Moment Matrix Research Articles

An Ideal Four-factor Components Mixture Experiment Design Based on Prominent Optimality Criteria at Level Two

Aim: Mixture experiment designs are essential tools that need to be determined prior to conducting any mixture experiment in any field of study. The primary goal of these types of experiments are optimization that is either maximizing the profit/produce or minimizing the cost of production. Study Design: Any researcher anticipating to do any research work on the mixture experiment will not evade to talk about the design that he or she is likely to use. The common of such designs being either Simplex Lattice Design (SLD) or Simplex Centroid Design (SCD). Methodology: The choice of such design is wholly based on the optimality criteria employed. The classical of these criteria include on D-Determinant criterion, A-Average variance criterion, E- Eigen value criterion and T- Trace optimality criterion usually denoted as D-, A-, E- T- criteria. They are also known as prominent criteria. Results: This paper considered four-factor components at order two. The penalized moment matrices obtained from the information matrices whose primary source was the design points gave the values of the criteria aforementioned. These values were ranked independently with the least average rank termed as the best design. Conclusion: The {4,4} Simplex Lattice Design had the lowest rank value of 1.0 as compared to the other designs. This design is therefore to be used in any research work considering four factors when only two of factors are to be employed.

On the matricial truncated moment problem. II

We continue the study of truncated matrix-valued moment problems begun in [12]. Let q∈N. Suppose that (X,X) is a measurable space and E is a finite-dimensional vector space of measurable mappings of X into Hq, the Hermitian q×q matrices. A linear functional Λ on E is called a moment functional if there exists a positive Hq-valued measure μ on (X,X) such that Λ(F)=∫X〈F,dμ〉 for F∈E.In this paper a number of special topics on the truncated matricial moment problem are treated. We restate a result from [11] to obtain a matricial version of the flat extension theorem. Assuming that X is a compact space and all elements of E are continuous on X we characterize moment functionals in terms of positivity and obtain an ordered maximal mass representing measure for each moment functional. The set of masses of representing measures at a fixed point and some related sets are studied. The class of commutative matrix moment functionals is investigated. We generalize the apolar scalar product for homogeneous polynomials to the matrix case and apply this to the matricial truncated moment problem.

Collisionless Relaxation from Near-equilibrium Configurations: Linear Theory and Application to Tidal Stripping

Placed slightly out of dynamical equilibrium, an isolated stellar system quickly returns toward a steady virialized state. We study this process of collisionless relaxation using the matrix method of linear response theory. We show that the full phase-space distribution of the final virialized state can be recovered directly from the disequilibrium initial conditions, without the need to compute the time evolution of the system. This shortcut allows us to determine the final virialized configuration with minimal computational effort. Complementing this result, we develop tools to model the system's full time evolution in the linear approximation. In particular, we show that moments of the velocity distribution can be efficiently computed using a generalized moment matrix. We apply our linear methods to study the relaxation of energy-truncated Hernquist spheres, mimicking the tidal stripping of a cuspy dark matter subhalo. Comparison of our linear predictions against controlled, isolated N-body simulations shows agreement at percent level for the parts of the system where a linear response to the perturbation is expected. We find that relaxation generates a tangential velocity anisotropy in the intermediate regions, despite the initial disequilibrium state having isotropic kinematics. Our results also strengthen the case for relaxation depleting the amplitude of the density cusp, without affecting its asymptotic slope. Finally, we compare the linear theory against an N-body simulation of tidal stripping on a radial orbit, confirming that the theory still accurately predicts density and velocity dispersion profiles for most of the system.

Sobolev orthogonal polynomials, Gauss–Borel factorization and perturbations

We present a comprehensive class of Sobolev bi-orthogonal polynomial sequences, which emerge from a moment matrix with an LU factorization. These sequences are associated with a measure matrix defining the Sobolev bilinear form. Additionally, we develop a theory of deformations for Sobolev bilinear forms, focusing on polynomial deformations of the measure matrix. Notably, we introduce the concepts of Christoffel–Sobolev and Geronimus–Sobolev transformations. The connection formulas between these newly introduced polynomial sequences and existing ones are explicitly determined.

Moments based entanglement criteria and measures

Quantum entanglement plays a key role in quantum computation and quantum information processing. It is of great significance to find efficient and experimentally friend separability criteria to detect entanglement. In this paper, we firstly propose two easily used entanglement criteria based on matrix moments. The first entanglement criterion only uses the first two realignment moments of a density matrix. The second entanglement criterion is based on the moments related to the partially transposed matrix. By detailed examples we illustrate the effectiveness of these criteria in detecting entanglement. Moreover, we provide an experimentally measurable lower bound of concurrence based on these moments. Finally, we present both bipartite and genuine tripartite entanglement measures based on the moments of the reduced states. By detailed examples, we show that our entanglement measures characterize the quantum entanglement in a more fine ways than the existing measures.

CLT for $$\beta $$-Ensembles at High Temperature and for Integrable Systems: A Transfer Operator Approach

AbstractIn this paper, we prove a polynomial central limit theorem for several integrable models and for the $$\beta $$ β -ensembles at high temperature with polynomial potential. Furthermore, we connect the mean values, the variances and the correlations of the moments of the Lax matrices of these integrable systems with the ones of the $$\beta $$ β -ensembles. Moreover, we show that the local functions’ space-correlations decay exponentially fast for the considered integrable systems. For these models, we also established a Berry–Esseen-type bound.

A Note on the Matrix Moment Problem and its Relation with Matrix Biorthogonal Polynomials

A Note on the Matrix Moment Problem and its Relation with Matrix Biorthogonal Polynomials

A matrix-based method for synthesizing planetary gear trains to facilitate the creation of transmission systems

The structural synthesis of planetary gear trains (PGTs) lays an important foundation for creating new mechanisms of transmission systems. Most existing studies focused on the synthesis of single degree-of-freedom (DOF) PGTs, while the synthesis of multi-DOF PGTs is very limited. This paper presents a matrix-based method for the automatic synthesis of non-fractionated PGTs with both single and multiple DOFs. First, the concept of a modified dual graph is proposed, based on which a new and simple method is presented to efficiently detect fractionated graphs. Then, the concepts of vertex adjacency index and moment matrix are defined, and a reliable algorithm is proposed to detect isomorphic graphs. Finally, a matrix-based method for the automatic synthesis of non-fractionated PGTs is presented, and the complete databases of non-fractionated PGTs with 3–9 links and 1–3 DOFs are generated for the first time. The creative design of mechanisms of 6-speed automatic transmission (AT) is taken for instance to illustrate how to design new transmission systems based on our databases.

Model Robust Optimal Designs for Kronecker Model for Mixture Experiments

In comparison to Scheffè’s canonical polynomial models (S-models), the Kronecker models (K-models) for mixture experiments are symmetric, compact in notation, and based on the Kronecker algebra of vectors and matrices. Further, there is a corresponding transition from S-models to K-models in the form of model re-parameterization. In the literature, it has been recommended to use second-degree K-models in practice compared to the widely used second-degree S-models especially when the moment matrix is of an ill-conditioning type. The motivation of the present article is to discriminate between K-models and S-models in terms of the model-robust D- and A-optimality criteria. These optimality criteria are discussed when there is uncertainty in selecting an appropriate model out of two rival models for a mixture experiment. International Journal of Statistical Sciences, Vol.24(1), March, 2024, pp 31-48

On the roles of strain gradient tensors in quasi‐3D nonlinear dynamics of microplate‐type piezoelectric systems of energy harvesting under triangular mechanical actuation

AbstractThe small‐scale‐dependent nonlinear dynamical attributes of microplate‐type laminated piezoelectric systems of harvesting energy subjected to a triangular mechanical actuation are explored in the current exploration. The core of harvesting systems is supposed to be made of nanocomposites reinforced by randomly oriented carbon nanotubes (CNTs) having various degrees of agglomeration coated by piezoelectric layers. In this regard, the microplate‐type systems of harvesting energy are modeled via a quasi‐3D plate theory combined with the modified strain gradient elasticity in the presence of various microsize‐dependent strain gradient tensors. Thereafter, the solution of the developed coupled electromechanical size‐dependent nonlinear problem is obtained numerically by utilizing the meshless collocation approach employing incorporation of the polynomial and radial basis functions to remove any feasible singularity for the associated moment matrices. It comes to the conclusion that by elevating the quantity of CNTs disclosed within the clusters, the significance of the microsize dependency in the achieved voltage enhances from 22.06% to 22.73% toward the simply supported bridge‐type context, and from 33.00% to 33.92% toward the clamped bridge‐type context.Highlights Development of a microstructural‐dependent quasi‐3D shear flexible energy harvester model. Incorporating the roles of various strain gradient tensors in the nonlinear dynamics of microharvesters. Size‐dependent time‐histories of achieved voltage from triangular mechanical‐loaded microharvesters. Influence of the CNT reinforcement clustering on the nonlinear dynamic performance.

On the Resolvent Matrix of the Truncated Hausdorff Matrix Moment Problem

On the Resolvent Matrix of the Truncated Hausdorff Matrix Moment Problem

A batch-filling method of VIE-MoM matrix for inhomogeneous dielectric target with full- and half-SWG function

A batch-filling method (BFM) for generating the volume-integral-equation-methods of moment (VIE-MoM) matrix for the scattering of inhomogeneous objects by using the full- and half-SWG basis function is proposed. The BFM is based on the summation of contributions of all integrals over tetrahedrons and boundary faces, and the contributions are arranged into a column vector that represents the interactions of the source basis function with all the test basis functions. The time of the BFM for filling VIE-MoM matrix can be reduced over 95% compared with the conventional basis-basis (CBB) matching filling scheme due to the vector filling scheme, which avoids the repeated traversal between the source tetrahedron and test tetrahedron to fill the impedance matrix elements. Details for the numerical process and the filling scheme of impedance matrix are elaborated. Numerical results show that the batch-filling method has high computational efficiency and accuracy for VIE-MoM.

Distribution of intensity and M2 factor for a partially coherent flat-topped beam in bidirectional turbulent atmosphere and plasma connection

This study investigates the bidirectional transmission of a partially coherent flat-topped beam in a turbulent atmosphere and plasma. Analytical formulas for the intensity distribution and M2 factor are derived based on the optical transmission matrix, Collins formula, and second moment theory with Wigner distribution function. Numerical results show that the beam order and transverse spatial coherence width can be selected appropriately to mitigate turbulence and plasma induced evolution properties. The partially coherent flat-topped beam propagation through a turbulent atmosphere and plasma of the forward transmission effect on the intensity distribution and M2 factor are smaller than that of the reverse transmission. Under the same conditions, the M2 factor of a partially coherent flat-topped beam is smaller than the Gaussian beam in bidirectional transmission. Our results can be used in long-distance free-space optical communications.

Charge-density wave mediated quasi-one-dimensional Kondo lattice in stripe-phase monolayer 1T-NbSe2

The heavy fermion physics is dictated by subtle competing exchange interactions, posing a challenge to their understanding. One-dimensional (1D) Kondo lattice model has attracted special attention in theory, because of its exact solvability and expected unusual quantum criticality. However, such experimental material systems are extremely rare. Here, we demonstrate the realization of quasi-1D Kondo lattice behavior in a monolayer van der Waals crystal NbSe2, that is driven into a stripe phase via Se-deficient line defects. Spectroscopic imaging scanning tunneling microscopy measurements and first-principles calculations indicate that the stripe-phase NbSe2 undergoes a novel charge-density wave transition, creating a matrix of local magnetic moments. The Kondo lattice behavior is manifested as a Fano resonance at the Fermi energy that prevails the entire film with a high Kondo temperature. Importantly, coherent Kondo screening occurs only in the direction of the stripes. Upon approaching defects, the Fano resonance exhibits prominent spatial 1D oscillations along the stripe direction, reminiscent of Kondo holes in a quasi-1D Kondo lattice. Our findings provide a platform for exploring anisotropic Kondo lattice behavior in the monolayer limit.

Value iteration for LQR control of unknown stochastic-parameter linear systems

This paper focuses on the linear-quadratic optimal control problem for unknown stochastic-parameter linear systems using reinforcement learning methods. Based on the second moments of random system matrices, a model-based value iteration algorithm is proposed to solve the problem and is proved to be convergent by using the contraction mapping theorem. For the case without knowing any information about the random system matrices, a normalized model-free value iteration algorithm is presented to learn the optimal control law by estimating the data for the next time. The collected data are normalized first in our model-free algorithm to reduce the error of the least squares method. It is proved that our algorithm can obtain an approximate optimal solution. Our algorithm is applicable for any distribution of the random system matrices and does not require an initial mean-square stabilizing control policy. Finally, an example illustrates that our algorithm converges to an approximate optimal control policy and that the normalization step can significantly reduce convergence errors.

Moderate deviation principle for guesswork

In this paper, we establish a moderate deviation principle for the logarithm of guesswork, which was initially studied by Massey to quantify the number of guesses needed to ascertain a discrete random variable. Our approach is based on an asymptotic analysis for Perron–Frobenius eigenvalue of some perturbed matrix and several moments estimates bounds for guesswork.

A Spectroscopic Investigation of the Lowest Electronic States of the I+ 2 Cation as a Candidate for Detecting the Time Variation of Fundamental Constants

The four lowest Ω substates (X2Π3/2,g, X2Π1/2,g, A2Π3/2,u and A2Π1/2,u) of the I+ 2 cation have been studied by high-precision ab initio calculations in comparison with experimental high-resolution absorption spectra. The potential energy curves were calculated using the multi-reference configuration interaction (MRCI) method and Dirac method, respectively. Rovibrational levels of these electronic states were derived by solving the radial Schrödinger rovibrational equation. Molecular constants were obtained in fitting energy levels to a spectroscopic model. Using the fit spectroscopic constants and newly calculated transition dipole moment matrix elements, line strengths of vibronic bands in the A2Π3/2,u- X2Π3/2,g system, as well as Einstein A coefficients for 45 of these bands with ν′ = 11−19 and ν′′ = 1−5, have been derived. The Einstein A coefficients were used to compute radiative lifetimes of the ν′ = 11−19 vibrational levels of the A2Π3/2,u state. Enhancement factors for detecting the variation of the fine-structure constant (α) and the proton-to-electron mass ratio(µ) using transitions between nearly degenerate rovibronic levels of these low-lying states have been calculated.

Matrix factorizations for the generalized Charlier and Meixner orthogonal polynomials

The Cholesky factorization of the moment matrix is considered for the generalized Charlier and generalized Meixner discrete orthogonal polynomials. For the generalized Charlier, we present an alternative derivation of the Laguerre–Freud relations found by Smet and Van Assche. Third-order and second-order nonlinear ordinary differential equations are found for the recursion coefficient γn, that happen to be forms of the Painlevé deg-PV in disguise. New Laguerre–Freud relations are also found for the generalized Meixner case, which are compared with those of Smet and Van Assche.

Phase-field modeling of anisotropic crack propagation based on higher-order nonlocal operator theory

This paper presents a novel higher-order nonlocal operator theory for the phase-field modeling of brittle fracture in anisotropic materials. Incorporating higher order nonlocal operators can enhance the accuracy of the phase-field model by effectively capturing long-range interactions that hold significance in numerous materials. The reproducing kernel particle method is employed to derive a nonlocal differential operator to enhance computational stability and accuracy. Moreover, the proposed method eliminates the need for direct computation of derivatives of the modified kernel function, which avoids the calculation of moment matrix derivatives and improves computational efficiency. The phase-field modeling of polycrystalline materials, considering the anisotropic fracture resistance of each grain, is implemented using this numerical framework. The present method is able to capture different scenarios intergranular and transgranular crack propagation patterns in polycrystalline materials. The proposed method involves a detailed representation of the complex process of crack initiation and propagation in 2D and 3D models of polycrystalline materials.

Laguerre–Freud Equations for the Gauss Hypergeometric Discrete Orthogonal Polynomials

The Cholesky factorization of the moment matrix is considered for the Gauss hypergeometric discrete orthogonal polynomials. This family of discrete orthogonal polynomials has a weight with first moment given by ρ0=2F1a,bc+1;η. For the Gauss hypergeometric discrete orthogonal polynomials, also known as generalized Hahn of type I, Laguerre–Freud equations are found, and the differences with the equations found by Dominici and by Filipuk and Van Assche are provided.