an adaptive generalized finite element method applied to free vibration analysis of straight bars and trusses in this paper a first application of an adaptive generalized finite element method to free longitudinal vibration analysis of straight bars and trusses is presented the generalized finite element method is developed by enriching the standard finite element method space whose basis performs a partition of unity with knowledge about the differential equation being solved the enrichment functions used are dependent on the geometric and mechanical properties of the element the proposed approach converges very fast and is able to approximate the frequency related to any vibration mode the variational problem of free vibration is formulated and the main aspects of the adaptive generalized finite element method are presented and discussed the efficiency and convergence of the proposed method in vibration analysis of uniform and non uniform straight bars are checked the application of this technique in a truss is also presented the frequencies obtained by the adaptive generalized finite element method are compared with those obtained by the analytical solution the composite element method and the h version of finite element method

natural frequencies and mode shapes of deterministic and stochastic non homogeneous rods and beams abstract natural frequencies and mode shapes of non homogeneous deterministic and stochastic rods and beams are studied the solution is based on the functional perturbation method fpm the frequencies and mode shapes are considered as functionals of the non homogeneous properties the natural frequency and mode shape of the k th order is obtained analytically to any desired degree of accuracy once the functional derivatives with respect to the non uniform property have been found the solution for any morphology is obtained by direct integration without resolving the differential equation several examples with different non homogeneous properties are solved and compared with exact solutions as an accuracy check the fpm accuracy range for the frequency and the mode shape is less than 1 even for high heterogeneities in the stochastic case the accuracy of the natural frequencies depends on the stochastic information used given on the correlation distance roughly the grain size on the function around which the perturbation is executed and on whether we are interested in the properties of or of 2 moreover all frequency modes have the same response to heterogeneity as long as their wave length is of the order of the heterogeneity s characteristic distance in addition the heterogeneity effect on the average natural frequencies is minimal for the fundamental mode and may serve as a design tool

exact solutions for the longitudinal vibration of non uniform rods abstract the objective of this paper is to present exact analytical solutions for the longitudinal vibration of rods with non uniform cross section using appropriate transformation the equation of motion of axial vibration of a rod with varying cross section is reduced to analytically solvable standard differential equations whose form depends upon the specific area variation solutions are obtained for a rod with a polynomial area vibration and for a sinusoidal rod the solutions are obtained in terms of special functions such as bessel and neumann as well as trigonometric functions simple formulas to predict the natural frequencies of non uniform rods with various end conditions are presented the natural frequencies of non uniform rods for these end conditions are calculated and their dependence on taper is discussed the governing equation for the problem is the same as that of wave propagation through ducts with non uniform cross sections therefore solutions presented here can be used to investigate such problems

a new approach for free vibration of axially functionally graded beams with non uniform cross section this paper studies free vibration of axially functionally graded beams with non uniform cross section a novel and simple approach is presented to solve natural frequencies of free vibration of beams with variable flexural rigidity and mass density for various end supports including simply supported clamped and free ends we transform the governing equation with varying coefficients to fredholm integral equations natural frequencies can be determined by requiring that the resulting fredholm integral equation has a non trivial solution our method has fast convergence and obtained numerical results have high accuracy the effectiveness of the method is confirmed by comparing numerical results with those available for tapered beams of linearly variable width or depth and graded beams of special polynomial non homogeneity moreover fundamental frequencies of a graded beam combined of aluminum and zirconia as two constituent phases under typical end supports are evaluated for axially varying material properties the effects of the geometrical and gradient parameters are elucidated the present results are of benefit to optimum design of non homogeneous tapered beam structures

quantum locally testable codes we initiate the study of quantum locally testable codes text qltc s classical text ltc s are very important in computational complexity these codes are defined as the linear subspace satisfying a set of local constraints with the additional requirement that their soundness r delta which is the probability that a randomly chosen constraint is violated is proportional to the proximity delta where delta n is the distance of a word from the code excellent text ltc s exist in the classical world and they are tightly related to the celebrated text pcp probabilistically checkable proof theorem in quantum complexity quantum error correcting codes provide central examples in the study of the illusive behavior of multiparticle entanglement and they have played a crucial role in many computational complexity results we provide a definition of the quantum analogue of text ltc s and motivate it by connecting its central notions in the study of both entanglement and quantum ha

property testing and its connection to learning and approximation in this paper we consider the question of determining whether a function f has property p or is e far from any function with property p a property testing algorithm is given a sample of the value of f on instances drawn according to some distribution in some cases it is also allowed to query f on instances of its choice we study this question for different properties and establish some connections to problems in learning theory and approximation in particular we focus our attention on testing graph properties given access to a graph g in the form of being able to query whether an edge exists or not between a pair of vertices we devise algorithms to test whether the underlying graph has properties such as being bipartite k colorable or having a p clique clique of density p with respect to the vertex set our graph property testing algorithms are probabilistic and make assertions that are correct with high probability while making a number of queries that is independent of the size of the graph moreover the property testing algorithms can be used to efficiently i e in time linear in the number of vertices construct partitions of the graph that correspond to the property being tested if it holds for the input graph

why can lda be performed in pca transformed space abstract pca plus lda is a popular framework for linear discriminant analysis lda in high dimensional and singular case in this paper we focus on building a theoretical foundation for this framework moreover we point out the weakness of the previous lda based methods and suggest a complete pca plus lda algorithm experimental results on orl face image database indicate that the proposed method is more effective than the previous ones

some optimal inapproximability results we prove optimal up to an arbitrary e 0 inapproximability results for max e k sat for k 3 maximizing the number of satisfied linear equations in an over determined system of linear equations modulo a prime p and set splitting as a consequence of these results we get improved lower bounds for the efficient approximability of many optimization problems studied previously in particular for max e2 sat max cut max di cut and vertex cover

probabilistic checking of proofs a new characterization of np we give a new characterization of np the class np contains exactly those languages l for which membership proofs a proof that an input x is in l can be verified probabilistically in polynomial time using logarithmic number of random bits and by reading sublogarithmic number of bits from the proof we discuss implications of this characterization specifically we show that approximating clique and independent set even in a very weak sense is np hard

expander codes using expander graphs we construct a new family of asymptotically good linear error correcting codes these codes have linear time sequential decoding algorithms and logarithmic time parallel decoding algorithms that use a linear number of processors we present both randomized and explicit constructions of these codes experimental results demonstrate the good performance of the randomly chosen codes

a wide class of analytical solutions of the webster equation abstract this paper aims at presenting closed form general analytical solutions of the webster equation describing plane elastic or acoustic waves the considered radius functions of non uniform cross sectioned rods or ducts are based on the triconfluent heun functions and contain some optional parameters enabling us to set various profiles of the radius functions in a relatively wide range while it is possible to employ the presented exact general analytical solution of the webster equation for all selected profiles if the radius functions are predetermined then the derived general analytical solution can also be employed for their triconfluent heun approximations including certain polynomial ones the applicability and correctness of the derived analytical solutions are demonstrated by calculations of natural frequencies and mode shapes for representative radius functions while the results based on approximate analytical solutions are verified numerically

testing reed muller codes a code is locally testable if there is a way to indicate with high probability that a vector is far enough from any codeword by accessing only a very small number of the vector s bits we show that the reed muller codes of constant order are locally testable specifically we describe an efficient randomized algorithm to test if a given vector of length n 2 sup m is a word in the rth order reed muller code r r m of length n 2 sup m for a given integer r spl ges 1 and real spl epsi 0 the algorithm queries the input vector spl upsi at o 1 spl epsi r2 sup 2r positions on the one hand if spl upsi is at distance at least spl epsi n from the closest codeword then the algorithm discovers it with probability at least 2 3 on the other hand if spl upsi is a codeword then it always passes the test our result is almost tight any algorithm for testing r r m must perform spl omega 1 spl epsi 2 sup r queries

acoustic waves in slender axisymmetric tubes acoustic waves in a rigid axisymmetric tube with a variable cross section are considered the governing helmholtz equation is solved using neumann series expansions in bessel functions of various orders with a stretched radial coordinate leading to a hierarchy of one dimensional ordinary differential equations in the longitudinal direction the lowest approximation for axisymmetric motion includes webster s horn equation as a special case fourth order differential equations are obtained at the next level of approximation good agreement with existing asymptotic theories for waves in slender tubes is found

bounding the distance of quantum surface codes homological quantum codes also called topological codes are low density parity check error correcting codes that come from surfaces and higher dimension manifolds homological codes from surfaces i e surface codes have also been suggested as a possible way to construct stable quantum memory and fault tolerant computation it has been conjectured that all homological codes have a square root bound on there distance and therefore cannot produce good codes this claim has been disputed in dimension four using the geometric property of systolic freedom we will show in this paper that the conjecture holds in dimension two due to the negation of systolic freedom i e systolic rigidity

free and forced vibrations of stepped rods and coupled systems abstract analytical and closed form frequency responses of the longitudinal vibration of n stepped rods as well as the coupled system of multi stepped rods and lumped elements with non classical boundary conditions are proposed in this paper the closed form frequency equation for these systems is also obtained in this article a new two way state flow model based on the dynamics of the rod is developed in addition to the state flow model developed for lumped elements concentrated mass and spring and damper the exact frequency response of both stepped rod and the coupled system can be directly written out according to the graph theory since the results are analytical and closed form they can be directly implemented for analytical and numerical analysis without resorting to complex operations such as inversion recursive calculation or iteration computation moreover the derived equations are feasible not only for the multi stepped and multi stage rods but also for the coupled system of multi stepped rods and lumped elements finally the results were checked using multi stepped rods with classical and non classical boundary conditions as well as a combination of stepped rods and lumped elements to demonstrate the efficiency of this method

an efficient approach for face recognition based on common eigenvalues in this paper a simple technique is proposed for face recognition among many human faces it is based on the polynomial coefficients covariance matrix and algorithm on common eigenvalues the main advantage of the proposed approach is that the identification of similarity between human faces is carried out without computing actual eigenvalues and eigenvectors a symmetric matrix is calculated using the polynomial coefficients based companion matrices of two compared images the nullity of a calculated symmetric matrix is used as similarity measure for face recognition the value of nullity is very small for dissimilar images and distinctly large for similar face images the feasibility of the propose approach is demonstrated on three face databases i e the orl database the yale database b and the feret database experimental results have shown the effectiveness of the proposed approach for feature extraction and classification of the face images having large variation in pose and illumination

the need for structure in quantum ldpc codes the existence of quantum ldpc codes with minimal distance scaling linearly in the number of qubits is a central open problem in quantum information despite years of research good quantum ldpc codes are not known to exist but at the very least it is known they cannot be defined on very regular topologies like low dimensional grids in this work we establish a complementary result showing that good quantum css codes which are sparsely generated require structure in the local terms that constrain the code space so as not to be too random in a well defined sense to show this we prove a weak converse to a theorem of krasikov and litsyn on weight distributions of classical codes due to which may be of independent interest subspaces for which the distribution of weights in the dual space is approximately binomial have very few codewords of low weight tantamount to having a non negligible approximate minimal distance while they may not have a large minimal non zero weight they still have very few words of low hamming weight

enriched finite elements for time harmonic webster s equation this work presents an enriched finite element method fem dedicated to the numerical resolution of webster s equation in the time harmonic regime which models many physical configurations e g wave propagation in acoustic waveguides or vibration of bars with varying cross section building on the wave based methods existing in the literature we present new enriched finite element bases that account for both the frequency of the problem and the heterogeneity of the coefficients of the equation the enriched method is compared to the classical fifth order polynomial fem and we show they share the same asymptotic convergence order present the same easiness of implementation and have similar computational costs the main improvement brought by these enriched bases relies on the convergence threshold the mesh size at which the convergence regime begins and the convergence multiplicative constant which are observed to be i better than the ones associated with polynomial bases and ii not only dependent on the resolution the number of elements per wavelength but also on the frequency for a fixed resolution making the method we propose well adapted to high frequency regimes moreover taking into account the heterogeneity of the coefficient in webster s equation by using an element dependent enrichment leads to a significant decrease of the approximation error on the considered examples compared to a uniform enrichment again with almost no additional cost several possible extensions of this work are finally discussed

closed form solutions for the free longitudinal vibration of inhomogeneous rods this paper aims at presenting a family of exact solutions for the longitudinal vibration of variable area rods area variations that give solutions in terms of the confluent hypergeometric function are being sought for and the governing differential equation is appropriately reduced to the confluent hypergeometric differential equation using a generic transformation the eigenfrequencies of rods with certain area variations subjected to classical boundary conditions are obtained and the parametric space of the solutions obtained is studied these solutions are also highly useful in other topics of study such as torsional vibration of rods and wave propagation in ducts with variable cross sectional areas since the governing differential equations are very similar

locally testable codes and pcps of almost linear length we initiate a systematic study of locally testable codes that is error correcting codes that admit very efficient membership tests specifically these are codes accompanied with tests that make a constant number of random queries into any given word and reject non codewords with probability proportional to their distance from the code locally testable codes are believed to be the combinatorial core of pcps however the relation is less immediate than commonly believed nevertheless we show that certain pcp systems can be modified to yield locally testable codes on the other hand we adapt techniques that we develop for the construction of the latter to yield new pcps our main results are locally testable codes and pcps of almost linear length specifically we prove the existence of the following constructs locally testable binary linear codes in which k information bits are encoded by a codeword of length k exp o log k this improves over previous results that either yield codewords of exponential length or obtained almost quadratic length codewords for sufficiently large nonbinary alphabet pcp systems of almost linear length for sat the length of the proof is n exp o log n and verification in performed by a constant number i e 19 of queries as opposed to previous results that used proof length n 1 p o 1 q for verification by q queries the novel techniques in use include a random projection of certain codewords and pcp oracles that preserves local testability an adaptation of pcp constructions to obtain linear pcp oracles for proving conjunctions of linear conditions and design of pcps with some new soundness properties a direct construction of locally testable linear codes of subexponential length

on propagation of plane symmetric waves in a periodically corrugated straight elastic layer abstarct the classical bernoulli euler model is employed for analysis of the plane wave propagation in an infinitely long elastic layer with a periodically varying thickness the floquet theory is applied to derive asymptotic formulas defining location and broadness of frequency stop bands for several corrugation shapes with different levels of discontinuity for a layer with the perfectly smooth periodic corrugation the equation of the axial wave motion is solved by the method of multiple scales in vicinity of critical frequencies then it is transformed to the canonical mathieu equation and its stability diagram is compared with predictions of the floquet theory the qualitative differences between the shapes of stop bands are discussed the wentzel kramers brillouin approximation is employed to solve the bernoulli euler equation for the layer with perfectly smooth periodic corrugation and the results are discussed in view of existence of pass and stop bands and in view of the levels of approximation used in alternative asymptotic methods

on webster s horn equation and some generalizations sound waves along a rigid axisymmetric tube with a variable cross section are considered the governing helmholtz equation is solved using power series expansions in a stretched radial coordinate leading to a hierarchy of one dimensional ordinary differential equations in the longitudinal direction the lowest approximation for axisymmetric motion turns out to be webster s horn equation fourth order differential equations are obtained at the next level of approximation comparisons with existing asymptotic theories for waves in slender tubes are made

mass lumping techniques in the spectral element method on the equivalence of the row sum nodal quadrature and diagonal scaling methods abstract in the context of wave propagation analysis the spectral element method sem in conjunction with a diagonal mass matrix is often the method of choice therefore it is of high importance to investigate the influence of different mass lumping schemes on the accuracy of the numerical results to this end we compare the performance of three established methods including the row sum method the nodal quadrature method and the diagonal scaling method the theoretical analysis of these methods reveals a close connection between them under certain conditions that are discussed in detail in this article we are able to show a direct equivalence between these three approaches in this regard the attainable accuracy of the numerical integration of the mass matrix plays an important role by means of several dynamic benchmark problems we verify the theoretical results and illustrate the convergence properties of the lumped mass matrix sem in comparison to a formulation based on the consistent mass matrix

the pcp theorem by gap amplification the pcp theorem arora and safra 1998 arora et al 1998 says that every language in np has a witness format that can be checked probabilistically by reading only a constant number of bits from the proof the celebrated equivalence of this theorem and inapproximability of certain optimization problems due to feige et al 1996 has placed the pcp theorem at the heart of the area of inapproximability in this work we present a new proof of the pcp theorem that draws on this equivalence we give a combinatorial proof for the np hardness of approximating a certain constraint satisfaction problem which can then be reinterpreted to yield the pcp theorem our approach is to consider the unsat value of a constraint system which is the smallest fraction of unsatisfied constraints ranging over all possible assignments for the underlying variables we describe a new combinatorial amplification transformation that doubles the unsat value of a constraint system with only a linear blowup in the size of the system the amplification step causes an increase in alphabet size that is corrected by a standard pcp composition step iterative application of these two steps yields a proof for the pcp theorem the amplification lemma relies on a new notion of graph powering that can be applied to systems of binary constraints this powering amplifies the unsat value of a constraint system provided that the underlying graph structure is an expander we also extend our amplification lemma towards construction of assignment testers alternatively pcps of proximity which are slightly stronger objects than pcps we then construct pcps and locally testable codes whose length is linear up to a polylog factor and whose correctness can be probabilistically verified by making a constant number of queries namely we prove sat pcp 1 2 1 log2 n poly log n o 1

exact solutions for free longitudinal vibrations of non uniform rods an exact approach for free longitudinal vibrations of one step non uniform rods with classical and non classical boundary conditions is presented in this paper the expression for describing the distribution of mass is arbitrary and the distribution of longitudinal stiffness is expressed as a functional relation with the mass distribution and vice versa using appropriate functional transformation the governing differential equations for free vibrations of one step non uniform rods are reduced to analytically solvable differential equations for several functional relations between stiffness and mass the fundamental solutions that satisfy the normalization conditions are derived and used to establish the frequency equations for one step rods with classical and non classical boundary conditions using the fundamental solutions of each step rod and a recurrence formula developed in this paper a new exact approach for determining the longitudinal natural frequencies and mode shapes of multi step non uniform rods is proposed numerical examples demonstrate that the calculated longitudinal natural frequencies and mode shapes are in good agreement with the experimental data and those determined by the finite element method and the proposed procedure is an efficient and exact method

wave evolution reflection and transmission along inhomogeneous waveguides abstract one dimensional wave propagation along an inhomogeneous waveguide such as an acoustic horn a plate strip a rod or a beam is considered the equations of motion of a generic waveguide are written in first order form and the properties of the position dependent eigenvalues and eigenvectors of the system are deduced by considering the symmetry and or the energetics of the system the equations of motion are then transformed to wave co ordinates based on the right eigenvectors of the system and a perturbation method is employed to study wave propagation along deterministic periodic and random waveguides attention is then turned to wave reflection at a cut off cross section and a general result for the phase of the reflection coefficient is derived the general theory is illustrated by application to rod and beam examples the method can be likened to a coupled modes approach in which the properties of the eigenvectors or modes arising from symmetry and or energetics are exploited