spread of information and infection on finite random networks the modeling of epidemic like processes on random networks has received considerable attention in recent years while these processes are inherently stochastic most previous work has been focused on deterministic models that ignore important fluctuations that may persist even in the infinite network size limit in a previous paper for a class of epidemic and rumor processes we derived approximate models for the full probability distribution of the final size of the epidemic as opposed to only mean values in this paper we examine via direct simulations the adequacy of the approximate model to describe stochastic epidemics and rumors on several random network topologies homogeneous networks erd os r enyi er random graphs barabasi albert scale free networks and random geometric graphs we find that the approximate model is reasonably accurate in predicting the probability of spread however the position of the threshold and the conditional mean of the final size for processes near the threshold are not well described by the approximate model even in the case of homogeneous networks we attribute this failure to the presence of other structural properties beyond degree degree correlations and in particular clustering which are present in any finite network but are not incorporated in the approximate model in order to test this hypothesis we perform additional simulations on a set of er random graphs where degree degree correlations and clustering are separately and independently introduced using recently proposed algorithms from the literature our results show that even strong degree degree correlations have only weak effects on the position of the threshold and the conditional mean of the final size on the other hand the introduction of clustering greatly affects both the position of the threshold and the conditional mean similar analysis for the barabasi albert scale free network confirms the significance of clustering on the dynamics of rumor spread for this network though with its highly skewed degree distribution the addition of positive correlation had a much stronger effect on the final size distribution than was found for the simple random graph

consensus dynamics on random rectangular graphs abstract a random rectangular graph rrg is a generalization of the random geometric graph rgg in which the nodes are embedded into a rectangle with side lengths a and b 1 a instead of on a unit square 0 1 2 two nodes are then connected if and only if they are separated at a euclidean distance smaller than or equal to a certain threshold radius r when a 1 the rrg is identical to the rgg here we apply the consensus dynamics model to the rrg our main result is a lower bound for the time of consensus i e the time at which the network reaches a global consensus state to prove this result we need first to find an upper bound for the algebraic connectivity of the rrg i e the second smallest eigenvalue of the combinatorial laplacian of the graph this bound is based on a tight lower bound found for the graph diameter our results prove that as the rectangle in which the nodes are embedded becomes more elongated the rrg becomes a large world i e the diameter grows to infinity and a poorly connected graph i e the algebraic connectivity decays to zero the main consequence of these findings is the proof that the time of consensus in rrgs grows to infinity as the rectangle becomes more elongated in closing consensus dynamics in rrgs strongly depend on the geometric characteristics of the embedding space and reaching the consensus state becomes more difficult as the rectangle is more elongated

products of three triangular matrices over commutative rings abstract we characterize commutative rings with 1 over which any n n matrix a can be written as lum where l m are lower triangular u is upper triangular and l and u have all their diagonal entries 1

uncertainty investigations in nonlinear aeroelastic systems in this paper the stochastic collocation method scm is applied to investigate the nonlinear behavior of an aeroelastic system with uncertainties in the system parameter and the initial condition numerical case studies for problems with uncertainties are carried out in particular the performance of the scm is compared with solutions based on other computational techniques such as monte carlo simulation wiener chaos expansion and wavelet chaos expansion from the computational results we conclude that the scm is an effective tool to study a nonlinear aeroelastic system with random parameters

random geometric graphs we analyze graphs in which each vertex is assigned random coordinates in a geometric space of arbitrary dimensionality and only edges between adjacent points are present the critical connectivity is found numerically by examining the size of the largest cluster we derive an analytical expression for the cluster coefficient which shows that the graphs are distinctly different from standard random graphs even for infinite dimensionality insights relevant for graph bipartitioning are included

uncertainty propagation using wiener haar expansions an uncertainty quantification scheme is constructed based on generalized polynomial chaos pc representations two such representations are considered based on the orthogonal projection of uncertain data and solution variables using either a haar or a legendre basis governing equations for the unknown coefficients in the resulting representations are derived using a galerkin procedure and then integrated in order to determine the behavior of the stochastic process the schemes are applied to a model problem involving a simplified dynamical system and to the classical problem of rayleigh benard instability for situations involving random parameters close to a critical point the computational implementations show that the wiener haar wha representation provides more robust predictions that those based on a wiener legendre wle decomposition however when the solution depends smoothly on the random data the wle scheme exhibits superior convergence suggestions regarding future extensions are finally drawn based on these experiences

multi resolution analysis of wiener type uncertainty propagation schemes a multi resolution analysis mra is applied to an uncertainty propagation scheme based on a generalized polynomial chaos pc representation the mra relies on an orthogonal projection of uncertain data and solution variables onto a multi wavelet basis consisting of compact piecewise smooth polynomial functions the coefficients of the expansion are computed through a galerkin procedure the mra scheme is applied to the simulation of the lorenz system having a single random parameter the convergence of the solution with respect to the resolution level and expansion order is investigated in particular results are compared to two monte carlo sampling strategies demonstrating the superiority of the mra for more complex problems however the mra approach may require excessive cpu times adaptive methods are consequently developed in order to overcome this drawback two approaches are explored the first is based on adaptive refinement of the multi wavelet basis while the second is based on adaptive block partitioning of the space of random variables computational tests indicate that the latter approach is better suited for large problems leading to a more efficient flexible and parallelizable scheme

bayesian inference of nonlinear unsteady aerodynamics from aeroelastic limit cycle oscillations a bayesian model selection and parameter estimation algorithm is applied to investigate the influence of nonlinear and unsteady aerodynamic loads on the limit cycle oscillation lco of a pitching airfoil in the transitional reynolds number regime at small angles of attack laminar boundary layer trailing edge separation causes negative aerodynamic damping leading to the lco the fluid structure interaction of the rigid but elastically mounted airfoil and nonlinear unsteady aerodynamics is represented by two coupled nonlinear stochastic ordinary differential equations containing uncertain parameters and model approximation errors several plausible aerodynamic models with increasing complexity are proposed to describe the aeroelastic system leading to lco the likelihood in the posterior parameter probability density function pdf is available semi analytically using the extended kalman filter for the state estimation of the coupled nonlinear structural and unsteady aerodynamic model the posterior parameter pdf is sampled using a parallel and adaptive markov chain monte carlo mcmc algorithm the posterior probability of each model is estimated using the chib jeliazkov method that directly uses the posterior mcmc samples for evidence marginal likelihood computation the bayesian algorithm is validated through a numerical study and then applied to model the nonlinear unsteady aerodynamic loads using wind tunnel test data at various reynolds numbers

an adaptive multi element generalized polynomial chaos method for stochastic differential equations we formulate a multi element generalized polynomial chaos me gpc method to deal with long term integration and discontinuities in stochastic differential equations we first present this method for legendre chaos corresponding to uniform random inputs and subsequently we generalize it to other random inputs the main idea of me gpc is to decompose the space of random inputs when the relative error in variance becomes greater than a threshold value in each subdomain or random element we then employ a generalized polynomial chaos expansion we develop a criterion to perform such a decomposition adaptively and demonstrate its effectiveness for odes including the kraichnan orszag three mode problem as well as advection diffusion problems the new method is similar to spectral element method for deterministic problems but with h p discretization of the random space

uncertainty quantification of limit cycle oscillations different computational methodologies have been developed to quantify the uncertain response of a relatively simple aeroelastic system in limit cycle oscillation subject to parametric variability the aeroelastic system is that of a rigid airfoil supported by pitch and plunge structural coupling with nonlinearities in the component in pitch the nonlinearities are adjusted to permit the formation of a either a subcritical or supercritical branch of limit cycle oscillations uncertainties are specified in the cubic coefficient of the torsional spring and in the initial pitch angle of the airfoil stochastic projections of the time domain and cyclic equations governing system response are carried out leading to both intrusive and non intrusive computational formulations non intrusive formulations are examined using stochastic projections derived from wiener expansions involving haar wavelet and b spline bases while wiener hermite expansions of the cyclic equations are employed intrusively and non intrusively application of the b spline stochastic projection is extended to the treatment of aerodynamic nonlinearities as modeled through the discrete euler equations the methodologies are compared in terms of computational cost convergence properties ease of implementation and potential for application to complex aeroelastic systems

commutators and companion matrices over rings of stable rank 1 abstract we consider the group gl n a of all invertible n by n matrices over a ring a satisfying the first bass stable range condition we prove that every matrix is similar to the product of a lower and upper triangular matrix and that it is also the product of two matrices each similar to a companion matrix we use this to show that when n 3 and a is commutative every matrix in sl n a is the product of two commutators

random rectangular graphs a generalization of the random geometric graph rgg model is proposed by considering a set of points uniformly and independently distributed on a rectangle of unit area instead of on a unit square 0 1 2 the topological properties of the random rectangular graphs rrgs generated by this model are then studied as a function of the rectangle sides lengths a and b 1 a and the radius r used to connect the nodes when a 1 we recover the rgg and when a the very elongated rectangle generated resembles a one dimensional rgg we obtain here analytical expressions for the average degree degree distribution connectivity average path length and clustering coefficient for rrg these results provide evidence that show that most of these properties depend on the connection radius and the side length of the rectangle usually in a monotonic way the clustering coefficient however increases when the square is transformed into a slightly elongated rectangle and after this maximum it decays with the increase of the elongation of the rectangle we support all our findings by computational simulations that show the goodness of the theoretical models proposed for rrgs

the operator factorization problems we survey various results concerning operator factorization problems more precisely we consider the following setting let h be a complex hilbert space and let g h be the algebra of all bounded linear operators on h for a given subset of 3 h we are interested in the characterization of operators in q h which are expressible as a product of finitely many operators in v and for each such operator the minimal number of factors in a factorization the classes of operators we consider include normal operators involutions partial isometries together with their various subclasses and other miscellaneous classes of operators most of the known results are for operators on finite dimensional spaces or finite matrices the paper concludes with some applications due to hochwald concerning the uniqueness of the adjoint operation on operators

random spherical graphs this work addresses a modification of the random geometric graph rgg model by considering a set of points uniformly and independently distributed on the surface of a d 1 sphere with radius r in a d dimensional euclidean space instead of on an unit hypercube 0 1 d then two vertices are connected by a link if their great circle distance is at most s in the case of d 3 the topological properties of the random spherical graphs rsgs generated by this model are studied as a function of s we obtain analytical expressions for the average degree degree distribution connectivity average path length diameter and clustering coefficient for rsgs by setting r sqrt pi 2 pi we also show the differences between the topological properties of rsgs and those of two dimensional rggs and random rectangular graphs rrgs surprisingly in terms of the average clustering coefficient rsgs look more similar to the analytical estimation for rggs than rggs themselves when their boundary effects are considered we support all our findings by computer simulations that corroborate the accuracy of the theoretical models proposed for rsgs

connectivity distribution of spatial networks we study spatial networks constructed by randomly placing nodes on a manifold and joining two nodes with an edge whenever their distance is less than a certain cutoff we derive the general expression for the connectivity distribution of such networks as a functional of the distribution of the nodes we show that for regular spatial densities the corresponding spatial network has a connectivity distribution decreasing faster than an exponential in contrast we also show that scale free networks with a power law decreasing connectivity distribution are obtained when a certain information measure of the node distribution integral of higher powers of the distribution diverges we illustrate our results on a simple example for which we present simulation results finally we speculate on the role played by the limiting case p k proportional variant k 1 which appears empirically to be relevant to spatial networks of biological origin such as the ones constructed from gene expression data

non intrusive generalized polynomial chaos for the robust stability analysis of uncertain nonlinear dynamic friction systems this paper is devoted to the stability analysis of uncertain nonlinear dynamic dry friction systems the stability property of dry friction systems is known to be very sensitive to the variations of friction laws moreover the friction coefficient admits dispersions due to the manufacturing processes therefore it becomes necessary to take this uncertainty into account in the stability analysis of dry friction systems to ensure robust predictions of stable and instable behaviours the generalized polynomial chaos formalism is proposed to deal with this challenging problem treated in most cases with the prohibitive monte carlo based techniques two equivalent methods presented here combine the non intrusive generalized polynomial chaos with the indirect lyapunov method both methods are shown to be efficient in the estimation of the stability and instability regions with high accuracy and high confidence levels and at lower cost compared with the classic monte carlo based method

factorization of transformations over a local ring abstract let v be a finitely generated free module over a local ring r and an invertible linear transformation for v then is a product of simple mappings in addition the determinant of each simple factor but one can be chosen to be any given unit in r the smallest number of factors needed is not greater than r 1 where r is the codimension of the vector space that is associated with the module of vectors fixed under

adaptive surrogate modeling by anova and sparse polynomial dimensional decomposition for global sensitivity analysis in fluid simulation the polynomial dimensional decomposition pdd is employed in this work for the global sensitivity analysis and uncertainty quantification uq of stochastic systems subject to a moderate to large number of input random variables due to the intimate connection between the pdd and the analysis of variance anova approaches pdd is able to provide a simpler and more direct evaluation of the sobol sensitivity indices when compared to the polynomial chaos expansion pc unfortunately the number of pdd terms grows exponentially with respect to the size of the input random vector which makes the computational cost of standard methods unaffordable for real engineering applications in order to address the problem of the curse of dimensionality this work proposes essentially variance based adaptive strategies aiming to build a cheap meta model i e surrogate model by employing the sparse pdd approach with its coefficients computed by regression three levels of adaptivity are carried out in this paper 1 the truncated dimensionality for anova component functions 2 the active dimension technique especially for second and higher order parameter interactions and 3 the stepwise regression approach designed to retain only the most influential polynomials in the pdd expansion during this adaptive procedure featuring stepwise regressions the surrogate model representation keeps containing few terms so that the cost to resolve repeatedly the linear systems of the least squares regression problem is negligible the size of the finally obtained sparse pdd representation is much smaller than the one of the full expansion since only significant terms are eventually retained consequently a much smaller number of calls to the deterministic model is required to compute the final pdd coefficients

spectral element method in time for rapidly actuated systems in this paper the spectral element se method is applied in time to find the entire time periodic or transient solution of time dependent differential equations the time periodic solution is computed by enforcing periodicity of the element set of particular interest are periodic forcing functions possessing high frequency content to maintain the spectral accuracy for such forcing functions an h refinement scheme is employed near the semi discontinuity without increasing the number of degrees of freedom time discretization by spectral elements is applied initially to a standard form of a set of linear first order differential equations subject to harmonic excitation and an excitation admitting rapid variation other case studies include the application of the se approach to parabolic and hyperbolic partial differential equations the first order form of these equations is obtained through semi discretization using conventional finite element spectral element and finite difference schemes element clustering h refinement is applied to maintain the high accuracy and efficiency in the region of the forcing function admitting rapid variation the convergence in time of the method is demonstrated in some cases machine precision is obtained with 25 degrees of freedom per cycle finally the method is applied to a weakly nonlinear problem with time periodic solution to demonstrate its future applicability to the analysis of limit cycle oscillations in aeroelastic systems

an adaptive least squares global sensitivity method and application to a plasma coupled combustion prediction with parametric correlation abstract we introduce an efficient non intrusive surrogate based methodology for global sensitivity analysis and uncertainty quantification modified covariance based sensitivity indices mcov si are defined for outputs that reflect correlated effects the overall approach is applied to simulations of a complex plasma coupled combustion system with disparate uncertain parameters in sub models for chemical kinetics and a laser induced breakdown ignition seed the surrogate is based on an analysis of variance anova expansion such as widely used in statistics with orthogonal polynomials representing the anova subspaces and a polynomial dimensional decomposition pdd representing its multi dimensional components the coefficients of the pdd expansion are obtained using a least squares regression which both avoids the direct computation of high dimensional integrals and affords an attractive flexibility in choosing sampling points this facilitates importance sampling using a bayesian calibrated posterior distribution which is fast and thus particularly advantageous in common practical cases such as our large scale demonstration for which the asymptotic convergence properties of polynomial expansions cannot be realized due to computation expense effort instead is focused on efficient finite resolution sampling standard covariance based sensitivity indices cov si are employed to account for correlation of the uncertain parameters magnitude of cov si is unfortunately unbounded which can produce extremely large indices that limit their utility alternatively mcov si are then proposed in order to bound this magnitude 0 1 the polynomial expansion is coupled with an adaptive anova strategy to provide an accurate surrogate as the union of several low dimensional spaces avoiding the typical computational cost of a high dimensional expansion it is also adaptively simplified according to the relative contribution of the different polynomials to the total variance the approach is demonstrated for a laser induced turbulent combustion simulation model which includes parameters with correlated effects

numerical analysis of the burgers equation in the presence of uncertainty the burgers equation with uncertain initial and boundary conditions is investigated using a polynomial chaos pc expansion approach where the solution is represented as a truncated series of stochastic orthogonal polynomials the analysis of well posedness for the system resulting after galerkin projection is presented and follows the pattern of the corresponding deterministic burgers equation the numerical discretization is based on spatial derivative operators satisfying the summation by parts property and weak boundary conditions to ensure stability similarly to the deterministic case the explicit time step for the hyperbolic stochastic problem is proportional to the inverse of the largest eigenvalue of the system matrix the time step naturally decreases compared to the deterministic case since the spectral radius of the continuous problem grows with the number of polynomial chaos coefficients an estimate of the eigenvalues is provided a characteristic analysis of the truncated pc system is presented and gives a qualitative description of the development of the system over time for different initial and boundary conditions it is shown that a precise statistical characterization of the input uncertainty is required and partial information e g the expected values and the variance are not sufficient to obtain a solution an analytical solution is derived and the coefficients of the infinite pc expansion are shown to be smooth while the corresponding coefficients of the truncated expansion are discontinuous

random geometric graphs with general connection functions in the original 1961 gilbert model of random geometric graphs nodes are placed according to a poisson point process and links formed between those within a fixed range motivated by wireless ad hoc networks soft or probabilistic connection models have recently been introduced involving a connection function h r that gives the probability that two nodes at distance r are linked directly connect in many applications not only wireless networks it is desirable that the graph is connected that is every node is linked to every other node in a multihop fashion here the connection probability of a dense network in a convex domain in two or three dimensions is expressed in terms of contributions from boundary components for a very general class of connection functions it turns out that only a few quantities such as moments of the connection function appear good agreement is found with special cases from previous studies and with numerical simulations

on products of matrices of a fixed order abstract we study products of matrices of fixed orders we show that if g is an upper triangular matrix finite or infinite over a field of q elements then g can be expressed as a product of at most four triangular matrices whose orders are divisors of q 1 this result can be applied to the general linear and to the vershik kerov group we also present some facts about conjugacy of elements of orders dividing q 1

stochastic nonlinear aeroelastic analysis of a supersonic lifting surface using an adaptive spectral method an adaptive stochastic spectral projection method is deployed for the uncertainty quantification in limit cycle oscillations of an elastically mounted two dimensional lifting surface in a supersonic flow field variabilities in the structural parameters are propagated in the aeroelastic system which accounts for nonlinear restoring force and moment by means of hardening cubic springs the physical nonlinearities promote sharp and sudden flutter onset for small change of the reduced velocity in a stochastic context this behavior translates to steep solution gradients developing in the parametric space a remedy is to expand the stochastic response of the airfoil on a piecewise generalized polynomial chaos basis accurate approximation andaffordable computational costs are obtained using sensitivity based adaptivity for various types of supersonic stochastic responses depending on the selected values of the mach number on the bifurcation map sensitivity analysis via sobol indices shows how the probability density function of the peak pitch amplitude responds to combined uncertainties e g the elastic axis location torsional stiffness and flap angle we believe that this work demonstrates the capability and flexibility of the approach for more reliable predictions of realistic aeroelastic systems subject to a moderate number of uncertainties

on products of three triangular matrices over associative rings abstract in this paper we characterize an associative ring over which any n n n 2 square matrix a can be written as lum where l m are lower triangular u is upper triangular and l and u have all their diagonal entries 1

the wiener askey polynomial chaos for stochastic differential equations we present a new method for solving stochastic differential equations based on galerkin projections and extensions of wiener s polynomial chaos specifically we represent the stochastic processes with an optimum trial basis from the askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error several continuous and discrete processes are treated and numerical examples show substantial speed up compared to monte carlo simulations for low dimensional stochastic inputs

hopf bifurcation analysis of an aeroelastic model using stochastic normal form abstract we investigate the effects of parameter uncertainties on the dynamical response of an aeroelastic model representing an oscillating airfoil in pitch and plunge with linear aerodynamics and cubic structural nonlinearities an approach based on the stochastic normal form is proposed to determine the effects due to the variations in the flow speed and the structural stiffness terms on the stability of the aeroelastic system near the hopf bifurcation point this approach allows us to study analytically the bifurcation scenario and to predict the amplitude and frequency of the limit cycle oscillation lco the results show that the amplitude of lco corresponding to the supercritical hopf bifurcation increases with the intensity of the noise perturbing the pitch and plunge cubic terms but there is almost no effect on the lco frequency uncertainties in the flow speed produce a shift in the bifurcation point and unstable subcritical behavior may occur for values of parameters for which the corresponding deterministic model is stable the stochastic normal form confirms and extends previously known numerical results regarding the effect of parameter variations and offers an effective way to perform sensitivity analysis of the system s response

degree correlations in random geometric graphs spatially embedded networks are important in several disciplines the prototypical spatial network we assume is the random geometric graph of which many properties are known here we present new results for the two point degree correlation function in terms of the clustering coefficient of the graphs for two dimensional space in particular with extensions to arbitrary finite dimensions

stochastic investigation of flows about airfoils at transonic speeds aerodynamics nonlinearities are reported in the uncertain probabilistic space the efficiency of the present methodology are evaluated for the propagation of random disturbances associated with the angle of attack and the freestreammachnumber anerroranalysisiscarriedoutinordertodetermineappropriatephysicalandstochastic discretization levels different stochastic flow regimes are analyzed in details by means of various postprocessing procedures including error bars probabilistic density function of the aerodynamic field and sobol s coefficients

stochastic collocation method for secondary bifurcation of a nonlinear aeroelastic system abstract a stochastic collocation method is proposed to investigate the secondary bifurcation of a two dimensional aeroelastic system with structural nonlinearity represented by cubic restoring forces and uncertainties expressed by random parameters in the cubic stiffness coefficient and in the initial pitch angle the accuracy of the stochastic collocation method is improved by incorporating higher order schemes such as piecewise cubic interpolation and piecewise cubic spline interpolation instead of a piecewise linear interpolation formula for an aeroelastic problem with the uncertainty expressed by a time dependent combination of five random variables an efficient collocation method is developed using a sparse grid approach with a dimension adaptive strategy numerical simulations are carried out to demonstrate the effectiveness of the proposed method for long term computation and discontinuous problems

enforcing positivity in intrusive pc uq methods for reactive ode systems we explore the relation between the development of a non negligible probability of negative states and the instability of numerical integration of the intrusive galerkin ordinary differential equation system describing uncertain chemical ignition to prevent this instability without resorting to either multi element local polynomial chaos pc methods or increasing the order of the pc representation in time we propose a procedure aimed at modifying the amplitude of the pc modes to bring the probability of negative state values below a user defined threshold this modification can be effectively described as a filtering procedure of the spectral pc coefficients which is applied on the fly during the numerical integration when the current value of the probability of negative states exceeds the prescribed threshold we demonstrate the filtering procedure using a simple model of an ignition process in a batch reactor this is carried out by comparing different observables and error measures as obtained by non intrusive monte carlo and gauss quadrature integration and the filtered intrusive procedure the filtering procedure has been shown to effectively stabilize divergent intrusive solutions and also to improve the accuracy of stable intrusive solutions which are close to the stability limits

uncertainty analysis near bifurcation of an aeroelastic system variations in structural and aerodynamic nonlinearities on the dynamic behavior of an aeroelastic system are investigated the aeroelastic system consists of a rigid airfoil that is supported by nonlinear springs in the pitch and plunge directions and subjected to nonlinear aerodynamic loads we follow two approaches to determine the effects of variations in the linear and nonlinear plunge and pitch stiffness coefficients of this aeroelastic system on its stability near the bifurcation the first approach is based on implementation of intrusive polynomial chaos expansion pce on the governing equations yielding a set of nonlinear coupled ordinary differential equations that are numerically solved the results show that this approach is capable of determining sensitivity of the flutter speed to variations in the linear pitch stiffness coefficient on the other hand it fails to predict changes in the type of the instability associated with randomness in the cubic stiffness coefficient in the second approach the normal form is used to investigate the flutter hopf bifurcation boundary that occurs as the freestream velocity is increased and to analytically predict the amplitude and frequency of the ensuing lco the results show that this mathematical approach provides detailed aspects of the effects of the different system nonlinearities on its dynamic behavior furthermore this approach could be effectively used to perform sensitivity analysis of the system s response to variations in its parameters

parabolic subgroups of groups of column finite infinite matrices abstract for a wide class of rings r that contains all local and semilocal rings we consider the group sl vk r of all matrices over r of the form a b 0 c in which a is a finite matrix with determinant 1 and c is an upper triangular infinite matrix with the main diagonal 1 1 we describe all subgroups of sl vk r that contain all its upper triangular matrices and study their properties

effect of stiffness uncertainties on the flutter of a cantilever wing this paper deals with the investigation of the influence of spanwise distribution of bending and torsion stiffness uncertainties on the flutter behavior of an aeroelastic wing using a stochastic finite element approach the analysis adopted a numerical algorithm to simulate unsteady nonlinear incompressible flow based on the unsteady vortex lattice method interacting with linear aeroelastic structure in the absence of uncertainties the airflow and wing structure are treated as elements of a single dynamic system parameter uncertainties are represented by a truncated karhunen love expansion both perturbation technique and monte carlo simulation are used to establish the boundary of stiffness uncertainty level at which the wing exhibits flutter in the form of limit cycle oscillations and above which the wing experiences dynamic instability the analysis also includes the limitation of perturbation solution for a relatively large level of stiffness uncertainty

a newton galerkin method for fluid flow exhibiting uncertain periodic dynamics the determination of limit cycles plays an important role in characterizing complex dynamical systems such as unsteady fluid flows in practice dynamical systems are described by models equations involving parameters which are seldom exactly known leading to parametric uncertainties these parameters can be suitably modeled as random variables so if the system possesses almost surely a stable time periodic solution limit cycles become stochastic too this paper introduces a novel numerical method for the computation of stable stochastic limit cycles based on the spectral stochastic finite element method with polynomial chaos pc expansions the method is designed to overcome the limitation of pc expansions related to convergence breakdown for long term integration first a stochastic time scaling of the model equations is determined to control the phase drift of the stochastic trajectories and allowing for accurate low order pc expansions second using the rescaled governing equations we aim at

random geometric graphs in high dimension many machine learning algorithms used for dimensional reduction and manifold learning leverage on the computation of the nearest neighbors to each point of a data set to perform their tasks these proximity relations define a so called geometric graph where two nodes are linked if they are sufficiently close to each other random geometric graphs where the positions of nodes are randomly generated in a subset of r d offer a null model to study typical properties of data sets and of machine learning algorithms up to now most of the literature focused on the characterization of low dimensional random geometric graphs whereas typical data sets of interest in machine learning live in high dimensional spaces d 10 2 in this work we consider the infinite dimensions limit of hard and soft random geometric graphs and we show how to compute the average number of subgraphs of given finite size k e g the average number of k cliques this analysis highlights that local observables display different behaviors depending on the chosen ensemble soft random geometric graphs with continuous activation functions converge to the naive infinite dimensional limit provided by erdos renyi graphs whereas hard random geometric graphs can show systematic deviations from it we present numerical evidence that our analytical results exact in infinite dimensions provide a good approximation also for dimension d 10

spectral and multiresolution wiener expansions of oscillatory stochastic processes abstract wiener chaos expansions are being evaluated for the representation of stochastic variability in the response of nonlinear aeroelastic systems which often exhibit limit cycles preliminary studies with a simple nonlinear aeroelastic computational model have shown that the standard non intrusive wiener hermite expansion fails to maintain time accuracy as the simulation evolves wiener hermite expansions faithfully reproduce the short term characteristics of the process but consistently lose energy after several mean periods of oscillation this energy loss remains even for very high order expansions to uncover the cause of this energy loss and to explore potential remedies the more elementary problem of a sinusoid with random frequency is used herein to simulate the periodic response of an uncertain system as time progresses coefficients of the higher order terms in both the wiener hermite and wiener legendre expansions successively gain and lose dominance over the lower order coefficients in a manner that causes any fixed order expansion in terms of global basis functions to fail over a simulation time of sufficient duration this characteristic behavior is attributed to the continually increasing frequency of the process in the random dimension the recently developed wiener haar expansion is found to almost entirely eliminate the loss of energy at large times both for the sinusoidal process and for the response of a two degree of freedom nonlinear system which is examined as a prelude to the stochastic simulation of aeroelastic limit cycles it is also found that mallat s pyramid algorithm is more efficient and accurate for evaluating wiener haar expansion coefficients than monte carlo simulation or numerical quadrature

effect of randomness on multi frequency aeroelastic responses resolved by unsteady adaptive stochastic finite elements the unsteady adaptive stochastic finite elements uasfe method resolves the effect of randomness in numerical simulations of single mode aeroelastic responses with a constant accuracy in time for a constant number of samples in this paper the uasfe framework is extended to multi frequency responses and continuous structures by employing a wavelet decomposition pre processing step to decompose the sampled multi frequency signals into single frequency components the effect of the randomness on the multi frequency response is then obtained by summing the results of the uasfe interpolation at constant phase for the different frequency components results for multi frequency responses and continuous structures show a three orders of magnitude reduction of computational costs compared to crude monte carlo simulations in a harmonically forced oscillator a flutter panel problem and the three dimensional transonic agard 445 6 wing aeroelastic benchmark subject to random fields and random parameters with various probability distributions

derper an algorithm for the continuation of periodic solutions in ordinary differential equations abstract an algorithm for the continuation of periodic solutions of ode was derived and tested the algorithm is based on the shooting method and a standard continuation algorithm variational variables are used to compute the jacobi matrix the starting point on the periodic orbit is adaptively changed in the algorithm by fixing one component of the solution stability of periodic solutions along the continuous branch of solutions is determined by computing characteristic multipliers without difficulty the algorithm crosses limit points and at bifurcation points double period bifurcation points and points of tori bifurcation it proceeds along the original branch of solutions the usefulness of the algorithm is demonstrated in two examples one of them describes two stirred tank reactors with mutual mass exchange the other one is the lorenz model of turbulence

sur la simplicite essentielle du groupe des inversibles et du groupe unitaire dans une c algebre simple let a be a simple approximately finite dimensional c algebra with unit let gl1 a be the group of invertible elements and let u1 a be that of unitaries in a in a previous work the abelianized groups gl1 a dgl1 a and u1 a du1 a have been described with grothendieck s group k0 a viewed as a dimension group for a here it is shown that the groups dgl1 a and du1 a are simple up to their centres the case of a stable c algebra and that of a c algebra with unit which is simple infinite are also studied in the last two cases moreover it is shown that there exists a natural homomorphism from milnor s group k2mil a onto k0 a

pairs of involutions in the general linear group in theorem 1 we classify all finite dimensional indecomposable representations of the infinite dihedral group over an arbitrary field k of char k 2 of course that result is equivalent to the problem of finding a normal form for a pair of involutions in gln k in theorem 2 we find necessary and sufficient conditions for a pair of conjugate involutions in gln k to be conjugate via an involution where k is assumed to be algebraically closed and char k 2

impact of boundaries on fully connected random geometric networks many complex networks exhibit a percolation transition involving a macroscopic connected component with universal features largely independent of the microscopic model and the macroscopic domain geometry in contrast we show that the transition to full connectivity is strongly influenced by details of the boundary but observe an alternative form of universality our approach correctly distinguishes connectivity properties of networks in domains with equal bulk contributions it also facilitates system design to promote or avoid full connectivity for diverse geometries in arbitrary dimension

on a question of m newman on the number of commutators abstract m newman asked whether there is an absolute constant c such that every matrix in slnr is the product of at most c commutators where r ranges over euclidean commutative rings and n 3 we give here a negative answer however if for the ring r every matrix in slmr is the product of a bounded number of commutators for some fixed m 3 then for all sufficiently large n every matrix in slnr is the product of six commutators

products of three triangular matrices abstract we prove that any square matrix over a field is the product of at most three triangular matrices we also give explicit lul factorizations for all 2 2 and 3 3 matrices over fields

an alternative unsteady adaptive stochastic finite elements formulation based on interpolation at constant phase abstract the unsteady adaptive stochastic finite elements method based on time independent parametrization uasfe ti is an efficient approach for resolving the effect of random parameters in unsteady simulations it achieves a constant accuracy in time with a constant number of samples in contrast with the usually fast increasing number of samples required by other methods in this paper an alternative unsteady adaptive stochastic finite elements formulation based on interpolation at constant phase uasfe cp is developed to further improve the accuracy and extend the applicability of uasfe ti in addition to achieving a constant number of samples in time interpolation at constant phase 1 eliminates the parametrization error of the time independent parametrization 2 resolves time dependent functionals which cannot be modeled by the parametrization and 3 captures transient behavior of the samples which is an important special case of time dependent functionals these three points are illustrated by the application of uasfe cp to random parameters in a mass spring damper system the damped nonlinear duffing oscillator and an elastically mounted airfoil with nonlinearity in the flow and the structure results for different types of probability distributions are compared to those of uasfe ti and monte carlo simulations