Two-dimensional Moment Research Articles

Two-Dimensional Half-Strong Real Moment Problem and the Corresponding Block Matrices. Part I

We generalize the relationship between the classical moment problem and the spectral theory of Jacobi matrices. We present the solution of the two-dimensional half-strong moment problem and suggest an analog of Jacobi-type matrices associated with the two-dimensional half-strong moment problem and the corresponding system of polynomials orthogonal with respect to a measure with compact support in the real plane.

Solution of an Integro-differential Equation with Dirichlet Conditions using Techniques of the Inverse Moments Problem

It will be shown that finding solutions from some integro-differential equation under Dirichlet conditions is equivalent to solving an integral equation, which can be treated as a generalized two-dimensional moment problem over a domain E = {(x, t), 0 > x > L; t > 0}. We will see that an approximate solution of the equation integro-differential can be found using the techniques of generalized inverse moments problem and bounds for the error of the estimated solution. First the problem is reduced to solving a hyperbolic or parabolic partial derivative equation considering the unknown source. The method consists of two steps. In each one an integral equation is solved numerically using the two-dimensional inverse moment problem techniques. We illustrate the different cases with examples.

Solution of Partial Derivative Equations of Poisson and Klein-Gordon with Neumann Conditions as a Generalized Problem of Two-Dimensional Moments

It will be shown that finding solutions from the Poisson and Klein-Gordon equations under Neumann conditions are equivalent to solving an integral equation, which can be treated as a generalized two-dimensional moment problem over a domain that is considered rectangular. The method consists to solve the integral equation numerically using the two-dimensional inverse moments problem techniques. We illustrate the different cases with examples.

Study of the Dimensionality Reduction Approach for the Efficient Simulation and Optimization Design of Terahertz Lens With Electrically-Large Dimension

Dielectric lenses are widely used in terahertz imaging and communication systems to focus or collimate the Gaussian beam by adjusting the phase distribution of wave front. However, due to the high frequency and short wavelength of the terahertz band, the focusing lenses used in the application systems are usually electrically large, which bring the extremely difficulty to carry out efficient electromagnetic (EM) simulation and optimization design. In this paper, a dimensionality reduction concept was introduced to achieve efficient design and optimization of electrically-large terahertz lenses, which have symmetric structures and are commonly used in quasi-optical systems, such as circular lenses and cylindrical lenses. To precisely solve the EM problem with reduced dimension, a two-dimensional moment method (2D-MOM) for homogeneous dielectric targets was studied and successfully developed by solving the surface coupled integral equation discretized with appropriate basis and test functions. Then, the dimensionality reduction approach with the combination of the ray-tracing method (RTM) and the 2D-MOM was developed for the shaped design of terahertz lens with high efficiency. A 0.3THz lens with diameter 10cm was designed with the proposed approach as an example, with its pattern measured by a terahertz field scanning platform. It's found that, with the dimensional reduction, the unknowns can be reduced more than 1500 times and the memory required can be reduced more than 2.5 million times, as compared to the traditional 3D-MOM simulation. And the simulation results agree well with the experiments, which both demonstrate the greatly improved performance of the lens designed by the proposed approach, as compared to the standard lens.

Methods of Moment and Maximum Entropy for Solving Nonlinear Expectation

In this paper, we consider a special nonlinear expectation problem on the special parameter space and give a necessary and sufficient condition for the existence of the solution. Meanwhile, we generalize the necessary and sufficient condition to the two-dimensional moment problem. Moreover, we use the maximum entropy method to carry out a kind of concrete solution and analyze the convergence for the maximum entropy solution. Numerical experiments are presented to compute the maximum entropy density functions.

Solution of an equation in Poisson partial derivatives with conditions of Dirichlet using techniques of the inverse moments problem

In this paper, it will be shown that finding solutions from the Helmholtz equation and the non-linear Poisson equation under Dirichlet conditions is equivalent to solving an integral equation, which can be treated as a generalized two-dimensional moment problem over a domain that is considered rectangular in principle. We will see that an approximate solution of the equation in partial derivatives can be found using the techniques of generalized inverse moments problem and bounds for the error of the estimated solution. The method consists of two steps.In each one an integral equation is solved numerically using the two-dimensional inverse moments problem techniques. We illustrate the different cases with examples.

On the truncated two-dimensional moment problem

We study the truncated two-dimensional moment problem (with rectangular data) to find a non-negative measure $\mu(\delta)$, $\delta\in\mathfrak{B}(\mathbb{R}^2)$, such that $\int_{\mathbb{R}^2} x_1^m x_2^n d \mu = s_{m,n}$, $0\leq m\leq M,\quad 0\leq n\leq N$, where $\{ s_{m,n} \}_{0\leq m\leq M, 0\leq n\leq N}$ is a prescribed sequence of real numbers; $M,N\in\mathbb{Z}_+$. For the cases $M=N=1$ and $M=1, N=2$ explicit numerical necessary and sufficient conditions for the solvability of the moment problem are given. In the cases $M=N=2$; $M=2, N=3$; $M=3, N=2$; $M=3, N=3$ some explicit numerical sufficient conditions for the solvability are obtained. In all the cases some solutions (not necessarily atomic) of the moment problem can be constructed.

Jacobi-Type Block Matrices Corresponding to the Two-Dimensional Moment Problem: Polynomials of the Second Kind and Weyl Function

We continue our investigations of the Jacobi-type symmetric matrices corresponding to the two-dimensional real power moment problem. We introduce polynomials of second kind and the corresponding analog of the Weyl function.

Computer simulation of phase transformation and plastic deformation in IN718 superalloy: Microstructural evolution during precipitation

Microstructural evolution during co-precipitation of γ′, γ″ and δ phases from a supersaturated γ matrix during aging of superalloy Inconel 718 (IN718) is investigated by computer simulation using the phase-field method. The precipitation model is quantitative, using as model inputs ab initio calculations of elastic constants, experimental data on lattice parameters, precipitate–matrix orientation relationship, interfacial energy of each individual precipitate phase and interdiffusivities, and a Ni–Nb–Al pseudo-ternary thermodynamic database specifically developed for IN718. In order to simulate statistically representative multiphase microstructures observed in the alloy, the Kim–Kim–Suzuki treatment of interfaces is employed. Simulation results show how alloy composition, lattice misfit, external stress, temperature and time affect precipitate microstructure and variant selection during isothermal aging, without any a priori assumptions about key microstructural features including size, shape, volume fraction and spatial distribution of different types of precipitates and different variants of the same precipitate phase. The shapes of precipitates and their coarsening kinetics are analyzed based on the two-dimensional moment invariant. The various multiphase microstructures generated by the simulations have been used as model inputs in a study to investigate how precipitate microstructure (in particular shape and spatial distribution) influences the strength of IN718.

Accelerated discrete velocity method for axial-symmetric gaseous flows

An accelerated discrete velocity method is presented for the simulation of axial-symmetric single-component gas flows in the whole range of gas rarefaction. The method relies on the Bhatnagar–Gross–Krook (BGK) kinetic equation. A diffusion-type moment equation is derived on the basis of the cylindrical kinetic one. Both equations are solved by discretizing the spatial and molecular velocity spaces in a joint iteration. The relationship between the cylindrical and the two-dimensional moment equations is shown. A stability analysis is developed to show the improved iteration performance of the accelerated method. Pressure-driven flow in a tube is simulated to study the method. The accelerated scheme is superior compared to the non-accelerated standard one at intermediate and high values of the rarefaction parameter in terms of the number of iterations required and the computational time. The flow rates and the velocity profiles are compared to the analytical discrete ordinate solution of the problems. There is a very good agreement between the results provided by the two methods.

Moment Problems on Bounded and Unbounded Domains

Using approximation results, we characterize the existence of the solution for a two-dimensional moment problem in the first quadrant, in terms of quadratic forms, similar to the one-dimensional case. For the bounded domain case, one considers a space of complex analytic functions in a disk and a space of continuous functions on a compact interval. The latter result seems to give sufficient (and necessary) conditions for the existence of a multiplicative solution.

Accurate pseudo Zernike moment invariants for grey-level images

Pseudo Zernike moments are orthogonal moments used to represent digital images with minimum amount of information redundancy. Pseudo Zernike moment invariants are image features that are invariant to translation, scaling and rotation which play an essential role in discriminating and classifying similar images where the performance and robustness of the classifiers are highly dependent on the accuracy of these futures. In this work, a new method is presented for accurate computation of two-dimensional pseudo Zernike moment invariants. Approximation errors are removed by using exact geometric and radial geometric moments. Invariance to translation and scaling are computed, while the rotation invariance is directly achieved as direct property of pseudo Zernike moments. Numerical experiments are conducted on a set of standard images. The obtained results demonstrate the efficiency of the proposed method.

Optimal precipitate shapes in nickel-base γ–γ′ alloys

Precipitate shapes in nickel-base superalloys vary substantially with alloy composition, partitioning of elements to the disordered gamma matrix and the ordered gamma prime precipitates and the degree of coarsening during elevated temperature exposures. These shapes, which vary from spherical to cuboidal to rod-like, have typically not been quantified in Ni-base alloys containing relatively high γ′ volume fractions, in spite of their importance to mechanical properties. Precipitate shapes in a series of new platinum group metal (PGM)-containing Ni-base alloys have been quantified by their two-dimensional moment invariants. Precipitate morphologies were characterized in a total of 17 PGM-containing alloys in the solution treated and aged condition. The average γ′ volume fraction was measured as 0.60, typical of highly creep resistant Ni-base alloys. PGM additions resulted in an unusually large range of precipitate shapes. Precipitate morphologies were quantitatively analyzed using a shape parameter derived from the absolute moment invariant. For the compositions examined, the shape parameter reaches a maximum value at a precipitate–matrix misfit magnitude of 0.4%. A large set of commercial single crystals exhibits the same range in shape parameter values around this misfit magnitude, suggesting that favorable high temperature properties are correlated with an optimum precipitate shape.

Site-response effects on RC buildings isolated by triple concave friction pendulum bearings

The main object of this study is to evaluate the seismic response effects on a reinforced concrete building isolated by triple concave friction pendulum (TCFP) bearings. The site-response effects arise from the difference in the local soil conditions at the support points of the buildings. The local soil conditions are, therefore, considered as soft, medium and firm; separately. The results on the responses of the isolated building are compared with those of the non-isolated. The building model used in the time history analysis, which is a two-dimensional and eight-storey reinforced concrete building with and without the seismic isolation bearings and/or the local soil conditions, is composed of two-dimensional moment resisting frames for superstructure and of plane elements featuring plane-stress for substructure. The TCFP bearings for isolating the building are modelled as of a series arrangement of the three single concave friction pendulum (SCFP) bearings. In order to investigate the efficiency of both the seismic isolation bearings and the site-response effects on the buildings, the time history analyses are elaborately conducted. It is noted that the site-response effects are important for the isolated building constructed on soft, medium or firm type local foundation soil. The results of the analysis demonstrate that the site-response has significant effects on the response values of the structure-seismic isolation-foundation soil system.

Two-Dimensional Péclet Numbers for Peak Quality Scoring

We have developed a tool we call Two-Dimensional Peclet analysis to help in determining optimum operational parameters for analytical systems that require automated analysis of large numbers of analyte data peaks. In this paper, we derive the technique from moment analysis of the peaks and single-dimensional Peclet theory. Two-Dimensional Peclet analysis allows automated comparison of response peaks with differing shapes and amplitudes to be compared simultaneously in multicomponent mixtures. In addition to peak resolution and fidelity, individual two-dimensional moment components can provide feedback to total mass (zero moment), centroid location in time space (first moment), and two-dimensional spread of data (second moment).

Vertically-averaged and moment model for meandering river morphology

We report the first numerical tests of the two-dimensional vertically-averaged and moment (VAM) equations to simulate scour and deposition along alluvial meandering rivers with fixed banks. The VAM hydrodynamic model assumes a linear vertical distribution of the velocity profile and provides estimates of the magnitude and direction of the both near-bed and surface velocity vectors in a quasi three-dimensional (3D) fashion, reproducing at a very basic level some of the 3D effects of the secondary flow in bends. By assuming that the bed shear stress direction along the bed was given by the VAM’s near-bed velocity direction, it was possible to successfully reproduce the scour and deposition observed in a curved experimental flume and a meandering river reach. Although the VAM model is more general and does not require a secondary flow correction, as needed by conventional depth-averaged models, the results were not significantly better for the two cases analyzed.

Characterizing the Variability and Extremes of the Stratospheric Polar Vortices Using 2D Moment Analysis

Abstract The mean state, variability, and extreme variability of the stratospheric polar vortices, with an emphasis on the Northern Hemisphere (NH) vortex, are examined using two-dimensional moment analysis and extreme value theory (EVT). The use of moments as an analysis tool gives rise to information about the vortex area, centroid latitude, aspect ratio, and kurtosis. The application of EVT to these moment-derived quantities allows the extreme variability of the vortex to be assessed. The data used for this study are 40-yr ECMWF Re-Analysis (ERA-40) potential vorticity fields on interpolated isentropic surfaces that range from 450 to 1450 K. Analyses show that the most extreme vortex variability occurs most commonly in late January and early February, consistent with when most planetary wave driving from the troposphere is observed. Composites around sudden stratospheric warming (SSW) events reveal that the moment diagnostics evolve in statistically different ways between vortex splitting events and vortex displacement events, in contrast to the traditional diagnostics. Histograms of the vortex diagnostics on the 850-K (~10 hPa) surface over the 1958–2001 period are fitted with parametric distributions and show that SSW events constitute the majority of data in the tails of the distributions. The distribution of each diagnostic is computed on various surfaces throughout the depth of the stratosphere; it shows that in general the vortex becomes more circular with higher filamentation at the upper levels. The Northern and Southern Hemisphere (SH) vortices are also compared through the analysis of their respective vortex diagnostics, confirming that the SH vortex is less variable and lacks extreme events compared to the NH vortex. Finally, extreme value theory is used to statistically model the vortex diagnostics and make inferences about the underlying dynamics of the polar vortices.

Reflecting Brownian Motion in Two Dimensions: Exact Asymptotics for the Stationary Distribution

We consider a two-dimensional semimartingale reflecting Brownian motion (SRBM) in the nonnegative quadrant. The data of the SRBM consists of a two-dimensional drift vector, a 2 × 2 positive definite covariance matrix, and a 2 × 2 reflection matrix. Assuming the SRBM is positive recurrent, we are interested in tail asymptotic of its marginal stationary distribution along each direction in the quadrant. For a given direction, the marginal tail distribution has the exact asymptotic of the form bxκ exp(−αx) as x goes to infinity, where α and b are positive constants and κ takes one of the values −3/2, −1/2, 0, or 1; both the decay rate α and the power κ can be computed explicitly from the given direction and the SRBM data. A key tool in our proof is a relationship governing the moment generating function of the two-dimensional stationary distribution and two moment generating functions of the associated one-dimensional boundary measures. This relationship allows us to characterize the convergence domain of the two-dimensional moment generating function. For a given direction c, the line in this direction intersects the boundary of the convergence domain at one point, and that point uniquely determines the decay rate α. The one-dimensional moment generating function of the marginal distribution along direction c has a singularity at α. Using analytic extension in complex analysis, we characterize the precise nature of the singularity there. Using that characterization and complex inversion techniques, we obtain the exact asymptotic of the marginal tail distribution.

Reflecting Brownian motion in two dimensions: Exact asymptotics for the stationary distribution

Open AccessOpen Access licenseAboutSectionsView PDF ToolsAdd to favoritesDownload CitationsTrack Citations ShareShare onFacebookTwitterLinked InEmail Go to SectionOpen AccessOpen Access license HomeStochastic SystemsVol. 1, No. 1 Reflecting Brownian Motion in Two Dimensions: Exact Asymptotics for the Stationary DistributionJ. G. Dai, Masakiyo MiyazawaJ. G. Dai, Masakiyo MiyazawaPublished Online:15 Jul 2011https://doi.org/10.1287/10-SSY022AbstractWe consider a two-dimensional semimartingale reflecting Brownian motion (SRBM) in the nonnegative quadrant. The data of the SRBM consists of a two-dimensional drift vector, a 2 × 2 positive definite covariance matrix, and a 2 × 2 reflection matrix. Assuming the SRBM is positive recurrent, we are interested in tail asymptotic of its marginal stationary distribution along each direction in the quadrant. For a given direction, the marginal tail distribution has the exact asymptotic of the form bxκ exp(−αx) as x goes to infinity, where α and b are positive constants and κ takes one of the values −3/2, −1/2, 0, or 1; both the decay rate α and the power κ can be computed explicitly from the given direction and the SRBM data.A key tool in our proof is a relationship governing the moment generating function of the two-dimensional stationary distribution and two moment generating functions of the associated one-dimensional boundary measures. This relationship allows us to characterize the convergence domain of the two-dimensional moment generating function. For a given direction c, the line in this direction intersects the boundary of the convergence domain at one point, and that point uniquely determines the decay rate α. The one-dimensional moment generating function of the marginal distribution along direction c has a singularity at α. Using analytic extension in complex analysis, we characterize the precise nature of the singularity there. Using that characterization and complex inversion techniques, we obtain the exact asymptotic of the marginal tail distribution. Previous Back to Top FiguresReferencesRelatedInformationCited BySplitting Algorithms for Rare Events of Semimartingale Reflecting Brownian MotionsKevin Leder, Xin Liu, Zicheng Wang2 September 2021 | Stochastic Systems, Vol. 11, No. 4Asymptotic Expansion of Stationary Distribution for Reflected Brownian Motion in the Quarter Plane via Analytic ApproachSandro Franceschi, Irina Kourkova8 May 2017 | Stochastic Systems, Vol. 7, No. 1Optimal Paths in Large Deviations of Symmetric Reflected Brownian Motion in the OctantZiyu Liang, John J. Hasenbein19 August 2013 | Stochastic Systems, Vol. 3, No. 1Wiener-Hopf Factorizations for a Multidimensional Markov Additive Process and their Applications to Reflected ProcessesMasakiyo Miyazawa, Bert Zwart7 May 2012 | Stochastic Systems, Vol. 2, No. 1 Volume 1, Issue 1June 2011Pages 1-208 Article Information Metrics Downloaded 67 times in the past 12 months Information Received:November 01, 2010Published Online:July 15, 2011 Copyright © 2011, The author(s)Cite asJ. G. Dai, Masakiyo Miyazawa (2011) Reflecting Brownian Motion in Two Dimensions: Exact Asymptotics for the Stationary Distribution. Stochastic Systems 1(1):146-208. https://doi.org/10.1287/10-SSY022 KeywordsDiffusion processheavy trafficqueueing networktail behaviorlarge deviationsPDF download

ChemInform Abstract: Photoluminescence from Single Semiconductor Nanostructures

We review some recent results in the spectroscopy of single CdSe nanocrystal quantum dots. By eliminating the effects of inhomogeneous broadening and ensemble averaging, single nanocrystal spectroscopy has revealed many new and previously unexpected physical phenomena. Among those discussed in this review are ultra-narrow emission lineshapes (∼600× narrower than ensemble spectra), a highly polarizable emitting state in the presence of strong local electric fields, line broadening as a result of environmental fluctuations, and shifting of the emission spectra over a wide range of energies (from less than 300 μeV to 80 meV). In addition, polarization spectroscopy of single nanocrystals has revealed the presence of a theoretically predicted two-dimensional transition dipole moment oriented in the xy plane of the nanocrystals. As a result, it is, in principle, possible to use polarization spectroscopy to determine the three-dimensional orientation of individual nanocrystals. These and other studies of single quantum dots have provided us with significant insight into the detailed physics and dynamics of this unique and fascinating physical system.