dimension reduction and coefficient estimation in multivariate linear regression summary we introduce a general formulation for dimension reduction and coefficient estimation in the multivariate linear model we argue that many of the existing methods that are commonly used in practice can be formulated in this framework and have various restrictions we continue to propose a new method that is more flexible and more generally applicable the method proposed can be formulated as a novel penalized least squares estimate the penalty that we employ is the coefficient matrix s ky fan norm such a penalty encourages the sparsity among singular values and at the same time gives shrinkage coefficient estimates and thus conducts dimension reduction and coefficient estimation simultaneously in the multivariate linear model we also propose a generalized cross validation type of criterion for the selection of the tuning parameter in the penalized least squares simulations and an application in financial econometrics demonstrate competitive performance of the new method an extension to the non parametric factor model is also discussed

joint continuum regression for multiple predictands abstract this article generalizes continuum regression cr in the hope that regressors jointly constructed for several predictands might improve on the separate prediction of individual predictands the generalization developed is a mixture of principal components regression and de jong s modification of partial least squares for multiple predictands the balance of ingredients can be chosen by cross validation as can the number of regressors constructed the new method has been tested on real and simulated data the indications are that conditions for the superiority of the joint approach may be rare in practice

convex optimization methods for dimension reduction and coefficient estimation in multivariate linear regression in this paper we study convex optimization methods for computing the nuclear or trace norm regularized least squares estimate in multivariate linear regression the so called factor estimation and selection method recently proposed by yuan et al j royal stat soc ser b statistical methodology 69 3 329 346 2007 conducts parameter estimation and factor selection simultaneously and have been shown to enjoy nice properties in both large and finite samples to compute the estimates however can be very challenging in practice because of the high dimensionality and the nuclear norm constraint in this paper we explore a variant due to tseng of nesterov s smooth method and interior point methods for computing the penalized least squares estimate the performance of these methods is then compared using a set of randomly generated instances we show that the variant of nesterov s smooth method generally outperforms the interior point method implemented in sdpt3 version 4 0 beta toh et al on the implementation and usage of sdpt3 a matlab software package for semidefinite quadratic linear programming version 4 0 manuscript department of mathematics national university of singapore 2006 substantially moreover the former method is much more memory efficient

smooth optimization approach for sparse covariance selection in this paper we first study a smooth optimization approach for solving a class of nonsmooth strictly concave maximization problems whose objective functions admit smooth convex minimization reformulations in particular we apply nesterov s smooth optimization technique y e nesterov dokl akad nauk sssr 269 1983 pp 543 547 y e nesterov math programming 103 2005 pp 127 152 to their dual counterparts that are smooth convex problems it is shown that the resulting approach has cal o 1 sqrt epsilon iteration complexity for finding an epsilon optimal solution to both primal and dual problems we then discuss the application of this approach to sparse covariance selection that is approximately solved as an l 1 norm penalized maximum likelihood estimation problem and also propose a variant of this approach which has substantially outperformed the latter one in our computational experiments we finally compare the performance of these approaches with other first order methods namely nesterov s cal o 1 epsilon smooth approximation scheme and block coordinate descent method studied in a d aspremont o banerjee and l el ghaoui siam j matrix anal appl 30 2008 pp 56 66 j friedman t hastie and r tibshirani biostatistics 9 2008 pp 432 441 for sparse covariance selection on a set of randomly generated instances it shows that our smooth optimization approach substantially outperforms the first method above and moreover its variant substantially outperforms both methods above

an optimal method for stochastic composite optimization this paper considers an important class of convex programming cp problems namely the stochastic composite optimization sco whose objective function is given by the summation of general nonsmooth and smooth stochastic components since sco covers non smooth smooth and stochastic cp as certain special cases a valid lower bound on the rate of convergence for solving these problems is known from the classic complexity theory of convex programming note however that the optimization algorithms that can achieve this lower bound had never been developed in this paper we show that the simple mirror descent stochastic approximation method exhibits the best known rate of convergence for solving these problems our major contribution is to introduce the accelerated stochastic approximation ac sa algorithm based on nesterov s optimal method for smooth cp nesterov in doklady an sssr 269 543 547 1983 nesterov in math program 103 127 152 2005 and show that the ac sa algorithm can achieve the aforementioned lower bound on the rate of convergence for sco to the best of our knowledge it is also the first universally optimal algorithm in the literature for solving non smooth smooth and stochastic cp problems we illustrate the significant advantages of the ac sa algorithm over existing methods in the context of solving a special but broad class of stochastic programming problems

heuristics of instability and stabilization in model selection in model selection usually a best predictor is chosen from a collection hat mu cdot s of predictors where hat mu cdot s is the minimum least squares predictor in a collection mathsf u s of predictors here s is a complexity parameter that is the smaller s the lower dimensional smoother the models in mathsf u s if mathsf l is the data used to derive the sequence hat mu cdot s the procedure is called unstable if a small change in mathsf l can cause large changes in hat mu cdot s with a crystal ball one could pick the predictor in hat mu cdot s having minimum prediction error without prescience one uses test sets cross validation and so forth the difference in prediction error between the crystal ball selection and the statistician s choice we call predictive loss for an unstable procedure the predictive loss is large this is shown by some analytics in a simple case and by simulation results in a more complex comparison of four different linear regression methods unstable procedures can be stabilized by perturbing the data getting a new predictor sequence hat mu cdot s and then averaging over many such predictor sequences

exponential stability for a class of boundary conditions on a euler bernoulli beam subject to disturbances via boundary control abstract this paper outlines the procedure for applying a sliding mode backstepping boundary control technique to a euler bernoulli beam subject to unknown bounded disturbances on the boundary for four different types of boundary conditions referred to in this paper as a class of boundary conditions these boundary conditions include all combinations of pinned and sliding types with control and disturbances on one boundary i e pinned pinned pinned sliding sliding pinned and sliding sliding with input on the second boundary the technique was developed for the specific case of a pinned pinned beam in the literature and in this paper the technique is generalized to the degree that it is possible to cover all four of the aforementioned cases furthermore a proof of the fully arbitrary exponential stability of the closed loop system is provided in this paper to this point only asymptotic stability was shown for the closed loop system for one case the technique outlined in this paper provides for a class of boundary conditions a controller that exponentially stabilizes the vibrations in a beam in the presence of bounded unknown disturbances the beam displacement returns to the origin and its decay is bounded by a known exponential decay function after a known finite time reaching phase

cohomological aspects of hypergraphs by a k graph we will mean a collection of k element subsets of some fixed set v a k graph can be regarded as a k 1 chain on 2 v the simplicial complex of all subsets of v over the coefficient group z 2 the additive group of integers modulo 2 the induced group structure on the k 1 chains leads to natural definitions of the coboundary of a chain the cochain complex of c c k and the usual cohomology groups h k c z 2 in particular it is possible to construct what could be called higher order coboundary operators i where i increases dimension by i rather than just 1

consistency of trace norm minimization regularization by the sum of singular values also referred to as the trace norm is a popular technique for estimating low rank rectangular matrices in this paper we extend some of the consistency results of the lasso to provide necessary and sufficient conditions for rank consistency of trace norm minimization with the square loss we also provide an adaptive version that is rank consistent even when the necessary condition for the non adaptive version is not fulfilled

approximating semidefinite packing programs in this paper we define semidefinite packing programs and describe an algorithm to approximately solve these problems semidefinite packing programs arise in many applications such as semidefinite programming relaxations for combinatorial optimization problems sparse principal component analysis and sparse variance unfolding techniques for dimension reduction our algorithm exploits the structural similarity between semidefinite packing programs and linear packing programs

invariant boundary value problems for a fourth order dynamic euler bernoulli beam equation we obtain the complete lie symmetry group classification of the dynamic fourth order euler bernoulli partial differential equation where the elastic modulus the area moment of inertia are constants and the applied load is a nonlinear function in the lie analysis the principal lie algebra which is two dimensional extends in three cases viz the linear the exponential and the general power law for each of the nontrivial cases we determine symmetry reductions to ordinary differential equations which are of order four in only one case related to the power law we are able to have a compatible initial boundary value problem for a clamped end and a free beam for these cases we deduce the corresponding fourth order ordinary differential equations with appropriate boundary conditions we provide an asymptotic solution for the reduced fourth order ordinary differential equation corresponding to a clamped or free beam

trace norm regularization reformulations algorithms and multi task learning we consider a recently proposed optimization formulation of multi task learning based on trace norm regularized least squares while this problem may be formulated as a semidefinite program sdp its size is beyond general sdp solvers previous solution approaches apply proximal gradient methods to solve the primal problem we derive new primal and dual reformulations of this problem including a reduced dual formulation that involves minimizing a convex quadratic function over an operator norm ball in matrix space this reduced dual problem may be solved by gradient projection methods with each projection involving a singular value decomposition the dual approach is compared with existing approaches and its practical effectiveness is illustrated on simulations and an application to gene expression pattern analysis

reduced rank regression via adaptive nuclear norm penalization we propose an adaptive nuclear norm penalization approach for low rank matrix approximation and use it to develop a new reduced rank estimation method for high dimensional multivariate regression the adaptive nuclear norm is defined as the weighted sum of the singular values of the matrix and it is generally non convex under the natural restriction that the weight decreases with the singular value however we show that the proposed non convex penalized regression method has a global optimal solution obtained from an adaptively soft thresholded singular value decomposition the method is computationally efficient and the resulting solution path is continuous the rank consistency of and prediction estimation performance bounds for the estimator are established for a high dimensional asymptotic regime simulation studies and an application in genetics demonstrate its efficacy

lyapunov approach to output feedback stabilization for the euler bernoulli beam equation with boundary input disturbance we propose a boundary output feedback control law for a one dimensional euler bernoulli beam equation with general external disturbance entering the control end a galerkin approximation scheme is constructed to show the existence of solution to the closed loop system the exponential stability of the closed loop system is obtained by the lyapunov functional method numerical simulations are presented for illustration

transient wave analysis of a cantilever timoshenko beam subjected to impact loading by laplace transform and normal mode methods abstract this study applies two analytical approaches laplace transform and normal mode methods to investigate the dynamic transient response of a cantilever timoshenko beam subjected to impact forces explicit solutions for the normal mode method and the laplace transform method are presented the durbin method is used to perform the laplace inverse transformation and numerical results based on these two approaches are compared the comparison indicates that the normal mode method is more efficient than the laplace transform method in the transient response analysis of a cantilever timoshenko beam whereas the laplace transform method is more appropriate than the normal mode method when analyzing the complicated multi span timoshenko beam furthermore a three dimensional finite element cantilever beam model is implemented the results are compared with the transient responses for displacement normal stress shear stress and the resonant frequencies of a timoshenko beam and bernoulli euler beam theories the transient displacement response for a cantilever beam can be appropriately evaluated using the timoshenko beam theory if the slender ratio is greater than 10 or using the bernoulli euler beam theory if the slender ratio is greater than 100 moreover the resonant frequency of a cantilever beam can be accurately determined by the timoshenko beam theory if the slender ratio is greater than 100 or by the bernoulli euler beam theory if the slender ratio is greater than 400

non linear coupled transverse and axial vibration of a compliant structure part 1 formulation and free vibration a compliant tower in the ocean environment is modelled as a beam undergoing coupled transverse and axial motion the equations of motion are non linear and coupled and are derived here the beam is assumed to be supported by a linear elastic torsional spring at one end and with a point mass at the other end such a model is representative of numerous applications the equations of motion and boundary conditions are obtained using hamilton s variational principle it is assumed that strains are small but the rotation is moderate compared to the strain so that the equations of motion for the axial and transverse motion are non linearly coupled the free response in vacua and the free response in water are considered in particular the fluid forces the added mass and drag forces are modelled using a semi empirical morison equation the resulting non linear coupled partial differential equations are solved numerically using the finite difference approach in part 2 of this work various forced responses are studied

variational derivation of governing differential equations for truncated version of bresse timoshenko beams abstract in this paper we provide a variational derivation of modified bresse timoshenko equations this process leads to the additional term to so called truncated set of bresse timoshenko equations this new term is associated with the modified slope inertia the two sets of truncated versions of the bresse timoshenko equations are contrasted with each other as well as with the original bresse timoshenko equations on the example of the a beam that is simply supported at its both ends and b the cantilever beam with end mass it is concluded that application of either truncated sets of equations is advantageous over the original bresse timoshenko equations as far as two truncated sets of equations are concerned the variationally derived set appears to be preferable

vibration confinement in a general beam structure during harmonic excitations abstract in the present paper the green s function is utilized to provide a simple exact and direct analytical method for analysis of a beam structure of generic boundary conditions with attached springs and or finite masses the aim is to confine the vibration in a certain part of the structure the beam equation is based on the timoshenko beam theory with corrections for shear deformation and rotary inertia effects in the analysis the beam is driven by a harmonic external excitation the attached springs are modeled as simple reactions that provide transverse forces to the beam while each added mass provides a transverse force in addition to a moment at its location these forces moments act as secondary forces moments that reduce the response caused by the external force numerical simulations are conducted to find the optimal masses and or springs that confine the vibration in a certain chosen region the results were compared to that obtained from euler bernoulli beam theory beams that are excited by a bi harmonic force as well as dual excitation forces are analyzed in addition the case when the beams are excited near resonances is discussed also a method is proposed to impose a node at any desired location along the structure

vibration of composite beams with deformable shear connection mathematical modelling and numerical solution using general purpose software for two point boundary value problems abstract in this paper we develop from first principles a linear one dimensional mathematical model for the dynamic response of planar composite beams with deformable shear connection in which the inertias associated with the axial motion of the components and with the bending rotations of their cross sections are consistently taken into account to the best of our knowledge the paper by auclair et al j sound vib 366 2016 230 247 is the only work which heretofore considered these two effects however we point out and correct a flaw in the derivations of these authors to solve the eigenproblem defining the natural vibration frequencies of the composite beam and the corresponding vibration modes we advocate an untried approach the eigenproblem is converted into a standard inhomogeneous boundary value problem which can be solved using freely available state of the art mathematical software for benchmarking purposes the formulation of a conventional displacement based conforming finite element is also briefly addressed the effect of rotational and axial inertias on the natural vibration frequencies is found to be marginal and considerably less pronounced than the predictions of auclair and coworkers in disagreement with the findings of these authors it appears legitimate to ignore the above mentioned effect in a full dynamic analysis of composite beams at least within the practical range of span to depth ratios

material point method enhanced by modified gradient of shape function a numerical scheme of computing quantities involving gradients of shape functions is introduced for the material point method mpm so that the quantities are continuous as material points move across cell boundaries the noise and instability caused by cell crossing of the material points are then eliminated in this scheme the formulas used to compute these quantities can be expressed in the same forms as in the original material point method but with the gradient of the shape function modified for one dimensional cases the gradient of the shape function used in the generalized interpolation material point gimp method is a special case of the modified gradient if the characteristic function of a material point is introduced the characteristic function of a material point is not otherwise needed in this scheme therefore difficulties in tracking its evolution are avoided although the support of the modified gradient of a shape function is enlarged from the cell containing the material point to also include the immediate neighbor cells all the non local effects of a material point can be accounted for by two consecutive local operations therefore this scheme can be used in calculations with unstructured grids this scheme is proved to satisfy mass and momentum conservations exactly the error in energy conservation is shown to be second order on both spatial and temporal discretizations although the error in energy conservation is the same order as that in the original material point method numerical examples show that this scheme has significantly better energy conservation properties than those of the original material point method

modes for the coupled timoshenko model with a restrained end abstract the modes of the second order timoshenko system for the displacement and rotation of a fixed beam with a restrained end at the left are formulated in terms of a fundamental spatial response this is done without decoupling the system into fourth order scalar equations the restrained end leads to time space boundary conditions which introduce the frequency as a parameter into the system of equations for determining the modes these equations involve first order derivatives and consequently the modes are determined by solving a non conservative differential system this modal differential equation is discussed in terms of a fundamental matrix response it is determined by applying a closed formula that was obtained by the first author and involves the characteristic polynomial of the modal differential equation

a new rectangular beam theory abstract a new theory for beams of rectangular cross section which includes warping of the cross sections is presented in the present work by satisfying the shear free conditions on the lateral surfaces of the beam a pair of coupled equations of motion are obtained such that no arbitrary shear coefficient is required it is shown that the uncoupled equation for the transverse displacement is the same as the corresponding equation in timoshenko beam theory provided that for the timoshenko equation the shear coefficient is taken to be 5 6 this value lies within the range of values 0 822 0 870 appearing in the literature for the beam of rectangular cross section results for two typical static examples are given for both the new theory and timoshenko beam theory these results are compared with the solutions of the comparable problems in the linear theory of elasticity for the end loaded cantilever beam the new theory predicts the same result for the neutral surface deflection as does the linear theory of elasticity while timoshenko beam theory underestimates the shear correction term by 20 for the uniformly loaded and simply supported case both beam theories provide the same overestimate of the central deflection when compared with the theory of elasticity solution

the equivalence problem for the euler bernoulli beam equation via cartan s method we completely solve the local point equivalence problem for the euler bernoulli beam equation using cartan s method of equivalence we obtain five equivalence classes for each equivalence class we establish the necessary and sufficient conditions for similarity and derive a basis of differential invariants as well as operators of invariant differentiation

stability of a flexible structure with destabilizing boundary conditions the euler bernoulli beam model with non dissipative boundary conditions of feedback control type is investigated components of the two dimensional input vector are shear and moment at the right end and components of the observation vector are time derivatives of displacement and slope at the right end the codiagonal matrix depending on two control parameters relates input and observation the paper contains five results first asymptotic approximation for eigenmodes is derived second the main identity is established it provides a relation between mode shapes of two systems one with non zero control parameters and the other one with zero control parameters third when one control parameter is positive and the other one is zero the main identity yields stability of all eigenmodes though the system is non dissipative fourth the stability of eigenmodes is extended to the case when one control parameter is positive and the other one is sufficiently small finally existence and properties of deadbeat modes are investigated

the second frequency spectrum of timoshenko beams abstract a second spectrum of frequencies was reported in early analytical work on the vibrations of timoshenko beams however in subsequent finite element modelling this phenomenon was either ignored or not definitively classified and recorded in fact from a recent finite element analysis with a high precision element it was even concluded that there is no separate second spectrum of frequencies except for the special case of hinged hinged beams and it was asserted that previous investigators had misinterpreted some frequencies thus introducing the notion of second frequencies in this paper a simple linear beam element with independent displacement fields and reduced integration to eliminate shear locking is used and enables one to detect the second spectrum accurately guidelines are provided which help to identify and classify the frequencies into two separate spectra

distance shrinkage and euclidean embedding via regularized kernel estimation summary although recovering a euclidean distance matrix from noisy observations is a common problem in practice how well this could be done remains largely unknown to fill in this void we study a simple distance matrix estimate based on the so called regularized kernel estimate we show that such an estimate can be characterized as simply applying a constant amount of shrinkage to all observed pairwise distances this fact allows us to establish risk bounds for the estimate implying that the true distances can be estimated consistently in an average sense as the number of objects increases in addition such a characterization suggests an efficient algorithm to compute the distance matrix estimator as an alternative to the usual second order cone programming which is known not to scale well for large problems numerical experiments and an application in visualizing the diversity of vpu protein sequences from a recent study of human immunodeficiency virus type 1 further demonstrate the practical merits of the method proposed

dynamics of transversely vibrating beams using four engineering theories abstract in this paper the full development and analysis of four models for the transversely vibrating uniform beam are presented the four theories are the euler bernoulli rayleigh shear and timoshenko first a brief history of the development of each beam model is presented second the equation of motion for each model and the expressions for boundary conditions are obtained using hamilton s variational principle third the frequency equations are obtained for four sets of end conditions free free clamped clamped hinged hinged and clamped free the roots of the frequency equations are presented in terms of normalized wave numbers the normalized wave numbers for the other six sets of end conditions are obtained using the analysis of symmetric and antisymmetric modes fourth the orthogonality conditions of the eigenfunctions or mode shape and the procedure to obtain the forced response using the method of eigenfunction expansion is presented finally a numerical example is shown for a non slender beam to signify the differences among the four beam models

classical jacobi polynomials closed form solutions for transverse vibrations this paper deals with transverse vibrations of nonuniform homogeneous beams and plates classes of beams and axisymmetrical circular plates whose boundary value problems of free transverse vibrations and free transverse axisymmetrical vibrations respectively can be reduced to an eigenvalue singular problem singularities occur at both ends of orthogonal polynomials are reported exact natural frequencies and jacobi polynomials as exact mode shapes which result directly from eigenvalues and eigenfunctions of eigenvalue singular problems of classical orthogonal polynomials are reported for these classes the above classes of beams and plates hereafter called jacobi classes are given by geometry and boundary conditions the geometry consists of parabolic thickness variation with respect to the axial coordinate for beams and with respect to the radius for plates beams belonging to this class have either one or two sharp ends singularities along with certain boundary conditions plates have zero thickness at zero and outer radii the boundary value problems associated with plates and beams of two sharp ends are free boundary problems two other boundary value problems hinged free and sliding free are reported for beams with one sharp end also exact natural frequencies and mode shapes for uniformly rotating beams with hinged free boundary are reported

vibration analysis of periodic cellular solids based on an effective couple stress continuum model abstract cellular solids are usually treated as homogeneous continuums with effective properties nevertheless these mechanical properties depend strongly on the ratio of the specimen size to the cell size these size effects may be accounted for according to preliminary static analysis of effective continuums based on couple stress theory in this paper an effective dynamic continuum model based on couple stress theory is proposed to analyze the behavior of free vibrations of periodic cellular solids in this continuum model the effective mechanical constants of the effective continuum are deduced by an equivalent energy method the cellular solid structure is then replaced with the equivalent couple stress continuum with same overall dimension and shape moreover the finite element formulation of the couple stress continuum for the generalized eigenvalue analysis is developed to implement the free vibration analysis the eigenfrequencies of the effective continuum are then obtained via the shear beam theory or the finite element method a conventional finite element analysis by discretizing each cell of the cellular solids is also carried out to serve as an exact solution several structural cases are calculated to demonstrate the accuracy and effectiveness of the proposed continuum model good agreement on structural eigenfrequencies between the effective continuum solutions and the exact solutions shows that the proposed continuum model can accurately simulate the dynamic behavior of the cellular solids

active disturbance rejection control for rejecting boundary disturbance from multidimensional kirchhoff plate via boundary control in this paper an algorithm is developed to reject time and spatially varying boundary disturbances from a multidimensional kirchhoff plate via boundary control the disturbance and control input are assumed to be matched the active disturbance rejection control approach is adopted for developing the algorithm a state feedback scheme is designed to estimate the disturbance based on an infinite number of ordinary differential equations obtained from the original multidimensional system using infinitely many time dependent test functions the proposed control law cancels the disturbance using its estimated value all subsystems in the closed loop are shown to be asymptotically stable simulation results are presented to validate the theoretical conclusions and to exhibit the reduction in the peaking phenomenon due to the use of time varying gains instead of constant high gains

a simple mathematical model of microtubules surrounded by an elastic matrix by nonlocal finite element method a size dependent finite element model is developed nonlocal elasticity theory is used finite element solution is presented for buckling of microtubules the effect of elastic matrix surrounded by microtubules is modeled a simple nonlocal beam model is proposed to study buckling response of protein microtubules the size effect for buckling model of microtubules is considered by using the nonlocal continuum theory finite element procedure is used for solution of nonlocal differential equation of microtubules for elastic stability the influence of the small length scale on the buckling value is examined for different geometric parameters the effect of elastic matrix surrounded of microtubules is also examined and some benchmark results are presented

the active disturbance rejection and sliding mode control approach to the stabilization of the euler bernoulli beam equation with boundary input disturbance in this paper we are concerned with the boundary feedback stabilization of a one dimensional euler bernoulli beam equation with the external disturbance flowing to the control end the active disturbance rejection control adrc and sliding mode control smc are adopted in investigation by the adrc approach the disturbance is estimated through an extended state observer and canceled online by the approximated one in the closed loop it is shown that the external disturbance can be attenuated in the sense that the resulting closed loop system under the extended state feedback tends to any arbitrary given vicinity of zero as the time goes to infinity in the second part we use the smc to reject the disturbance by removing the condition in adrc that the derivative of the disturbance is supposed to be bounded the existence and uniqueness of the solution for the closed loop via smc are proved and the monotonicity of the reaching condition is presented without the differentiation of the sliding mode function for which it may not always exist for the weak solution of the closed loop system the numerical simulations validate the effectiveness of both methods

quasi random set systems there are many properties of mathematical objects that satisfy what is sometimes called a 0 1 law in the following sense under some natural probability measure on the set of objects the measure of the subset of objects having the given property is either 0 or 1 in the latter case we can say that almost all the objects have the property familiar examples of this phenomenon are the following almost all real numbers are transcendental or normal to every base almost all integers are composite almost all continuous real functions are nondifferentiable etc it is often the case that the objects under consideration can be partitioned into a countable number of finite classes cn with the probability assigned to an object in cn being just 1 icn i in this case we say that a property pn satisfies a 0 1 law if the fraction of the number of objects in cn that satisfy pn either tends to 0 or tends to 1 as n x 0o for example almost all graphs on n vertices have maximum cliques and maximum independent sets of size at most 2 log n almost all boolean functions with n variables have circuit complexity 1 o 1 2n and almost all binary codes of length n with at most 2nr codewords with r less than the binary symmetric channel capacity c have arbitrarily small error probability a special case of shannon s coding theorem see s48 one of the first general results of this type was the theorem of fagin f76 and glebskii et al gklt69 which asserts that every property of graphs that can be expressed in first order logic satisfies a 0 1 law see ss88 for recent striking developments in this topic one obvious method for finding explicit objects having some property pn shared by almost all objects in cn is simply to select one at random with overwhelming probability tending to 1 as n x oc the selected object will have property pn unfortunately it may be and often is extremely difficult to prove that any particular object does indeed satisfy pn it is our purpose in this paper to describe a method that can to a certain extent circumvent this difficulty we will show that for a variety of families it is possible to identify a natural hierarchy of equivalence classes of properties all of which are shared by almost all objects in the family any object satisfying

an implementable proximal point algorithmic framework for nuclear norm minimization the nuclear norm minimization problem is to find a matrix with the minimum nuclear norm subject to linear and second order cone constraints such a problem often arises from the convex relaxation of a rank minimization problem with noisy data and arises in many fields of engineering and science in this paper we study inexact proximal point algorithms in the primal dual and primal dual forms for solving the nuclear norm minimization with linear equality and second order cone constraints we design efficient implementations of these algorithms and present comprehensive convergence results in particular we investigate the performance of our proposed algorithms in which the inner sub problems are approximately solved by the gradient projection method or the accelerated proximal gradient method our numerical results for solving randomly generated matrix completion problems and real matrix completion problems show that our algorithms perform favorably in comparison to several recently proposed state of the art algorithms interestingly our proposed algorithms are connected with other algorithms that have been studied in the literature

a series expansion approach to interpreting the spectra of the timoshenko beam abstract the timoshenko beam model results in two fourth order partial differential equations in time and space consequently solving the boundary value problem yields two independent sequences of natural frequencies and two corresponding sequences of mode shapes a particular natural frequency and its corresponding mode shape describe one particular solution to the boundary value problem of the timoshenko beam from an eigenfunction expansion sense all these possible solutions have to be considered in the complete series expansion of the solution however the question of whether these two independent sequences of natural frequencies implies the existence of two distinct spectra of frequencies has been a long standing topic of debate and hitherto has not been resolved completely the object of this paper is to provide answers to some of the issues raised by this debate in this context the complete solution in a series form to the timoshenko beam is investigated and it is shown for the first time that a particular mode shape of the solution is naturally expressed by an ordered pair of characteristic values rather than a single characteristic value this representation facilitates the progressive ordering of all the natural frequencies of the system and their respective mode shapes in a single set and eliminates the remaining argument for the two spectra interpretation

parametric resonance voltage response of electrostatically actuated micro electro mechanical systems cantilever resonators abstract this paper investigates the parametric resonance voltage response of nonlinear parametrically actuated micro electro mechanical systems mems cantilever resonators a soft ac voltage of frequency near natural frequency is applied between the resonator and a parallel ground plate this produces an electrostatic force that leads the structure into parametric resonance the model consists of an euler bernoulli thin cantilever under the actuation of electrostatic force to include fringe effect and damping force two methods of investigation are used namely the method of multiple scales mms and reduced order model rom method rom convergence of the voltage response and the limitation of mms to small to moderate amplitudes with respect to the gap gap amplitudes are reported mms predicts accurately both hopf supercritical and supercritical bifurcation voltages however mms overestimates the large gap amplitudes of the resonator and misses completely or overestimates the saddle node bifurcation occurring at large gap amplitudes rom produces valid results for small and or large gap amplitudes for a sufficient number of terms vibration modes as the voltage is swept up at constant frequency the resonator maintains zero amplitude until reaches the subcritical hopf bifurcation voltage where it loses stability and jumps up to large gap amplitudes next the gap amplitude decreases until it reaches the supercritical hopf bifurcation point and after that the gap amplitude remains zero for the voltage range considered in this work as the voltage is swept down at constant frequency the zero gap amplitude of the resonator starts increasing continuously after reaching the supercritical hopf bifurcation voltage until it reaches the saddle node bifurcation voltage when a sudden jump to zero gap amplitude occurs effects of frequency damping and fringe parameters on the voltage response show that 1 the supercritical hopf bifurcation is shifted to lower voltage values with the increase of any of the mentioned parameters 2 the subcritical hopf bifurcation is shifted to larger voltage values with the increase of damping shifted to lower voltage values with the increase of the fringe parameter and not significantly altered by the change in frequency 3 the saddle node bifurcation voltage decreases with the increase of frequency and damping and decrease of fringe parameter and 4 the saddle node bifurcation gap amplitude decreases with the increase of frequency and damping and it is not significantly altered by the change of the fringe parameter

matching experimental and three dimensional numerical models for structural vibration problems with uncertainties abstract the simulation model which examines the dynamic behavior of real structures needs to address the impact of uncertainty in both geometry and material parameters this article investigates three dimensional finite element models for structural dynamics problems with respect to both model and parameter uncertainties the parameter uncertainties are determined via laboratory measurements on several beam like samples the parameters are then considered as random variables to the finite element model for exploring the uncertainty effects on the quality of the model outputs i e natural frequencies the accuracy of the output predictions from the model is compared with the experimental results to this end the non contact experimental modal analysis is conducted to identify the natural frequency of the samples the results show a good agreement compared with experimental data furthermore it is demonstrated that geometrical uncertainties have more influence on the natural frequencies compared to material parameters and material uncertainties are about two times higher than geometrical uncertainties this gives valuable insights for improving the finite element model due to various parameter ranges required in a modeling process involving uncertainty

on the degrees of freedom of reduced rank estimators in multivariate regression we study the effective degrees of freedom of a general class of reduced rank estimators for multivariate regression in the framework of stein s unbiased risk estimation a finite sample exact unbiased estimator is derived that admits a closed form expression in terms of the thresholded singular values of the least squares solution and hence is readily computable the results continue to hold in the high dimensional setting where both the predictor and the response dimensions may be larger than the sample size the derived analytical form facilitates the investigation of theoretical properties and provides new insights into the empirical behaviour of the degrees of freedom in particular we examine the differences and connections between the proposed estimator and a commonly used naive estimator the use of the proposed estimator leads to efficient and accurate prediction risk estimation and model selection as demonstrated by simulation studies and a data example

non ambiguous recovery of biot poroelastic parameters of cellular panels using ultrasonicwaves abstract the inverse problem of the recovery of the poroelastic parameters of open cell soft plastic foam panels is solved by employing transmitted ultrasonic waves usw and the biot johnson koplik champoux allard bjkca model it is shown by constructing the objective functional given by the total square of the difference between predictions from the bjkca interaction model and experimental data obtained with transmitted usw that the inverse problem is ill posed since the functional exhibits several local minima and maxima in order to solve this problem which is beyond the capability of most off the shelf iterative nonlinear least squares optimization algorithms such as the levenberg marquadt or nelder mead simplex methods simple strategies are developed the recovered acoustic parameters are compared with those obtained using simpler interaction models and a method employing asymptotic phase velocity of the transmitted usw the retrieved elastic moduli are validated by solving an inverse vibration spectroscopy problem with data obtained from beam like specimens cut from the panels using an equivalent solid elastodynamic model as estimator the phase velocities are reconstructed using computed measured resonance frequencies and a time frequency decomposition of transient waves induced in the beam specimen these confirm that the elastic parameters recovered using vibration are valid over the frequency range ofstudy

a gradient elasticity theory for second grade materials and higher order inertia abstract second grade elastic materials featured by a free energy depending on the strain and the strain gradient and a kinetic energy depending on the velocity and the velocity gradient are addressed an inertial energy balance principle and a virtual work principle for inertial actions are envisioned to enrich the set of traditional theoretical tools of thermodynamics and continuum mechanics the state variables include the body momentum and the surface momentum related to the velocity in a nonstandard way as well as the concomitant mass accelerations and inertial forces which do intervene into the motion equations and into the force boundary conditions the boundary traction is the sum of two parts i e the cauchy traction and the gurtin murdoch traction whereas the traction boundary condition exhibits the typical format of the equilibrium equation of a material surface as known from the principles of surface mechanics whereby the gurtin murdoch traction incorporating the inertial surface force plays the role of applied surfacial force density the body s boundary surface constitutes a thin boundary layer which is in global equilibrium under all the external forces applied on it a feature that makes it possible to exploit the traction cauchy theorem within second grade materials this means that a second grade material is formed up by two sub systems that is the bulk material operating as a classical cauchy continuum and the thin boundary layer operating as a gurtin murdoch material surface the classical linear and angular momentum theorems are suitably extended for higher order inertia from which the local motion equations and the moment equilibrium equations stress symmetry can be derived for an isotropic material featured by four constants i e the lame constants and two length scale parameters aifantis model the dynamic evolution problem is characterized by a hamilton type variational principle and a solution uniqueness theorem closed form solutions of the wave dispersion analysis problem for beam models are presented and compared with known results from the literature the paper indicates a correct thermodynamically consistent way to take into account higher order inertia effects within continuum mechanics

on the two frequency spectra of timoshenko beams abstract this paper is concerned with the question of whether there are indeed two distinct spectra of frequencies for the transverse vibrations of timoshenko beams as has been claimed by a number of prior authors for the case of the simply supported beam and more recently for beams supported in an arbitrary manner elementary analysis leads to the conclusion that there is only a single frequency spectrum in the particular case of the simply supported beam the two frequency spectra viewpoint may be expedient as a device to compute frequencies but does not serve otherwise to explain the complex dynamical behavior of timoshenko beams