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For a degree 2n finite sequence of real numbers \(\beta \equiv \beta ^{(2n)}= \{ \beta _{00},\beta _{10},\)\(\beta _{01},\ldots , \beta _{2n,0}, \beta _{2n-1,1},\ldots , \beta _{1,2n-1},\beta _{0,2n} \}\) to have a representing measure \(\mu \), it is necessary for the associated moment matrix \(\mathcal {M}(n)\) to be positive semidefinite, and for the algebraic variety associated to \(\beta \), \(\mathcal {V}_{\beta } \equiv \mathcal {V}(\mathcal {M}(n))\), to satisfy \({\text {rank}} \mathcal {M}(n)\le {\text {card}} \mathcal {V}_{\beta }\) as well as the following consistency condition: if a polynomial \(p(x,y)\equiv \sum _{ij}a_{ij}x^{i}y^j\) of degree at most 2n vanishes on \(\mathcal {V}_{\beta }\), then the Riesz functional\(\Lambda (p) \equiv p(\beta ):=\sum _{ij}a_{ij}\beta _{ij}=0\). Positive semidefiniteness, recursiveness, and the variety condition of a moment matrix are necessary and sufficient conditions to solve the quadratic (\(n=1\)) and quartic (\(n=2\)) moment problems. Also, positive semidefiniteness, combined with consistency, is a sufficient condition in the case of extremal moment problems, i.e., when the rank of the moment matrix (denoted by r) and the cardinality of the associated algebraic variety (denoted by v) are equal. For extremal sextic moment problems, verifying consistency amounts to having good representation theorems for sextic polynomials in two variables vanishing on the algebraic variety of the moment sequence. We obtain such representation theorems using the Division Algorithm from algebraic geometry. As a consequence, we are able to complete the analysis of extremal sextic moment problems.

With the advance of quantum information technology, the question of how to most efficiently test quantum circuits is becoming of increasing relevance. Here we introduce the statistics of lengths of measurement sequences that allows one to certify entanglement across a given bi-partition of a multi-qubit system over the possible sequence of measurements of random unknown states, and identify the best measurement strategies in the sense of the (on average) shortest measurement sequence of (multi-qubit) Pauli measurements. The approach is based on the algorithm of truncated moment sequences that allows one to deal naturally with incomplete information, i.e. information that does not fully specify the quantum state. We find that the set of measurements corresponding to diagonal matrix elements of the moment matrix of the state are particularly efficient. For symmetric states their number grows only like the third power of the number $N$ of qubits. Their efficiency grows rapidly with $N$, leaving already for $N=4$ less than a fraction $10^{-6}$ of randomly chosen entangled states undetected., 12 pages, 9 figures

We propose a new solver for the sparse spikes superresolution problem over the space of Radon measures. A common approach to off-the-grid deconvolution considers semidefinite relaxations of the tot...

Low-rank compression of moment matrix blocks, representing interactions between clusters of vector basis/testing functions, is often done by a rank-revealing analysis of a simplified grid-to-grid interaction matrix, followed by inter/anterpolation and recompression. While this can provide a close estimate of the rank, it can lead to large errors in the compressed representation due to misrepresentation by grids of the underlying relative orientations of the vector functions. The potential hazard is demonstrated via a simplified example for the dyadic formulation.

The major problem in applying a meshless (RPIM) method is that when a large number of sampling nodes is used, the condition number of the moment matrix increases, and numerical solution becomes unstable due to the required inversion of the matrix. To address the problem, in this paper, with the radial point interpolation meshless (RPIM) method as an example, the moment matrix is first diagonalized and then the associated singular eigenvalues that cause the instability are truncated. As a result, the dependence of the stability on the number of the sampling nodes is removed. Numerical experiments are conducted, and the results have shown that the proposed algorithm are stable irrespective of number of sampling nodes; they pay the way forward for the meshless method to be applied to practical structures.

We first discuss the spherical Aluthge and spherical Duggal transforms for commuting pairs of operators on Hilbert space. Second, we study the fixed points of these transforms, which are the spherically quasinormal commuting pairs. In the case of commuting 2-variable weighted shifts, we prove that spherically quasinormal pairs are intimately related to spherically isometric pairs. We show that each spherically quasinormal 2-variable weighted shift is completely determined by a subnormal unilateral weighted shift (either the 0-th row or the 0-th column in the weight diagram). We then focus our attention on the case when this unilateral weighted shift is recursively generated (which corresponds to a finitely atomic Berger measure). We show that in this case the 2-variable weighted shift is also recursively generated, with a finitely atomic Berger measure that can be computed from its 0-th row or 0-th column. We do this by invoking the relevant Riesz functionals and the functional calculus for the columns of the associated moment matrix.

In this work, our aim is to obtain conditions to assure polynomial approximation in Hilbert spaces L 2 ( &mu, ) , with &mu, a compactly supported measure in the complex plane, in terms of properties of the associated moment matrix with the measure &mu, To do it, in the more general context of Hermitian positive semidefinite matrices, we introduce two indexes, &gamma, ( M ) and &lambda, ( M ) , associated with different optimization problems concerning theses matrices. Our main result is a characterization of density of polynomials in the case of measures supported on Jordan curves with non-empty interior using the index &gamma, and other specific index related to it. Moreover, we provide a new point of view of bounded point evaluations associated with a measure in terms of the index &gamma, that will allow us to give an alternative proof of Thomson&rsquo, s theorem, by using these matrix indexes. We point out that our techniques are based in matrix algebra tools in the framework of Hermitian positive definite matrices and in the computation of certain indexes related to some optimization problems for infinite matrices.

In the present work, we are interested to develop a meshless approach, based on the strong form MLS approximation with additional constraints, to solve the nonlinear elastic and elsto-plastic problems for regular and irregular distribution of points. We adopt a plastic behavior law based on the total deformation theory, which is very convenient when the physical nonlinearity is more important than the effect of irreversible process and the loading history. In plasticity, one encounters discontinuities of rigidity where the application of asymptotic developments seems difficult or impossible. To apply the Taylor series expansion, regularization methods have been adapted. The strong form MLS approximation allows us to avoid the inconvenient of the numerical integration, while the asymptotic developments help us to reduce the computation cost observed in the incremental law of plasticity and the iterative methods. For irregular points distribution, we can get an ill-posed least squares problems due to a singular moment matrix of MLS approximation. To avoid this difficulty, we propose a modified MLS approximation by introducing additional constraints which allows to increase the error functional used in the derivation of the shape functions.

To reduce the dimensionality of the second-order response surface design model, variance component indices under imposing and non imposing restrictions on the moment matrix toward the orthogonality are derived and presented and the same is illustrated with suitable examples in this article.

For an ideal I with a positive-dimensional real variety V R ( I ) , based on moment relaxations, we study how to compute a Pommaret basis which is simultaneously a Grobner basis of an ideal J generated by the kernel of a truncated moment matrix and satisfying I ⊆ J ⊆ I ( V R ( I ) ) , V R ( I ) = V C ( J ) ∩ R n . We provide a certificate consisting of a condition on coranks of moment matrices for terminating the algorithm. For a generic δ-regular coordinate system, we prove that the condition is satisfiable in a large enough order of moment relaxations.

Recent breakthroughs have been made in the use of semi-definite programming and its application to real polynomial solving. For example, the real radical of a zero dimensional ideal, can be determined by such approaches. Some progress has been made on the determination of the real radical in positive dimension by Ma, Wang and Zhi[5, 4]. Such work involves the determination of maximal rank semidefinite matrices. Existing methods are computationally expensive and have poorer accuracy on larger examples. In previous work we showed that regularity in the form of the Slater constraint qualification (strict feasibility) fails for the moment matrix in the SDP feasibility problem[6]. We used facial reduction to obtain a smaller regularized problem for which strict feasibility holds. However we did not have a theoretical guarantee that our methods, based on facial reduction and Douglas-Rachford iteration ensured the satisfaction of the maximum rank condition. Our work is motivated by the problems above. We discuss how to compute the moment matrix and its kernel using facial reduction techniques where the maximum rank property can be guaranteed by solving the dual problem. The facial reduction algorithms on the primal form is presented. We give examples that exhibit for the first time additional facial reductions beyond the first which are effective in practice with much better accuracy than SeDuMi(CVX).

Based on the continuum mechanics principles, a rigorous formulation is presented for the linearized stiffness equation of three-dimensional beam elements with account taken of the joint moment equilibrium in the deformed configuration C 2 . By sticking to the Bernoulli–Euler hypothesis of plane sections and elasticity definitions for stress resultants, the bending moments and torque of the element are shown to be quasi- and semi-tangential, respectively, in the updated Lagrangian formulation. Further, by invoking the moment equilibrium conditions for structural nodes at C 2 , the induced moment matrix that first appears to be antisymmetric on the element level turns out to be symmetric upon assembly of all elements on the structural level. The joint equilibrium conditions at C 2 , as represented by the induced moment matrix, are central not only to the out-of-plane buckling analysis of angled frames, but also to the simulation of curved beams by the straight-beam elements. Examples on the buckling of angled frames and curved beams are provided to support the theory presented.

The domain of influence (doi) plays a very important role in the Element Free Galerkin (EFGM) method. It contributes to the calculation of the weight function and so influences the accuracy of the approximation. The radius of the (doi) determines the number of nodes contained in the support. The compromise is that the radius must not be large to not lose the local aspect of the moving least square method (MLS) used in (EFGM) and not small to have enough node so a regular moment matrix. This paper compares several theories for calculating the radius of (doi).

We present an alternative solution to nonsingular cubic moment problems, using techniques that are expected to be useful for higher-degree truncated moment problems. In particular, we apply the theory of recursively determinate moment matrices to deal with a case of rank-increasing moment matrix extensions.

A three dimensional implementation of the recently introduced balanced electromagnetic absorber (BEMA) method is presented herein. In the BEMA, a balanced (Weston-type) absorber, characterized by both electric and magnetic loss mechanisms, is placed within the null field inside the equivalent surface currents replacing a perfectly conducting or homogeneous scatterer. The absorber, referred to as the filler, substantially reduces interactions between pairs of opposing basis/testing functions. The resultant moment matrix, formulated with the filler Green’s function, is thinned accordingly. Moreover, most annulled elements need not be computed at all, thereby reducing substantially the matrix fill time. The lossy nature of the Green’s function also serves to eliminate spurious internal resonances and thus makes the electric or magnetic field integral equation matrix well conditioned without resorting to a combined field integral equation.

We consider T-optimal experiment design problems for discriminating multi-factor polynomial regression models where the design space is defined by polynomial inequalities and the regression parameters are constrained to given convex sets. Our proposed optimality criterion is formulated as a convex optimization problem with a moment cone constraint. When the regression models have one factor, an exact semidefinite representation of the moment cone constraint can be applied to obtain an equivalent semidefinite program. When there are two or more factors in the models, we apply a moment relaxation technique and approximate the moment cone constraint by a hierarchy of semidefinite-representable outer approximations. When the relaxation hierarchy converges, an optimal discrimination design can be recovered from the optimal moment matrix, and its optimality is confirmed by an equivalence theorem. The methodology is illustrated with several examples., Comment: 22 pages, 8 figures

In this paper, a new measurement to compare two large-scale graphs based on the theory of quantum probability is proposed. An explicit form for the spectral distribution of the corresponding adjacency matrix of a graph is established. Our proposed distance between two graphs is defined as the distance between the corresponding moment matrices of their spectral distributions. It is shown that the spectral distributions of their adjacency matrices in a vector state includes information not only about their eigenvalues, but also about the corresponding eigenvectors. Moreover, we prove that the proposed distance is graph invariant and sub-structure invariant. Examples with various graphs are given, and distances between graphs with few vertices are checked. Computational results for real large-scale graphs show that its accuracy is better than any existing methods and time cost is extensively cheap.

This article proposes 2 classes of ridge generalized least squares (GLS) procedures for structural equation modeling (SEM) with unknown population distributions. The weight matrix for the first class of ridge GLS is obtained by combining the sample fourth-order moment matrix with the identity matrix. The weight matrix for the second class is obtained by combining the sample fourth-order moment matrix with its diagonal matrix. Empirical results indicate that, with data from an unknown population distribution, parameter estimates by ridge GLS can be much more accurate than those by either GLS or normal-distribution-based maximum likelihood; and standard errors of the parameter estimates also become more accurate in predicting the empirical ones. Rescaled and adjusted statistics are proposed for overall model evaluation, and they also perform much better than the default statistic following from the GLS method. The use of the ridge GLS procedures is illustrated with a real data set.

Non-convex of an optimal power dispatch problem makes it difficult to guarantee the global optimum. This paper presents a convex relaxation approach, called the Moment Semidefinite Programming (MSDP) method, to facilitate the search for deterministic global optimal solutions. The method employs a sequence of moments, which can linearize polynomial functions and construct positive semidefinite moment matrices, to form an SDP convex relaxation for power dispatch problems. In particular, the rank of the moment matrix is used as a sufficient condition to ensure the global optimality. The same condition can also be leveraged to estimate the number of global optimal solution(s). This method is effectively applied to {0,1}-economic dispatch (ED) problems and optimal power flow (OPF) problems. Simulation results showed that the MSDP method is capable of solving {0,1}-ED problems with integer values directly, and is able to identify if more than one global optimal solutions exist. In additional, the method can obtain rank-1 moment matrices for OPF’s counterexamples of existing SDP method, this ensures the global solution and overcomes the problem that existing SDP method cannot meet the rank-1 condition sometimes.

Motivated by [8] we study the existence of the inverse of an infinite Hermitian positive definite matrix (in short, HPD matrix) from the point of view of the asymptotic behaviour of the smallest eigenvalues of the finite sections. We prove a sufficient condition to assure the inversion of an HPD matrix with square summable rows. For infinite Toeplitz matrices we introduce the notion of asymptotic Toeplitz matrix and we show that, under certain assumptions, the inverse of an infinite Toeplitz positive definite matrix is asymptotic Toeplitz. Such inverses are computed in terms of the limits of the coefficients of the associated orthogonal polynomials. We apply these results in the context of the theory of orthogonal polynomials. In particular, we show that for measures on the unit circle T verifying that the smallest eigenvalue of the finite sections of the corresponding moment matrix are away from zero in the limit we may assure the existence of all the limits of the coefficients of the orthonormal polynomials with respect to such measures.

A new parameterization of the effective dipole moment matrix elements for the asymmetric top molecules is suggested. It is based on the introduction of the Herman-Wallis type factors for the matrix elements. The advantage of this new parameterization consists in the utilization of the matrix element parameters which describe explicitly the dependence of the principal matrix element parameter on the rotational quantum numbers J and K. The application of this approach to the open-shell NO2 molecule is discussed.

Person re-identification (re-id) has drawn significant attention in the recent decade. The design of view-invariant feature descriptors is one of the most crucial problems for this task. Covariance descriptors have often been used in person re-id because of their invariance properties. More recently, a new state-of-the-art performance was achieved by also including first-order moment and two-level Gaussian descriptors. However, using second-order or lower moments information might not be enough when the feature distribution is not Gaussian. In this paper, we address this limitation, by using the empirical (symmetric positive definite) moment matrix to incorporate higher order moments and by applying the on-manifold mean to pool the features along horizontal strips. The new descriptor, based on the on-manifold mean of a moment matrix (moM), can be used to approximate more complex, non-Gaussian, distributions of the pixel features within a mid-sized local patch. We have evaluated the proposed feature on five widely used re-id datasets. The experiments show that the moM and hierarchical Gaussian descriptor (GOG) [30] features complement each other and that using a combination of both features achieves a comparable performance with the state-of-the-art methods.

Chained correlation inequalities involving pairwise correlations of qubit observables in the equatorial plane are constructed based on the positivity of a sequence of moment matrices. When a jointly measurable set of fuzzy POVMs is employed in first measurement of every pair of sequential measurements, the chained pairwise correlations do not violate the classical bound imposed by the moment matrix positivity. We identify that incompatibility of measurements is only necessary, but not sufficient, in general, for the violation of the inequality. On the other hand, there exists a one-to-one equivalence between the degree of incompatibility (which quantifies the joint measurability) of the equatorial qubit observables and the optimal violation of a non-local steering inequality, proposed by Jones and Wiseman (Phys. Rev. A, 84, 012110 (2011)). To this end, we construct a local analogue of this steering inequality in a single qubit system and show that its violation is a mere reflection of measurement incompatibility of equatorial qubit POVMs, employed in first measurements in the sequential unsharp-sharp scheme., Comment: 12 pages, three tables; restructured the contents slightly

In this article we study the large $N$ asymptotics of complex moments of the absolute value of the characteristic polynomial of a $N\times N$ complex Ginibre random matrix with the characteristic polynomial evaluated at a point in the unit disk. More precisely, we calculate the large $N$ asymptotics of $\mathbb{E}|\det(G_N-z)|^{\gamma}$, where $G_N$ is a $N\times N$ matrix whose entries are i.i.d and distributed as $N^{-1/2}Z$, $Z$ being a standard complex Gaussian, $\mathrm{Re}(\gamma)>-2$, and $|z, Comment: Result and proof extended to include complex-valued gamma; some reference added

The bivariate moment problem for a sequence β ≡ β ( 6 ) \beta \equiv \beta ^{(6)} of degree 6 6 remains unsolved, but we prove that if the associated 10 × 10 10 \times 10 moment matrix M 3 ( β ) M_{3}(\beta ) satisfies M 3 ⪰ 0 M_{3}\succeq 0 and r a n k M 3 ≤ 6 rank~M_{3}\le 6 , then β \beta admits a sequence of approximate representing measures, and β ( 5 ) \beta ^{(5)} has a representing measure. More generally, let F d ¯ \overline {\mathcal {F}_{d}} denote the closure of the positive flat moment matrices of degree 2 d 2d in n n variables. Each matrix in F d ¯ \overline {\mathcal {F}_{d}} admits computable approximate representing measures, and in 2013, Jiawang Nie and the author began to study concrete conditions for membership in this class. Let β ≡ β ( 2 d ) = { β i } i ∈ Z + n , | i | ≤ 2 d \beta \equiv \beta ^{(2d)}=\{\beta _{i}\}_{ i\in \mathbb {Z}_{+}^{n},|i| \leq 2d } , β 0 > 0 \beta _{0}>0 , denote a real n n -dimensional sequence of degree 2 d 2d . If the corresponding moment matrix M d ≡ M d ( β ) M_{d}\equiv M_{d}(\beta ) is the limit of a sequence of positive flat moment matrices { M d ( k ) } \{M_{d}^{(k)}\} , i.e., M d ( k ) ⪰ 0 M_{d}^{(k)}\succeq 0 and r a n k M d ( k ) = r a n k M d − 1 ( k ) rank~M_{d}^{(k)} = rank~M_{d-1}^{(k)} , then i) M d ⪰ 0 M_{d}\succeq 0 , ii) r a n k M d ≤ ρ d − 1 ≡ d i m R [ x 1 , … , x n ] d − 1 rank~M_{d} \le \rho _{d-1} \equiv dim~\mathbb {R}[x_{1},\ldots ,x_{n}]_{d-1} , and iii) β ( 2 d − 1 ) \beta ^{(2d-1)} admits a representing measure. We extend our earlier results by proving, conversely, that for n = 2 n=2 , if M d M_{d} satisfies certain positivity and rank conditions related to i)-iii), then M d M_{d} is the limit of positive flat moment matrices.

In dynamic panel data (DPD) models, thegeneralized method of moments (GMM) estimation gives efficient estimators. However, this efficiency is affected by the choice of the initial weighting matrix. In practice, the inverse of the moment matrix of the instruments has been used as an initial weighting matrix which led to a loss of efficiency. Therefore, we will present new GMM estimators based on optimal or suboptimal weighting matrices in GMM estimation. Monte Carlo study indicates that the potential efficiency gain by using these matrices. Moreover, the bias and efficiency of the new GMM estimators are more reliable than any other conventional GMM estimators

An improved MUSIC algorithm based on third-order cyclic moment is proposed to estimate the bearing and range parameters of near-field cyclostationary sources. The algorithm adopts the uniform linear array, structures the third-order cyclic moment matrix by the array outputs, and utilizes the propagator method to replace the singular value decomposition, calculate the signal noise subspace directly. Compared with the traditional MUSIC algorithm, the proposed algorithm has high estimation precision and low computational complexity, and effectively solves the two-dimensional parameters estimation problems of near-field cyclostationary sources in the case of interfering signals and non-Gaussian white noise. The performance of the proposed method can be verified by computer simulations.

Linear canonical transform (LCT) form a three-parameter family of intergral transforms with wide application in optics. In this paper, we investigate the linear canonical Wigner distribution (LCWD) which is based on the LCT and the classical Wigner distribution (WD). Firstly, the definition of LCWD is discussed. Moreover, the transformation law for the LCWD through a first-order optical system is derived. This new phase-space distribution provides analysis of signals in both space and LCT domains simultaneously. Then, the main properties of LCWD are investigated in detail. Finally, the application of the LCWD is presented. The LCWD is found to be the appropriate phase-space distribution function for light-beam characterization in first-order optical system. Moreover, the moment matrix formalism for beam characterization is studied.

We show that the planar orthogonal polynomials with $l$ logarithmic singularities in the potential are the multiple orthogonal polynomials (Hermite-Pad\'e polynomials) of Type II with $l$ measures. We also find the ratio between the determinant of the moment matrix corresponding to the multiple orthogonal polynomials and the determinant of the moment matrix from the original planar measure.

Controlling non-linear dynamic model of quadrotor is a challenging task since the quadrotor is considered as an under-actuated system with highly coupled dynamics due to both the six Degrees Of Freedom (DOF) motion and four control input variables. The objective of this research is to investigate the implementation of fuzzy logic concepts using linearised inverted moment matrix in order to design Multi-Inputs-Multi-Outputs (MIMO) fuzzy flight control system for the quadratic non-linear dynamic model. Non-linear mathematical model of the quadrotor is derived and MIMO attitude and position fuzzy controllers are designed. The invert of moment matrix is linearised and differential actions of PD-fuzzy controllers are considered as well. The quadrotor dynamic behaviour is simulated and evaluated using M-file programs, the results have shown that using PD-fuzzy controller type is capable to achieve the design requirements and significantly improves the performance of the quadrotor stability characteristics.

A complete classification of SL(n) covariant matrix-valued valuations on functions with finite second moments is obtained. It is shown that there is a unique homogeneous such valuation. This valuation turns out to be the moment matrix.

Let y y (2d) =fyigi2Z n,jij62d denote a real n-dimensional mul- tisequence of degree 2d, y0 > 0. Let Ly : R2d(x1, . . . , xn) 7! R denote the Riesz functional, defined by Ly(Âjij62d ai x i ) = Â aiyi, and let Md(y) denote the corresponding moment matrix. Positivity of Ly plays a significant role in the Truncated Moment Problem and in the Polynomial Optimization Prob- lem, but concrete conditions for positivity are unknown in general. Md(y) is flat if rankMd(y) = rankMd 1(y); it is known that if Md(y) is positive semi- definite and flat, then y has a representing measure (and Ly is positive). Let Fd := fy y (2d) : Md(y) 0 is flatg. If y 2 F d (the closure), then y does not necessarily have a representing measure, but Ly is positive, so Md(y) 0 and, moreover, rankMd(y) 6 dimRd 1(x1, . . . , xn). We prove, conversely, that these positivity and rank conditions for Md(y) are sufficient for membership inF d in two basic cases: when n = 1, d > 1, and when n = d = 2.

In this paper, we study optimal experimental design problems with a broad class of smooth convex optimality criteria, including the classical A-, D-, and $p$th mean criterion. In particular, we propose an interior point (IP) method for them and establish its global convergence. Further, by exploiting the structure of the Hessian matrix of the optimality criteria, we derive an explicit formula for computing its rank. Using this result, we then demonstrate that the Newton direction arising in the IP method can be computed efficiently via the Sherman--Morrison--Woodbury formula when the size of the moment matrix is small relative to the size of the design space. Finally, we compare our IP method with the widely used multiplicative algorithm introduced by [S. D. Silvey, D. M. Titterington, and B. Torsney, Commun. Statist. Theory Methods, 7 (1978), pp. 1379--1389] and the standard IP solver SDPT3 [K. C. Toh, M. J. Todd, and R. H. Tutuncu, Optim. Methods Softw., 11/12 (1999), pp. 545--581], [R. H. Tutuncu, K. C...

Positive semidefiniteness, recursiveness, and the variety condition of a moment matrix are necessary and sufficient conditions to solve the quadratic and quartic moment problems. Also, positive semidefiniteness, combined with another necessary condition, consistency, is a sufficient condition in the case of extremal moment problems, i.e., when the rank of the moment matrix (denoted by r ) and the cardinality of the associated algebraic variety (denoted by v ) are equal. However, these conditions are not sufficient for non-extremal sextic or higher-order truncated moment problems. In this paper we settle three key instances of the non -extremal (i.e., r v ) sextic moment problem, as follows: when r = 7 , positive semidefiniteness, consistency and the variety condition guarantee the existence of a 7-atomic representing measure; when r = 8 we construct two determining algorithms, corresponding to the cases v = 9 and v = + ∞ . To accomplish this, we generalize the rank-reduction technique developed in previous work, where we solved the nonsingular quartic moment problem and found an explicit way to build a representing measure.

A new arbitrary Polynomial Chaos (aPC) method is presented for moderately high-dimensional problems characterised by limited input data availability. The proposed methodology improves the algorithm of aPC and extends the method, that was previously only introduced as tensor product expansion, to moderately high-dimensional stochastic problems. The fundamental idea of aPC is to use the statistical moments of the input random variables to develop the polynomial chaos expansion. This approach provides the possibility to propagate continuous or discrete probability density functions and also histograms (data sets) as long as their moments exist, are finite and the determinant of the moment matrix is strictly positive. For cases with limited data availability, this approach avoids bias and fitting errors caused by wrong assumptions. In this work, an alternative way to calculate the aPC is suggested, which provides the optimal polynomials, Gaussian quadrature collocation points and weights from the moments using only a handful of matrix operations on the Hankel matrix of moments. It can therefore be implemented without requiring prior knowledge about statistical data analysis or a detailed understanding of the mathematics of polynomial chaos expansions. The extension to more input variables suggested in this work, is an anisotropic and adaptive version of Smolyak's algorithm that is solely based on the moments of the input probability distributions. It is referred to as SAMBA (PC), which is short for Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos. It is illustrated that for moderately high-dimensional problems (up to 20 different input variables or histograms) SAMBA can significantly simplify the calculation of sparse Gaussian quadrature rules. SAMBA's efficiency for multivariate functions with regard to data availability is further demonstrated by analysing higher order convergence and accuracy for a set of nonlinear test functions with 2, 5 and 10 different input distributions or histograms.

This paper presents numerical estimation for the capacitance of an arbitrarily shaped conducting bodies based on the moment method with triangular patch modeling. The integral equation is evaluated numerically using a Gaussian quadrature method for triangles. The singular elements of the moment matrix are evaluated by using the Duffy transformation method. The Numerical data on the capacitance of conducting objects such as cuboid is presented. The computed results are compared with the previously published results. The findings of our study show that the Method of Moments is a more accurate method for the computation of free space capacitance.

We present a new method for computing the impedance matrix (IM) elements in the method of moments for geometries described by bilinear quadrilaterals (BQ) and for higher-order basis functions. Our method is restricted to the Electric Field Integral Equation and focuses on the self-elements of the IM and elements for which the observation point is near the integration BQ. The method is based on the simple idea of analyti-cal integration along one of the BQs parameters and numerical integration along the remaining one. For the singular (or nearly so) part of the integral, we show through analysis and examples that our method can provide precision to 15 significant digits.

Navier-Stokes (N-S) equations, in their primitive variable (u-v-p) formulation, are numerically solved using the Implicit Potential (IPOT) numerical scheme in the context of strong form Meshless Point Collocation (MPC) method. The unknown field functions are computed using the Discretization Correction Particle Strength Exchange (DC PSE) approximation method. The latter makes use of discrete moment conditions to derive the operator kernels, which leads to low condition number for the moment matrix compared to other meshless interpolation methods and increased stability for the numerical solution. The proposed meshless scheme is applied on 2D and 3D spatial domains, using uniform or irregular set of nodes to represent the domain. The numerical results obtained are compared against those obtained using well-established methods.

This chapter discusses the representation of uncertain homogeneous. We introduce a representation of the uncertainty which is minimal, thus does not contain singular covariance matrices, and develop methods for the estimation of geometric elements and transformation parameters.

An iterative least squares parameter estimation algorithm is developed for controlled moving average systems based on matrix decomposition. The proposed algorithm avoids repeatedly computing the inverse of the data product moment matrix with large sizes at each iteration and has a high computational efficiency. A numerical example indicates that the proposed algorithm is effective.