global solvability on two step compact nilmanifolds we apply the methods of representation theory of nilpotent lie groups to study the convergence of fourier series of smooth global solutions to first order invariant partial differential equations df g in c of a two step compact nilmanifold we show that under algebraically well defined conditions on d in the complexified lie algebra smooth infinite dimensional irreducible solutions when they exist satisfy estimates strong enough to guarantee uniform convergence of the irreducible or primary fourier series to a smooth global solution such strong estimates are not possible on multidimensional tori

consistency of the group lasso and multiple kernel learning we consider the least square regression problem with regularization by a block l1 norm that is a sum of euclidean norms over spaces of dimensions larger than one this problem referred to as the group lasso extends the usual regularization by the l1 norm where all spaces have dimension one where it is commonly referred to as the lasso in this paper we study the asymptotic group selection consistency of the group lasso we derive necessary and sufficient conditions for the consistency of group lasso under practical assumptions such as model mis specification when the linear predictors and euclidean norms are replaced by functions and reproducing kernel hilbert norms the problem is usually referred to as multiple kernel learning and is commonly used for learning from heterogeneous data sources and for non linear variable selection using tools from functional analysis and in particular covar iance operators we extend the consistency results to this infinite dimensional case and also propose an adaptive scheme to obtain a consistent model estimate even when the necessary condition required for the non adaptive scheme is not satisfied

a new method for evaluating singular integrals in stress analysis of solids by the direct boundary element method the purpose of this paper is to report on a new and efficient method for the evaluation of singular integrals in stress analysis of elastic and elasto plastic solids respectively by the direct boundary element method bem triangle polar co ordinates are used to reduce the order of singularity of the boundary integrals by one degree and to carry out the integration over mappings of the boundary elements onto plane squares the method was subsequently extended to the cubature of singular integrals over three dimensional internal cells as occur in applications of the bem to three dimensional elasto plasticity for this purpose so called tetrahedron polar co ordinates were introduced singular boundary integrals stretching over either linear triangular or quadratic quadilateral isoparametric boundry elements and singular volume integrals extending over either linear tetrahedral or quadratic hexahedral isoparametric internal cells are treated in case of higher order isoparametric boundary elements and internal cells division into a number of subelements and subcells respectively is necessary the analytical investigation is followed by a numerical study restricted to the use of quadratic quadrilateral isoparametric boundary elements this is justified by the fact that such elements as opposed to linear elements yield singular boundary integrals which cannot be integrated analytically the results of the numerical investigation demonstrate the potential of the developed concept

global solvability on compact nilmanifolds of three or more steps we apply the methods of representation theory of nilpotent lie groups to study the convergence of fourier series of smooth global solutions to first order invariant partial differential equations df g in coo of a compact nilmanifold of three or more steps we investigate which algebraically welldefined conditions on d in the complexified lie algebra imply that smooth infinite dimensional irreducible solutions when they exist satisfy estimates strong enough to guarantee uniform convergence of the irreducible or primary fourier series to a smooth global solution this extends and improves the results of an earlier two step paper 0 introduction and preliminaries let n be a nilpotent lie group f n a compact nilmanifold and d a differential operator left invariant on n and viewed on f n if g e cco f n and if g is the orthogonal component of g in some irreducible subspace corresponding to the irreducible unitary representation ir then g7 e c f n too 2 modulo unitary equivalence we may think of g as being a c vector in any concrete realization or model of ir we will determine algebraically well defined conditions on first order and suitable higher order d under which the global solvability of df g in c f n is equivalent to the solvability of 7r d f g in the coo vectors for each ir in the spectrum of f n in one sense we will be presenting algebraic conditions on d for the reduction of a global geometrical problem on f n to a collection of purely group representation theoretic problems none of which needs to be regarded as living on the manifold f n operators d admitting such a reduction are called globally regular definition 0 1 in effect we will prove global regularity for suitable operators by showing that if the smooth solutions f of 7r d f g exist for each ir then they can be summed uniformly to a smooth global solution of df g globally regular operators are usually neither locally solvable nor onto c o f n 12 in order to make the necessary estimates on f7r we construct a suitable schrodinger model of ir which for convenience is not in the hilbert space l2 f n in this paper we extend to nilmanifolds of three or more steps and correct some of the results for two step nilmanifolds presented in 7 in order to describe the main results it will be helpful to review the classical situation on a torus t2 of two dimensions the situation being similar for tn with n 2 let d acl cx 3c cy and suppose for simplicity that a and 3 are real received by the editors july 24 1984 and in revised form april 10 1986 1980 mathematics subject classification 1985 revision primary 22e27 22e30 secondary 22e25 this research was supported in part by the national science foundation 1987 american mathematical society 0002 9947 87 1 00 25 per page

multiple emitter location and signal parameter estimation processing the signals received on an array of sensors for the location of the emitter is of great enough interest to have been treated under many special case assumptions the general problem considers sensors with arbitrary locations and arbitrary directional characteristics gain phase polarization in a noise interference environment of arbitrary covariance matrix this report is concerned first with the multiple emitter aspect of this problem and second with the generality of solution a description is given of the multiple signal classification music algorithm which provides asymptotically unbiased estimates of 1 number of incident wavefronts present 2 directions of arrival doa or emitter locations 3 strengths and cross correlations among the incident waveforms 4 noise interference strength examples and comparisons with methods based on maximum likelihood ml and maximum entropy me as well as conventional beamforming are included an example of its use as a multiple frequency estimator operating on time series is included

a representation theoretic criterion for local solvability of left invariant differential operators on nilpotent lie groups let l be a left invariant differential operator on the nilpotent lie group n it is shown that if 7r l is invertible for all irreducible representations ir in general position and if the inverses satisfy some mild technical conditions then l is locally solvable this result generalizes a theorem of l rothschild 1 let n be a connected simply connected nilpotent lie group with lie algebra ol and let l be an element of the enveloping algebra u ol of ol we may regard l as a left invariant operator on n acting by convolution that is we consider l as a distribution whose support is the identity and let 4 l be the operator associated with l similarly 4 l 4 is the right invariant operator associated with l this is not the standard convention see 4 for a discussion if x is any unitary representation of n then we may define 7rr l for l e ol 7rr l v d dt q7 exp tl v 1 10 in general 7r l is an unbounded operator one advantage of the definitions given above is that on appropriate domains g 4 l 4 t l l 4 l t 4 in 10 rothschild gave a representation theoretic condition that implied local solvability for certain left invariant operators l on the 2n 1 dimensional heisenberg group h l was required to be homogeneous with respect to certain dilations on the group her main theorem said that l was locally solvable if izj l had a bounded right inverse an operator b with 7rr l b i for all infinite dimensional representations xr of h the theorem refers to left inverses but the proof makes it clear that right inverses are required rockland had conjectured such a result in 9 in this note we generalize this theorem to arbitrary connected simply connected nilpotent lie groups in this generality we need to make two further hypotheses one about the smoothness of the inverse as the representations vary and one giving bounds on the norms of the inverses both hypotheses are trivially satisfied by the operators in 10 in the next section we give some results about plancherel measure and about central measures in u ol they are used in the proof of the main theorem given in 3 4 is devoted to assorted remarks received by theneditors january 24 1980 and in revised form march 11 1980 1980 mathematics subject classification primary 22e30 58g15 secondary 22e25 22e45 35a05 supported by nsf contract nsf mcs 78 02715 1981 american mathematical society 0002 9947 81 0000 01 07 0 3 00 113 this content downloaded from 157 55 39 17 on fri 02 sep 2016 04 58 50 utc all use subject to http about jstor org terms

advanced development of the boundary element method for steady state heat conduction considerable progress has been made in recent years toward advancing the state of the art in solid mechanics boundary element technology in the present work much of this new technology is applied in the development of a general purpose boundary element method bem for steady state heat conduction in particular the bem implementation involves the use of higher order conforming elements self adaptive integration and multi region capability two and three dimensional as well as axisymmetric analyses are incorporated within a unified framework in addition techniques are introduced for the calculation of boundary flux and for the inclusion of thermal resistance across interfaces as a final extension an efficient formulation is developed for the analysis of solid three dimensional bodies with embedded holes for this last class of problems the new bem formulation is particularly attractive since use of the alternatives i e finite element or finite difference methods is not practical a number of detailed examples illustrate the suitability and robustness of the present approach for steady state heat conduction

a general and effective way for evaluating the integrals with various orders of singularity in the direct boundary element method a general and efficient technique is developed for the evaluation of the integrals with various orders of singularity such as occur in the three dimensional boundary element method bem generalized extended triangle tetrahedron polar co ordinate mappings together with two conditions are used to remove the singularity of the integrals and to evaluate the corresponding non singular ones in a new numerical space triangle and tetrahedron polar co ordinates in reference 1 are proved to be a special case of the generalized ones in this paper with the developed idea boundary element results converge rapidly towards the analytical solutions for the strongly singular integrals evaluated directly and the analytical solutions can be gained in principle even when employing higher order triangular boundary elements and tetrahedral cells the generality and practicability of the method are demonstrated in the case of higher order elements discontinuous elements and large engineering problems

quadrature over a pyramid or cube of integrands with a singularity at a vertex a simple transformation is introduced which facilitates the evaluation of integrals over certain square based pyramids or cubes in cases where the integrand has a singularity at a vertex

asymptotic performance analysis of direction finding algorithms based on fourth order cumulants in the narrow band array processing context the use of higher order statistics has been often advocated because consistent and asymptotically unbiased parameter estimates can be obtained without it being necessary to know to model or to estimate the spatial covariance of the noise as long as it is normally distributed however experimentation shows that this noise insensitivity is traded for increased variability of the parameter estimates the main purpose of this contribution is to derive and work out closed form expressions of the asymptotic covariance of music like direction of arrival estimates based on two fourth order cumulant matrices the diagonal slice and the contracted quadricovariance this is compared with the standard covariance based music estimate establishing on a rational basis the domain of applicability of higher order statistics for doa estimation in particular the actual impact of the noise variance and of the dynamic range of the sources is investigated this asymptotic performance analysis is achieved within a general framework which we believe to be of general interest

global regularity on 3 dimensional solvmanifolds let m be any 3 dimensional nonabelian compact solvmanifold we apply the methods of representation theory to study the convergence of fourier series of smooth global solutions to first order invariant partial differential equations df g in c m we show that smooth infinite dimensional irreducible solutions when they exist satisfy estimates strong enough to guarantee uniform convergence of the irreducible or primary fourier series to a smooth global solution

accurate numerical integration of singular boundary element kernels over boundaries with curvature use of the green function for the solution of boundary value problems frequently results in singular integral equations algorithms are presented for the accurate and efficient treatment of singular kernels frequently encountered in the boundary element method bem they are based upon the use of appropriately weighted gaussian quadrature formulae together with numerical geometrical transformations of the region of integration the use of high order subdomain expansion functions for interpolation over nonplanar elements allows boundary curvature to be accommodated in particular the handling of green functions with logarithmic and r 1 behaviour are detailed volume integrals with r 2 singularity are outlined operations are performed on a simplex thus resulting in generality and ease of automation this scheme has been incorporated into boundary element method software and successfully applied to a variety of problems

globally hypoelliptic systems of vector fields on nilmanifolds abstract we show that a system l of real vector fields on a general compact nilmanifold n induced by the lie algebra n of n is globally hypoelliptic gh iff 1 the symbols of the vector fields of l projected onto the associated torus t n n n as functions on the integral lattice t collectively decrease at infinity not faster than a reciprocal of a polynomial and 2 the lie subalgebra of n that l generates is not annihilated by any non zero integral linear functional on any n j n j 1 j 0 1 n j 1 n n j n o n it follows that gh is equivalent to injectivity of the system l on the dual on the space of c vectors of all the non trivial representations in the spectrum of n a rockland type condition plus a number theoretic condition on l on the associated torus to avoid small divisors

compressive music revisiting the link between compressive sensing and array signal processing the multiple measurement vector mmv problem addresses the identification of unknown input vectors that share common sparse support even though mmv problems have been traditionally addressed within the context of sensor array signal processing the recent trend is to apply compressive sensing cs due to its capability to estimate sparse support even with an insufficient number of snapshots in which case classical array signal processing fails however cs guarantees the accurate recovery in a probabilistic manner which often shows inferior performance in the regime where the traditional array signal processing approaches succeed the apparent dichotomy between the probabilistic cs and deterministic sensor array signal processing has not been fully understood the main contribution of the present article is a unified approach that revisits the link between cs and array signal processing first unveiled in the mid 1990s by feng and bresler the new algorithm which we call compressive music identifies the parts of support using cs after which the remaining supports are estimated using a novel generalized music criterion using a large system mmv model we show that our compressive music requires a smaller number of sensor elements for accurate support recovery than the existing cs methods and that it can approach the optimal bound with finite number of snapshots even in cases where the signals are linearly dependent

a variable stiffness type boundary element formulation for axisymmetric elastoplastic media a new direct elastoplastic boundary element formulation for axisymmetric bodies is developed the procedure exploits certain features of the constitutive relationship to express the unknwon non linear initial stress rate tensor as a scalar quantity which can then be eliminated from the boundary equation system though a back substitution of the modified stress rate equations this formulation is implemented in a multi region system that utilizes quadratic isoparametric shape functions to model the geometry and field variables of the boundary and domain of a body the kernel functions are presented in appendix i and the singularities associated with these kernels are discussed new techniques are described to calculate the coefficients associated with the singular nodes

some notes on singular integral techniques in boundary element analysis the paper presented here reviews the numerical techniques used currently to calculate the singular integrals and nearly singular integrals in the boundary element analysis some incorrect algorithms published before are discussed and a new numerical technique to calculate the nearly singular integral is developed the numerical results show a significant improvement in both accuracy and efficiency compared to the traditional adaptive gaussian quadrature and subdivision techniques

singular cubature over triangles integration over triangles where the integrand contains a singularity at one of the vertices is considered in this paper the applications for such problems arise in using the boundary element method low order formulae are presented and compared to previous methods where the singularity was not considered

global solvability on compact heisenberg manifolds we apply the methods of primary and irreducible fourier series on compact nilmanifolds to determine the ranges of all first order invariant operators on the compact heisenberg manifolds we show that the sums of primary solutions behave better on these manifolds than on any multidimensional torus

reusable intrinsic sample point risp algorithm for the efficient numerical integration of three dimensional curved boundary elements this paper presents an algorithm that provides an order of magnitude gain in the computational performance of the numerical integration of the boundary integral equations for three dimensional analysis existing algorithms for numerical integration have strategically clustered integration sample points based on the relative proximity of the load points to the boundary element being integrated using element subdivision or element co ordinate transformation the emphasis in these techniques has been on minimizing the number of sample points required to obtain a given level of accuracy the present algorithm while closely following the spirit of these earlier approaches employs a discrete number of sets of predetermined customized near optimum sample point quantities associated with the intrinsic boundary element the ability created by this approach to reuse sample point geometric information of the actual element allows for the realization of substantive computational economy this algorithm provides accurate and efficient numerical results both when load points are far from and when they are on the boundary element being integrated numerical results are provided to demonstrate the substantial economy achieved through the use of the present algorithm

singular and near singular integrals in the bem a global approach a unified approach for analyzing and evaluating singular and near singular integrals relevant to the boundary element method bem is obtained the original n dimensional integral is mapped to one that is performed exclusively on the boundary of the integration domain therefore the dimensionality of the integration is reduced by l and integrand evaluations close to the singularity are avoided the new formulae are obtained through a straightforward one dimensional analytical integration regardless of the dimensionality of the problem and can be used generally for numerical integration the formulae for evaluating near singular integrals are a generalization of those obtained by the invariant imbedding technique for singular integrals and reduce to the singular integral formulae in the limit in this sense the singular integrals are treated as limits or continuations of nonsingular ones this leads to a general interpretation that includes the singular integrals as a special case in particular t