closed forms of the green s function and the generalized green s function for the helmholtz operator on the n dimensional unit sphere the green s function for the helmholtz differential operator 2 n 1 on the n dimensional with n 1 hyperspherical surface s n of unit radius is investigated its closed form is shown to be g n n n n 1 sn sin c n 1 2 n n where sn is the area of s n c x is the gegenbauer function of the first

numerical calculation of the diffraction coefficients for an arbitrary shaped perfectly conducting cone a method for numerical calculation of the diffraction coefficients for electromagnetic diffraction by arbitrarily shaped perfectly conducting cones is proposed the approach makes an extensive use of the analytic formulas of smyshlyaev in combination with further developments including a use of the potential theory adapted to the laplace beltrami operator on a subdomain of unit sphere this reduces the problem to a fredholm integral equation on the closed curve of the unit sphere defining the cone s geometry which can be solved numerically this strategy permits us to implement a numerical code for calculation of the diffraction coefficients for cones of rather general cross sections results of sample calculations for the circular and elliptic cones are given

knots and links in steady solutions of the euler equation given any possibly unbounded locally nite link we show that there exists a smooth dieomorphism transforming this link into a set of stream or vortex lines of a vector eld that solves the steady incompressible euler equation in r 3 furthermore the dieomorphism can be chosen arbitrarily close to the identity in any c r norm

closed form of the generalized green s function for the helmholtz operator on the two dimensional unit sphere the closed representation of the generalized known also as reduced or modified green s function for the helmholtz partial differential operator on the surface of the two dimensional unit sphere is derived in its compact form the derived formula contains a legendre polynomial and a derivative of the legendre function of the first kind with respect to its index an explicit expression for that derivative is found and used to obtain an expanded and potentially more suitable in applications form of the generalized green s function for the operator in question the related problem of constructing the closed form of the generalized green s function for the legendre ordinary differential operator on the segment 1 x 1 with the boundary conditions of finiteness at x 1 is also solved

canonical representations of complex vibratory subsystems time domain dirichlet to neumann maps large scale dynamic simulations can often be simplified by appropriately replacing large portions of the domain by a dirichlet to neumann or dtn map givoli 1992 numerical methods for problems on infinite domains 1st ed elsevier amsterdam here we consider the problem of representing a linear dynamical subsystem by such a map the exact dtn map is computed as a modal summation and its properties are studied bounds on the symbol of the dtn map in the laplace domain are obtained the exact map is then approximated in particular in the high modal density regime in the high modal density limit we obtain the result that a subsystem can be accurately represented with just three parameters within such an approximation we obtain representations based on a maximum entropy representation self similar or fractal representation and a rational function representation the rational function representation leads to the interesting result that any complicated dynamical subsystem with a large number of degrees of freedom is asymptotically equivalent in the limit of infinite modal density to a single mass dashpot spring system we end with numerical examples showing the efficiency of the rational function approximation

boundary integral equation technique for the singular behavior of electromagnetic fields at the common tip of metallic and bi isotropic cones with arbitrary cross section a boundary integral equation technique is developed to determine the singular field behavior at the common tip of lossless bi isotropic and perfectly electrically and or magnetically conducting cones with arbitrary cross section the kernel of the set of boundary integral equations is a green s function defined on a spherical surface this green s function is the associated legendre function of the first kind the integral equations are solved with the galerkin method of moments the theory is illustrated with a number of examples that show the effects of bi isotropy on the singular field behavior

existence of knotted vortex tubes in steady euler flows we prove the existence of knotted and linked thin vortex tubes for steady solutions to the incompressible euler equation in mathbb r 3 more precisely given a finite collection of possibly linked and knotted disjoint thin tubes in mathbb r 3 we show that they can be transformed with a c m small diffeomorphism into a set of vortex tubes of a beltrami field that tends to zero at infinity the structure of the vortex lines in the tubes is extremely rich presenting a positive measure set of invariant tori and infinitely many periodic vortex lines the problem of the existence of steady knotted thin vortex tubes can be traced back to lord kelvin

equipartition and mean square responses in large undamped structures it is shown that the smoothed spectral energy density at one point in a large complex structure may be approximated in a simple manner as proportional to the frequency smoothed admittance at the receiver and at the source and inversely proportional to the global modal density thus the mean square response may be estimated with little computational burden knowing only the local properties at source and receiver and the size of the system the approximate predictions are compared with the results of direct numerical simulations and found to be accurate except in the presence of anderson localization work supported by nsf

beltrami fields with nonconstant proportionality factor we consider the question raised by enciso and peralta salas arch ration mech anal 220 1 243 260 2016 what nonconstant functions f can occur as the proportionality factor for a beltrami field mathbf u on an open subset u subset mathbb r 3 we also consider the related question for any such f how large is the space of associated beltrami fields by applying cartan s method of moving frames and the theory of exterior differential systems we are able to improve upon the results given in peralta salas 2016 in particular the answer to the second question depends crucially upon the geometry of the level surfaces of f we conclude by giving a complete classification of beltrami fields that possess either a translation symmetry or a rotation symmetry

computations of the complex error function rational expansions for computing the complex error function w z e z 2 text erfc iz are presented these expansions have the following attractive properties 1 they can be evaluated using a polynomial evaluation routine such as homer s method 2 the polynomial coefficients can be computed once and for all by a single fast fourier transform fft and 3 high accuracy is achieved uniformly in the complex plane with only a small number of terms comparisons reveal that in some parts of the complex plane certain competitors may be more efficient however the difference in efficiency is never great and the new algorithms are simpler than existing ones a complete program takes eight lines of matlab code

high modal density approximations for equipment in the time domain this article considers modeling the effect of complex subsystems on the dynamics of the main structure to which they are attached it is proposed that the substructure say a piece of equipment be replaced by an equivalent set of forces which react back on the main structure these forces are given as time convolutions of the displacements at the equipment attachment points the convolution integral which represents a time domain dtn dirichlet to neumann map is approximated in the high modal density limit with determined error bounds this approximation leads to a family of equipment representations the simplest requires few measured equipment properties though more information can lead to greater accuracy our approximate dtns are demonstrated numerically in finite element simulations

a theoretical formulation of the dynamical response of a master structure coupled with elastic continuous fuzzy subsystems with discrete attachments we present the formulation of the dynamic response of a master structure coupled with a locally homogeneous and orthotropic structural fuzzy with discrete attachment composed of elastic continuous fuzzy subsystems as introduced by soize the master structure is the part of the coupled system which is accessible by classical modeling whereas the structural fuzzy represents systems connected to the master structure whose characteristics are imprecisely known a deterministic formulation of the boundary impedance of a general continuous structural fuzzy which models its action on the master structure is derived it is shown that the formulation is different from the solution proposed by soize in the context of the type i fuzzy law established from the deterministic model of a linear oscillator excited by its support finally the general boundary impedance is applied to the special situation of a structural fuzzy composed of elastic bars whose geometrical parameters are randomly defined and numerical results are presented

the computation of conical diffraction coefficients in high frequency acoustic wave scattering when a high frequency acoustic or electromagnetic wave is scattered by a surface with a conical point the component of the asymptotics of the scattered wave corresponding to diffraction by the conical point can be represented as an asymptotic expansion valid as the wave number k rightarrow infty the diffraction coefficient is the coefficient of the principal term in this expansion and is of fundamental interest in high frequency scattering it can be computed by solving a family of homogeneous boundary value problems for the laplace beltrami helmholtz equation parametrized by a complex wave number like parameter nu on a portion of the unit sphere bounded by a simple closed contour ell and then integrating the resulting solutions with respect to nu in this paper we give the numerical analysis of a method for carrying out this computation in the case of acoustic waves via the boundary integral method applied on ell emphasizing the practically important case when the conical scat

induced damping by a nearly continuous distribution of nearly undamped oscillators linear analysis abstract the induced damping in a master oscillator contributed by a set of satellite oscillators is obtained in terms of a summation over a discrete distribution of the set the distribution is with respect to the resonance frequencies of the satellite oscillators in the set the distribution is cast in an ascending order and is assumed to be centered about the resonance frequency of the master oscillator in isolation if the modal overlap parameters are less than unity significant undulations are present in the induced damping the less the modal overlap parameters are compared with unity the more prominent are the undulations the undulations are largely suppressed when the local modal overlap parameters exceed unity moreover appropriately averaging the undulations yields values for the induced damping that coincide with those obtained when the modal overlap parameters exceed unity further it transpires that these common values are independent of the individual modal overlap parameters when the summation is replaced by an integration the first order results are undulations free and the values so obtained again coincide with those pertaining to modal overlap parameters that exceed unity without defining the excursions in the undulations the transition from a discrete to a continuous distribution that is implied by a summation to an integration must assume that the modal overlap parameters exceed unity in particular without careful and meaningful qualifications it may be misleading to assume a priori that the modal overlap parameters are equal to zero

beltrami fields with a nonconstant proportionality factor are rare we consider the existence of beltrami fields with a nonconstant proportionality factor f in an open subset u of mathbb r 3 by reformulating this problem as a constrained evolution equation on a surface we find an explicit differential equation that f must satisfy whenever there is a nontrivial beltrami field with this factor this ensures that there are no nontrivial regular solutions for an open and dense set of factors f in the c k topology k geqq 7 in particular there are no nontrivial beltrami fields whenever f has a regular level set diffeomorphic to the sphere this provides an explanation of the helical flow paradox of morgulis et al commun pure appl math 48 571 582 1995

trax an approach for the time rational analysis of complex dynamic systems this paper introduces a theoretical and algorithmic reduced model approach to efficiently evaluate time responses of complex dynamic systems the proposed approach combines four main components analytical expressions of the average of the system s transfer functions in the frequency domain precise and convergent rational approximations of these exact expressions exact evaluation of these approximations through model reduction in rational krylov subspaces and semi analytical interpolation at just a few frequency points the resulting algorithmic principles to evaluate the time response of a particular system are relatively straightforward one first evaluates the response of the system with slight additional damping at a few frequencies and one then projects or reduces the system in the subspace spanned by these responses the time response of the reduced model implicitly provides a precise evaluation of that of the original system the properties of the reduced models and the precision of the proposed approach are studied and applications on complex matrix systems are presented and discussed while the theory and numerical algorithms are presented in a matrix context they are also transposable in a continuous functional context

fuzzy structure theory modeling of sound insulation layers in complex vibroacoustic uncertain systems theory and experimental validation the fuzzy structure theory was introduced 20 years ago in order to model the effects of complex subsystems imprecisely known on a master structure this theory was only aimed at structural dynamics in this paper an extension of that theory is proposed in developing an elastoacoustic element useful to model sound insulation layers for computational vibroacoustics of complex systems the simplified model constructed enhances computation time and memory allocation because the number of physical and generalized degrees of freedom in the computational vibroacoustic model is not increased however these simplifications introduce model uncertainties in order to take into account these uncertainties the nonparametric probabilistic approach recently introduced is used a robust simplified model for sound insulation layers is then obtained this model is controlled by a small number of physical and dispersion parameters first the extension of the fuzzy structure theory to elastoacoustic element is presented

mean and mean square responses of a prototypical master fuzzy structure analytical estimates are made for mean and mean square transient responses of a single degree of freedom master oscillator attached to an undamped n degree of freedom random substructure the ensemble averaged response is found to be asymptotically for a large number of substructural degrees of freedom in accord with the predictions of the pierce sparrow russell theory in particular the mean response of the master oscillator manifests an apparent damping corrections to this behavior are also predicted but are found to lie well below the root mean square level of the fluctuations the corrections are therefore for practical purposes unimportant the ensemble average of the square of the response is also investigated and found to take a value at later times that is in accord with simple equipartition arguments advanced in an earlier paper the power spectral density of the late time behavior is shown to be proportional to the square of the power spectrum of the early time response and thus to be a n

on the derivative of the legendre function of the first kind with respect to its degree we study the derivative of the legendre function of the first kind p z with respect to its degree at first we provide two contour integral representations for p z then we proceed to investigate the case of p z n with n being an integer this case is met in some physical and engineering problems since it holds that we focus on the sub case of n being a non negative integer we show that where rn z is a polynomial in z of degree n we present alternative derivations of several known explicit expressions for rn z and also add some new a generating function for rn z is also constructed properties of the polynomials vn z rn z 1 nrn z 2 and wn 1 z rn z 1 nrn z 2 are also investigated it is found that wn 1 z is the christoffel polynomial well known from the theory of the legendre function of the second kind qn z as examples of applications of the results obtained we present non standard derivations of some representations of qn z sum to closed forms some legendre series evaluate some definite integrals involving legendre polynomials and also derive an explicit representation of the indefinite integral of the legendre polynomial squared

distance to ck hypersurfaces abstract it is shown by elementary means that a ck hypersurface m of positive reach in r n 1 has the property that the signed distance function to it is ck k 1 this extends and complements work of federer gilbarg and trudinger and serrin

divergence free fields energy and asymptotic crossing number in more than one context the equations of classical physics lead to a closed 2 form co on 3 dimensional space which is a conserved quantity in the sense that it pulls back under time evolution in the physics literature co is written as a divergence free vector field w appendix a describes two settings in which a divergence free w is conserved or frozen one is the magnetic field in ideal magneto hydrodynamics and the other is curl velocity in incompressible inviscid fluid flow if ht denotes the transport diffeomorphisms of a moving fluid in which the field is imbedded with ho id wt denotes the field in the fluid at time t and x0 is the space variable at time t 0 then

contact topology and hydrodynamics iii knotted orbits we employ the relationship between contact structures and beltrami fields derived in part i of this series to construct a steady nonsingular solution to the euler equations on a riemannian s3 whose flowlines trace out closed curves of all possible knot and link types using careful contacttopological controls we can make such vector fields real analytic and transverse to the tight contact structure on s3 sufficient review of concepts is included to make this paper independent of the previous works in this series

existence and uniqueness for a coupled parabolic elliptic model with applications to magnetic relaxation we prove the existence uniqueness and regularity of weak solutions of a coupled parabolic elliptic model in 2d and the existence of weak solutions in 3d we consider the standard equations of magnetohydrodynamics with the advective terms removed from the velocity equation despite the apparent simplicity of the model the proof in 2d requires results that are at the limit of what is available including elliptic regularity in l 1 and a strengthened form of the ladyzhenskaya inequality f l 4 leqq c f l 2 infty 1 2 nabla f l 2 1 2 which we derive using the theory of interpolation the model potentially has applications to the method of magnetic relaxation introduced by moffatt j fluid mech 159 359 378 1985 to construct stationary euler flows with non trivial topology

linear system approximation via covariance equivalent realizations abstract a truncation technique for model reduction to simultaneously match a specified number of output covariance derivatives is described these covariance equivalent realizations which match q 1 covariance derivatives are called q covers in general q covers are not unique the additional freedom is used herein so that the q cover obtained also matches q markov parameters the truncation technique uses a form of the observability matrices of the full order system to determine a priori the order required of the reduced order model to match a specified number of output covariance derivatives and markov parameters the resulting realization is shown to be independent of the basis of the complete model to within a unitary transformation stability conditions for the reduced order model are also described and the relationships are established between stochastically equivalent realizations and the q covers

uncertainty in structural dynamics experimental validation of a wishart random matrix model in the modeling of complex dynamical systems high resolution finite element models are routinely adopted to reduce the discretization error such an approach may fail to enhance the confidence in simulation based predictions when the dynamical systems exhibit significant variability in the parameterization and description of the mathematical model due to parametric data uncertainty and model uncertainty in contrast to parametric uncertainty model uncertainty poses significant challenges as no explicit parameter is available a priori to characterize its behavior in the current investigation a set of sprung mass oscillators randomly attached to a baseline system namely a homogeneous cantilever plate gives rise to model uncertainty contrary to model parametric uncertainty traditionally tackled using stochastic finite element method sfem the model uncertainty arising from randomly attached sprung masses gives rise to a new variant of dynamical system for every sample the feasibility of adopting a wishart random matrix to represent such ensemble of dynamical systems derived from model perturbation is investigated in this paper the experiments conducted on a vibrating plate with randomly attached sprung mass oscillators demonstrate that the wishart random matrix model may provide a reasonable representation of model uncertainty in linear dynamical systems in the medium and high frequency region

a model and numerical method in the medium frequency range for vibroacoustic predictions using the theory of structural fuzzy in linear dynamical analysis of complex mechanical systems the structural fuzzy is defined as the set of minor subsystems that are connected to the master structure but are not accessible by classical modeling the notion of master structure is presently extended to others elements such as an external dense compressible fluid strongly coupled with the primary structure for the low frequency lf dynamical analysis the modeling of the structural fuzzy is commonly made with a system of masses if the lf modeling of the structural fuzzy is applied in the medium frequency mf domain there are some large differences between calculations and experiment it is therefore necessary to take into account internal degrees of freedom of the structural fuzzy a global probabilistic modeling of the structural fuzzy is proposed to improve the calculated estimates of the mf vibrations into the master structure and of the far field radiated by itself in or out of context of the acoustic scattering this paper reviews the author s previous work and introduces the type ii probabilistic constitutive law in this paper 1 a probabilistic modeling of the structural fuzzy is presented 2 two probabilistic constitutive laws of the structural fuzzy are constructed 3 the modeling in the mf range for vibroacoustic predictions using theory of structural fuzzy is developed 4 the fuzzy solution in the mf range is studied and 5 numerical simulations on standard structures and on a submerged complex industrial structure are shown

an exact solution for the static and dynamic analysis of fe discretized uncertain structures the probabilistic analysis of fe discretized uncertain linear structures in the static and dynamic domain is the goal of this paper in particular a procedure able to give the exact relationship between the response and the random variables representing the structural uncertainties is presented under the assumption that a point discretization method is used for the representation of the uncertain random field this procedure is based on the properties of the structural deformation modes and in particular on the number of principal deformation modes of the fe type used for the structural discretization the use of these relationships leads to the exact probabilistic characterization of the response once that the probabilistic description of the uncertainty is fixed moreover they give the approximate relationship when only some of the random variables chosen by a sensitivity procedure are included in the analysis this allows to obtain an optimum level of accuracy through a reduced computational effort

vortex reconnection in the three dimensional navier stokes equations abstract we prove that the vortex structures of solutions to the 3d navier stokes equations can change their topology without any loss of regularity more precisely we construct smooth high frequency solutions to the navier stokes equations where vortex lines and vortex tubes of arbitrarily complicated topologies are created and destroyed in arbitrarily small times this instance of vortex reconnection is structurally stable and in perfect agreement with the existing computer simulations and experiments we also provide a non structurally stable scenario where the destruction of vortex structures is instantaneous

interpolatory projection methods for structure preserving model reduction abstract we present a framework for interpolatory model reduction that treats systems having a generalized coprime factorization c s s 1 b s d this includes rational krylov based interpolation methods as a special case the broader framework allows retention of special structure in reduced models such as symmetry second and higher order structure state constraints internal delays and infinite dimensional subsystems two numerical examples are provided for systems associated with viscoelastic dynamics and with an internal state delay that demonstrate the effectiveness and flexibility of the approach

energy equipartition and frequency distribution in complex attachments as reported in several recent publications an undamped simple oscillator with a complex attachment that consists of a set of undamped parallel resonators can exhibit unusual energy sharing properties the conservative set of oscillators of the attachment can absorb nearly all the impulsive energy applied to the primary oscillator to which it is connected the key factor in the ability of the attachment to absorb energy with near irreversibility correlates with the natural frequency distribution of the resonators within it the reported results also show that a family of optimal frequency distributions can be determined on the basis of a variational approach minimizing a certain functional related to the system response the present paper establishes a link between these optimal frequency distributions and the energy equipartition principle optimal frequency distributions are those that spread the injected energy as uniformly as possible over the degrees of freedom or over the modes of the system theoretical as well as numerical results presented support this point of view

on evaluation of the diffraction coefficients for arbitrary nonsingular directions of a smooth convex cone diffraction coefficients are fundamental objects determining principal amplitudes in asymptotic expansions for a general high frequency diffraction problem and are to be found from an associated canonical problem for a class of canonical problems of plane wave diffraction by an arbitrarily shaped smooth convex cone formulated as a boundary value problem for the helmholtz equation with dirichlet or neumann boundary conditions we develop a method for evaluating the diffraction coefficients in arbitrary nonsingular directions i e in all the directions where they are well defined the approach builds further on the analytic representations of v p smyshlyaev wave motion 12 1990 pp 329 339 and the constructions of v m babich d b dement ev and b a samokish wave motion 21 1995 pp 203 207 who have previously developed a method for calculating the diffraction coefficients in certain restricted directions for which an integral over a complex contour determining the diffraction

approximation theorems for parabolic equations and movement of local hot spots the authors were supported by european research council starting grants 633152 enciso and garcia ferrero and 335079 peralta salas this work was also supported by instituto de ciencias matematicas severo ochoa grant sev 2015 0554

the method of fundamental solutions applied to boundary value problems on the surface of a sphere abstract in this work we propose using the method of fundamental solutions mfs to solve boundary value problems for the helmholtz beltrami equation on a sphere we prove density and convergence results that justify the proposed mfs approximation several numerical examples are considered to illustrate the good performance of the method

two dimensional magnetohydrodynamic equilibria with prescribed topology we establish existence of weak solutions to the equilibrium equations of magnetohydrodynamics with prescribed topology this is carried out in two settings in the first we consider the variational problem of minimizing total energy in a torus under the assumption of axisymmetry and prescription of mass and flux profiles existence of weak solutions implies that the prescription of topology is a natural constraint both the compressible and incompressible cases are considered in the second setting we adapt examples of b c low and r wolfson 13 and j j aly and t amari 1 2 3 associated with parker s explanation of the extreme heating of the solar corona and other solar phenomena existence of solutions with fixed topology is a first crucial step in rigorously examining the relationship between topology and the existence of current sheets we use a decomposition introduced in 8 9 11 that captures much of the topology of level sets for certain classes of sobolev functions this decomposition is preserved under weak limits and so is useful for prescribing topological constraints the approach is especially suited to the use of domain deformations 2000 john wiley sons inc

on the existence of stationary splash singularities for the euler equations abstract in this paper we discuss the existence of stationary incompressible fluids with splash singularities specifically we show that there are stationary solutions to the euler equations with two fluids whose interfaces are arbitrarily close to a splash and that there are stationary water waves with splash singularities

efficient high order frequency interpolation of structural dynamic response a method is presented for interpolating structural dynamic response within a finite frequency band it is based on a condensation of the equations of motion using the response vectors at interpolation points as a basis this basis set naturally aligns itself with the eigenvectors whose eigenvalues lie inside the band resulting in a small condensed model that yields the interpolated response combining the sylvester law of inertia with eigenvalue analysis of the condensed system indicates when the condensation captures the eigenvalues that lie in the band of interest the high accuracy of the interpolation results from a combination of features an accurate estimation of eigenvalues and eigenvectors as well as matching the value and slope of the displacement vector projected on the force vector

higher order commutator estimates and local existence for the non resistive mhd equations and related models this paper establishes the local in time existence and uniqueness of strong solutions in hs for s n 2 to the viscous non resistive magnetohydrodynamics mhd equations in rn n 2 3 as well as for a related model where the advection terms are removed from the velocity equation the uniform bounds required for proving existence are established by means of a new estimate which is a partial generalisation of the commutator estimate of kato ponce comm pure appl math 41 7 891 907 1988

moduli spaces of vortex knots for an exact fluid flow the moduli spaces s d of non isotopic vortex knots are introduced for the ideal fluid flows in invariant domains d the analogous moduli spaces of the magnetic fields b knots are defined we derive and investigate new exact fluid flows and analogous plasma equilibria satisfying the beltrami equation which have nested invariant balls b k 3 with radii rk k 1 k the first flow is z axisymmetric the other ones do not possess any rotational symmetries the axisymmetric flow has an invariant plane z 0 due to an involutive symmetry of the flow its vortex knots in the invariant half spaces z 0 and z 0 are equivalent it is demonstrated that the moduli space x1d4ae 3 for the derived fluid flow in 3 is naturally isomorphic to the set of all rational numbers p q in the interval j 1 0 25 q m 1 0 5847 where q is the safety factor for the fluid flow in the first invariant ball b 1 3 it is shown that all values of the safety factor q belong to a small interval of length l 0

transient energy exchange between a primary structure and a set of oscillators return time and apparent damping in this paper we examine the conditions that influence the return time the time it takes before energy returns from a set of satellite oscillators attached to a primary structure two methods are presented to estimate the return time one estimate is based on an analysis of the reaction force on a rigid base by a finite number of oscillators as compared with an infinite number of continuously distributed oscillators the result gives a lower bound estimate for the return time a more accurate estimation results from considering the dynamic behavior of a set of oscillators as waves in a waveguide such an analogy explains energy flow between a primary structure and the oscillators in terms of pseudowaves and shows that a nonlinear frequency distribution of the oscillators leads to pseudodispersive waves the resulting approximate expressions show the influence of the natural frequency distribution within the set of oscillators and of their number on the return time as compared with the asymptotic case of a continuous set with infinite oscillators in the paper we also introduce a new method based on a hilbert envelope to estimate the apparent damping loss factor of the primary structure during the return time considering transient energy flow from the primary structure before any energy reflects back from the attached oscillators the expressions developed for return time and damping factor show close agreement with direct numerical simulations the paper concludes with a discussion of the return time and its relation to apparent damping and optimum frequency distribution within a set of oscillators that maximize these quantities

submanifolds that are level sets of solutions to a second order elliptic pde abstract we consider the problem of characterizing which noncompact hypersurfaces in r n can be regular level sets of a harmonic function modulo a c diffeomorphism as well as certain generalizations to other pdes we prove a versatile sufficient condition that shows in particular that any nonsingular algebraic hypersurface whose connected components are all noncompact can be transformed onto a union of components of the zero set of a harmonic function via a diffeomorphism of r n the technique we use combines robust but not explicit local constructions with appropriate global approximation theorems in view of applications to a problem posed by berry and dennis intersections of level sets are also studied

some properties of hyperspherical harmonics a general formula is given for the canonical decomposition of a homogeneous polynomial of order in m variables into a sum of harmonic polynomials this formula which involves successive applications of the generalized laplace operator is proved in the appendix it is shown that the group theoretical method for constructing irreducible cartesian tensors follows from the general formula for canonical decomposition the relationship between harmonic polynomials and hyperspherical harmonics is discussed and an addition theorem for hyperspherical harmonics is derived an expansion of a many dimensional plane wave in terms of gegenbauer polynomials and bessel functions is derived and used to construct bicenter expansions of arbitrary functions in many dimensional spaces finally a formula is derived for the 3 coefficients of hyperspherical harmonics these coefficients give the values of integrals involving the products of three harmonics

estimation of fuzzy structure parameters for continuous junctions the fuzzy structure theory introduced 15 years ago is designed to predict the frequency response functions of structures and structural acoustic systems with structural complexity in the low and medium frequency ranges this paper constitutes a first validation of the fuzzy structure theory for continuous junctions between the master structure and the fuzzy substructures in addition a method to estimate the fuzzy structure parameters introduced in the fuzzy structure theory is presented and it is validated for the case of continuous junctions this validation obtained by numerical simulations opens the field of experimental identifications

local representation and construction of beltrami fields abstract a beltrami field is an eigenvector of the curl operator beltrami fields describe steady flows in fluid dynamics and force free magnetic fields in plasma turbulence by application of the lie darboux theorem of differential geometry we prove a local representation theorem for beltrami fields we find that locally a beltrami field has a standard form amenable to an arnold beltrami childress flow with two of the parameters set to zero furthermore a beltrami flow admits two local invariants a coordinate representing the physical plane of the flow and an angular momentum like quantity in the direction across the plane as a consequence of the theorem we derive a method to construct beltrami fields with given proportionality factor this method based on the solution of the eikonal equation guarantees the existence of beltrami fields for any orthogonal coordinate system such that at least two scale factors are equal we construct several solenoidal and non solenoidal beltrami fields with both homogeneous and inhomogeneous proportionality factors

beltrami fields with hyperbolic periodic orbits enclosed by knotted invariant tori abstract we prove that there exist beltrami fields in euclidean space with sharp decay at infinity which have a prescribed set of invariant tori possibly knotted or linked that enclose an arbitrarily large number of hyperbolic periodic orbits these hyperbolic orbits are cablings over the core curve of each torus moreover the domain bounded by each invariant torus is covered by an almost full measure set of invariant tori we show that an analogous result holds for high frequency beltrami fields on the flat torus t 3

near irreversibility in a conservative linear structure with singularity points in its modal density through two complementary approaches using modal response and wave propagation the analyses presented here show the conditions under which a decaying impulse response or a nearly irreversible energy trapping takes place in a linear conservative continuous system the results show that the basic foundation of near irreversibility or apparent damping rests upon the presence of singularity points in the modal density of dynamic systems or analogously in the wave stopping properties associated with these singularities to illustrate the concept of apparent damping in detail a simple undamped beam is modified to introduce a singularity point in its modal density distribution simulations show that a suitable application of a compressive axial force to an undamped beam placed on an elastic foundation attenuates its impulse response with time and develops the characteristics of a nearly irreversible energy trap

vortex knots for the spheromak fluid flow and their moduli spaces abstract new exact solutions to the euler hydrodynamics equations are constructed a method for the study of vortex knots is developed for a special class of ideal fluid flows the axisymmetric ones satisfying the beltrami equation curl v x v x the method is based on a construction of the moduli spaces of vortex knots s r 3 applying the method to the spheromak fluid flow we demonstrate that only those torus knots k p q are realized as vortex knots for which p q belongs to the interval i 1 0 5 m 1 0 8252 we prove that each torus knot k p q with 1 2 p q 1 2 is realized on countably many invariant tori t 2 r 3 while torus knots with 1 2 p q m 1 are realized only on finitely many tori the moduli spaces of vortex knots s m b a 3 m 1 2 are constructed for the spheromak fluid flows inside a ball b a 3 of radius a

a new approach for the stochastic analysis of finite element modelled structures with uncertain parameters a method for evaluating the static response of uncertain finite element fe discretised structures is presented the method is comparable with the perturbation procedures from a computational point of view but it overcomes the drawbacks related to these procedures in fact the present method gives excellent level of accuracy even for high amount of uncertainties moreover it produces the exact solution for statically determinate structures the accuracy is remarkable even in the evaluation of high order statistics and probability density function of the response so that reliability consideration can be drawn

liouville theorems for the navier stokes equations and applications we study bounded ancient solutions of the navier stokes equations these are solutions with bounded velocity defined in rn 1 0 in two space dimensions we prove that such solutions are either constant or of the form u x t b t depending on the exact definition of admissible solutions the general 3 dimensional problem seems to be out of reach of existing techniques but partial results can be obtained in the case of axisymmetric solutions we apply these results to some scenarios of potential singularity formation for axi symmetric solutions and obtain extensions of results in a recent paper by chen strain tsai and yau 4

a variational principle for three dimensional water waves over beltrami flows abstract we consider steady three dimensional gravity capillary water waves with vorticity propagating on water of finite depth we prove a variational principle for doubly periodic waves with relative velocities given by beltrami vector fields under general assumptions on the wave profile

exact statistics of systems with uncertainties an analytical theory of rank one stochastic dynamic systems this paper concerns the response of uncertain vibro acoustic and structural dynamic systems here exact expressions are presented for the statistics of systems with a random rank one component the expressions are derived using the sherman morrison update formula that gives the exact expression of the disturbed response for any magnitude of the disturbance it is shown that the probability density function pdf of any transfer function is a simple function of the pdf of the disturbance magnitude of the random component the expressions for the mean variance and covariance of any transfer function and at any frequency of a random system necessitate non trivial integrals exact including closed form expressions of these integrals are derived in the particular cases of a real or complex normal disturbance magnitude and qualitative differences between these two cases are highlighted the theoretical and practical advantages of the theory are discussed and applied to a model of a bladed disk subjected to random damage the comparison with monte carlo simulations demonstrates that the statistics can be evaluated efficiently and precisely the theory derived formally in the context of discretised systems is directly applicable to continuous systems

a hybrid method for the vibration analysis of complex structural acoustic systems a new method is presented for the analysis of complex dynamic systems which is based on partitioning the system degrees of freedom into a global set and a local set the global equations of motion are formulated and solved in a standard deterministic manner although due account is taken of the presence of the local degrees of freedom via an approach which is analogous to fuzzy structure theory the local equations of motion are formulated and solved by using statistical energy analysis sea with due account being taken of power input from the global degrees of freedom the method can be considered to encompass a number of existing analysis techniques and to form a flexible framework within which a number of detailed modeling strategies can be devised a simple rod system provides an initial application and it is shown that the method yields very good results from low to high frequency excitation