rank deficient submatrices of fourier matrices we consider the maximal rank deficient submatrices of fourier matrices with order a power of a prime number we do this by considering a hierarchical subdivision of these matrices into low rank blocks we also explore some connections with the fast fourier transform fft and with an uncertainty principle for fourier transforms over finite abelian groups

regular multivariate sampling and approximation in lp shift invariant spaces abstract in this paper some results about regular multivariate generalized sampling in the l p setting 1 p are proven thus stable multivariate regular sampling formulas are derived for l p shift invariant spaces v p i e the l p closure of the linear span of the shifts of a finite set of generators these sampling formulas include regular samples of the function its derivatives and or some filtered versions of the function itself taken at a lattice of r d approximation schemes using these generalized sampling formulas are also included

a new inequality in discrete fourier theory discrete fourier theory has been applied successfully in digital communication theory in this correspondence we prove a new inequality linking the number of nonzero components of a complex valued function defined on a finite abelian group to the number of nonzero components of its fourier transform we characterize the functions achieving equality finally we compare this inequality applied to boolean functions to the inequality arising from the minimal distance property of reed muller codes

on the controllability properties of circulant networks this paper examines the controllability of a group of first order agents adopting a weighted consensus type coordination protocol over a circulant network specifically it is shown that a circulant network with laplacian eigenvalues of maximum algebraic multiplicity q is controllable from q nodes our approach leverages on the cauchy binet formula which in conjunction with the popov belevitch hautus test leads to new insights on structural aspects of network controllability

analysis of kinematic linked interpolation methods for reissner mindlin plate problems the approximation to the solution of reissner mindlin plate problem is considered in the framework of finite element techniques a general strategy involving a linking operator between rotations and vertical displacements is analyzed an abstract convergence result is provided examples of elements falling into this framework are presented and shown to be stable and locking free numerical tests detail the performances of the elements email lovadina ing unitn it

nonuniform sampling and reconstruction in shift invariant spaces this article discusses modern techniques for nonuniform sampling and reconstruction of functions in shift invariant spaces it is a survey as well as a research paper and provides a unified framework for uniform and nonuniform sampling and reconstruction in shift invariant subspaces by bringing together wavelet theory frame theory reproducing kernel hilbert spaces approximation theory amalgam spaces and sampling inspired by applications taken from communication astronomy and medicine the following aspects will be emphasized a the sampling problem is well defined within the setting of shift invariant spaces b the general theory works in arbitrary dimension and for a broad class of generators c the reconstruction of a function from any sufficiently dense nonuniform sampling set is obtained by efficient iterative algorithms these algorithms converge geometrically and are robust in the presence of noise d to model the natural decay conditions of real signals and images the sampling theory is developed in weighted l p spaces

an existence theorem for a nonlinear difference equation where p 1 and s p is a real sequence we establish conditions under which 1 1 has a positive nondecreasing solution here a solution of 1 1 is a real sequence y y which satisfies 1 1 since 1 1 is a recurrence relation given real initial values y and y it is clear that we can inductively obtain y y since p is a real number greater than 1 the sequence yk may not be real that is not a solution note however that any constant multiple of a solution is again a solution the existence question originated in l where it was shown that hardy s inequality for a series 2 can be viewed as a necessary condition for the existence of a positive nondecreasing solution of 1 1 our existence theorem reduces to a nonoscillation criterion in the case p 2 since then 1 1 changes to a second order linear difference equation see e g 3 a brief outline of the paper is as follows section 2 introduces a riccati type transformation and then develops various necessary conditions for the existence of a positive nondecreasing solution section 3 discusses some of the functional analytic background which is needed in order to apply schauder s theorem section 4 contains the proof of our existence theorem and section 5 describes a comparison theorem which allows us to drop the assumption that the coefficients sk in 1 1 are nonnegative

generation of a section of conjugation classes and lyndon word tree of limited length resume etant donne un alphabet ordonne un mot de lyndon est un mot primitif qui est le plus petit dans sa classe de conjugaison nous donnons ici un algorithme qui construit dans l ordre la liste des mots de lyndon de longueur donnee cet algorithme est optimal en ce sens que le calcul du mot de lyndon suivant de la liste se fait en temps lineaire a partir du mot precedent et sans utilisation d une memoire auxiliaire cet algorithme trouve son application dans la construction d une section des classes de conjugaison d une longueur donnee c est une response a une question posee a l auter par m p schutzenberger

distributed field reconstruction in wireless sensor networks based on hybrid shift invariant spaces we use the theory and algorithms developed for so called shift invariant spaces to develop a novel distributed architecture for sampling and reconstructing time varying non bandlimited physical fields in wireless sensor networks we introduce hybrid shift invariant spaces that generalize conventional shift invariant spaces and can adapt to local smoothness properties of the field using shift invariant spaces with compactly supported generator functions allows us to split the global reconstruction into several smaller local problems that can be solved independently capitalizing on the sparsity of the matrices involved in the reconstruction we propose direct and iterative reconstruction algorithms whose complexity per time slot scales only linearly with the number of sensor nodes we furthermore analyze the impact of sensor localization errors on the mean square error of the reconstructed field numerical simulations illustrate that the proposed field reconstruction scheme performs close to bandlimited reconstruction and is less sensitive to sensor location errors while providing a significant reduction in computational complexity

the patch test for mixed formulations this paper presents a simple extension of the patch test to mixed formulations to provide the necessary and sufficient conditions for convergence the general algebraic conditions of babuska and brezzi are given a simple form and a conceptual application of the patch tests serves to point out the instability of several well known formulations for incompressible problems the test is also applied to mixed displacement strain formulations of elasticity and limiting formulations are noted

universal spatiotemporal sampling sets for discrete spatially invariant evolution processes let i be a finite abelian group and mathbf a be a circular convolution operator on ell 2 i the problem under consideration is how to construct minimal omega subset i and l i such that y mathbf e i mathbf a mathbf e i cdots mathbf a l i mathbf e i i in omega is a frame for ell 2 i where mathbf e i i in i is the canonical basis of ell 2 i this problem is motivated by the spatiotemporal sampling problem in discrete spatially invariant evolution processes we will show that the cardinality of omega should be at least equal to the largest geometric multiplicity of eigenvalues of mathbf a and consider the universal spatiotemporal sampling sets omega l i for convolution operators whose eigenvalues subject to the same largest geometric multiplicity we will give an algebraic characterization for such sampling sets and show how this problem is linked with sparse signal processing theory and polynomial interpolation theory

a relationship between stabilized finite element methods and the galerkin method with bubble functions abstract a relation between stabilized finite element methods and the galerkin method employing interpolations with bubble functions is established for the advective diffusive model and for the linearized compressible navier stokes equations the bubble functions are shown to help in stabilizing the advective operator without recourse to upwinding or any other numerical strategy in particular for the advective diffusive model the galerkin method employing piecewise linears with bubble functions is shown to be equivalent to the streamline upwind petrov galerkin supg method in the diffusive limit

performance of linear field reconstruction techniques with noise and uncertain sensor locations we consider a wireless sensor network sampling a bandlimited field described by a limited number of harmonics sensor nodes are irregularly deployed over the area of interest or subject to random displacement in addition sensors measurements are affected by noise our goal is to obtain a high quality reconstruction of the field with the mean square error mse of the estimate as performance metric in particular we analytically derive the performance of several reconstruction estimation techniques based on linear filtering for each technique we obtain the mse as well as its asymptotic expression in the case where the number of field harmonics and the number of sensors grow to infinity while their ratio is kept constant through numerical simulations we show the validity of the asymptotic analysis even for a small number of sensors we provide some novel guidelines for the design of sensor networks when many parameters such as field bandwidth number of sensors reconstruction quality and sensor displacement characteristics to be traded off

estimating localized sources of diffusion fields using spatiotemporal sensor measurements we consider diffusion fields induced by a finite number of spatially localized sources and address the problem of estimating these sources using spatiotemporal samples of the field obtained with a sensor network within this framework we consider two different time evolutions the case where the sources are instantaneous as well as the case where the sources decay exponentially in time after activation we first derive novel exact inversion formulas for both source distributions through the use of green s second theorem and a family of sensing functions to compute generalized field samples these generalized samples can then be inverted using variations of existing algebraic methods such as prony s method next we develop a novel and robust reconstruction method for diffusion fields by properly extending these formulas to operate on the spatiotemporal samples of the field finally we present numerical results using both synthetic and real data to verify the algorithms proposed herein

some combinatorial properties of jack symmetric functions if 1 l n then write 2 n or 1 n if p is another partition then write p c j if pi 6 for all i i e if the diagram of 1 contains the diagram of p if ipl ii 1 then write p 2 1 reverse lexicographic order if either j i or the first nonvanishing difference e pi is positive for instance writing 2 1 i for ai i it 5 5 41 32 5 311 221 2111 11111 finally write p i if ipl lal and p1 pl2 pli a1 a2 li for all i macdonald m p 63 call the partial ordering the natural ordering but we will call it the dominance ordering let x x x2 be an infinite set of indeterminates as in mi we

augmented lagrangian finite elements for plate contact problems the present work investigates the unilateral frictionless contact between a plate and a rigid obstacle two sets of problems are studied a plate constrained through unilateral edge supports and a plate seating in its undeformed configuration at a given distance from a rigid support the attention is concentrated on two augmented lagrangian formulations the algorithmic implementation within a finite element scheme is presented and discussed the importance of using appropriate plate elements for the discretization of the structure is stressed new gap elements compatible with a robust plate element are derived computational aspects are emphasized a simple and effective numerical integration for the determination of the gap stiffnesses in partial contact with the support is proposed numerical results are carried out and compared with analytical solutions the convergence to the solution of the perfectly constrained problem is numerically investigated the inadequacy of the penalty method and the satisfactory performance obtained from only one augmented lagrangian procedure are emphasized

linked interpolation for reissner mindlin plate elements part ii a simple triangle the formulation and shape functions given in part i are extended to develop a simple plate bending triangle with good performance in both thin and thick situations indeed its performance is better than that of other nine dof elements and its computer implementation simpler when used with selective reduced integration the element produces identical results as that of xu 1 2 to save space details given in part i are not repeated and some results are also presented in the figures of part i

an entropy based uncertainty principle for a locally compact abelian group abstract we classify all functions on a locally compact abelian group giving equality in an entropy inequality generalizing the heisenberg uncertainty principle in particular for functions on a real line we proof a conjecture of hirschman published in 1957

analytic sequential weiss weinstein bounds in this paper we explore a sequential bayesian bound for state space models focusing on hybrid continuous and discrete random states we provide an analytic recursion for the sequential weiss weinstein sww bound for linear state space models with solutions for gaussian uniform and exponential distributions as derived as well as for a combination of these we compare the sww bound for discretized states with the corresponding bound for the continuous states the sww bound is contrasted with the sequential cramer rao bound for gaussian distributions practical issues of sww bounds are discussed and numerical simulation results provide insights into their behavior

kronecker product approximations for some function related matrices abstract a new approximation tool such as sums of kronecker products is recently found to provide a superb compression property on a series of numerical examples of quite a general nature the purpose of the paper is explanation of this phenomenon in the form of existence theorems for matrix approximations of low kronecker rank for some classes of function related matrices including important specimens from potential theory this lays the grounds for development of new approximation algorithms for example in the cases when a matrix is associated with a shift invariant function on the cartesian product of nonunform grids which is of great practical interest in the solution of integral equations on plates or screens

oscillation theorems for second order nonlinear partial difference equations 479 abstract the oscillation of the second order nonlinear partial difference equation t 1 2 c mn 1 y mn i 1 s a i m n f i y m 1 n 1 y mn 0 is investigated some sufficient conditions for the oscillation of all solutions of the above equation are obtained

an edge based smoothed finite element method es fem with stabilized discrete shear gap technique for analysis of reissner mindlin plates abstract an edge based smoothed finite element method es fem for static free vibration and buckling analyses of reissner mindlin plates using 3 node triangular elements is studied in this paper the calculation of the system stiffness matrix is performed by using the strain smoothing technique over the smoothing domains associated with edges of elements in order to avoid the transverse shear locking and to improve the accuracy of the present formulation the es fem is incorporated with the discrete shear gap dsg method together with a stabilization technique to give a so called edge based smoothed stabilized discrete shear gap method es dsg the numerical examples demonstrated that the present es dsg method is free of shear locking and achieves the high accuracy compared to the exact solutions and others existing elements in the literature

finite sampling in multiple generated u invariant subspaces the relevance in a sampling theory of u invariant subspaces of a hilbert space mathcal h where u denotes a unitary operator on mathcal h is nowadays a recognized fact indeed shift invariant subspaces of l 2 mathbb r become a particular example periodic extensions of finite signals also provide a remarkable example as a consequence the availability of an abstract u sampling theory becomes a useful tool to handle these problems in this paper we derive a sampling theory for finite dimensional multiple generated u invariant subspaces of a hilbert space mathcal h as the involved samples are identified as frame coefficients in a suitable euclidean space the relevant mathematical technique is that of the finite frame theory since finite frames are nothing but spanning sets of vectors the used technique naturally meets matrix analysis

oscillation criteria for nonlinear partial difference equations with delays abstract we offer sufficient conditions for the oscillation of all solutions of the partial difference equations y m 1 n m n y m n 1 m n y m n p m n y m k n l q m n y m k n l and y m 1 n m n y m n 1 m n y m n i 1 p i m n y m k i n l i i 1 q i m n y m k i n l i several examples which dwell upon the importance of our results are also included

an isostatic assumed stress triangular element for the reissner mindlin plate bending problem the paper describes a new assumed stress triangular element for reissner mindlin plates called tip3 with three nodes and three degrees of freedom per node the kinematics is constructed by means of the so called linked interpolation ruled by technically significant degrees of freedom i e one transversal displacement and two rotations per node without using additional bubble modes the static representation starts from a moment shear uncoupled polynomial approximation and is constrained to satisfy some equilibrium conditions in order to reduce the stress parameters to a minimum number the resulting element is locking free presents correct rank and passes the bending patch test even for very thin plates the good performances of the element are demonstrated by several comparisons with other triangular plate elements available in the literature copyright 2007 john wiley sons ltd

discrete sampling and interpolation universal sampling sets for discrete bandlimited spaces we study the problem of interpolating all values of a discrete signal f of length n when d n values are known especially in the case when the fourier transform of the signal is zero outside some prescribed index set j these comprise the generalized bandlimited spaces bj the sampling pattern for f is specified by an index set i and is said to be a universal sampling set if samples in the locations i can be used to interpolate signals from bjfor any j when n is a prime power we give several characterizations of universal sampling sets some structure theorems for such sets an algorithm for their construction and a formula that counts them there are also natural applications to additive uncertainty principles

a posteriori error analysis of finite element methods for reissner mindlin plates this paper develops an a posteriori error control theory of finite element methods for reissner mindlin plates which states that one can derive a uniformly reliable and efficient a posteriori error estimate for a given scheme by only 1 checking three conditions 2 designing three functions and one parameter 3 bounding the last three terms of the abstract estimator we apply this theory to two classes of methods and achieve robust a posteriori error controls for them

linked interpolation for reissner mindlin plate elements part i a simple quadrilateral in most plate elements using the reissner mindlin assumptions the interpolations used for the lateral displacements w and the rotation involve the independent representation of each variable by its nodal values usually with identical interpolations to ensure a higher order of expansion for displacement w its representation is linked in the present paper with both sets of nodal variables conditions necessary for the use of such expansions are established here and the paper shows the development of a linear quadrilateral element q4bl whose performance and robustness are good although it possesses one singularity if only three degrees of freedom are prescribed in part ii we apply the identical formulation to develop a triangular element t3bl which performs equally well and is fully robust

iterative actions of normal operators abstract let a be a normal operator in a hilbert space h and let g h be a countable set of vectors we investigate the relations between a g and l that make the system of iterations a n g g g 0 n l g complete bessel a basis or a frame for h the problem is motivated by the dynamical sampling problem and is connected to several topics in functional analysis including frame theory and spectral theory it also has relations to topics in applied harmonic analysis including wavelet theory and time frequency analysis

accurate and efficient evaluation of schur and jack functions we present new algorithms for computing the values of the schur s x 1 x 2 x n and jack j x 1 x 2 x n functions in floating point arithmetic these algorithms deliver guaranteed high relative accuracy for positive data x i a 0 and run in time that is only linear in n

existence of monotone positive solution of neutral partial difference equation abstract this paper is concerned with a class of neutral partial difference equations the conditions for the existence of monotone eventually positive solutions are established which improve and extend some of the criteria existing in the literature comparison theorems are also derived results are obtained on the existence of a monotone eventually positive solution of dual equation

perfect reconstruction formulas and bounds on aliasing error in sub nyquist nonuniform sampling of multiband signals we examine the problem of periodic nonuniform sampling of a multiband signal and its reconstruction from the samples this sampling scheme which has been studied previously has an interesting optimality property that uniform sampling lacks one can sample and reconstruct the class spl bscr spl fscr of multiband signals with spectral support spl fscr at rates arbitrarily close to the landau 1969 minimum rate equal to the lebesgue measure of spl fscr even when spl fscr does not tile r under translation using the conditions for exact reconstruction we derive an explicit reconstruction formula we compute bounds on the peak value and the energy of the aliasing error in the event that the input signal is band limited to the span of spl fscr the smallest interval containing spl fscr which is a bigger class than the valid signals spl bscr spl fscr band limited to spl fscr we also examine the performance of the reconstruction system when the input contains additive sample noise

multidimensional multichannel fir deconvolution using grobner bases we present a new method for general multidimensional multichannel deconvolution with finite impulse response fir convolution and deconvolution filters using grobner bases previous work formulates the problem of multichannel fir deconvolution as the construction of a left inverse of the convolution matrix which is solved by numerical linear algebra however this approach requires the prior information of the support of deconvolution filters using algebraic geometry and grobner bases we find necessary and sufficient conditions for the existence of exact deconvolution fir filters and propose simple algorithms to find these deconvolution filters the main contribution of our work is to extend the previous grobner basis results on multidimensional multichannel deconvolution for polynomial or causal filters to general fir filters the proposed algorithms obtain a set of fir deconvolution filters with a small number of nonzero coefficients a desirable feature in the impulsive noise environment and do not require the prior information of the support moreover we provide a complete characterization of all exact deconvolution fir filters from which good fir deconvolution filters under the additive white noise environment are found simulation results show that our approaches achieve good results under different noise settings

linearized oscillation theorems for certain nonlinear delay partial difference equations this paper is concerned with the nonlinear delay partial difference equation am 1 n am n 1 am n pm nf am k n 1 0 where k and l are nonnegative integers and m n 0 1 some linearized oscillation theorems for this equation are obtained i e we will prove that under some conditions on f this equation has the same oscillatory character as an associated linear equation an existence result for positive solutions of this equation is obtained also

the uncertainty principle on groups the classical uncertainty principle asserts that both a function and its fourier transform cannot be largely concentrated on intervals of small measure donoho and stark siam j app math 49 1989 pp 906 931 have shown recently that both cannot be largely concentrated on any sets of small measure in the case of functions on the line or functions on finite cyclic groups and with concentrations measured in l 2 the purpose of this note is to extend these results to functions on sigma finite locally compact abelian groups with concentrations measured in l p 1 leqq p leqq 2 the first uncertainty principle on groups due to matolcsi and szucs d r acad sci paris 277 1973 pp 841 843 deals with full rather than large concentration asserting that if a function and its fourier transform are supported by sets t and w then the product of the haar measures of t and w must be at least 1 for the case of full concentration in r n benedicks j math anal appl 106 1985 pp 1

rank deficient submatrices of kronecker products of fourier matrices abstract we provide a set of maximal rank deficient submatrices of a kronecker product of two matrices a b and in particular the kronecker product of fourier matrices f f n 1 f n k we show how in the latter case maximal rank deficient submatrices can be constructed as tilings of rank one blocks several such tilings may be associated to any subgroup of the abelian group z n 1 z n k that corresponds to the matrix f the maximal rank deficient submatrices of f are also related to an uncertainty principle for fourier transforms over finite abelian groups for which we can then obtain stronger versions

a low order nonconforming finite element for reissner mindlin plates we propose a locking free element for plate bending problems based on the use of nonconforming piecewise linear functions for both rotations and deflections we prove optimal error estimates with respect to both the meshsize and the analytical solution regularity

finite element methods for a modified reissner mindlin free plate model the solution of the reissner mindlin plate problem with free boundary conditions presents a strong layer effect near the free edges as a consequence the solution is not even uniformly bounded even in h3 2 which implies that at most an o h1 2 uniform convergence rate can be reached by finite element methods in the h1 norm following instead the modified free boundary model presented by beirao da veiga and brezzi which gives more regular solutions better error estimates can be obtained in principle in this paper we present and analyze the extension of different families of well known optimal plate methods to this new model all the modified methods presented are proved to be optimal and free of locking

interpolatory filter banks and interpolatory wavelet packets abstract in classic wavelet sampling research is generally focused on interpolatory bases in individual spaces consideration is seldom given to relations among different interpolatory bases including those potentially related by filter banks in this paper we show that when an interpolatory basis exists in a space with a multiresolution analysis this space can be decomposed orthogonally into two subspaces in which interpolatory bases also exist furthermore similarly to standard orthogonal bases these interpolatory bases can be related by pairs of filters called interpolatory filter banks for such interpolatory filter banks it is shown that existence of interpolatory scaling functions generally leads to that of interpolatory wavelets and interpolatory wavelet packets with this result we further propose new algorithms for constructing such interpolatory wavelets wavelet packets from corresponding scaling functions in examples our algorithms are applied to some typical wavelet spaces demonstrating our theorems for interpolatory wavelets interpolatory wavelet packets and interpolatory filter banks

convolution idempotents with a given zero set we investigate the structure of n length discrete signals h satisfying h h h that vanish on a given set of indices we motivate this problem from examples in sampling fuglede s conjecture and orthogonal interpolation of bandlimited signals when n p m is a prime power we characterize all such h with a prescribed zero set in terms of base p expansions of nonzero indices in mathcal f 1 h

voronoi splines we introduce a framework for construction of non separable multivariate splines that are geometrically tailored for general sampling lattices voronoi splines are b spline like elements that inherit the geometry of a sampling lattice from its voronoi cell and generate a lattice shift invariant spline space for approximation in rd the spline spaces associated with voronoi splines have guaranteed approximation order and degree of continuity by exploiting the geometric properties of voronoi polytopes and zonotopes we establish the relationship between voronoi splines and box splines which are used for a closed form characterization of the former for cartesian lattices voronoi splines coincide with tensor product b splines and for the 2 d hexagonal lattice the proposed approach offers a reformulation of hex splines in terms of multi box splines while the construction is for general multidimensional lattices we particularly characterize bivariate and trivariate voronoi splines for all 2 d and 3 d lattices and specifically study them for body centered cubic and face centered cubic lattices

on the oscillation of two partial difference equations abstract this paper is concerned with two partial difference equations of the form a m 1 n a m n 1 a m n p m n a m k n l 0 and a m 1 n a m n 1 a m n i 1 u p i m n f i a m k i n l i 0 respectively we obtain sufficient conditions for the oscillation of all solutions of these equations

virtual bubbles and galerkin least squares type methods ga l s abstract the equivalence between stabilized finite element methods or galerkin least squares tyoe methods ga l s and the standard galerkin method with bubble functions is established in an abstract framework the results are applicable to various finite element spaces including high order elements and applications to the advective diffusive model and to the stokes problem are presented illustrating the potential of the abstract theory introduced here

spectral properties of sums of certain kronecker products we introduce two kinds of sums of kronecker products and their induced operators we study the algebraic properties of these two kinds of matrices and their associated operators the properties include their eigenvalues their eigenvectors and the relationships between their spectral radii or spectral abscissae furthermore two projected matrices of these kronecker products and their induced operators are also studied

multivariate generalized sampling in shift invariant spaces and its approximation properties abstract nowadays the topic of sampling in a shift invariant space is having a significant impact it avoids most of the problems associated with classical shannon s theory under appropriate hypotheses any multivariate function in a shift invariant space can be recovered from its samples at z d however in many common situations the available data are samples of some convolution operators acting on the function itself this leads to the problem of multivariate generalized sampling in shift invariant spaces this extra information on the functions in the shift invariant space will allow to sample in an appropriate sub lattice of z d in this paper an l 2 r d theory involving the frame theory is exhibited sampling formulas which are frame expansions for the shift invariant space are obtained in the case of overcomplete frame formulas the search of reconstruction functions with prescribed good properties is allowed finally approximation schemes using these generalized sampling formulas are included

discrete sampling a graph theoretic approach to orthogonal interpolation we study the problem of finding unitary submatrices of the n times n discrete fourier transform matrix in the context of interpolating a discrete bandlimited signal using an orthogonal basis this problem is related to a diverse set of questions on idempotents on mathbb z n and tiling mathbb z n in this work we establish a graph theoretic approach and connections to the problem of finding maximum cliques we identify the key properties of these graphs that make the interpolation problem tractable when n is a prime power and we identify the challenges in generalizing to arbitrary n finally we investigate some connections between graph properties and the spectral tile direction of the fuglede conjecture

distortion immune nine node displacement based quadrilateral thick plate finite elements that satisfy constant bending patch test we employ the linked interpolation concept to develop a higher order 9 node quadrilateral plate finite element with curved sides which passes the constant bending patch test for arbitrary node positions the linked interpolation for the plate displacements is expanded with three bubble parameters to get polynomial completeness necessary to satisfy the patch test in contrast to some other techniques the elements developed in this way retain a symmetric stiffness matrix at a marginal computational expense at the element level two new elements are generated using this concept and tested on several examples with curved sides or some kind of geometric distortion

a shear deformable plate element with an exact thin limit abstract we present a quadrilateral finite element developed within the framework of a shear deformable plate theory the interpolation for the rotation takes advantage of internal rotational degrees of freedom through the use of bubble functions while the interpolation for the transverse displacement is linked to the nodal rotations a careful study of the element behavior is performed using an extensive set of mixed patch tests results from several numerical examples are also presented the element has proper rank and excellent interpolating capacity moreover without using any ad hoc assumption e g energy balancing schemes the element presents no locking effects at all in fact the shear energy may be set identically equal to zero without introducing any ill conditioning in the problem thus recovering a proper thin plate limit

a four node plate bending element based on mindlin reissner plate theory and a mixed interpolation this communication discusses a 4 node plate bending element for linear elastic analysis which is obtained as a special case from a general nonlinear continuum mechanics based 4 node shell element formulation the formulation of the plate element is presented and the results of various example solutions are given that yield insight into the predictive capability of the plate and shell element

an improved treatment of transverse shear in the mindlin type four node quadrilateral element abstract an improved displacement methodology based on mindlin theory is developed and applied to a four node twelve degrees of freedom quadrilateral element special attention is directed toward enhancement of the transverse shear energy representation particularly a non uniform order kinematic interpolation scheme is accompanied by what is termed element appropriate shear correction factors the element matrices are consistently derived from hamilton s variational principle using exact numerical integration throughout the element stiffness matrix possesses correct rank and is well conditioned over the entire mindlin theory range of thickness to length ratios exceptionally good element performance is demonstrated by solutions to a set of static and vibrational problems

generalized sampling theorems in multiresolution subspaces it is well known that under very mild conditions on the scaling function multiresolution subspaces are reproducing kernel hilbert spaces rkhss this allows for the development of a sampling theory in this paper we extend the existing sampling theory for wavelet subspaces in several directions we consider periodically nonuniform sampling sampling of a function and its derivatives oversampling multiband sampling and reconstruction from local averages all these problems are treated in a unified way using the perfect reconstruction pr filter bank theory we give conditions for stable reconstructions in each of these cases sampling theorems developed in the past do not allow the scaling function and the synthesizing function to be both compactly supported except in trivial cases this restriction no longer applies for the generalizations we study here due to the existence of fir pr banks in fact with nonuniform sampling oversampling and reconstruction from local averages we can guarantee compactly supported synthesizing functions moreover local averaging schemes have additional nice properties robustness to the input noise and compression capabilities we also show that some of the proposed methods can be used for efficient computation of inner products in multiresolution analysis after this we extend the sampling theory to random processes we require autocorrelation functions to belong to some subspace related to wavelet subspaces it turns out that we cannot recover random processes themselves unless they are bandlimited but only their power spectral density functions we consider both uniform and nonuniform sampling

a sampling framework for solving physics driven inverse source problems partial differential equations are central to describing many physical phenomena in many applications these phenomena are observed through a sensor network with the aim of inferring their underlying properties leveraging from certain results in sampling and approximation theory we present a new framework for solving a class of inverse source problems for physical fields governed by linear partial differential equations specifically we demonstrate that the unknown field sources can be recovered from a sequence of so called generalized measurements by using multidimensional frequency estimation techniques next we show that for physics driven fields this sequence of generalized measurements can be estimated by computing a linear weighted sum of the sensor measurements whereby the exact weights of the sums correspond to those that reproduce multidimensional exponentials when used to linearly combine translates of a particular prototype function related to the green s function of our underlying field explicit formulas are then derived for the sequence of weights which map sensor samples to the exact sequence of generalized measurements when the green s function satisfies the generalized strang fix condition otherwise the same mapping yields a close approximation of the generalized measurements based on this new framework we develop practical noise robust sensor network strategies for solving the inverse source problem and then present numerical simulation results to verify their performance

the patch test a condition for assessing fem convergence this paper discusses various forms of the irons patch test and shows again that its satisfaction is a necessary convergence condition being equivalent to consistency further it is shown that the test can be applied for verification of stability requirements thus providing also a sufficient condition of convergence the patch test is useful in testing the order of asymptotic convergence and the satisfaction of the test gives a guide for development of certain incompatible element forms

generating bracelets in constant amortized time a bracelet is the lexicographically smallest element in an equivalence class of strings under string rotation and reversal we present a fast simple recursive algorithm for generating i e listing k ary bracelets using simple bounding techniques we prove that the algorithm is optimal in the sense that the running time is proportional to the number of bracelets produced this is an improvement by a factor of n where n is the length of the bracelets being generated over the fastest previously known algorithm to generate bracelets

oscillation criteria of a class of partial difference equations with delays this paper is concerned with the linear delay partial difference equation aa m 1 n 1 ba m 1 n ca n m n l da m n p m na m s n t 0 where s and t are two nonnegative integers a b c and d are positive constants and p i j i j e n 0 is a double real sequence sufficient conditions for this equation to be oscillatory are derived

completing a matrix when certain entries of its inverse are specified abstract suppose a a 11 a 12 a 21 a 22 is an n n matrix where a 11 has order p q if a is nonsingular let a 1 have the transposed partitioning b 11 b 12 b 21 b 22 first we show that the nullities of a 11 and b 22 are equal the remainder of our paper deals with the following completion problem suppose m 1 m 2 n 1 n 2 are nonnegative integers such that m 1 m 2 n 1 n 2 n 0

a class of mixed assumed strain methods and the method of incompatible modes a three field mixed formulation in terms of displacements stresses and an enhanced strain field is presented which encompasses as a particular case the classical method of incompatible modes within this frame work incompatible elements arise as particular compatible mixed approximations of the enhanced strain field the conditions that the stress interpolation contain piece wise constant functions and be l2 ortho gonal to the enhanced strain interpolation ensure satisfaction of the patch test and allow the elimination of the stress field from the formulation the preceding conditions are formulated in a form particularly convenient for element design as an illustration of the methodology three new elements are developed and shown to exhibit good performance a plane 3d elastic plastic quad an axisymmetric element and a thick plate bending quad the formulation described herein is suitable for non linear analysis

the finite element patch test revisited a computer test for convergence validation and error estimates abstract the patch test was introduced in the mid 1960s first as a check of satisfactory performance for elements violating continuity requirements and later simply as a necessary condition which all elements had to satisfy and which frequently checked simple programming errors in its original form the patch test simply verified that the minimum local consistency error was achievable and that the programming was correct later it was established that the test could also be used to find the convergence order of any finite element type while consistency is one of the necessary requirements for convergence the second is of course that of stability in the original early work only displacement type structural elements were involved and thus stability was ensured by rejecting elements which could form mechanisms in a 1986 reformulation of the patch test this stability or solvability requirement was formalized to all element assemblies and the patch test extended to become a generally applicable sufficient and necessary condition for convergence of all element forms 1 2 although some misinterpretation of the patch test has occurred in literature 3 4 the authors believe that with certain restrictions on regularity of the problem dealt with the satisfaction of the patch test is equivalent to the satisfaction of the more mathematical babuska brezzi criteria 5 6 in recent years another important use of the patch test has been found by babuska and his coworkers 7 9 this work tests the asymptotic effectivity of various error estimating procedures used with various element patterns

the minimum rank problem for circulants abstract the minimum rank problem is to determine for a graph g the smallest rank of a hermitian or real symmetric matrix whose off diagonal zero nonzero pattern is that of the adjacency matrix of g here g is taken to be a circulant graph and only circulant matrices are considered the resulting graph parameter is termed the minimum circulant rank of the graph this value is determined for every circulant graph in which a vertex neighborhood forms a consecutive set and in this case is shown to coincide with the usual minimum rank under the additional restriction to positive semidefinite matrices the resulting parameter is shown to be equal to the smallest number of dimensions in which the graph has an orthogonal representation with a certain symmetry property and also to the smallest number of terms appearing among a certain family of polynomials determined by the graph this value is then determined when the number of vertices is prime the analogous parameter over r is also investigated

tensor properties of multilevel toeplitz and related matrices a general proposal is presented for fast algorithms for multilevel structured matrices it is based on investigation of their tensor properties and develops the idea recently introduced by kamm and nagy in the block toeplitz case we show that tensor properties of multilevel toeplitz matrices are related to separation of variables in the corresponding symbol present analytical tools to study the latter expose truncation algorithms preserving the structure and report on some numerical results confirming advantages of the proposal

linked interpolation for reissner mindlin plate elements part iii an alternative quadrilateral an alternative quadrilateral element for reissner mindlin plates which extends the mixed formulation described in part 1 using linked interpolations is presented differently from element q4bl presented in part i this element q4bla employs four internal rotation nodes and a linear approximation for transverse shear forces it is shown that the element passes various patch tests it does not possess any zero energy mode and its performance is good for both thick and thin plates

a triangular element based on reissner mindlin plate theory a new triangular plate blending element based on the reissner mindlin theory is developed through a mixed formulation emanating from the hu washizu variational principle a main feature of the formulation is the use of a linear transverse shear interpolation scheme with discrete constraint conditions on the edges