear poincare maps for the non linear oscillator in this scheme the characteristic time interval for the map is arbitrarily fixed the methodology is found to be quite versatile for handling non linear dynamical systems in particular it is verified that these schemes are capable of accurately predicting a wide spectrum of typically non linear response characteristics such as limit cycles multi periodicity almost periodicity and chaos for a limited class of response patterns the principle proposed has another advantage in that the time step for integrating the non linear odes need not be small an improved and a more general higher order version of the ltl method is also considered a distinct advantage of this higher version is in its improved ability for a closer simulation of phase dependent time histories where a difference in the initial conditions leads to a phase difference in the realized solution history

generalization properties of multilayered neural networks generalization properties of multilayered neural networks with binary couplings are studied in the high temperature limit the transition to the perfect generalization phase is evaluated for systems with an arbitrary number of layers it is found that the thermodynamic transition occurs for a number of examples lower than for the perceptron but the opposite occurs for the transition in which the poor generalization solution disappears the generalization error is also decomposed according to the contributions coming from different numbers of hidden neurons that have a wrong sign in the internal field this allows the authors to describe the generalization behaviour of the hidden neurons

a new numeric analytical principle for nonlinear deterministic and stochastic dynamical systems a new implicit local linearization method termed as the locally transversal linearization ltl is developed in the present paper for a numeric analytical integration of nonlinear dynamical systems under deterministic and or stochastic excitations the ltl procedure is developed in such a way that at a chosen time instant the ltl solution manifold transversally intersects the nonlinear solution manifold at that particular point or cross section in the state space where the solution vector needs to be obtained in general construction of the ltl equation is non unique and thus it is generally needed to substitute the conditionally linear ltl based solution into the given nonlinear ordinary differential equation ode at a specified time instant to ensure a transversal intersection in the present study however it is demonstrated that a transversal intersection at a given time instant may as well be ensured by just choosing a few specific forms of the ltl equation thereby eliminating the need for substitution this new variant of ltl is particularly suitable for discretization of stochastic differential equations sdes in that one simultaneously avoids dealing with formal derivatives of wiener processes while satisfying the nonlinear sde at discrete time points the stochastic ltl procedure is adaptable for obtaining strong pathwise solutions of a general nonlinear dynamical system with continuous but not necessarily differentiable vector fields under additive and or multiplicative excitations a specific advantage of the ltl method is that for a class of response regime herein referred to as the phase independent regime it can obtain an accurate description of response irrespective of the choice of the time step and without a propagation of local errors presently the method has been numerically illustrated for strong solutions of the hardening duffing oscillator under deterministic sinusoidal or additive stochastic white noise excitations or their combination limited numerical results on periodic solutions chaos with or without noise and stochastic resonance are included in this study the results presented are indicative of the sample path accuracy consistency stochastic numerical stability and a partial insensitivity to time step sizes of the proposed method

application of the incremental harmonic balance method to cubic non linearity systems abstract in this paper the formulation of the incremental harmonic balance ihb method is derived for a general system of differential equations with cubic non linearity which governs a wide range of engineering problems such as large amplitude vibration of beams or plates an incremental arc length method combined with a cubic extrapolation technique is adopted to trace the response curve automatically the stability of its periodic solutions can also be evaluated from the ihb formulation by multi variable floquet theory hsu s method is adopted for computing the transition matrix at the end of one period two numerical examples are presented which demonstrate the effectiveness of the ihb method and hsu s method in treating the non linear vibration of a strongly non linear multiple degree of freedom system

the response of two degree of freedom systems with quadratic non linearities to a combination parametric resonance abstract the response of two degree of freedom systems with quadratic non linearities to a combination parametric resonance in the presence of two to one internal resonances is investigated the method of multiple scales is used to construct a first order uniform expansion yielding four first order non linear ordinary differential equations governing the modulation of the amplitudes and the phases of the two modes steady state responses and their stability are computed for selected values of the system parameters the effects of detuning the internal resonance detuning the parametric resonance the phase and magnitude of the second mode parametric excitation and the initial conditions are investigated the first order perturbation solution predicts qualitatively the trivial and non trivial stable steady state solutions and illustrates both the quenching and saturation phenomena in addition to the steady state solutions other periodic solutions are predicted by the perturbation amplitude and phase modulation equations these equations predict a transition from constant steady state non trivial responses to limit cycle responses hopf bifurcation some limit cycles are also shown to experience period doubling bifurcations the perturbation solutions are verified by numerically integrating the governing differential equations

scattering from a cylindrically conformal slotted waveguide array antenna a numerical method is developed to investigate electromagnetic scattering by a cylindrically conformal waveguide fed slot array the problem is first formulated in terms of integral equations using the equivalence principle the integral equations are then solved using the method of moments mom in conjunction with global sinusoidal basis functions and galerkin s testing procedure the mom solution requires the evaluation of the generalized admittance matrices involving various dyadic green s functions the slow convergence of the series associated with the summation of waveguide modes is accelerated using the kummer transformation and the slow convergence of the series associated with the summation of the exterior modes is avoided by using the asymptotic solutions with proper treatment for singular integrals the evaluation of the excitation vector and scattered field is also accelerated using watson s transformation and asymptotic solutions numerical results are presented to illustrate the scattering characteristics of the cylindrically conformal waveguide fed slot arrays such as the effects of curvature slot thickness and waveguide termination on the radar cross section rcs of the arrays

two degree of freedom systems with quadratic non linearities subjected to parametric and self excitation abstract the method of multiple scales is used to study the response of two degree of freedom systems with quadratic non linearities under the simultaneous effects of a harmonic parametric excitation and self excitation the principal parametric resonance of the first mode and a three to one internal resonance is considered followed by the case of internal and parametric resonance of the second mode in both cases the stability of the system is also studied amplitude and frequency response curves are presented for both cases the character of stability and the mode in which the system loses its stability is also discussed

generalization and selection of examples in feedforward neural networks in this work we study how the selection of examples affects the learning procedure in a boolean neural network and its relationship with the complexity of the function under study and its architecture we analyze the generalization capacity for different target functions with particular architectures through an analytical calculation of the minimum number of examples needed to obtain full generalization i e zero generalization error the analysis of the training sets associated with such parameter leads us to propose a general architecture independent criterion for selection of training examples the criterion was checked through numerical simulations for various particular target functions with particular architectures as well as for random target functions in a nonoverlapping receptive field perceptron in all cases the selection sampling criterion lead to an improvement in the generalization capacity compared with a pure random sampling we also show that for the parity problem one of the most used problems for testing learning algorithms only the use of the whole set of examples ensures global learning in a depth two architecture we show that this difficulty can be overcome by considering a tree structured network of depth 2 log2 n 1

a fully connected committee machine learning unrealizable rules we study generalization in a large fully connected committee machine with continuous weights trained on patterns with outputs generated by a teacher of the same structure but corrupted by noise the corruption is due to additive gaussian noise applied in the input layer or the hidden layer of the teacher contrary to related cases in the presence of input noise the generalization error epsilon g is not minimized by the teacher s weights for small values of the load parameter alpha the student is in a permutation symmetric phase as alpha increases three additional phases emerge the large alpha theory of the stable phase is similar to the tree committee machine in particular at zero temperature in the presence of noise epsilon g does not approach its minimal value epsilon min and the student s weights do not converge to those of the teacher for a positive temperature epsilon g epsilon min decays as a power of alpha the exponent being the same as in the corresponding case of the tree however for all values of alpha an at least metastable phase exists which is permutation symmetric with respect to the teacher

scattering by cavity backed antennas on a circular cylinder conformal arrays are popular antennas for aircraft spacecraft and land vehicle platforms due to their inherent low weight drag and observables however to date there has been a dearth of rigorous analytical or numerical solutions to aid the designer in fact it has been common practice to use limited measurements and planar approximations in designing such nonplanar antennas we extend the finite element boundary integral method to scattering by cavity backed structures in an infinite metallic cylinder in particular we discuss the formulation specifics such as weight functions dyadic green s function implementation details and particular difficulties inherent to cylindrical structures special care is taken to ensure that the resulting computer program has low memory demand and minimal computational requirements scattering results are presented and validated as much as possible

accelerating the convergence of series representing the free space periodic green s function a method for improving the convergence of the series representing the doubly periodic free space green s function is presented the method consists of successively applying three different transformations to the green s function spectral representation kummer s transformation is first applied to convert the slowly converging spectral representation into the sum of a rapidly converging series and a slowly converging series the latter series is recognized as the spectral representation of the original periodic source distribution radiating in a medium with an imaginary wavenumber application of the poisson transformation to this series renders it exponentially convergent since it effectively represents propagation of point source contributions through a medium with imaginary wavenumber finally shanks transform is plotted versus the number of terms taken in the series numerical results confirm that an improvement in the convergence rate of the series is achieved for a particular convergence criterion

computational capabilities of multilayer committee machines we obtained an analytical expression for the computational complexity of many layered committee machines with a finite number of hidden layers l 8 using the generalization complexity measure introduced by franco et al 2006 ieee trans neural netw 17 578 although our result is valid in the large size limit and for an overlap synaptic matrix that is ultrametric it provides a useful tool for inferring the appropriate architecture a network must have to reproduce an arbitrary realizable boolean function

clustering of neural code words revealed by a first order phase transition a network of neurons in the central nervous system collectively represents information by its spiking activity states typically observed states i e code words occupy only a limited portion of the state space due to constraints imposed by network interactions geometrical organization of code words in the state space critical for neural information processing is poorly understood due to its high dimensionality here we explore the organization of neural code words using retinal data by computing the entropy of code words as a function of hamming distance from a particular reference codeword specifically we report that the retinal code words in the state space are divided into multiple distinct clusters separated by entropy gaps and that this structure is shared with well known associative memory networks in a recallable phase our analysis also elucidates a special nature of the all silent state the all silent state is surrounded by the densest cluster of code words and located within a reachable distance from most code words this code word space structure quantitatively predicts typical deviation of a state trajectory from its initial state altogether our findings reveal a non trivial heterogeneous structure of the code word space that shapes information representation in a biological network

a pole zero matching method for ebg surfaces composed of a dipole fss printed on a grounded dielectric slab a method is presented for the efficient derivation of the dispersion equation associated with electromagnetic bandgap ebg structures composed by lossless frequency selective surfaces fss printed on stratified dielectric media the method valid for the range of frequency where a single propagating floquet mode occurs is based on foster s reactance theorem applied to an equivalent transmission line network this theorem implies that the admittance functions of frequency which represent the fss satisfy the pole zero analytical properties of the driving point lc admittance functions by these basic properties and by the full wave identification of the fss resonances an analytical form of the dispersion equation is obtained this equation is next solved for both surface wave and leaky wave modes by a conventional numerical technique the results are successfully compared with those from a full wave analysis

the response of two degree of freedom systems with quadratic non linearities to a parametric excitation the method of multiple scales is used to analyze the response of two degree of freedom systems with quadratic non linearities to a parametric harmonic excitation having the frequency four ordinary differential equations are derived to describe the modulation of the amplitudes and the phases when 2 2 1 and either 2 1 or 2 2 where 1 and 2 are the linear undamped natural frequencies of the system two critical values 1 and 2 of the amplitude f of the excitation are identified in the analysis when f 2 the amplitude of the directly excited mode grows exponentially with time according to the linear analysis whereas the amplitudes of both modes achieve steady state constant values irrespective of the initial amplitudes according to the non linear analysis when f 1 the motion decays to zero according to both the linear and non linear analyses when 1 f 2 the motion decays to zero according to the linear analysis whereas it achieves a periodic steady state or decays to zero depending on the initial amplitudes according to the non linear analysis this is an example of subcritical instability when 2 2 the steady state value of the higher mode which is directly excited is a constant that is independent of the excitation of amplitude f whereas the amplitude of the lower mode which is indirectly excited through internal resonance grows with the excitation amplitude f this is another example of saturation

simultaneous combination principal parametric and internal resonances in a slender beam with a lumped mass three mode interactions abstract the non linear dynamics of a parametrically base excited slender beam carrying a lumped mass with an excitation frequency nearly equal to twice the second mode natural frequency is examined the mass and the position of the attached element are so adjusted that the system exhibits 1 3 5 internal resonance by virtue of which a combination resonance of type 1 3 occurs simultaneously in the system the method of multiple scales mms is applied to reduce the second order temporal differential equations of motion of the system to a set of first order differential equations which is used to study the steady state periodic quasi periodic and chaotic responses of the system for different control parameters namely frequency and amplitude of base excitation damping internal detuning etc the results of perturbation analysis are compared with those obtained by numerically integrating the original temporal equations of motion poincare section and lyapunov exponents are used to characterize the chaotic responses

mutual admittance between slots in cylinders of arbitrary shape a numerical formulation involving the fourier transform and the combined field integral equation is presented to calculate the mutual admittance between axial slots or circumferential slots in arbitrarily shaped cylinders two cases are considered coupling between two circumferential slots and coupling between two axial slots the slots are assumed to be waveguide fed apertures excited with the te sub 10 waveguide mode with no higher order modes included in the model the combined field integral equation cfie formulation was used in conjunction with the magnetic field integral equation mfie to solve the resulting two dimensional integral equation formulations for the axial slot case are discussed results concerning the discretization and truncation of the spectrum as well as the accuracy of the procedure are presented for both axial and circumferential cases the accuracy of the procedure is demonstrated by comparison with eigenfunction data for circular cylinders

learning in ultrametric committee machines the problem of learning by examples in ultrametric committee machines ucms is studied within the framework of statistical mechanics using the replica formalism we calculate the average generalization error in ucms with l hidden layers and for a large enough number of units in most of the regimes studied we find that the generalization error as a function of the number of examples presented develops a discontinuous drop at a critical value of the load parameter we also find that when l 1 a number of teacher networks with the same number of hidden layers and different overlaps induce learning processes with the same critical points

period one motions of a mechanical oscillator with periodically time varying piecewise nonlinear stiffness the dynamic behavior of a piecewise nonlinear mechanical oscillator with parametric and external excitations is investigated the viscously damped single degree of freedom oscillator is subjected to a periodically time varying piecewise nonlinear restoring function defined by a clearance surrounded by continuously nonlinear quadratic and cubic regions typical applications represented by this oscillator are highlighted a multi term harmonic balance formulation is used in conjunction with discrete fourier transforms and a parametric continuation scheme to determine steady state period 1 motions of the system due to both parametric and external excitations the accuracy of the analytical solutions is demonstrated by comparing them to direct numerical integration solutions floquet theory is applied to determine the stability of the steady state harmonic balance solutions at the end detailed parametric studies are presented to quantify the combined influence of clearance quadratic and cubic nonlinearities within reasonable ranges of all other system parameters a comparison between time varying and time invariant systems is also provided to demonstrate the influence of the parametric and external excitations on a piecewise nonlinear system

the response of a single degree of freedom system with quadratic and cubic non linearities to a fundamental parametric resonance abstract the response of a one degree of freedom system with quadratic and cubic non linearities to a principal parametric resonance is investigated the method of multiple scales is used to determine the equations that describe to second order the modulation of the amplitude and phase with time about one of the foci these equations are used to determine the fixed points and their stability the perturbation results are verified by integrating the governing equation with use of a digital computer and an analogue computer for small excitation amplitudes the analytical results are in excellent agreement with the numerical solutions the large amplitude responses are investigated by using both a digital and an analogue computer the cases of single and double well potentials are investigated systems with double well potentials exhibit complicated dynamic behaviors including period multiplying and demultiplying bifurcations and chaos in some cases a bifurcated response coexists with another periodic attractor and a chaotic attractor coexists with a periodic attractor long time histories phase planes poincare maps fractal basin maps and spectra of the response are presented a bifurcation diagram of many solutions in the excitation amplitude excitation frequency plane is also presented

finite connectivity attractor neural networks we study a family of diluted attractor neural networks with a finite average number of symmetric connections per neuron as in finite connectivity spin glasses their equilibrium properties are described by order parameter functions for which we derive an integral equation in replica symmetric approximation a bifurcation analysis of this equation reveals the locations of the paramagnetic to recall and paramagnetic to spin glass transition lines in the phase diagram the line separating the retrieval phase from the spin glass phase is calculated at zero temperature all phase transitions are found to be continuous

combination parametric resonance leading to periodic and chaotic response in two degree of freedom systems with quadratic non linearities the dynamic response of a parametrically excited non linear system with two degrees of freedom is studied the system of equations which describes the compliant motion of a robotic manipulator exhibits internal resonance and is excited at a frequency near a primary resonance the effect of coupling is studied in detail non zero periodic motions are observed to co exist with a stable equilibrium state over a range of detuning near the resonant frequency loss of stability of the periodic response is observed at points of vertical tangency corresponding to a jump in the response while for some of the cases hopf bifurcation in the response corresponding to amplitude modulated motions is observed in the latter case period doubling of the amplitude of response is observed with a monotonic change in detuning eventually leading to chaotic motion in the system near exact tuning to the parametric resonance a quenching of the chaotic motion is found leaving only the stable zero equilibrium state for the system over some detuning interval

learning and generalization in a two layer neural network the role of the vapnik chervonvenkis dimension bounds for the generalization ability of neural networks based on vapnik chervonenkis vc theory are compared with statistical mechanics results for the case of the parity machine for fixed phase space dimension the vc dimension can grows arbitrarily by increasing the number k of hidden units generalization is impossible up to a critical number of training examples that grows with the vc dimension the asymptotic decrease of the generalization error mathrm ensuremath varepsilon mathit g comes out independent of k and the vc bounds strongly overestimate mathrm ensuremath varepsilon mathit g this shows that phase space dimension and vc dimension can play independent and different roles for the generalization process

resonant parametric perturbations of the hopf bifurcation abstract the effects of periodic parametric perturbations on a system undergoing hopf bifurcation are studied in detail of primary interest is the case of resonance between the hopf bifurcation frequency and the perturbation frequency method of alternative problems is used to obtain the small nonlinear periodic solutions it is shown that for a certain range of parameters the hopf solution is modified to give rise to jump response as well as isolated solutions for some parameter combinations stable solutions can get unstable and may bifurcate into aperiodic or amplitude modulated motions

learning a rule in a multilayer neural network the problem of learning from examples in multilayer networks is studied within the framework of statistical mechanics using the replica formalism we calculate the average generalization error of a fully connected committee machine in the limit of a large number of hidden units if the number of training examples is proportional to the number of inputs in the network the generalization error as a function of the training set size approaches a finite value if the number of training examples is proportional to the number of weights in the network we find first order phase transitions with a discontinuous drop in the generalization error for both binary and continuous weights

noisy regression and classification with continuous multilayer networks we investigate zero temperature gibbs learning for two classes of unrealizable rules which play an important role in practical applications of multilayer neural networks with differentiable activation functions classification problems and noisy regression problems considering one step of replica symmetry breaking we surprisingly find that for sufficiently large training sets the stable state is replica symmetric even though the target rule is unrealizable furthermore the classification problem is shown to be formally equivalent to the noisy regression problem

the response of two degree of freedom systems with quadratic and cubic non linearities to multifrequency parametric excitations abstract the response of two d o f systems with quadratic and cubic non linearities to multi frequency parametric excitations is determined by using the method of multiple scales four first order ordinary differential equations are derived to describe the modulation of the amplitudes and the phases when principal parametric resonances of both modes and combination resonances of the additive and difference type occur simultaneously in all cases the steady state solutions and their stability are determined numerical results depicting the various resonances are presented

on the convergence of mom and mode matching solutions for infinite array and waveguide problems it is shown that the method of moment mom fails to analyze an infinite array of strip discontinuities if the number of basis functions and the number of expansion modes are set equal however large they individually may be using spectral analysis and linear vector space theory it is found that for a given number of basis functions there exists a minimum number of expansion modes required for a reasonable accuracy subsequently for a given number of expansion modes there is an upper limit for the number of basis functions the mode matching mm analysis is viewed as a special case of the mom analysis and the accuracy of the analysis is discussed with respect to the number of modes in the waveguides

statistical mechanics of an error correcting code using monotonic and nonmonotonic treelike multilayer perceptrons an error correcting code using a treelike multilayer perceptron is proposed an original message mathbit s 0 is encoded into a codeword mathbit y 0 using a treelike committee machine committee tree or a treelike parity machine parity tree based on these architectures several schemes featuring monotonic or nonmonotonic units are introduced the codeword mathbit y 0 is then transmitted via a binary asymmetric channel where it is corrupted by noise the analytical performance of these schemes is investigated using the replica method of statistical mechanics under some specific conditions some of the proposed schemes are shown to saturate the shannon bound at the infinite codeword length limit the influence of the monotonicity of the units on the performance is also discussed

element pattern of an axial dipole in a cylindrical phased array part i theory cylindrical array antennas are attractive for applications requiring uniform 360 deg azimuthal pattern performance for very low sidelobe or precise angle measurement designs an accurate knowledge of the mutually coupled element pattern becomes important a modal solution is presented for an element pattern in a periodic array of axial dipole radiators placed coaxially over a circular cylindrical ground this solution avoids the problems associated with asymptotic appproaches and displays a number of advantages 1 it is exact subject only to truncation errors 2 it is uniformly valid in all spatial regions for both rectangular and with a simple modification for triangular isosceles lattice configurations and 3 it provides absolute gain information the analysis is presented along with numerical results for representative parameter values selected to illustrate the various trade offs comparison with published data employing the asymptotic approach and with planar array results is given

an asymptotic solution for the surface magnetic field within the paraxial region of a circular cylinder with an impedance boundary condition it is well known that the high frequency asymptotic evaluation of surface fields by the conventional geometrical theory of diffraction gtd usually becomes less accurate within the paraxial close to axial region of a source excited electrically large circular cylinder uniform versions of the gtd based solution for the surface field on a source excited perfect electrically conducting pec circular cylinder were published earlier to yield better accuracy within the paraxial region of the cylinder however efficient and sufficiently accurate solutions are needed for the surface field within the paraxial region of a source excited circular cylinder with an impedance boundary condition ibc in this work an alternative approximate asymptotic closed form solution is proposed for the accurate representation of the tangential surface magnetic field within the paraxial region of a tangential magnetic current excited circular cylinder with an ibc similar to the treatment for the pec case hankel functions are asymptotically approximated by a two term debye expansion within the spectral integral representation of the relevant green s function pertaining to the ibc case although one of the two integrals within the spectral representation is evaluated in an exact fashion the other integral for which an exact analytical evaluation does not appear to be possible is evaluated asymptotically unlike the pec case in which both integrals were evaluated analytically in an exact fashion validity of the proposed asymptotic solution is investigated by comparison with the exact eigenfunction solution for the surface magnetic field

learning from examples in fully connected committee machines the problem of learning from examples in two layer neural networks is studied within the framework of statistical mechanics a fully connected committee machine is trained to implement a task which is not linearly separable the generalization error as a function of the number of training examples per adjustable weight is calculated in the annealed approximation for both binary and continuous weights we find a first order transition with a discontinuous drop in the generalization error the transitions occur due to a specialization of the hidden units from a symmetric state to one with each hidden unit in the student network specialized on a corresponding unit in the teacher network the symmetric states of poor generalization remain metastable even for large training sets

the response of multidegree of freedom systems with quadratic non linearities to a harmonic parametric resonance abstract an analysis is presented of the response of multidegree of freedom systems with quadratic non linearities to a harmonic parametric excitation in the presence of an internal resonance of the combination type 3 2 1 where the n are the linear natural frequencies of the systems in the case of a fundamental resonance of the third mode i e 3 where is the frequency of the excitation one can identify two critical values 1 and 2 where 2 1 of the amplitude f of the excitation the value f 2 corresponds to the transition from stable to unstable solutions when f 2 the motion grows exponentially with time according to the linear theory but the non linearity limits the motion to a finite amplitude steady state the amplitude of the third mode which is directly excited is independent of f whereas the amplitudes of the first and second modes which are indirectly excited through the internal resonance are functions of f when 1 f 2 the motion decays or achieves a finite amplitude steady state depending on the initial conditions according to the non linear theory whereas it decays to zero according to the linear theory this is an example of subcritical instability in the case of a fundamental resonance of either the first or second mode the trivial response is the only possible steady state when f 2 the motion decays to zero according to both linear and non linear theories when f 2 the motion grows exponentially with time according to the linear theory but it is aperiodic according to the non linear theory experiments are being planned to check these theoretical results

periodic structure ray method for analysis of coupling coefficients in large concave arrays i theory a description is given of the periodic structure ray method psrm an efficient variant of the geometric theory of diffraction gtd for the determination of coupling scattering coefficients in large arrays on concave surfaces with slowly varying curvature and or periodicity the method is particularly suitable for applications involving low side lobe lens arrays and feed through radomes in such arrays the coefficients of coupling between array elements can be evaluated as a superposition of a small number of contributions from the periodic structure a typical contribution consists of a number of constituents each expressing a specific physical process occurring during the ray s travel from the transmitting to the receiving element the periodic structure ray formalism applies equally well to two and three dimensional array geometries and to a broad class of radiating elements the final results are presented in an application oriented form

on line learning in the committee machine the dynamics of learning from examples in the k 3 non overlapping committee machine with single presentation of examples is studied the optimal algorithm in the sense of mean generalization is obtained from a variational analysis of the differential equations which describe the dynamics the agreement of the theoretical predictions and the results of numerical simulations is excellent the optimized dynamics has the extra advantage with respect to the non optimized cases in that it uncouples the differential equations which describe the evolution of the relevant parameters i e the student teacher overlap and the norm of the student synaptic vector this in turn translates into the possibility of constructing useful practical optimized on line algorithms for the optimal algorithm the generalization error decays as approximately 0 88 alpha 1 the same nominal error as for the simple perceptron with optimized dynamics

efficient computation of nonparaxial surface fields excited on an electrically large circular cylinder with an impedance boundary condition an alternative numerical approach is presented for the evaluation of the fock type integrals that exist in the uniform geometrical theory of diffraction utd based asymptotic solution for the nonparaxial surface fields excited by a magnetic or an electric source located on the surface of an electrically large circular cylinder with an impedance boundary condition ibc this alternative approach is based on performing numerical integration of the fock type integrals on a deformed path on which the integrands are nonoscillatory and rapidly decaying comparison of this approach with the previously developed one presented in which is based on invoking the cauchy s residue theorem by finding the pole singularities numerically reveals that the alternative approach is considerably more efficient

storage capacity of the fully connected committee machine the storage capacity that is the number of patterns which can be stored per weight is calculated for the fully connected committee machine with real couplings and k hidden units from the vanishing of the entropy of the internal representations and it is found to diverge as

storage capacity of ultrametric committee machines the problem of computing the storage capacity of a feed forward network with l hidden layers n inputs and k units in the first hidden layer is analyzed using techniques from statistical mechanics we found that the storage capacity strongly depends on the network architecture c log k 1 1 2l and that the number of units k limits the number of possible hidden layers l through the relationship 2l 1 2log k

parallel strategy for optimal learning in perceptrons we developed a parallel strategy for learning optimally specific realizable rules by perceptrons in an online learning scenario our result is a generalization of the caticha kinouchi ck algorithm developed for learning a perceptron with a synaptic vector drawn from a uniform distribution over the n dimensional sphere so called the typical case our method outperforms the ck algorithm in almost all possible situations failing only in a denumerable set of cases the algorithm is optimal in the sense that it saturates bayesian bounds when it succeeds

phased arrays of rectangular apertures on conformal cylindrical surfaces a multimode equivalent network approach in this paper we present an efficient integral equation formulation based on a multimode equivalent network men approach for the analysis of phased arrays on cylindrical surfaces using open ended waveguide radiators in addition to theory experimental data are also presented showing very good agreement with simulated results

phase transitions in the generalization behaviour of multilayer neural networks we study the generalization ability of multilayer neural networks with tree architecture for the case of random training sets the first layer consists of k spherical perceptrons with binary output a boolean function b computes the final output from the k values produced by the first layer we first calculate the learning behaviour of gibbs learning in the case of learnable rules where teacher and student have the same architecture and the boolean function b is permutation symmetric with respect to the hidden units in the asymptotic case of high loading alpha to infinity as usual alpha is the loading parameter we find that the generalization error vanishes in the same way for all b the reason being is that there are two effects cancelling each other in the opposite limit for small alpha we find qualitatively different behaviour i e some networks undergo a phase transition we show how these differences in the behaviour depend on certain characteristics of the boolean function b we then study the bayes algorithm which generalizes according to the majority decision of a certain ensemble of machines and find that the parity function plays a special role if the teacher is a parity machine there always exists a single student parity machine that generalizes as well as the bayes algorithm

a new low side lobe pattern synthesis technique for conformal arrays the application of the new nonlinear optimization algorithms to array design is demonstrated by the low side lobe pattern synthesis of conformal arrays by adopting the information of array element realized gain patterns the authors formulate synthesis problems as nonlinearly constrained optimization problems and propose a direct method to solve them the technique allows them to find a set of array coefficients that yield a pattern meeting a specified side lobe level and achieving the maximum directivity if such a set of array coefficients exists if the side lobe specifications cannot be met with the given array the technique will result in a set of coefficients that yield a pattern meeting the best attainable side lobe level and having directivity as high as possible simplified synthesis problems for an axial dipole array in an infinite perfectly conducting cylinder are discussed and numerical results are given the synthesis technique works for general array patterns

parametric excitation of two internally resonant oscillators abstract the response of two degree of freedom systems with quadratic non linearities to a principal parametric resonance in the presence of two to one internal resonances is investigated the method of multiple scales is used to construct a first order uniform expansion yielding four first order non linear ordinary differential averaged equations governing the modulation of the amplitudes and the phases of the two modes these equations are used to determine steady state responses and their stability when the higher mode is excited by a principal parametric resonance the non trivial steady state value of its amplitude is a constant that is independent of the excitation amplitude whereas the amplitude of the lower mode which is indirectly excited through the internal resonance increases with the amplitude of the excitation however in addition to poincare type bifurcations this response exhibits a hopf bifurcation leading to amplitude and phase modulated motions when the lower mode is excited by a principal parametric resonance the averaged equations exhibit both poincare and hopf bifurcations in some intervals of the parameters the periodic solutions of the averaged equations in the latter case experience period doubling bifurcations leading to chaos

parametrically excited non linear multidegree of freedom systems with repeated natural frequencies abstract a method for analyzing multidegree of freedom systems having a repeated natural frequency subjected to a parametric excitation is presented attention is given to the ordering of the various terms linear and non linear in the governing equations the analysis is based on the method of multiple scales as a numerical example involving a parametric resonance panel flutter is discussed in detail in order to illustrate the type of results one can expect to obtain with this analysis some of the analytical results are verified by a numerical integration of the governing equations

small correlation expansions for the inverse ising problem we present a systematic small correlation expansion to solve the inverse ising problem and find a set of couplings and fields corresponding to a given set of correlations and magnetizations couplings are calculated up to the third order in the correlations for generic magnetizations and to the seventh order in the case of zero magnetizations in addition we show how to sum some useful classes of diagrams exactly the resulting expansion outperforms existing algorithms on the sherrington kirkpatrick spin glass model

nonlinear and chaotic vibration and stability analysis of an aero elastic piezoelectric fg plate under parametric and primary excitations abstract in this study in the presence of supersonic aerodynamic loading the nonlinear and chaotic vibrations and stability of a simply supported functionally graded piezoelectric fgp rectangular plate with bonded piezoelectric layer have been investigated it is assumed that the plate is simultaneously exposed to the effects of harmonic uniaxial in plane force and transverse piezoelectric excitations and aerodynamic loading it is considered that the potential distribution varies linearly through the piezoelectric layer thickness and the aerodynamic load is modeled by the first order piston theory the von karman nonlinear strain displacement relations are used to consider the geometrical nonlinearity based on the classical plate theory cpt and applying the hamilton s principle the nonlinear coupled partial differential equations of motion are derived the galerkin s procedure is used to reduce the equations of motion to nonlinear ordinary differential mathieu equations the validity of the formulation for analyzing the limit cycle oscillation lco aero elastic stability boundaries is accomplished by comparing the results with those of the literature and the convergence study of the fgp plate is performed by applying the multiple scales method the case of 1 2 internal resonance and primary parametric resonance are taken into account and the corresponding averaged equations are derived and analyzed numerically the results are provided to investigate the effects of the forcing piezoelectric detuning parameter amplitude of forcing piezoelectric excitation and dynamic pressure on the nonlinear dynamics and chaotic behavior of the fgp plate it is revealed that under the certain conditions due to the existence of bi stable region of non trivial solutions system shows the hysteretic behavior moreover in absence of airflow it is observed that variation of control parameters leads to the multi periodic and chaotic motions

multimode equivalent networks for the design and analysis of frequency selective surfaces a modular technique originally proposed for waveguide junctions the multimode equivalent network approach based on the integral equation formulation iemen is extended to the analysis of multilayer frequency selective surfaces integrated with waveguide array antennas this technique represents each layer and transition between layers in terms of a generalized impedance or admittance matrix obtained directly from the solution of an integral equation with reduced kernel thanks to the adopted formulation the integral equation needs to be solved only in a limited set of frequency points the iemen method is validated by comparison with results available in literature

the space of interactions in neural network models the typical fraction of the space of interactions between each pair of n ising spins which solve the problem of storing a given set of p random patterns as n bit spin configurations is considered the volume is calculated explicitly as a function of the storage ratio alpha p n of the value kappa 0 of the product of the spin and the magnetic field at each site and of the magnetisation m here m may vary between 0 no correlation and 1 completely correlated the capacity increases with the correlation between patterns from alpha 2 for correlated patterns with kappa 0 and tends to infinity as m tends to 1 the calculations use a saddle point method and the order parameters at the saddle point are assumed to be replica symmetric this solution is shown to be locally stable a local iterative learning algorithm for updating the interactions is given which will converge to a solution of given kappa provided such solutions exist

response of two degree of freedom systems to multifrequency parametric excitations abstract the method of multiple scales is used to analyze the response of two degree of freedom systems to multifrequency parametric excitations the equations describing the modulation of the amplitudes with time are derived for the case of four simultaneous resonances main resonances of the two modes combination resonance of the difference type and combination resonance of the summed type when only two simultaneous resonances exist the modulation equations can be transformed into a system of four real first order differential equations with constant coefficients whose solution depends on the eigenvalues of a four by four constant coefficient matrix when three or more simultaneous resonances exist the modulation equations cannot be made autononous in general and hence one may not be able to solve them except numerically

a semi analytical locally transversal linearization method for non linear dynamical systems an numeric analytical implicit and local linearization methodology called the locally transversal linearization ltl is developed in the present paper for analyses and simulations of non linear oscillators the ltl principle is based on deriving the locally linearized equations in such a way that the tangent space of the linearized equations transversally intersects that of the given non linear dynamical system at that particular point in the state space where the solution vector is sought for purposes of numerical implementation two different numerical schemes namely ltl 1 and ltl 2 schemes based on the ltl methodology are presented both ltl 1 and ltl 2 procedures finally reduce the given set of non linear ordinary differential equations odes to a set of transcendental algebraic equations valid over a short interval of time or over a short segment of the evolving trajectories as projected on the phase space while in the ltl 1 scheme the desired solution vector at a forward time point enters the linearized differential equations as an unknown parameter in the ltl 2 scheme a set of unknown residues enters the linearized system as parameters a limited set of examples involving a few well known single degree of freedom sdof non linear oscillators indicate that the ltl methodology is capable of accurately predicting many complicated non linear response patterns including limit cycles quasi periodic orbits and even strange attractors copyright 2001 john wiley sons ltd

hybrid finite element boundary integral method for cavities recessed in an elliptic cylinder the highly versatile finite element boundary integral method is presented for accurate modeling of fields within cavities recessed in a perfectly conducting infinite elliptic cylinder properties of flush mounted cavities and antennas are explored and compared with prior developments this formulation represents a significant advancement over prior techniques since surface curvature variation either across a single element or across an array of elements is now accurately included into the antenna model an advantage of this approach is the ability to model cavities with curvature varying from planar to the constant curvature of a circular cylinder computed eigenvalues for planar rectangular and cylindrical rectangular closed cavities are compared with theoretical values eigenvalues are presented for elliptic rectangular cavities furthermore the input impedance of a conformal cavity backed patch antenna mounted on an elliptic cylinder is presented and compared with a similar antenna on a circular cylinder

efficient learning in boltzmann machines using linear response theory the learning process in boltzmann machines is computationally very expensive the computational complexity of the exact algorithm is exponential in the number of neurons we present a new approximate learning algorithm for boltzmann machines based on mean field theory and the linear response theorem the computational complexity of the algorithm is cubic in the number of neurons in the absence of hidden units we show how the weights can be directly computed from the fixed point equation of the learning rules thus in this case we do not need to use a gradient descent procedure for the learning process we show that the solutions of this method are close to the optimal solutions and give a significant improvement when correlations play a significant role finally we apply the method to a pattern completion task and show good performance for networks up to 100 neurons

asymptotic analysis of mutual coupling in concave circular cylindrical arrays ii near zone for pt i see ibid vol 41 no 2 p 121 36 1993 this work evaluates the near zone coupling coefficients in large concave arrays a canonical model of a circular cylindrical concave array of open ended rectangular waveguides is analyzed it is shown that in the paraxial region which also includes the area close to the excited element the coupling coefficients are identical to lowest asymptotic order to those in an equivalent infinite planar array in the transition region which lies between the paraxial and the far zone or the ray region two representations were developed one a transition function expressed in terms of a canonical integral which must be numerically evaluated and the other a uniform representation which consists of a superposition of a planar array contribution and a few periodic structure rays the uniform representation is valid in the near zone which includes both the paraxial and the transition region as well as in the far zone this form is simple and may be immediately generalized to concave arrays with slowly varying curvature and periodicity

inference and learning in sparse systems with multiple states we discuss how inference can be performed when data are sampled from the nonergodic phase of systems with multiple attractors we take as a model system the finite connectivity hopfield model in the memory phase and suggest a cavity method approach to reconstruct the couplings when the data are separately sampled from few attractor states we also show how the inference results can be converted into a learning protocol for neural networks in which patterns are presented through weak external fields the protocol is simple and fully local and is able to store patterns with a finite overlap with the input patterns without ever reaching a spin glass phase where all memories are lost

the influence of oppositely classified examples on the generalization complexity of boolean functions in this paper we analyze boolean functions using a recently proposed measure of their complexity this complexity measure motivated by the aim of relating the complexity of the functions with the generalization ability that can be obtained when the functions are implemented in feed forward neural networks is the sum of a number of components we concentrate on the case in which we use the first two of these components the first is related to the average sensitivity of the function and the second is in a sense a measure of the randomness or lack of structure of the function in this paper we investigate the importance of using the second term in the complexity measure and we consider to what extent these two terms suffice as an indicator of how difficult it is to learn a boolean function we also explore the existence of very complex boolean functions considering in particular the symmetric boolean functions

inference of kinetic ising model on sparse graphs based on dynamical cavity method we propose an approach to the inference of kinetic ising model which asks to reconstruct couplings and external fields from given time dependent output of original system our approach gives an exact result on tree graphs and a good approximation on sparse graphs it can be seen as an extension of belief propagation inference of static ising model to kinetic ising model while existing mean field methods to the kinetic ising inference e g naive mean field tap equation and simply mean field use approximations which calculate magnetizations and correlations at time t from statistics of data at time t 1 dynamical cavity method can use statistics of data at times earlier than t 1 to capture more correlations at different time steps extensive numerical experiments show that our inference method is superior to existing mean field approaches on diluted networks

stability and primary simultaneous resonance of harmonically excited non linear spring pendulum system the considered system is a harmonically excited non linear spring pendulum which simulates the ship roll motion the method of multiple time scale perturbation is applied to solve the non linear differential equations describing the system up to and including the fourth order approximation all possible resonance cases were extracted at this approximation order this work is limited only to the primary resonance cases the stability of the system is investigated using both frequency response equations and phase plane method the effects of the different parameters on system behavior are studied numerically variation of some equations parameters leads to the bending of the frequency response curves and hence to the jump phenomenon it is quite clear that some of the simultaneous primary resonance cases are undesirable in the design of such system as they represent some of the worst behavior of the system such cases should be avoided as working conditions for the system some recommendations regarding the different parameters of the system are reported

admittance of rectangular waveguide radiating from a conducting cylinder theory and experiment are compared for the admittance presented to a rectangular waveguide which terminates at an arbitrary angle in the surface of a conducting cylinder two methods are described for calculating the admittance a single mode approximation from the modal solution and an alternative asymptotic solution using these methods the effect of aperture orientation on admittance and the results obtained are compared with the experiment the single mode approximation is in good agreement with experimental results for the case when the input waveguide supports the te sub 10 mode only the accuracy of the approximate formula for waveguide admittance was also verified the admittance is shown to be weakly dependent on aperture orientation for moderately large cylinders

accurate asymptotic solution for the surface field due to apertures in a conducting cylinder asymptotic solutions for the surface field due to a magnetic point source on a cylinder can simplify the computational task in the analysis of mutual coupling e g in conformal arrays although several asymptotic solutions are at present available their accuracy is limited the aim here is to obtain a new approximate solution with improved accuracy an asymptotic solution of the exact modal solution for the field on a cylinder which contains terms up to k 0 t 2 where t is the distance from the source is derived approximate terms up to k 0 t 3 are included from the solution for a source on a plane conductor and an earlier asymptotic solution the new approximate solution consists of the exact solution for a plane conductor modulated by a fock function plus a contribution associated with the cylinder which vanishes in the limit of infinite radius typically for cylinders of radius greater than one and half wavelengths it agrees to within 0 5 db in magnitude and 1 deg in phase of the modal solution results for dominant mode mutual and self admittances of rectangular apertures on a cylinder obtained from this solution are presented a formula is also given for calculating self admittances from an asymptotic solution there is excellent agreement between admittances computed from the asymptotic solution and the modal solution one exception is the self susceptance of an axial slot where the error is larger than expected although the error is low enough to be acceptable in most practical cases

mean field theory of boltzmann machine learning i present a mean field theory for boltzmann machine learning derived by employing thouless anderson palmer free energy formalism to a full extent using the plefka expansion an extended theory that takes higher order correction to mean field free energy formalism into consideration is presented from which the mean field approximation of general orders along with the linear response correction are derived by truncating the plefka expansion up to desired orders a theoretical foundation for an effective trick of using diagonal weights introduced by kappen and rodr i guez is also given because of the finite system size and a lack of scaling assumptions on interaction coefficients the truncated free energy formalism cannot provide an exact description in the case of boltzmann machines accuracies of mean field approximations of several orders are compared by computer simulations

internal resonances effects simulation versus analytical methods results abstract in this study of internal resonance effects in a two degree of freedom system with cubic non linearity attention is focussed on behaviour of the system as examined with the aid of an analog computer and on an adequacy of approximate analytical methods the averaging method and the ritz method it is shown that while the averaging method fails to predict the behaviour of the system the ritz method gives results surprisingly close to those obtained by the simulation method the results indicate that the internal resonance can be responsible for occurrence of two frequency oscillations with the amplitude of the subharmonic tone nearly 10 times higher than the amplitude of the fundamental harmonic component

uniform asymptotic solution for the surface magnetic field valid both within and outside the paraxial region of a perfect electrically conducting circular cylinder high frequency methods employed for the perfect electrically conducting circular cylinder form two sets the uniform theory of diffraction utd based ones are valid outside the paraxial region while there are others that are accurate close to the axis of the cylinder here we present a uniform asymptotic solution for the surface magnetic field valid both within and outside the paraxial region of a perfect electrically conducting circular cylinder this is achieved by approximating the hankel functions by a uniform asymptotic expansion within the spectral integral representation of the relevant green s function all the resulted integrals are evaluated analytically in an exact fashion such a solution is of interest both for theoretical and for practical reasons from a theoretical point of view it is important to have a closed form solution that copes with the canonical problem of a perfect electrically conducting circular cylinder from a practical point of view such a solution provides mutual coupling results and thus predicts compatibility and interference between conformal slot antennas mounted on constant radius cylindrically shaped perfectly conducting hosts validity of the proposed asymptotic solution is provided by comparison with published results

learning from examples in large neural networks a statistical mechanical theory of learning from examples in layered networks at finite temperature is studied when the training error is a smooth function of continuously varying weights the generalization error falls off asymptotically as the inverse number of examples by analytical and numerical studies of single layer perceptrons we show that when the weights are discrete the generalization error can exhibit a discontinuous transition to perfect generalization for intermediate sizes of the example set the state of perfect generalization coexists with a metastable spin glass state