k double schur functions and equivariant co homology of the affine grassmannian the schubert bases of the torus equivariant homology and cohomology rings of the affine grassmannian of sl k 1 are realized by new families of symmetric functions called k double schur functions and affine double schur functions

the ensemble statistics of the vibrational energy density of a random system subjected to single point harmonic excitation this paper is concerned with the ensemble statistics of the energy density of a random system subjected to a harmonic point load both the mean and variance of the reverberant energy density i e the total response minus the direct field are investigated it is shown that the ensemble average of the reverberant energy density is not spatially homogeneous but rather the value at the drive point is between two and three times the spatially averaged value depending on the modal overlap factor this result is closely analogous to established results regarding the case of transient excitation expressions are also derived for the relative variance of the reverberant energy density both remote from the drive point and at the drive point and a number of anomalies are found in existing results a comparison is made with simulation results for a randomized plate and this comparison highlights the importance of the ensemble size considered in the simulations the present analysis is based on a random point process model of the system natural frequencies and both gaussian orthogonal ensemble goe statistics and poisson statistics are considered

schubert polynomials for the affine grassmannian of the symplectic group we study the schubert calculus of the affine grassmannian gr of the symplectic group the integral homology and cohomology rings of gr are identified with dual hopf algebras of symmetric functions defined in terms of schur s p and q functions an explicit combinatorial description is obtained for the schubert basis of the cohomology of gr and this is extended to a definition of the affine type c stanley symmetric functions a homology pieri rule is also given for the product of a special schubert class with an arbitrary one

reformulation of the stochastic bem to improve the computational efficiency in the prediction of the vibro acoustic behaviour of structures with uncertainties abstract non determinism plays a crucial role in the prediction of the vibro acoustic behaviour of structures the stochastic boundary element method sbem allows computing the average response of systems with geometrical uncertainties the method consists in including such perturbations in the standard bem formulation and adding a set of equations representative of the energetic behaviour due to the presence of unknown cross products the addition of a set of auxiliary equation is necessary to solve the problem as a consequence the system dimension becomes very large and the application of the methodology is limited to small models this paper deals with a reformulation of the sbem approach such that the system dimension is drastically decreased and the methodology can be applied to more complex cases moreover the approach is extended to predict the variance of the field variable at no additional computational cost after a theoretical introduction the sbem is applied to two dimensional examples with pressure and velocity boundary conditions the comparison with monte carlo simulations shows a good agreement of the results and a reduction in the computational load is achieved

boundary effects on the vibration statistics of a random plate abstract this paper considers the estimation of statistics of displacement of a vibrating rectangular plate with random wave scatterers the influence of uncertainty is investigated using point impedance theory coherent boundary effects are seen which decrease when the number of scatterers increases the boundary effect is investigated using images and the first side and corner reflections are found to be a minimum requirement to estimate the spatial correlation statistics for point driven response are investigated under the assumption that the statistics of the natural frequencies follow those of the gaussian orthogonal ensemble goe the estimates are compared with monte carlo simulation results and they show good agreement

ensemble energy average and energy flow relationships for nonstationary vibrating systems abstract this paper attempts to introduce a new point of view on energy analysis in structural dynamics with particular emphasis to its link with uncertainty and complexity a linear elastic system undergoing free vibrations is considered the system is subdivided into two subsystems and their respective energies together with the shared energy flow are analysed first the ensemble energy average of the two subsystems assuming uncertain the natural frequencies is investigated it is shown how the energy averages follow a simple law when observing the long term response of the system obtained by a suitable asymptotic expansion the second part of the analysis shows how the ensemble energy average of a set of random samples is representative even of the single case if the system is complex enough the two previous points combined produce a result that applies to the energy sharing between two subsystems even independently of uncertainty for complex systems a simple energy sharing law is indeed stated moreover in the case of absence of damping a nonlinear relation between the energy flow and the energy weighted difference between the two subsystems is derived on the other hand when damping is present this relationship becomes linear including two terms one is proportional to the energy weighted difference between the two subsystems the other being proportional to its time derivative therefore the approach suggests a way for deriving a general approach to energy sharing in vibration with results that in some cases are reminiscent of those met in statistical energy analysis finally computational experiments performed on systems of increasing complexity validate the theoretical results

quantum and affine schubert calculus and macdonald polynomials abstract we definitively establish that the theory of symmetric macdonald polynomials aligns with quantum and affine schubert calculus using a discovery that distinguished weak chains can be identified by chains in the strong bruhat order poset on the type a affine weyl group we construct two one parameter families of functions that respectively transition positively with hall littlewood and macdonald s p functions and specialize to the representatives for schubert classes of homology and cohomology of the affine grassmannian our approach leads us to conjecture that all elements in a defining set of 3 point genus 0 gromov witten invariants for flag manifolds can be formulated as strong covers

schubert polynomials and the nilcoxeter algebra abstract schubert polynomials were introduced and extensively developed by lascoux and schutzenberger after an earlier less combinatorial version had been considered by bernstein gelfand and gelfand and demazure we give a new development of the theory of schubert polynomials based on formal computations in the algebra of operators u1 u2 satisfying the relations u2i 0 uiuj ujui if i j 2 and uiui 1ui ui 1uiui 1 we call this algebra the nilcoxeter algebra of the symmetric group s n our development leads to simple proofs of many standard results in particular a symmetry of the stable schubert polynomials fw b an explicit combinatorial formula for schubert polynomials due to billey jockusch and stanley c the cauchy formula for schubert polynomials and d a formula of macdonald for s w 1 1 our main new result is a proof of a conjectured q analogue of d due to macdonald which gives a formula for s w 1 q q2

rules of three for commutation relations abstract we study the phenomenon in which commutation relations for sequences of elements in a ring are implied by similar relations for subsequences involving at most three indices at a time

on the applicability of the lognormal distribution in random dynamical systems this paper is concerned with the probability density function of the energy of a random dynamical system subjected to harmonic excitation it is shown that if the natural frequencies and mode shapes of the system conform to the gaussian orthogonal ensemble then under common types of loading the distribution of the energy of the response is approximately lognormal providing the modal overlap factor is high typically greater than two in contrast it is shown that the response of a system with poisson natural frequencies is not approximately lognormal numerical simulations are conducted on a plate system to validate the theoretical findings and good agreement is obtained simulations are also conducted on a system made from two plates connected with rotational springs to demonstrate that the theoretical findings can be extended to a built up system the work provides a theoretical justification of the commonly used empirical practice of assuming that the energy response of a random system is lognormal

code construction and decoding algorithms for semi quantitative group testing with nonuniform thresholds we analyze a new group testing scheme termed semi quantitative group testing which may be viewed as a concatenation of an adder channel and a discrete quantizer our focus is on non uniform quantizers with arbitrary thresholds for the most general semi quantitative group testing model we define three new families of sequences capturing the constraints on the code design imposed by the choice of the thresholds the sequences represent extensions and generalizations of b h and certain types of super increasing and lexicographically ordered sequences and they lead to code structures amenable for efficient recursive decoding we describe the decoding methods and provide an accompanying computational complexity and performance analysis

the combinatorics of c fixed points in generalized calogero moser spaces and hilbert schemes abstract in this paper we study the combinatorial consequences of the relationship between rational cherednik algebras of type g l 1 n cyclic quiver varieties and hilbert schemes we classify and explicitly construct c fixed points in cyclic quiver varieties and calculate the corresponding characters of tautological bundles furthermore we give a combinatorial description of the bijections between c fixed points induced by the etingof ginzburg isomorphism and nakajima reflection functors we apply our results to obtain a new proof as well as a generalization of the q hook formula

the variational theory of complex rays an answer to the resolution of mid frequency 3d engineering problems the variational theory of complex rays vtrc is an approach to the simulation of mid frequency phenomena whose wavelengths are relatively small compared to the dimensions of the domain this is a wave based computational technique which involves a nonclassical variational formulation this paper focuses on the development of the approach for 3d engineering problems and shows that it is a mature technique illustrations are given to show the capabilities of this method

cylindric skew schur functions cylindric skew schur functions which are a generalisation of skew schur functions arise naturally in the study of p partitions also recent work of a postnikov shows they have a strong connection with a problem of considerable current interest that of finding a combinatorial proof of the non negativity of the 3 point gromov witten invariants after explaining these motivations we study cylindric skew schur functions from the point of view of schur positivity using a result of i gessel and c krattenthaler we generalise a formula of a bertram i ciocan fontanine and w fulton thus giving an expansion of an arbitrary cylindric skew schur function in terms of skew schur functions while we show that no non trivial cylindric skew schur functions are schur positive we conjecture that this can be reconciled using the new concept of cylindric schur positivity

affine approach to quantum schubert calculus this paper presents a formula for products of schubert classes in the quantum cohomology ring of the grassmannian we introduce a generalization of schur symmetric polynomials for shapes that are naturally embedded in a torus then we show that the coefficients in the expansion of these toric schur polynomials in terms of the regular schur polynomials are exactly the 3 point gromov witten invariants which are the structure constants of the quantum cohomology ring this construction implies three symmetries of the gromov witten invariants of the grassmannian with respect to the groups s3 z nz 2 and z 2z the last symmetry is a certain curious duality of the quantum cohomology which inverts the quantum parameter q our construction gives a solution to a problem posed by fulton and woodward about the characterization of the powers of the quantum parameter q which occur with nonzero coefficients in the quantum product of two schubert classes the curious duality switches the smallest such power of q with the highest power we also discuss the affine nil temperley lieb algebra that gives a model for the quantum cohomology

total positivity of a kind of generalized toric bernstein basis abstract the normalized totally positive bases are widely used in many fields based on the generalized vandermonde determinant the normalized total positivity of a kind of generalized toric bernstein basis is proved which is defined on a set of real points by this result then the progressive iterative approximation property of the generalized toric bezier curve is obtained and some examples are provided to illustrate this property

affine stanley symmetric functions we define a new family inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i of generating functions for w inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i which are affine analogues of stanley symmetric functions we establish basic properties of these functions including symmetry dominance and conjugation we conjecture certain positivity properties in terms of a subfamily of symmetric functions called affine schur functions as applications we show how affine stanley symmetric functions generalize the dual of the k schur functions of lapointe lascoux and morse as well as the cylindric schur functions of postnikov conjecturally affine stanley symmetric functions should be related to the cohomology of the affine flag variety

accurate extreme value based frequency response bounding for structures with a small number of highly random parameters abstract a modified extreme value based methodology is discussed for computing statistical bounds associated with the magnitude of the frequency response of a specified number of structures with high levels of random parameter uncertainty the methodology intended for small numbers of uncertain parameters is capable of constructing accurate statistical bounds in terms of quantiles associated with the extreme value distribution quantiles can be constructed for an ensemble of structural responses across the entire frequency range without using monte carlo simulation to test the methodology statistical bounds for the energy of an l shaped structure with low and high levels of uniformly distributed length and thickness variability are obtained i via direct integration using an ansys finite element model and ii via statistical energy analysis sea comparisons are shown with bounds obtained using monte carlo simulation the merit of direct integration for computing bounds associated with the responses of an ensemble of structures with high levels of random parameter uncertainty is demonstrated by its simplicity high accuracy and absence of statistical scatter

classical wave experiments on chaotic scattering we review recent research on the transport properties of classical waves through chaotic systems with special emphasis on microwaves and sound waves inasmuch as these experiments use antennas or transducers to couple waves into or out of the systems scattering theory has to be applied for a quantitative interpretation of the measurements most experiments concentrate on tests of predictions from random matrix theory and the random plane wave approximation in all studied examples a quantitative agreement between experiment and theory is achieved to this end it is necessary however to take absorption and imperfect coupling into account concepts that were ignored in most previous theoretical investigations classical phase space signatures of scattering are being examined in a small number of experiments

superimposed codes in r sup n the authors introduce the concept of superimposed codes in euclidean n space r sup n an asymptotic existence bound is derived for such codes the proof uses the idea of random coding in particular the asymptotic properties of long codes are studied it is shown that the derived existence bound differs only by a factor of four from a nonexistence bound obtained by a simple sphere packing argument

energy flow models from finite element analysis computationally efficient methods are described by which the results of a finite element analysis of a system may be post processed to form energy flow models yielding time and perhaps frequency average subsystem energies as well as input and dissipated powers the methods are particularly efficient for excitation which is spatially distributed or broadband e g rain on the roof or if the frequency average response is required first a method based on a global finite element analysis is presented this involves a global modal decomposition and a reordering of the subsequent numerical calculations the properties of the distribution of the excitation and the system s mass and stiffness lead to subsystem force distribution mass distribution and stiffness distribution matrices the response is given by a sum of terms involving the interaction of a pair of global modes the contribution of each pair depending on the appropriate elements of the distribution matrices frequency averaging is performed by separating the resulting frequency dependent terms and integrating in most practical cases this integration can be done analytically next an alternative method involving component mode synthesis is described in this individual finite element analyses are performed for each subsystem using here the fixed interface method these are then assembled to perform a global modal analysis with the order of the model being much reduced the consequent results are then post processed in the same way finally a system comprising three coupled plates is presented as a numerical example

amplituhedron cells and stanley symmetric functions the amplituhedron was recently introduced in the study of scattering amplitudes in n 4 super yang mills we compute the cohomology class of a tree amplituhedron subvariety of the grassmannian to be the truncation of an affine stanley symmetric function

pieri rule for the affine flag variety abstract we prove the affine pieri rule for the cohomology of the affine flag variety conjectured by lam lapointe morse and shimozono we study the cap operator on the affine nilhecke ring that is motivated by kostant and kumar s work on the equivariant cohomology of the affine flag variety we show that the cap operators for pieri elements are the same as pieri operators defined by berg saliola and serrano this establishes the affine pieri rule we also discuss properties of cap operators which are not necessarily affine a type

equivariant k theory of affine flag manifolds and affine grothendieck polynomials we study the equivariant k group of the affine flag manifold with respect to the borel group action we prove that the structure sheaf of the infinite dimensional schubert variety in the k group is represented by a unique polynomial which we call the affine grothendieck polynomial

a general derivation of the statistical energy analysis equations for coupled dynamic systems abstract the response of a multi coupled dynamic system to random excitation is considered with the aim of providing a theoretical derivation of the statistical energy analysis sea approach to vibration prediction it is shown that the general form of the principal sea equation is widely applicable provided that the coupling between the subsystems is conservative and a suitable definition of the subsystem energy is used under the assumption of weak coupling the standard sea equations are recovered expressions for the coupling loss factors are obtained in terms of the frequency and space averaged green functions of the coupled system

superimposed codes for the multiaccess binary adder channel superimposed codes for multiple access communication in a binary adder channel are analyzed the superposition mechanism used in this correspondence is ordinary addition each user is assigned a codeword from a superimposed code it is proved that every constant weight code c of weight w and maximal correlation c corresponds to a subclass of a disjunctive code d of order m

the center of the affine niltemperley lieb algebra we give a description of the center of the affine niltemperley lieb algebra based on a certain grading of the algebra and on a faithful representation of it on fermionic particle configurations we present a normal form for monomials hence construct a basis of the algebra and use this basis to show that the affine niltemperley lieb algebra is finitely generated over its center as an application we obtain a natural embedding of the affine niltemperley lieb algebra on n generators into the affine niltemperley lieb algebra on n 1 generators

an efficient wave based prediction technique for plate bending vibrations the finite element method is a commonly used prediction method for dynamic simulations of plate bending problems a major disadvantage of this method is its practical frequency limitation as its application involves prohibitively large computational efforts at higher frequencies a newly developed wave based prediction technique pushes up the frequency limit through an enhanced computational efficiency this paper discusses the principles of the wave based method for convex plate problems the beneficial convergence rate of the wave based method as compared with the finite element method is verified for various validation examples which include the most commonly encountered boundary conditions

an efficient probabilistic approach to vibro acoustic analysis based on the gaussian orthogonal ensemble vibro acoustic analysis of complex systems at higher frequencies faces two challenges how to compute the response without using an excessive number of degrees of freedom dofs and how to quantify the uncertainty of the response due to small spatial variations in geometry material properties and boundary conditions which have a wave scattering effect in this study a general method of analysis is presented that provides an answer to both questions while overcoming most limitations of statistical energy analysis the fundamental idea is to numerically compute an artificial ensemble of realizations for the components of the built up system that are highly sensitive to small random wave scatterers this can be efficiently performed because their eigenvalue spacings and mode shapes conform to gaussian orthogonal ensemble spacings and gaussian random fields respectively the dofs of the overall system are therefore limited to those of the deterministic components and the interface dofs of the random components the method is extensively validated by application to plate structures good agreement between the predicted response probability distributions and the results of detailed parametric probabilistic models is obtained also for cases of low modal overlap single point loading and strong subsystem coupling

an approach to the theoretical background of statistical energy analysis applied to structural vibration rayleigh s classical approach to the study of vibration of systems having a finite number of degrees of freedom is applied to the problem of coupling of subsystems in a complicated structure in order to probe the regions of applicability of the approach to vibration analysis usually known as statistical energy analysis sea the classical method has advantages of simplicity and rigor over previous approaches to the background of sea in certain cases and provides extensions and simplifications in several areas of the theory it also suggests modifications to sea modeling strategy depending on the type of coupling involved even when that coupling is weak so that earlier analyses might be thought to apply

uncertainty propagation in sea for structural acoustic coupled systems with non deterministic parameters abstract considering limited available information on uncertainties in structural acoustic coupled systems two methods namely the vertex method and the legendre orthogonal polynomial based method for predicting their dynamic behavior are developed based on the statistical energy analysis sea approach for the vertex method an efficient program for determining coordinates of all vertices of the rectangular spanned by entries of the involved interval input vector is coded which is well suited for an interval input vector in arbitrary dimension instead of calculating the extremum of the response of interest a method for determining its minimal and maximal point vectors dimension by dimension with respect to uncertain parameters is proposed based on the legendre orthogonal polynomial approximation following the theoretical analysis of the accuracy and efficiency of the proposed methods their validation is performed by one numerical example and two applications

a k tableau characterization of k schur functions we study k schur functions characterized by k tableaux proving combinatorial properties such as a k pieri rule and a k conjugation this new approach relies on developing the theory of k tableaux and includes the introduction of a weight permuting involution on these tableaux that generalizes the bender knuth involution this work lays the groundwork needed to prove that the set of k schur littlewood richardson coefficients contains the 3 point gromov witten invariants structure constants for the quantum cohomology ring

poisson group testing a probabilistic model for boolean compressed sensing we introduce a novel probabilistic group testing framework termed poisson group testing in which the number of defectives follows a right truncated poisson distribution the poisson model has a number of new applications including dynamic testing with diminishing relative rates of defectives we consider both nonadaptive and semi adaptive identification methods for nonadaptive methods we derive a lower bound on the number of tests required to identify the defectives with a probability of error that asymptotically converges to zero in addition we propose test matrix constructions for which the number of tests closely matches the lower bound for semiadaptive methods we describe a lower bound on the expected number of tests required to identify the defectives with zero error probability in addition we propose a stage wise reconstruction algorithm for which the expected number of tests is only a constant factor away from the lower bound the methods rely only on an estimate of the average number of defectives rather than on the individual probabilities of subjects being defective

physical consequences of a nonparametric uncertainty model in structural dynamics abstract one of the main claims of the nonparametric model of random uncertainty introduced by soize 2000 3 is its ability to account for model uncertainty the present paper investigates this claim by examining the statistics of natural frequencies total energy and underlying dispersion equation yielded by the nonparametric approach for two simple systems a thin plate in bending and a one dimensional finite periodic mass spring chain results for the plate show that the average modal density and the underlying dispersion equation of the structure are gradually and systematically altered with increasing uncertainty the findings for the mass spring chain corroborate the findings for the plate and show that the remote coupling of nonadjacent degrees of freedom induced by the approach suppresses the phenomenon of mode localization this remote coupling also leads to an instantaneous response of all points in the chain when one mass is excited in the light of these results it is argued that the nonparametric approach can deal with a certain type of model uncertainty in this case the presence of unknown terms of higher or lower order in the governing differential equation but that certain expectations about the system such as the average modal density may conflict with these results

response variance prediction in the statistical energy analysis of built up systems in the statistical energy analysis sea of high frequency noise and vibration a complex engineering structure is represented as an assembly of subsystems the response of the system to external excitation is expressed in terms of the vibrational energy of each subsystem and these energies are found by employing the principle of power balance strictly the computed energy is an average taken over an ensemble of random structures and for many years there has been interest in extending the sea prediction to the variance of the energy a variance prediction method for a general built up structure is presented here closed form expressions for the variance are obtained in terms of the standard sea parameters and an additional set of parameters alpha k that describe the nature of the power input to each subsystem k and alpha ks that describe the nature of the coupling between subsystems k and s the theory is validated by comparison with monte carlo simulations of plate networks and structural acoustic systems

gaussian orthogonal ensemble spacing statistics and the statistical overlap factor applied to dynamic systems abstract this paper examines the extent to which gaussian orthogonal ensemble goe statistics is applicable to the natural frequencies of a dynamic system the natural frequencies of a simply supported plate or a rectangular room tend to have an exponential spacing distribution however any disruption of the system symmetries has been shown to promote goe statistics for which the modal spacing distribution is rayleigh in this paper the effect of a range of uncertainties on the modal statistics of structures is numerically characterised this is achieved by examining the modal statistics of mass and or spring loaded plates and plates coupled by springs the natural frequencies of the aforementioned structures have been derived using the lagrange rayleigh ritz technique the degree of uncertainty required to effect the transition from an exponential to a rayleigh distribution and to achieve universality of the statistical properties is investigated a further measure of the randomness required to produce goe statistics can be obtained by examining the amount of mixing and veering between the modes of a dynamic system the statistical overlap factor is a non dimensional parameter related to the random variation in an individual natural frequency from its mean value and is useful to quantify the frequency beyond which the resonant behaviour of individual modes no longer dominates the response statistics using a first order perturbation analysis an approximate expression for the statistical overlap factor has been developed for the randomised plates to estimate the modal range for the occurrence of goe statistics

a family of k 1 ary signature codes for noisy multiple access adder channel a coding scheme of k 1 ary error correcting signature codes for a noisy multiple access adder channel is proposed given a signature matrix a and a difference matrix d d d a priori a larger signature matrix is obtained by replacing each element in the hadamard matrix with a or d or d depending on the values of the elements and their locations in the hadamard matrix the set of rows in the proposed matrix gives an error correcting signature code introducing a difference matrix makes it possible to construct an error correcting signature code whose sum rate is increased with an increase in the order of the hadamard matrix either binary or non binary signature codes are constructed when the pairs of matrices a and d are given

random matrix theory and non parametric model of random uncertainties in vibration analysis recently a new approach called a non parametric model of random uncertainties has been introduced for modelling random uncertainties in linear and non linear elastodynamics in the low frequency range this non parametric approach differs from the parametric methods for random uncertainties modelling and has been developed in introducing a new ensemble of random matrices constituted of symmetric positive definite real random matrices this ensemble differs from the gaussian orthogonal ensemble goe and from the other known ensembles of the random matrix theory the present paper has three main objectives the first one is to study the statistics of the random eigenvalues of random matrices belonging to this new ensemble and to compare with the goe the second one is to compare this new ensemble of random matrices with the goe in the context of the non parametric approach of random uncertainties in structural dynamics for the low frequency range the last objective is to give a new validation for the non parametric model of random uncertainties in structural dynamics in comparing in the low frequency range the dynamical response of a simple system having random uncertainties modelled by the parametric and the non parametric methods these three objectives will allow us to conclude about the validity of the different theories

schubert polynomials for the affine grassmannian let g be a complex simply connected simple group and k a maximal compact subgroup let f c t denote the field of formal laurent series and o c t the ring of formal power series the quotient qr g f g 0 is known as the affine grassmannian of g it is known that qr is homotopy equivalent to the space of based loops k into the compact group k see 9 30 the multiplication on qk endows the homology h qr and cohomology h qr with structures of dual hopf algebras these hopf algebras were first identified by bott 4 another descrip tion is given by ginzburg 10 which we shall not use the bruhat decomposition of g f induces a stratification of qr by schubert cells ilw indexed by grassman nian elements w of the affine weyl group waffwe denote by aw g h qr and aw g h qr the schubert classes in homology and cohomology when g sl n c bott identifies h qr and h qr with a subring an and a quotient an of the ring of symmetric functions our main theorem theorem 7 1 identifies the schubert classes aw g qr and aw g h qr as explicit symmet ric functions under this isomorphism these symmetric functions are combinatori ally defined in homology the aw g h qr are represented by lascoux lapointe morse s fc schur functions sw x g an and in cohomology the aw g h qr are represented by the dual fc schur functions or affine schur functions fw x g an see 21 24 17 our theorem was originally conjectured by mark shimozono the conjecture was made explicit in the cohomology case by jennifer morse kostant and kumar 14 studied the topology of homogeneous spaces for arbi trary kac moody groups and in particular calculated the structure constants of h qr in the schubert basis using the algebraic construction of the nilhecke ring our connection between the topology and the combinatorics proceeds via the study of three subalgebras of the nilhecke ring the first subalgebra is the nilcoxeter al gebra ao which is the algebra generated by the divided difference operators the second algebra is a certain centralizer algebra aaff s which we call the peterson subalgebra peterson 28 constructs an isomorphism j from the t equivariant ho mology hr qr of the affine grassmannian to z ff s the third algebra b is a combinatorially defined subalgebra of a0 which we call the affine fomin stanley

a derivation of the coupling loss factors used in statistical energy analysis abstract the conditions under which an exact theory concerning the energy power flow relationships for a complex dynamic system may be reduced to the standard statistical energy analysis sea equations are considered a definition of weak coupling is introduced which encompasses a number of previous definitions and it is shown that the presence of weak coupling does not guarantee the validity of the standard sea assumption that the coupling loss factor clf between two subsystems is zero unless the subsystems are directly connected the standard assumption is likely to be reasonable for systems which are both weakly coupled and reverberant attention is then turned to the clf between two connected subsystems and a general result is obtained which involves a space and frequency averaged green function it is shown that under the appropriate assumptions a number of existing results may be derived from this expression including those produced by both the wave and the modal approaches to sea the exact theory may be used as a basis for developing improved models of the system dynamics

totally positive matrices and cyclic polytopes abstract we show that totally positive matrices from linear algebra and cyclic polytopes from convexity are essentially equivalent structures and we give several applications of the resulting correspondence

the level crossing rates and associated statistical properties of a random frequency response function abstract this work is concerned with the statistical properties of the frequency response function of the energy of a random system earlier studies have considered the statistical distribution of the function at a single frequency or alternatively the statistics of a band average of the function in contrast the present analysis considers the statistical fluctuations over a frequency band and results are obtained for the mean rate at which the function crosses a specified level or equivalently the average number of times the level is crossed within the band results are also obtained for the probability of crossing a specified level at least once the mean rate of occurrence of peaks and the mean trough to peak height the analysis is based on the assumption that the natural frequencies and mode shapes of the system have statistical properties that are governed by the gaussian orthogonal ensemble goe and the validity of this assumption is demonstrated by comparison with numerical simulations for a random plate the work has application to the assessment of the performance of dynamic systems that are sensitive to random imperfections

statistical energy analysis with fuzzy parameters to handle populations of structures statistical modal energy distribution analysis smeda approach was developed to enlarge the application field of statistical energy analysis sea when equipartition of modal energies is not achieved smeda gives more precise results than standard sea when compared to exact energy response of a deterministic system in the case of low modal overlap heterogeneous systems or point excitation the present paper was initiated by this question when considering a population of similar structures each of them being described by smeda do the ensemble averaged energies of subsystem and injected power tend to satisfy sea equations in other terms despite the non equipartition of energy observed on each element of the population of structures does the ensemble averaging leads to sea equation where equipartition of energy is assumed the response to that question that rises from this paper is yes if the terms of the sea equation are fuzzy numbers it results that the energy response given by the model can be interpreted using fuzzy numbers theory

numerical and experimental validation of variance prediction in the statistical energy analysis of built up systems in the statistical energy analysis sea approach to vibration modeling a complex system is represented as an assembly of coupled subsystems and the method leads to the prediction of the vibrational energy level of each subsystem the averaging procedures implicit in the technique imply that the predicted energy is the mean value taken over an ensemble of random structures such as a set of vehicles leaving a production line recently a new method has been developed to allow the ensemble variance in addition to the mean to be predicted within the context of sea and the present paper concerns further extension and validation of this work the theoretical extension concerns the variance of the energy density at a single point in any of the subsystems and the validation includes both simulation and experimental studies the simulation results concern plate assemblies while experimental results are presented both for a single plate and for a cylinder plate structure in each case an ensemble of random structures has been generated by adding small point masses at random locations on the structure in general good agreement between the predictions and the validation results is observed

response variance prediction for uncertain vibro acoustic systems using a hybrid deterministic statistical method imperfections during the manufacturing process can cause significant variations in the noise and vibration levels exhibited by nominally identical structures any response calculations employed during the design process should ideally take account of these uncertainties and predict the expected range in performance recently a hybrid method has been developed to predict the ensemble average response of a built up system by combining a deterministic model of parts of the system with a statistical model of other components shorter p j and langley r s 2005 j sound vib 288 669 700 in this paper the method is extended to predict the ensemble variance of the response expressions are derived for the variance of the vibrational energies in the statistical components and for the variance of the cross spectrum of the response of the deterministic components which augment the mean values of these quantities predicted by the original theory the method employs a nonparametric model of uncertainty in the sense that the statistical components are taken to carry diffuse wave fields and this obviates the requirement for a detailed description of the system uncertainties the method is validated by application to a range of coupled plate structures and good agreement with detailed monte carlo simulations is found

on the diffuse field reciprocity relationship and vibrational energy variance in a random subsystem at high frequencies a recent paper has shown that under certain conditions the cross spectral matrix of the forces exerted by a vibrational or acoustic wave field on its surrounding boundaries can be expressed in terms of i the energy of the wave field and ii the direct field dynamic stiffness matrix of the boundary this diffuse field reciprocity relation was derived using wave mechanics and it is not immediately clear how the required wave field properties translate to conditions on the vibrational modes of the system or the applied forcing this issue is addressed here by deriving an extended version of the reciprocity relation using modal methods and the conditions required for the extended version to reduce to the existing relation are delineated it is shown that the existing diffuse field reciprocity relation leads to an anomalous result when used to predict the energy variance of a subsystem and that this anomaly is resolved by using the present extended version of the relation a supplementary result arising from the analysis is that for systems with a sufficient degree of randomness the ensemble average of the dynamic stiffness matrix of a random subsystem is equal to the inverse of the ensemble average of the receptance matrix

pattern characterization of rationally smooth affine schubert varieties of type a abstract schubert varieties in finite dimensional flag manifolds g p are a well studied family of projective varieties indexed by elements of the corresponding weyl group w in particular there are many tests for smoothness and rational smoothness of these varieties one key result due to lakshmibai and sandhya is that in type a the smooth schubert varieties are precisely those that are indexed by permutations that avoid the patterns 4231 and 3412 recently there has been a flurry of research related to the infinite dimensional analogs of flag manifolds corresponding with g being a kac moody group and w being an affine weyl group or parabolic quotient in this paper we study the case when w is the affine weyl group of type a or the affine permutations we develop the notion of pattern avoidance for affine permutations our main result is a characterization of the rationally smooth schubert varieties corresponding to affine permutations in terms of the patterns 4231 and 3412 and the twisted spiral permutations

resonant beam vibration a wave evolution analysis abstract in an attempt to verify the concept of wave mode duality and to understand the wave train closure principle in resonant beam vibration an elastically supported beam subjected to a swept sinusoidal force is investigated through a simulation of wave evolution from initial to final steady state response wave evolution is simulated in the superposition and the degenerate states according to the nature of wave reflection at the boundaries of the beam from the results resonance condition is established and the natural frequencies and mode shapes are calculated using a simply supported beam and a cantilever beam as examples the concept of wave mode duality at resonance is verified an examination of the wave train closure principle shows that although the frequency equations obtained by using the principle are identical to those derived from the wave evolution results the principle does not describe the realistic process of the formulation of vibration modes in general the waves described in the wave train closure principle are not physical but virtual waves that represent final steady state waves which can only be asymptotically approached in reality the principle can be used as a convenient technique to obtain vibration modes from the point of view of wave propagation

spreading set with error correction for multiple access adder channel the necessary and sufficient condition for constructing a spreading set with decodability is investigated it is proved that for a given delta decodable spreading set and a qtimesq square matrix h with components 1 or 1 a qdelta decodable spreading set s is obtained if and only if h is a hadamard matrix in addition a decoding rule with error correction and message data detection is provided

the effect of generalised force correlations on the response statistics of a harmonically driven random system if the physical properties of a structural component are sufficiently random then the statistical distribution of the natural frequencies and mode shapes tends to a universal distribution associated with the gaussian orthogonal ensemble goe of random matrices previous work has exploited this result to yield expressions for the relative variance of the energy of the response of a random system to harmonic excitation the derivation of these expressions employed random point process theory and in the theoretical development it was assumed that the modal generalised forces were uncorrelated although this assumption is often valid there are cases in which correlations between the generalised forces can significantly affect the response variance and in the present work the existing theory is extended to include correlations of this type the extended theory is applicable to both single frequency responses and to band average responses and the developed closed form expressions are validated by comparison with direct simulations for a random plate structure

quantum cohomology and the k schur basis we prove that structure constants related to hecke algebras at roots of unity are special cases of k littlewood richardson coefficients associated to a product of k schur functions as a consequence both the 3 point gromov witten invariants appearing in the quantum cohomology of the grassmannian and the fusion coefficients for the wzw conformal field theories associated to su l are shown to be k littlewood richardson coefficients from this mark shimozono conjectured that the k schur functions form the schubert basis for the homology of the loop grassmannian whereas k schur coproducts correspond to the integral cohomology of the loop grassmannian we introduce dual k schur functions defined on weights of k tableaux that given shimozono s conjecture form the schubert basis for the cohomology of the loop grassmannian we derive several properties of these functions that extend those of skew schur functions

statistical energy analysis modelling of complex structures as coupled sets of oscillators ensemble mean and variance of energy expressions are derived for the ensemble means and variances of the subsystem energies of built up systems comprising two subsystems the approach is based on the statistical energy analysis of two spring coupled oscillators and sets of oscillators or coupled continuous subsystems described by mace and ji the statistical energy analysis of coupled sets of oscillators proceedings of the royal society a 1824 2007 the paper focuses on spring coupling although similar results hold for more general forms of conservative coupling randomness is introduced into the system by assuming that the natural frequency spacings in each subsystem conform to certain statistical distributions a coupling coefficient parameter is introduced which together with the coupling strength parameter defined by mace and ji 2007 accounts for the statistics of the coupling stiffness various approximations and assumptions are made it is seen that the variance of the excited subsystem depends primarily on the variance of the input power which in turn depends on the variance of the number of modes of the excited subsystem in the frequency band of excitation and their mode shapes the variance of the undriven subsystem on the other hand depends primarily on the variance of the intermodal coupling coefficients which in turn depend on the variances of the number of in band modes of both subsystems and their mode shapes the cases of poisson and gaussian orthogonal ensemble natural frequency spacing statistics are considered numerical examples of two plates coupled by one or a number of springs are presented

linear transformations that preserve certain positivity classes of matrices abstract for each of several s r n n those linear transformations l r n n r n n which map s onto s are characterized each class is a familiar one which generalizes the notion of positivity to matrices the classes include the matrices with nonnegative principal minors the m matrices the totally nonnegative matrices the d stable matrices the matrices with positive diagonal lyapunov solutions and the h matrices as well as other related classes the set of transformations is somewhat different from case to case but the strategy of proof while differing in detail is similar

combinatorics of the k theory of affine grassmannians abstract we introduce a family of tableaux that simultaneously generalizes the tableaux used to characterize grothendieck polynomials and k schur functions we prove that the polynomials drawn from these tableaux are the affine grothendieck polynomials and k k schur functions schubert representatives for the k theory of affine grassmannians and their dual in the nil hecke ring we prove a number of combinatorial properties including pieri rules

a quantitative criterion validating coupling power proportionality in statistical energy analysis the response of two general spring coupled elements is investigated to develop a unifying approach to the weak coupling criterion in statistical energy analysis sea first the coupled determinis