exact solutions to the acoustic source reconstruction problem in this paper we generalize the recently developed analytical solutions of the radiation modes problem to the determination of closed form expressions for the singular value expansion of a number of integral operators that map the boundary velocity of a baffled planar structure onto the acoustic pressure radiated in far field or intermediate regions exact solutions to this problem involve prolate spheroidal wave functions that correspond to a set of independent distributions with finite spatial support and maximal energy concentration in a given bandwidth of the transform domain a stable solution to the inverse source reconstruction problem is obtained by decomposing the unknown boundary velocity into a number of efficiently radiating singular velocity patterns that correspond to the number of degrees of freedom of the radiated field it is found that the degree of ill posedness of the inverse problem is significantly reduced when considering a hemi circular observation arc with respect to a linear array of sensors by a factor scaling on the small angular aperture subtended by the observation line estimates are derived from the spatial resolution limits that can be achieved in the source reconstruction problem from the dimension of the efficiently radiating subspace

continuation of acoustic near fields this paper deals with the analytic continuation of a coherent pressure field specified on a finite sheet located close to and conformal to the surface of a vibrator this analytic continuation is an extension or extrapolation of the given measured field into a region outside and tangential to the original finite sheet and is based on the green s function the transfer function relating acoustic quantities on the two conformal surfaces the continuation of the measured pressure field is an inverse problem that requires the use or regularization theory especially when noise is present in the data an iteration algorithm is presented that is successful in continuing the pressure field into the tangential sheet the results are accurate close to the original boundary and taper decay toward zero with distance away from it the algorithm is tested on numerical and experimental data from a point driven rectangular plate results show the successful extrapolation continuation of this data into an area nearly double that of the original pressure field this algorithm is not limited to planar surfaces and can be applied to arbitrarily shaped surfaces

bicyclic graphs with small positive index of inertia abstract the positive index of inertia of a graph g denoted by i g is the number of the positive eigenvalues of the adjacency matrix of g in this paper we investigate the minimal positive index of inertia among all bicyclic graphs of order n with pendant vertices and characterize the bicyclic graphs with positive index 1 or 2 among all bicyclic graphs of order n

a unified presentation of certain families of non fuchsian differential equations via fractional calculus operators abstract in recent years many authors demonstrated the usefulness of fractional calculus operators in the derivation of explicit particular solutions of a number of linear ordinary and partial differential equations of the second and higher orders the main object of the present paper is to show how readily some recent contributions on this subject by several workers involving various interesting classes of non fuchsian differential equations including for example the fukuhara and tricomi equations and the celebrated bessel and whittaker equations can be obtained in a unified manner by suitably applying some general theorems on explicit particular solutions of a certain family of linear ordinary fractional differintegral equations

a polynomial algorithm of edge neighbor scattering number of trees abstract the edge neighbor scattering number ens is an alternative invulnerability measure of networks such as the vertices represent spies or virus carriers let g v e be a graph and e be any edge in g the open edge neighborhood of e is n e f e g f e e and f are adjacent and the closed edge neighborhood of e is n e n e e an edge e in g is said to be subverted when n e is deleted from g an edge set x e g is called an edge subversion strategy of g if each of the edges in x has been subverted from g the survival subgraph is denoted by g x an edge subversion strategy x is called an edge cut strategy of g if the survival subgraph g x is disconnected or is a single vertex or is the ens of a graph g is defined as e n s g max x e g g x x where x is any edge cut strategy of g g x is the number of the components of g x it is proved that the problem of computing the ens of a graph is np complete in this paper we give a polynomial algorithm of ens of trees

extension of measurement surface in near field acoustic holography in near field acoustic holography when measurements are made over a limited surface there has been recent interest in numerically enlarging the measurement surface tangentially current algorithms use iterative methods to extrapolate the field tangentially outward an algorithm based on the method of superposition is applied here which may be used to either extrapolate the field to enlarge the measurement surface or to interpolate in case there is a hole in the measurement surface where data are not available

3 d sound source reconstruction and field reprediction using the helmholtz integral equation abstract a method for reconstruction of the boundary conditions of an arbitrarily shaped three dimensional 3 d sound source from a finite number of field pressures is presented and the associated field is repredicted by using the helmholtz integral equation to minimize the reconstruction error due to noise in the field pressures a filtering method is proposed and it is verified that numerically singular values which correspond to the higher order modes of the evanescent like components of the source velocities are eliminated for arbitrarily shaped sources with complex boundary conditions the reconstruction is performed accurately when the size of each element is less than a quarter wavelength and the field measurement positions are equally or randomly distributed on the surface of a sphere in the vicinity of the source our technique can be used in computer aided loudspeaker design that reconstructs loudspeaker diaphragm velocities from the finite number of field pressures and repredicts the variation of the sound field procuced by a redesigned loudspeaker cabinet on which the reconstructed diaphragm is mounted

a note on extremal trees with degree conditions abstract a fundamental question in the study of graph invariants asks for the extremal structures under certain constraints that maximize or minimize a graph invariant in this note we summarize some recent work on the extremal trees of distance based and degree based graph invariants under various degree conditions we note that many of such extremal structures turned out to be identical for different but similar invariants such common extremal structures are investigated through the greedy trees and majorization between degree sequences we show that many of the known extremal results can be obtained through this line of arguments we also introduce some new extremal results as immediate consequences

on distances in vertex weighted trees abstract the study of extremal problems on various graph invariants has received great attention in recent years among the most well known graph invariants is the sum of distances between all pairs of vertices in a graph this is also known as the wiener index for its applications in chemical graph theory many interesting properties related to this concept have been established for extremal trees that maximize or minimize it recently a vertex weighted analogue of sum of distances is introduced for vertex weighted trees some extremal results on vertex weighted trees were obtained by goubko for trees with a given degree sequence in this note we first analyze the behavior of vertex weighted distance sum in general identifying the middle part of a tree analogous to that with respect to the regular distance sum we then provide a simpler approach than that of goubko s to obtain a stronger result regarding the extremal tree with a given degree sequence questions and directions for potential future study are also discussed

analytic solutions to the acoustic source reconstruction problem in this paper we generalize the recently developed analytical solutions of the radiation modes problem to the determination of closed form expressions for the singular value expansion of a number

conformal generalized near field acoustic holography for axisymmetric geometries the normal velocity field on the boundary of a radiating source is reconstructed with high resolution by using pressure data measured in the very near field to include evanescent components fast decaying with the distance the measurement points belong to a surface conformal to the reconstruction is performed by identifying the boundary normal velocity distribution generating the pressure field best mean square fitting the measurement data the ensuing normal equations are solved by resorting to the singular value decomposition svd of the transformation from the boundary normal velocity to the pressure at the measurement points the reconstructed boundary normal velocity is represented as a linear combination of basis functions each associated with a singular value sv the errors in the estimation of the coefficients of the linear combination are proportional to the inverses of the corresponding svs to achieve robustness the svd is truncated to retain the terms corresponding to a limited dynamic range in turn related to the dynamic range of the measurement system for axisymmetric but otherwise general surfaces the numerical complexity is reduced because separate svds of greatly reduced dimensionality must be performed on a number of matrices each associated with one of the harmonics of the fourier expansion of the field around the symmetry axis in a numerical experiment making use of an available computer model by processing very near field pressure samples the normal velocity was reconstructed with excellent accuracy on the surface of a finite compliant cylinder with flat endcaps excited by point forces the viability of the method was then confirmed in a measurement tank experiment in which the distribution of the boundary normal velocity was reconstructed on a finite compliant cylinder driven by point shakers comparison with accelerometer data shows good agreement on a wide frequency band

on edge rupture degree of graphs the edge rupture degree of an incomplete connected graph g is defined as r g m a x g s s m g s s e g g s 1 where g s and m g s respectively denote the number of components and the order of a largest component in g s this is a reasonable parameter to measure the vulnerability of networks as it takes into account both the amount of work done to damage the network and how badly the network is damaged in this paper firstly the relationships between the edge rupture degree and some other graph parameters namely the edge connectivity edge integrity edge toughness edge tenacity diameter the algebraic connectivity and the minimum degree are established after that the edge rupture degree of the middle graphs of path and cycle are given then we introduced the concept of r maximal graph and give some basic results of such graphs finally we introduce the concept of edge ruptured and strictly edge ruptured graph and we establish necessary and sufficient conditions for a graph to be edge ruptured and strictly edge ruptured respectively

basic theory and properties of statistically optimized near field acoustical holography to avoid the requirement set by standard near field acoustical holography nah to measure an area that fully covers the source a set of so called patch nah methods has been introduced one such method is the statistically optimized nah sonah in this method the acoustic quantities on a mapping surface near the measurement surface are calculated by using a transfer matrix defined in such a way that all propagating waves and a weighted set of evanescent waves are projected with optimal average accuracy the present paper gives an overview of the basic theory of sonah including a description of phenomena such as spatial aliasing and wave number domain leakage a revised and generalized mathematical formulation is given covering the calculation of all three components of particle velocity and the use of up to six virtual source planes a set of formulas for the inherent estimation error level of the method is derived and used to visualize the regions of validity of the sonah predictions for some typical microphone array geometries the sensitivity of the inherent error level distribution to changes in the parameters of the sonah algorithm is also investigated

a note on the first reformulated zagreb index let g v e v 1 2 n e e 1 e 2 e m be a simple graph with n vertices and m edges denote by d ei i 1 2 m an edge degree and by e m 1 i 1 m d e i 2 first reformulated zagreb index of graph g upper and lower bounds of graph invariant em1 are obtained

estimation of acoustic source strength by inverse methods part i conditioning of the inverse problem this paper deals with the discrete inverse problem in acoustics it is assumed that a number of acoustic sources are located at known spatial positions and that the acoustic pressure is measured at a number of spatial positions in the radiated field the transfer functions relating the strengths of the acoustic sources to the radiated pressures are also assumed known in principle the strengths of the acoustic sources can be deduced from the measured acoustic pressures by inversion of this matrix of transfer functions the accuracy of source strength reconstruction in the presence of noise which contaminates the measured pressures is crucially dependent on the conditioning of the matrix to be inverted this paper examines the conditioning of this inverse problem particularly with regard to the geometry and number of sources and measurement positions and the non dimensional frequency a preliminary investigation is also presented of methods such as tikhonov regularization and singular value discarding which can improve the accuracy of source strength reconstruction in poorly conditioned cases results are also presented which enable the solution of the inverse problem when the time histories of the acoustic sources are time stationary random processes and the spectra and cross spectra are measured at a number of positions in the radiated field the paper illustrates the possibilities and limitations of the use of inverse methods in the deduction of acoustic source strength from radiated field measurements

krylov subspace iterative methods for boundary element method based near field acoustic holography the reconstruction of the acoustic field for general surfaces is obtained from the solution of a matrix system that results from a boundary integral equation discretized using boundary element methods the solution to the resultant matrix system is obtained using iterative regularization methods that counteract the effect of noise on the measurements these methods will not require the calculation of the singular value decomposition which can be expensive when the matrix system is considerably large krylov subspace methods are iterative methods that have the phenomena known as semi convergence i e the optimal regularization solution is obtained after a few iterations if the iteration is not stopped the method converges to a solution that generally is totally corrupted by errors on the measurements for these methods the number of iterations play the role of the regularization parameter we will focus our attention to the study of the regularizing properties from the krylov subspace methods like conjugate gradients least squares qr and the recently proposed hybrid method a discussion and comparison of the available stopping rules will be included a vibrating plate is considered as an example to validate our results

a one step patch near field acoustical holography procedure when performing holography measurements over a limited area of a source the hologram pressure typically remains finite at the edge of the measurement aperture patch near field acoustical holography nah has been developed specifically to mitigate the effects related to that windowing in iterative patch nah the source distribution is reconstructed in two steps first the partially measured sound field is extended iteratively and then the extended pressure is projected onto the source surface by using conventional nah procedures in the present work a one step procedure for performing that combined task is described in this approach the acoustical property to be reconstructed on a surface of interest is related to the partially measured pressure on the hologram surface in terms of sampling and bandlimiting matrices and the reconstructed result is obtained by finding the regularized least squares solution of the latter relation a procedure for determining the cutoff wave number of the bandlimiting matrix without a priori knowledge of the signal bandwidth is suggested the proposed procedure was validated by using a synthetic sound field created by a point driven simply supported plated

an application of the n fractional calculus operator method to a modified whittaker equation by means of fractional calculus techniques we find explicit solutions of modified whittaker equations we use the n fractional calculus operator n m method to derive the solutions of these equations

acoustic source property prediction based on near field measurements in planar coordinate abstract near field acoustical holography nah can be used to predict the acoustical properties of noise sources by starting from measurements of near field pressures in particular the sound pressure particle velocity and intensity can all be estimated on a source plane in the present work an equation relating acoustical source properties and the pressure radiated from a source is derived from the kirchoff helmholtz integral equation and is here referred to as the inverse rayleigh method the inverse rayleigh method reduces to the rayleigh integral when the source is located on an infinite rigid baffle however the same equation can also be used to predict the acoustic properties of a source in a more general environment i e without assuming the presence of an infinite rigid baffle in this article the inverse rayleigh method is described and then its relative accuracy is compared with other procedures that can be used to predict source particle velocity from near field pressure measurements on a planar hologram by comparing the accuracy of the present nah method with other procedures in the case of a large hologram it was found that the inverse rayleigh method is accurate and computationally efficient for predicting the source particle velocity from near field pressure measurements both with and without a rigid baffle

wavelet preprocessing for lessening truncation effects in nearfield acoustical holography the goal of planar nearfield acoustical holography nah is to recover the sound field at the sound source from pressure measurements made close to the source plane the theory requires the pressure to be measured over a complete plane because experimentation consists of acquiring only a finite measurement aperture of the pressure field it naturally causes erroneous values in the reconstructed field wavelet preprocessing applied to the pressure measurements in the nearfield provides a solution to lessen the effects due to the truncation of the hologram the approach is based on a multiresolution analysis of the field from different wave number bands followed by selective spatial filtering of effects highlighted by the first analysis experimental results show the relevance of the method by comparison to standard nah involving exponential filtering in the wave number domain the computation of objective indicators based on distance measurements between wave number spectra and comparisons between pattern

measurement of structural intensity using boundary element method based nearfield acoustical holography the regularization method for measurement of structural intensity si using boundary element method bem based nearfield acoustical holography nah is proposed spatial derivatives of normal displacement are necessary to obtain the structural intensity the derivative operations amplify high wavenumber component of measurement noise and contaminate the measurement result of si to overcome this difficulty regularization method for measurement of si using fast fourier transform based nah has been introduced in this paper this regularization method is modified for the bem based nah the bem based nah avoids the aperture replication problem therefore measurement aperture for bem based nah can be set smaller than that for fft based nah the effectiveness of the proposed method is demonstrated by experiments

extremality of degree based graph entropies abstract many graph invariants have been used for the construction of entropy based measures to characterize the structure of complex networks based on shannon s entropy we study graph entropies which are based on vertex degrees by using so called information functionals when considering shannon entropy based graph measures there has been very little work to find their extremal values the main contribution of this paper is to prove some extremal values for the underlying graph entropy of certain families of graphs and to find the connection between the graph entropy and the sum of degree powers further conjectures to determine extremal values of graph entropies are given

the extremal values of some topological indices in bipartite graphs with a given matching number let i g be a topological index of a graph if i g e i g or i g e i g respectively for each edge e g then i g decreases or increases respectively with addition of edges in this paper we determine the extremal values of some topological indices which decrease or increase with addition of edges and characterize the corresponding extremal graphs in bipartite graphs with a given matching number

fast fourier transform and singular value decomposition formulations for patch nearfield acoustical holography nearfield acoustical holography nah requires the measurement of the pressure field over a complete surface in order to recover the normal velocity on a nearby concentric surface the latter generally coincident with a vibrator patch nah provides a major simplification by eliminating the need for complete surface pressure scans only a small area needs to be scanned to determine the normal velocity on the corresponding small area concentric patch on the vibrator the theory of patch nah is based on 1 an analytic continuation of the patch pressure which provides a spatially tapered aperture extension of the field and 2 a decomposition of the transfer function pressure to velocity and or pressure to pressure between the two surfaces using the singular value decomposition svd for general shapes and the fast fourier transform fft for planar surfaces inversion of the transfer function is stabilized using tikhonov regularization and the morozov discrepancy principle experimental results show that root mean square errors of the normal velocity reconstruction for a point driven vibrator over 200 2700 hz average less than 20 for two small concentric patch surfaces 0 4 cm apart reconstruction of the active normal acoustic intensity was also successful with less than 30 error over the frequency band

use of nonsingular boundary integral formulation for reducing errors due to near field measurements in the boundary element method based near field acoustic holography in the conformal near field acoustic holography nah using the boundary element method bem the transfer matrix relating the vibro acoustic properties of source and field depends solely on the geometrical condition of the problem this kind of nah is known to be very powerful in dealing with the sources having irregular shaped boundaries when the vibro acoustic source field is reconstructed by using this conformal nah one tends to position the sensors as close as possible to the source surface in order to get rich information on the nonpropagating wave components the conventional acoustic bem based on the kirchhoff helmholtz integral equation has the singularity problem in the close near field of the source surface this problem stems from the singular kernel of the green function of the boundary integral equation bie and the singularity can influence the reconstruction accuracy greatly in this paper the nonsingular bie is introduced to the nah calculation and the holographic bie is reformulated the effectiveness of nonsingular bem has been investigated for the reduction of reconstruction error through interior and exterior examples it is shown that the resolution of predicted field pressure could be improved in the close near field by employing the nonsingular bie because the bem based nah inevitably requires the field pressure measured in the close proximity to the source surface the present approach is recommended for improving the resolution of the reconstructed source field

graphs with fixed number of pendent vertices and minimal zeroth order general randic index we investigate the graph with minimal zeroth order general randic index in terms of its order n pendent number n1 and cyclomatic number 0 extremal graphs were completely characterized for cases of 0 1 2 which can be directly extended for graphs with cyclomatic number 3

nordhaus gaddum type results for graph irregularities abstract a graph whose vertices have the same degree is called regular otherwise the graph is irregular in fact various measures of irregularity have been proposed and examined for a given graph g v e with v v 1 v 2 v n and edge set e g di is the vertex degree where 1 i n the irregularity of g is defined by i r r g v i v j e g d i d j a similar measure can be defined by i r r 2 g v i v j e g d i d j 2 the total irregularity of g is defined by i r r t g 1 2 v i v j v g d i d j the variance of the vertex degrees is defined v a r g 1 n i 1 n d i 2 2 m n 2 in this paper we present some nordhaus gaddum type results for these measures and draw conclusions

patch near field acoustical holography in cylindrical geometry spatial discrete fourier transform based near field acoustical holography is known to suffer from windowing effects since the measurement aperture is necessarily finite the latter effect can be mitigated by using patch holography in which the measured field is extended beyond the measurement aperture based on successive smoothing operations in this article the application of a patch holography algorithm to cylindrical geometries is described in planar geometry initial zero padding can be applied to the hologram pressure to an arbitrary degree in both in plane directions since the pressure magnitude is expected to limit ultimately towards zero in both directions in a cylindrical geometry zero padding can be implemented axially in the same way but the number of zeros added in the circumferential direction is necessarily determined by the angular sample spacing owing to the periodic nature of the field in this direction by using both numerical simulation and experimental results it is shown that the patch extrapolation procedure works well in the cylindrical case and a discussion of rigid reflecting boundaries is also included finally reconstruction results obtained by the combined use of pressures measured in the finite aperture and two different regularization parameter selection techniques are compared and the effect of projection distance is discussed

reduction methods of the reconstruction error for large scale implementation of near field acoustical holography new extrapolation methods in real and wave number spaces are proposed to extend near field acoustical holography nah toward a more quantitative technique applicable to the vibration measurement of actual large scale structures the finiteness of the measurement aperture is a serious impediment to such large scale implementation of nah because the measurement aperture sufficiently larger than the vibrating structure of interest is usually needed we should thus investigate how to reduce the reconstruction error when the measurement aperture is restricted to a fraction of the vibrating structure the following practical suggestions are derived from simulations and underwater experiments 1 a wave number space extrapolation method can reduce the reconstruction error to about 25 for the measurement aperture corresponding to 1 16th of the vibrating structure when 10 000 iterations of the extrapolation process are applied 2 a real space extrapolation method can reduce the reconstruction error to negligible degree 10 10 for the measurement aperture corresponding to four times as large as the vibration structure when ten iterations of the extrapolation process are applied 3 the former method may be widely and safely applied but many iterations are necessary the latter method can reduce the reconstruction error very quickly but a wider measurement aperture than that in the former is needed therefore when the measurement aperture cannot be larger than the vibrating structure the former method is recommended while when the measurement aperture is larger than the structure the latter method is recommended summing up the accuracy of our proposed methods which is attributed to the property of data extrapolation method that the data inside the measurement aperture is conserved after adequately extrapolating the data outside the aperture will be relevant to a more quantitative measurement and analysis of real large scale structures

patch near field acoustic holography regularized extension and statistically optimized methods the patch holography method allows one to make measurements on an extended structure using a small microphone array increased attention has been paid to the two techniques which are quite different at first glance one is to extrapolate the pressure field measured on the hologram plane while the other is to use statistically optimized processing a singular value decomposition formulation of the latter is proposed in this paper the similarity of the two techniques is shown here both use a convolution of the measured pressure patch to obtain a better estimate of the wavenumber spectrum backward propagated on the structure by using the morozov discrepancy principle to compute the regularization parameter the two methods lead to very close results

the general randic index of trees with given number of pendent vertices the general randic index of a graph g is defined as r g u v e g d u d v where d u denotes the degree of a vertex u in g and is a real number in this paper we determine the maximum general randic indices of trees and chemical trees with n vertices and k pendent vertices for 4 k n 2 3 and 0 0 where 0 0 5122 is the unique non zero root of the equation 6 4 20 9 10 12 16 5 24 0 the corresponding extremal graphs are also characterized

certain classes of ordinary and partial differential equations solvable by means of fractional calculus recently many workers demonstrated the usefulness of fractional calculus in the derivation of particular solutions of a considerably large number of linear ordinary and partial differential equations of the second and higher orders the main object of the present paper is to show how readily some recent contributions on this subject by s t tu k nishimoto s j jaw and s d lin hiroshima math j 23 1993 63 77 and by other workers can be obtained by suitably applying some general theorems on particular solutions of a certain family of linear ordinary and partial fractional differintegral equations

application of bem boundary element method based acoustic holography to radiation analysis of sound sources with arbitrarily shaped geometries a method employing holograms that conform with arbitrarily shaped sources has been developed for enhancing conventional near field acoustic holography which has been limited to sources with simple geometries e g planar or cylindrical surfaces four holography transformation algorithms have been developed based on acoustic holography theory and the boundary element method bem singular value decomposition svd has been incorporated into the algorithms in order to alleviate the ill posed nature frequently encountered in backward reconstruction of a field a pulsating sphere a cylinder with spherical endcaps and a vibrating piston set in a rigid sphere have been adopted in a numerical simulation for verifying the algorithms satisfactory agreement has been achieved between the holographically transformed results and the analytical solutions

the inertia of weighted unicyclic graphs let gw be a weighted graph the inertia of gw is the triple in gw i gw i gw i0 gw where i gw i gw i0 gw are the numbers of the positive negative and zero eigenvalues of the adjacency matrix a gw of gw including their multiplicities respectively i gw i gw are called the positive negative indices of inertia of gw respectively in this paper we present a lower bound for the positive negative indices of weighted unicyclic graphs of order n with fixed girth and characterize all weighted unicyclic graphs attaining this lower bound moreover we characterize the weighted unicyclic graphs of order n with two positive two negative and at least n 6 zero eigenvalues respectively

degree based topological indices optimal trees with given number of pendents abstract a dynamic programming method is elaborated enabling the characterization of trees with a given number of pendent vertices for which a vertex degree based invariant topological index achieves its extremal value the method is applied to the chemically interesting and earlier much studied such invariants the first and second zagreb index and the atom bond connectivity index

on zagreb indices zagreb polynomials of some nanostar dendrimers dendrimers are highly branched organic macromolecules with successive layers or generations of branch units surrounding a central core ashrafi and mirzargar 2008 3 chen et al 2014 7 klajnert and bryszewska 2001 26 yamamoto et al 2002 36 these are key molecules in nanotechnology and can be put to good use topological indices are numbers associated with molecular graphs for the purpose of allowing quantitative structure activity property toxicity relationships these topological indices correlate certain physico chemical properties like boiling point stability strain energy etc of chemical compounds in this paper we determine hyper zagreb index first multiple zagreb index second multiple zagreb index and zagreb polynomials for some nanostar dendrimers

reconstruction of the acoustic field over a limited surface area on a vibrating cylinder the acoustic field on a small region on the surface of a radiating cylindrical shell is reconstructed from measurements of pressure made on a limited nearby surface the boundary element method is applied to compute an operator relating the pressure on a small measurement surface located near the structure to the normal velocity at all points on the structure surface since the inverse problem of the reconstruction of the surface field is ill posed singular value decomposition is applied to invert the operator it is demonstrated that the surface field may be reconstructed over the portion of the structure surface located under the measurement surface

regularization methods for near field acoustical holography the reconstruction of the pressure and normal surface velocity provided by near field acoustical holography nah from pressure measurements made near a vibrating structure is a linear ill posed inverse problem due to the existence of strongly decaying evanescentlike waves regularization provides a technique of overcoming the ill posedness and generates a solution to the linear problem in an automated way we present four robust methods for regularization the standard tikhonov procedure along with a novel improved version landweber iteration and the conjugate gradient approach each of these approaches can be applied to all forms of interior or exterior nah problems planar cylindrical spherical and conformal we also study two parameter selection procedures the morozov discrepancy principle and the generalized cross validation which are crucial to any regularization theory in particular we concentrate here on planar and cylindrical holography these forms of nah which rely on the discrete fourier transform are important due to their popularity and to their tremendous computational speed in order to use regularization theory for the separable geometry problems we reformulate the equations of planar cylindrical and spherical nah into an eigenvalue problem the resulting eigenvalues and eigenvectors couple easily to regularization theory which can be incorporated into the nah software with little sacrifice in computational speed the resulting complete automation of the nah algorithm for both separable and nonseparable geometries overcomes the last significant hurdle for nah

a survey of fractional calculus approaches to the solutions of the bessel differential equation of general order in a remarkably large number of recent works one can find the emphasis upon and demonstrations of the usefulness of fractional calculus operators in the derivation of explicit particular solutions of significantly general families of linear ordinary and partial differential equations of the second and higher orders the main object of this presentation is to survey some earlier investigations of this simple fractional calculus approach to the solutions of the classical bessel differential equation of general order and to show how it would lead naturally to several interesting consequences which include for example an alternative derivation of the complete power series solutions obtainable usually by the frobenius method the underlying analysis presented here is based chiefly upon some of the general theorems on explicit particular solutions of a certain family of linear ordinary fractional differintegral equations with polynomial coefficients

reconstructing the vibro acoustic quantities on a highly non spherical surface using the helmholtz equation least squares method this paper presents helpful guidelines and strategies for reconstructing the vibro acoustic quantities on a highly non spherical surface by using the helmholtz equation least squares hels this study highlights that a computationally simple code based on the spherical wave functions can produce an accurate reconstruction of the acoustic pressure and normal surface velocity on planar surfaces the key is to select the optimal origin of the coordinate system behind the planar surface choose a target structural wavelength to be reconstructed set an appropriate stand off distance and microphone spacing use a hybrid regularization scheme to determine the optimal number of the expansion functions etc the reconstructed vibro acoustic quantities are validated rigorously via experiments by comparing the reconstructed normal surface velocity spectra and distributions with the benchmark data obtained by scanning a laser vibrometer over the plate surface results confirm that following the proposed guidelines and strategies can ensure the accuracy in reconstructing the normal surface velocity up to the target structural wavelength and produce much more satisfactory results than a straight application of the original hels formulations experiment validations on a baffled square plate were conducted inside a fully anechoic chamber

a simple fractional calculus approach to the solutions of the bessel differential equation of general order and some of its applications in many recent works several authors demonstrated the usefulness of fractional calculus operators in the derivation of explicit particular solutions of a significantly large number of linear ordinary and partial differential equations of the second and higher orders the main object of the present paper is to show how this simple fractional calculus approach to the solutions of the classical bessel differential equation of general order would lead naturally to several interesting consequences which include for example an alternative investigation of the power series solutions obtainable usually by the frobenius method the methodology presented here is based largely upon some of the general theorems on explicit particular solutions of a certain family of linear ordinary fractional differintegral equations

probabilistic inequalities for evaluating structural network measures proving interrelations between structural graph measures analytically has been intricate generally relations between structural graph measures describe the interplay between measures which turned out to be useful for better understanding the properties of such quantities the results which have been achieved so far are restricted to simple measures or special graph classes such as trees in this paper we introduce a probabilistic approach for establishing inequalities between quantitative network measures on random networks those inequalities between different graph measures lead to a deeper understanding of the mathematical apparatus and in particular to a discussion of quality aspects of structural graph measures which is a major contribution of this paper

on rainbow domination numbers of graphs given a graph g and a set of t colors assume that we assign an arbitrary subset of these colors to each vertex of g if we require that each vertex to which an empty set is assigned has in its neighborhood all t colors then this assignment is called a t rainbow dominating function of the graph g the corresponding invariant c r t g which is the minimum sum of numbers of assigned colors over all vertices of g is called the t rainbow domination number of g in this paper bounds for the t rainbow domination number of an arbitrary graph for an arbitrary positive integer t are given the 3 rainbow domination numbers of several classes of graphs such as paths cycles and the generalized petersen graphs p n k are investigated the 3 rainbow domination number of p n 1 is determined and the upper bounds for p n 2 and p n 3 are provided by computer search we obtain that the upper bounds for p n 2 match their exact values for n 17

regularization method for the application of k space data extrapolation to near field acoustical holography a regularization method for applying the k space data extrapolation method to near field acoustical holography nah is proposed the influence of the finiteness of measurement aperture is a serious impediment to the nah reconstruction of structural vibration when the size of the measurement aperture is smaller than that of the structure the k space data extrapolation method has been proposed to reduce the influence of this finiteness an appropriate regularization method is necessary to apply the k space data extrapolation method if the measured pressure is contaminated by noise some regularization methods for nah have been proposed however it is difficult to determine an appropriate k space filter if the finiteness of the measurement aperture becomes an issue an estimation method of appropriate regularization filter for applying the k space data extrapolation method is proposed to solve this difficulty the effectiveness of this method is demonstrated by simulations the simulation results confirm

approximations of inverse boundary element methods with partial measurements of the pressure field boundary element methods bems based near field acoustic holography nah requires the measurement of the pressure field over a closed surface in order to recover the normal velocity on a nearby conformal surface there are practical cases when measurements are available over a patch from the measurement surface in which conventional inverse bem based nah ibem cannot be applied directly but instead as an approximation in this work two main approximations based on the indirect implicit methods are considered patch ibem and ibem with cauchy data patch ibem can be applied with a continuation procedure which as its predecessor patch nah a well known technique that can be used on separable geometries of the wave equation continues the pressure field using an iterative procedure or it can be applied by a direct procedure on the other hand ibem with cauchy data requires measurements over two conformal patches and it will be shown that this technique will be reliable regardless of the position of the source the theory behind each method will be justified and validated using a cylindrical surface with numerical data generated by point sources and using experimental data from a cylindrical fuselage excited by a point force

on the wiener polarity index of trees with maximum degree or given number of leaves the wiener polarity index w p g of a graph g v e is the number of unordered pairs of vertices u v of g such that the distance d g u v 3 in this paper the minimum resp maximum wiener polarity index of trees with n vertices and maximum degree d are given and the corresponding extremal trees are determined where email protected email protected n 1 moreover the trees minimizing w p t among all trees t of order n and k leaves are characterized where email protected email protected n 1

note on two generalizations of the randic index for a given graph g the well known randic index of g introduced by milan randic in 1975 is defined as r g u v e g d u d v 1 2 where the sum is taken over all edges uv and du denotes the degrees of u bollobas and erdos generalized this index by replacing 1 2 with any real number which is called the general randic index dvo ak et al introduced a modified version of randic index r g u v e g max d u d v 1 based on this recently knor et al introduced two generalizations r g u v e g min d u d v and r g u v e g max d u d v for any real number clearly the former is a lower bound for the general randic index and the latter is its upper bound knor et al studied extremal values of r g and r g and concluded some open problems in this paper we consider the open problems and give some comments and results some results for chemical trees are obtained

sqbc an efficient subgraph matching method over large and dense graphs recent progress in biology and computer science have generated many complicated networks most of which can be modeled as large and dense graphs developing effective and efficient subgraph match methods over these graphs is urgent meaningful and necessary although some excellent exploratory approaches have been proposed these years they show poor performances when the graphs are large and dense this paper presents a novel subgraph query technique based on clique feature called sqbc which integrates the carefully designed clique encoding with the existing vertex encoding 40 as the basic index unit to reduce the search space furthermore sqbc optimizes the subgraph isomorphism test based on clique features extensive experiments over biological networks rdf dataset and synthetic graphs have shown that sqbc outperforms the most popular competitors both in effectiveness and efficiency especially when the data graphs are large and dense

on the reconstruction of the vibro acoustic field over the surface enclosing an interior space using the boundary element method the vibrational velocity sound pressure and acoustic power on the vibrating boundary comprising an enclosed space are reconstructed by the boundary element method based on the measured field pressures the singular value decomposition is used to obtain the inverse solution in the least square sense and to express the acoustic modal expansion between the measurement and source fields in general such an inverse operation has been considered an ill posed problem having a divergence phenomenon involved with extremely small measurement errors the ill conditioned nature of the acoustic inverse problem is caused by the singularity of the transfer matrix which produces nonradiating wave components in order to minimize the singularity and to also reduce the number of measurement points optimal measurement positions are determined by the effective independence method regularization methods are used to stabilize the reconstructed field by suppressing nonradiating components resulting in the singular transfer

integral trees with diameter four a graph is called integral if all eigenvalues of its adjacency matrix consist entirely of integers in this paper we investigate integral trees s r m i s a 1 a 2 a s m 1 m 2 m s of diameter 4 with s 3 4 5 6 such integral trees are found by using a computer search or solving the diophantine equations new sufficient conditions for a construction of infinite families of integral trees s r m i s b 1 b s m 1 ms of diameter 4 from given integral trees s r m i s a 1 a s m 1 ms of diameter 4 are given further using these conditions we construct infinitely many new classes of integral trees s r m i s b 1 b s m 1 ms of diameter 4 with s 3 4 5 6 finally we propose two basic open problems about integral trees of diameter 4 for further study

on extremal zagreb indices of trees with given domination number first m1 and second m2 zagreb indices are graph invariants that originate from chemical researches on total electron energy of conjugated molecules there is a legion of articles dealing with these two indices this paper presents upper bounds on zagreb indices of trees in terms of domination number these are strict bounds and extremal trees are characterized in addition a lower bound for the first zagreb index of trees with a given domination number is determined and the extremal trees are characterized as well finally using previously known upper bound for harary index h in terms of m1 and m2 a unique tree with given domination number that maximizes h is characterized

convolution formulations for non negative intensity previously unknown spatial convolution formulas for a variant of the active normal intensity in planar coordinates have been derived that use measured pressure or normal velocity near field holograms to construct a positive only outward intensity distribution in the plane quantifying the areas of the vibrating structure that produce radiation to the far field this is an extension of the outgoing only unipolar intensity technique recently developed for arbitrary geometries by steffen marburg the method is applied independently to pressure and velocity data measured in a plane close to the surface of a point driven unbaffled rectangular plate in the laboratory it is demonstrated that the sound producing regions of the structure are clearly revealed using the derived formulas and that the spatial resolution is limited to a half wavelength a second set of formulas called the hybrid intensity formulas are also derived which yield a bipolar intensity using a different spatial convolution operator again using either the measured pressure or velocity it is demonstrated from the experiment results that the velocity formula yields the classical active intensity and the pressure formula an interesting hybrid intensity that may be useful for source localization computations are fast and carried out in real space without fourier transforms into wavenumber space

acoustical inverse problems regularization direct definition of filter factors using signal to noise ratio abstract acoustic imaging aims at localization and characterization of sound sources using microphone arrays in this paper a new regularization method for acoustic imaging by inverse approach is proposed the method first relies on the singular value decomposition of the plant matrix and on the projection of the measured data on the corresponding singular vectors in place of regularization using classical methods such as truncated singular value decomposition and tikhonov regularization the proposed method involves the direct definition of the filter factors on the basis of a thresholding operation defined from the estimated measurement noise the thresholding operation is achieved using modified filter functions the originality of the approach is to propose the definition of a filter factor which provides more damping to the singular components dominated by noise than that given by the tikhonov filter this has the advantage of potentially simplifying the selection of the best regularization amount in inverse problems theoretical results show that this method is comparatively more accurate than tikhonov regularization and truncated singular value decomposition

integral inequalities of systems and the estimate for solutions of certain nonlinear two dimensional fractional differential systems this paper generalizes the results for the constructions of explicit bounds and the qualitative properties for the solutions of certain two dimensional fractional differential systems established in a recent paper of the authors the main generalizations come from an elementary inequality and by means of the modification of medved s de singular approach

a computational approach to construct a multivariate complete graph invariant in this paper we present a computational approach for finding complete graph invariants specifically we generate exhaustive sets of connected non isomorphic graphs with 9 and 10 vertices and demonstrate that a 97 dimensional multivariate graph invariant is capable to distinguish each of the non isomorphic graphs furthermore in order to tame the computational complexity of the problem caused by the vast number of graphs e g involving over 10 million networks with 10 vertices we suggest a low dimensional iterative procedure that is based on highly discriminative individual graph invariants we show that also this computational approach leads to a perfect discrimination overall our numerical results prove the existence of such graph invariants for networks with 9 and 10 vertices furthermore we show that our iterative approach has a polynomial time complexity

surface decomposition method for near field acoustic holography near field acoustic holography reconstruction of the acoustic field at the surface of an arbitrarily shaped radiating structure from pressure measurements at a nearby conformal surface is obtained from the solution of a boundary integral equation this integral equation is discretized using the equivalent source method and transformed into a matrix system that can be solved using iterative regularization methods that counteract the effect of noise on the measurements this work considers the case when the resultant matrix system is so large that it cannot be explicitly formed and iterative methods of solution cannot be directly implemented in this case the method of surface decomposition is proposed where the measurement surface is divided into smaller nonoverlapping subsurfaces each subsurface is used to form a smaller matrix system that is solved and the result joined together to generate a global solution to the original matrix system numerically generated data are used to study the use of subsurface extensions to increase the continuity of the global solution and investigate the size of the subsurfaces as well as the distance between the measurement and the vibrating surface finally a vibrating ship hull structure is considered as a physical example to apply and validate the proposed methodology

helmholtz equation least squares method for reconstructing the acoustic pressure field a method using spherical wave expansion theory to reconstruct acoustic pressure field from a vibrating object is developed the radiated acoustic pressures are obtained by means of an expansion of independent functions generated by the gram schmidt orthonormalization with respect to the particular solutions to the helmholtz equation on the vibrating surface under consideration the coefficients associated with these independent functions are determined by requiring the assumed form of solution to satisfy the pressure boundary condition at the measurement points the errors incurred in this process are minimized by the least squares method once these coefficients are specified the acoustic pressure at any point including the source surface is completely determined in this paper this method is used to reconstruct the surface acoustic pressures based on the measured acoustic pressure signals in the field it is shown that this method can be applied to both separable and nonseparable geometries and the

a method for the efficient construction of acoustic pressure cross spectral matrices abstract a method is presented for constructing with the minimum number of physical measurements the full cross spectral matrix of acoustic pressures associated with a number of measurement positions it is necessary to evaluate the elements of the matrix in question when using inverse methods for the reconstruction of acoustic source strength spectra the method presented uses the concept of reference microphones the relation between the rank of the cross spectral matrix of acoustic pressures and the number of uncorrelated acoustic sources is discussed and used to determine the required number of reference microphones a method is proposed for selecting this number in the inverse problem in which information regarding acoustic sources is unknown the results of computer simulations are presented which explore the main features of the technique under various conditions experimental results are also presented which validate the technique

estimation of acoustic source strength by inverse methods part ii experimental investigation of methods for choosing regularization parameters two regularization methods tikhonov regularization and singular value discarding are used to improve the accuracy of reconstruction of acoustic source strength by inverse techniques in this paper some methods are investigated for choosing the tikhonov regularization parameter and the singular values to be discarded of these we concentrate on the use of ordinary cross validation and generalized cross validation these methods can provide an appropriate regularization parameter without prior knowledge of either the acoustic source strength or the contaminating measurement noise some experimental results obtained using a randomly vibrating simply supported plate mounted in a baffle are presented to illustrate the performance of the methods for choosing the regularization parameters

further results regarding the sum of domination number and average eccentricity the average eccentricity of a graph g denoted by ecc g is the mean value of eccentricities of all vertices of g let dn i be the n vertex tree obtained from a path p n 1 v 1 v 2 v n 1 by attaching a pendent vertex to vi in 13 it was shown that the maximum value for the sum of domination number and average eccentricity among n vertex connected graphs is attained by dn 3 when n 0 mod 3 and attained by the path pn when n 0 mod 3 in this paper we will further determine the second maximum value for the sum of domination number and average eccentricity among n vertex connected graphs it is interesting that the graphs attaining that second maximum value have three cases which is dn 6 when n 0 mod 3 dn 3 when n 1 mod 3 and tn when n 2 mod 3 where tn is the n vertex tree obtained from a path p n 2 v 1 v 2 v n 2 by attaching a pendent vertex to v3 and a pendent vertex to v n 4

ordering chemical graphs by randic and sum connectivity numbers abstract let g be a graph with edge set e g the randic and sum connectivity indices of g are defined as r g u v e g 1 d e g g u d e g g v and s c i g u v e g 1 d e g g u d e g g v respectively where degg u denotes the vertex degree of u in g in this paper the extremal randic and sum connectivity index among all n vertex chemical trees n 13 connected chemical unicyclic graphs n 7 connected chemical bicyclic graphs n 6 and connected chemical tricyclic graphs n 8 were presented

on the connective eccentricity index of trees and unicyclic graphs with given diameter abstract the connective eccentricity index cei of a graph g is defined as ce g v v g d v e v where e v d v denote the eccentricity and the degree of v respectively in this paper we investigate the connective eccentricity index of trees and unicyclic graphs on n vertices with diameter d we also characterize the extremal graphs

design of an optimal wave vector filter for enhancing the resolution of reconstructed source field by near field acoustical holography nah in near field acoustical holography using the boundary element method the reconstructed field often diverges due to the presence of small measurement errors in order to handle this instability in the inverse problem the reconstruction process should include some form of regularization for enhancing the resolution of source images the usual method of regularization has been the truncation of wave vectors associated with small singular values although the determination of an optimal truncation order is difficult in this article an iterative inverse solution technique is suggested in which the mean square error prediction is used a statistical estimation of the minimum mean square error between measured pressures and the model solution is required for yielding the optimal number of iterations the continuous curve of an optimal wave vector filter is designed for suppressing the high order modes that can produce large reconstruction errors experimental results from a baffled radiator reveal that the reconstruction errors can be reduced by this form of regularization by at least 48 compared to those without any regularization in comparison to results using the optimal truncation method of regularization the new scheme is shown to give further reductions of truncation error of between 7 and 39 for the example in this article

implicit methods of solution to integral formulations in boundary element method based nearfield acoustic holography two integral equation methods are considered which can be inverted to provide the surface velocity and or pressure given a measurement of the pressure on an imaginary surface in the nearfield of a vibrating or scattering body this problem is central to nearfield acoustical holography nah the integral equations are discretized using boundary element methods bems the integral equation methods considered are 1 an indirect formulation method based on the single layer integral equation and 2 a direct formulation method based on a system of equations derived from the helmholtz kirchhoff integral equation the formation of integral equations from the mentioned methods will not involve the explicit inversion of matrices but instead will require this inversion to be done implicitly since these methods back track the sound field from the measurement surface to the surface of the source vibrator they are ill posed in nature and tikhonov regularization are used to stabilize the reconstruction problems a

laplacian coefficients of trees with given number of leaves or vertices of degree two abstract let g be a simple undirected n vertex graph with the characteristic polynomial of its laplacian matrix l g det i l g k 0 n 1 k c k n k it is well known that for trees the laplacian coefficient c n 2 is equal to the wiener index of g while c n 3 is equal to the modified hyper wiener index of graph using a result of zhou and gutman on the relation between the laplacian coefficients and the matching numbers in subdivided bipartite graphs we characterize the trees with k leaves pendent vertices which simultaneously minimize all laplacian coefficients in particular this extremal balanced starlike tree s n k minimizes the wiener index the modified hyper wiener index and recently introduced laplacian like energy we prove that graph s n n 1 p has minimal laplacian coefficients among n vertex trees with p vertices of degree two in conclusion we illustrate on examples of these spectrum based invariants that the opposite problem of simultaneously maximizing all laplacian coefficients has no solution and pose a conjecture on extremal unicyclic graphs with k leaves

near field acoustic holography for an axisymmetric geometry a new formulation near field acoustic holography has recently been extended to nonseparable coordinate systems by applying the boundary elements method the formulation currently implemented applies singular value decomposition to treat the presence of evanescent waves in the near field of the structure a new method is introduced for an axisymmetric geometry that is better suited for broadband data than the singular value decomposition algorithm enabling reduced computation time and providing a new expansion for the surface field with a set of functions that can be expressed as real functions independent of frequency and independent of the choice of a measurement surface the method is tested on numerically generated data

reconstruction of source distributions from sound pressures measured over discontinuous regions multipatch holography and interpolation in the present work a method of alternating orthogonal projections is described in the context of near field acoustical holography it allows missing or not measured data to be recovered thus relieving the strictness of measurement requirements related to the use of the discrete fourier transform the method described here provides the detailed foundation for the patch holography procedure that has previously been introduced to mitigate finite measurement aperture effects by allowing the sound field to be iteratively extended beyond the measurement aperture it is also shown that the latter iterative algorithm can be used regardless of the spatial distribution of measured data i e patches can be discontinuous numerical simulations performed by using a synthetic sound field created by a point driven simply supported plate were used to demonstrate the latter point in particular a multipatch holography procedure is described that allows a source distribution to be reconstructed from the hologram

source visualization by using statistically optimized near field acoustical holography in cylindrical coordinates nearfield acoustical holography nah is a useful tool for visualizing noise sources however to avoid spatial fourier transform related truncation effects the measurement or hologram surface must extend beyond the source to a region where the sound pressure drops to a level significantly lower than the peak level within the measurement aperture statistically optimized nearfield acoustical holography sonah first derived by steiner and hald in planar geometry is based on a formulation similar to that of nah however in sonah surface to surface projection of the sound field is performed by using a transfer matrix defined in such a way that all propagating waves and a weighted set of evanescent waves are projected with optimal average accuracy i e no spatial fourier transforms are performed thus the requirement that the measurement surface be extended is eliminated without compromising the accuracy of the procedure in the present work sonah was re formulated in cylindrical coordinates and wa

network entropies based on independent sets and matchings entropy based measures have been used to characterize the structure of complex networks various graph entropies based on different graph parameters have been proposed and studied so it has been intricate to decide which measure is the right one for solving a particular problem in this paper we introduce graph entropy measures based on independent sets and matchings of graphs the values of entropies of some special graphs are calculated and we draw several conclusions regrading the usability of the measures based on our findings we conclude that it should be demanding studying extremal values of the novel entropies

interior sound field control using generalized singular value decomposition in the frequency domain the problem of controlling a sound field inside a region surrounded by acoustic control sources is considered inspired by the kirchhoff helmholtz integral the use of double layer source arrays allows such a control and avoids the modification of the external sound field by the control sources by the approximation of the sources as monopole and radial dipole transducers however the practical implementation of the kirchhoff helmholtz integral in physical space leads to large numbers of control sources and error sensors along with excessive controller complexity in three dimensions the present study investigates the potential of the generalized singular value decomposition gsvd to reduce the controller complexity and separate the effect of control sources on the interior and exterior sound fields respectively a proper truncation of the singular basis provided by the gsvd factorization is shown to lead to effective cancellation of the interior sound field at frequencies below the spatial nyquist freq

a bayesian approach to sound source reconstruction optimal basis regularization and focusing the reconstruction of acoustical sources from discrete field measurements is a difficult inverse problem that has been approached in different ways classical methods beamforming near field acoustical holography inverse boundary elements wave superposition equivalent sources etc all consist implicitly or explicitly in interpolating the measurements onto some spatial functions whose propagation are known and in reconstructing the source field by retropropagation this raises the fundamental question as whether for a given source topology and array geometry there exists an optimal interpolation basis which minimizes the reconstruction error this paper provides a general answer to this question by proceeding from a bayesian formulation that is ideally suited to combining information of physical and probabilistic natures the main findings are the followings 1 the optimal basis functions are the m eigen functions of a specific continuous discrete propagation operator with m being the number of microphones in the array 2 the a priori inclusion of spatial information on the source field causes super resolution according to a phenomenon coined bayesian focusing 3 the approach is naturally endowed with an internal regularization mechanism and results in a robust regularization criterion with no more than one minimum 4 it admits classical methods as particular cases

reconstructing interior acoustic pressure fields via helmholtz equation least squares method this paper extends the helmholtz equation least squares hels method previously developed by wang and wu j acoust soc am 102 2020 2032 1997 to reconstruction of acoustic pressure fields inside the cavity of a vibrating object the acoustic pressures are reconstructed through an expansion of the acoustic modes generated by the gram schmidt orthonormalization with respect to the particular solutions to the helmholtz equation such an expansion is uniformly convergent because the selected acoustic modes consist of a uniformly convergent series of legendre functions the coefficients associated with these acoustic modes are determined by requiring the assumed form solution to satisfy the pressure boundary condition at the measurement points the errors incurred in this process are minimized by the least squares method numerical examples of partially vibrating spheres and cylinders with various half length to radius aspect ratios subject to different frequency excitations are demonstrated the recon

method of superposition applied to patch near field acoustic holography the method of superposition may be applied to reconstruct the field on a partial surface on a radiating structure from measurements made on a nearby limited surface unlike conformal near field holography where the measurement surface surrounds the entire structure in patch holography the measurement surface need only be approximately as large as the patch on the structure surface where the reconstruction is required using the method of superposition the field on and near the measurement surface may be approximated by the field produced by a source distribution placed on a surface inside the structure the source strengths are evaluated by applying boundary conditions on the measurement surface the algorithm requires the inversion of the green s function matrix which may be ill conditioned truncated singular value decomposition is used to invert it the field on the structure surface is then approximated by the field produced by the source distribution the algorithm is easier to implement than the