Diagonal Terms Research Articles

Improved Solution of Equations by Regularizing Ill-Conditioned Coefficient Matrix for System Identification

A numerical method is proposed for solving a set of simultaneous equations with an ill-conditioned coefficient matrix to apply system identification. To find an approximate solution, the coefficient matrix is regularized by adding a small positive value to its diagonal terms. A regularized matrix is provided in different expressions that depend on the coefficient matrix. This paper regularizes a rectangular coefficient matrix and then a square coefficient matrix. Improvement of solution accuracy is possible by removing the very small singular values. Therefore, rank estimation of the coefficient matrix is a key to obtaining an accurate solution. This paper gives a method that estimates the rank by setting an appropriate value of . Numerical examples show that the proposed method is effective for system identification.

Measurement of the magneto-optical response of Fe and CrO2epitaxial films by pump-probe spectroscopy: Evidence for spin-charge separation

We have investigated the magneto-optical response of Fe and CrO${}_{2}$ epitaxial films by pump-probe polarimetry, showing that charge and spin dynamics can be unambiguously disentangled. The time-resolved Kerr ellipticity and rotation in the metallic sample are essentially identical after the initial transient (shorter than a picosecond), but they considerably differ in the oxide film, even tens of picoseconds past the optical excitation. These differences are determined by the combined effects of photoexcited charge carriers and spins on the Kerr signal, but a detailed polarimetric analysis can explicitly unravel these contributions. In addition, the diagonal and off-diagonal terms of the dielectric tensor can be retrieved, providing the complete dynamical characterization of magnetic and optical properties in a ferromagnet, which is of utmost importance to understand spin evolution in magnetically correlated complex oxide materials.

Diagonalizability of nonautonomous linear systems with bounded continuous coefficients

A linear ODE system with bounded continuous coefficients is studied. Provided that the diagonal terms are “separated” and the off-diagonal terms satisfy a smallness threshold, we present some sufficient conditions for the diagonalizability of the system. The results are valid for any couple of Banach subspaces V,W of BC(R,Cn) admissible for a family of reduced (n−1 dimensional) linear homogeneous systems having exponential dichotomy.

Differential matrix physically admissible for depolarizing media: the case of diagonal matrices

In this Letter, we address the question of the physical validity of a depolarizing differential matrix. A parameterization of the diagonal terms of these depolarizing differential matrices is proposed. It ensures that the generators associated with diagonal depolarization terms lead to physical Mueller matrices. The validity of this parameterization is discussed. A condition is derived and related to the spatial extension of inhomogeneities with respect to the optical path length as proposed by Ossikovski [Opt. Lett. 36, 2330 (2011)].

The Møller–Plesset perturbation revisited: origin of high-order divergences

Contrarily to what happens with the Epstein–Nesbet (EN) zeroth-order Hamiltonian, the Møller–Plesset (MP) perturbation operator has diagonal matrix elements, the expression of which is recalled. It is a balance between hole–hole and particle–particle repulsions on one hand and of hole–particle attractions on the other hand. For the double excitations, which dominate the correlation effects, the attractive terms prevail and the second-order MP energy is underestimated, at least for atoms of the first rows of the periodic table. It will be shown that when the perturbation expansion reaches multiple excitations, the diagonal terms of the MP perturbation operator may become larger than the zeroth-order MP excitation energy and creates an oscillating divergence of the series. Several situations of this type will be presented. This divergence is linked to the non-additivity of excitation energies, while this additivity is an underlying assumption for the linked cluster theorem and the coupled cluster method. This analysis may also explain why for heavy atoms the second-order MP energies overshoot the exact correlation energies.

Numerical simulation of plagioclase rim growth during magma ascent at Bezymianny Volcano, Kamchatka

Slow CaAl-NaSi interdiffusion in plagioclase crystals preserves chemical zoning of plagioclase in detail, which, along with strong dependence of anorthite content in plagioclase on melt composition, pressure, and temperature, make this mineral an important source of information on magma processes. A numerical model of zoned crystal growth is developed in the paper. The model is based on equations of multicomponent diffusion with diagonal cross-component diffusion terms and accounts for mass conservation on the melt–crystal interface and growth rate controlled by undercooling. The model is applied to the data of plagioclase rim zoning from several recent Bezymianny Volcano (Kamchatka) eruptions. We show that an equilibrium growth model cannot explain crystallization of naturally observed plagioclase during magma ascent. The developed non-equilibrium model reproduced natural plagioclase zoning and allowed magma ascent rates to be constrained. Matching of natural and simulated zoning suggests ascent from 100 to 50MPa during 15–20days. Magma ascent rate from 50MPa to the surface varies from eruption to eruption: plagioclase zoning from the December 2006 eruption suggests ascent to the surface in less than 1day, whereas plagioclase zoning from March 2000 and May 2007 eruptions are better explained by magma ascent over periods of more than 30days). Based on comparison of diffusion coefficients for individual elements a mechanism of atomic diffusion during plagioclase crystallization is proposed.

Modeling diffusion and gas–oil mass transfer in fractured reservoirs

Diffusion may play a key role in a number of oil recovery processes such as heavy oil and naturally fractured reservoirs. In fractured media, several laboratory experiments and numerical studies demonstrated that CO2 injection can improve recovery. Molecular diffusion, gravity drainage, and oil swelling are the main contributing mechanisms. Proper modeling of diffusion at the reservoir PVT and geological conditions is not a trivial task. The challenge is in computing the diffusion coefficients for the non-ideal multicomponent mixtures in the oil and gas phases, and in physically accurate representation of the diffusion driving forces. One common approach in most numerical models is to use the classical Fick's law which simplifies the multicomponent diffusion fluxes by only considering the main-diffusion (diagonal) terms and neglecting the cross-diffusion (off-diagonal) terms in the diffusion matrix. The diffusion fluxes are assumed independent and the diffusion driving force of each component is proportional to the component self concentration gradient. In this work, we demonstrate analytically and numerically that this simplified approach may not honor the total flux balance and, in some applications, fails to capture the right direction of the diffusion flux. We propose an alternative model based on the generalized Fick's law. The proposed model can be seen equivalent to the Maxwell–Stefan model. The diffusion fluxes take into account the species interactions and the diffusion coefficients are dependent on temperature, pressure, and composition. We also tackle another problem related to cross-phase mass transfer of components in the gas and oil phases. This mechanism may occur in fractured media when fractures saturated with gas get in contact with under-saturated oil in the surrounding matrix. Intra-phase gas (gas-in-gas) and oil (oil-in-oil) diffusions cannot be initiated due to phase discontinuities between the fracture and the rock matrix. Some approaches in the literature that allow for direct gas-to-oil diffusion may not have a sound basis. We propose an accurate and numerically efficient model to describe the gas–oil transfer mechanism. The proposed model, based on irreversible thermodynamics, uses the chemical potential gradient as the driving force. The gas–oil transfer coefficients are functions of the gas and oil diffusion coefficients. In this model, the component fluxes are assumed continuous across the gas–oil contact and the interface fluid is in thermodynamic equilibrium. Several numerical and experimental cases are provided to validate the proposed model.

Oblique Incidence Diffraction by Edges in Anisotropic Impedance Surfaces: A Collection of Exact Solutions for Specific Non-Axial Tensor Boundary Conditions

The electromagnetic scattering of an arbitrarily polarized plane wave obliquely impinging on the edge of anisotropic impedance half-planes and planar junctions featuring inclined anisotropy axes is analyzed. Exact spectral representations for the total field are derived when the wedge faces are characterized by a set of specific homogeneous anisotropic impedance boundary conditions exhibiting an impedance matrix with no vanishing diagonal terms, as implied by the arbitrary orientation of the principal anisotropy axes with respect to the edge. The proposed solutions, which extend previous formulations valid for direction of the anisotropy axes either parallel or perpendicular to the edge, are derived by resorting to the Sommerfeld–Maliuzhinets method and exploiting the properties of the special function χΦ, originally introduced in the study of the scattering from wedges in gyroelectric media. Explicit uniform asymptotic expressions for the fields are provided along with samples of numerical results proving their validity.

Quadratic coupling treatment of the Jahn-Teller effect in the triply-degenerate electronic state of $\bf CH_4^+$CH4+: Can one account for floppiness?

The Jahn-Teller (JT) coupling effects in the triply degenerate ground electronic state of methane radical cation are investigated theoretically within a quadratic vibronic coupling approach. The underlying potential energy surfaces over the two-dimensional space of nuclear coordinates, subject to the T(2) ⊗ (e + t(2) + t(2)) Jahn-Teller effect, are established from extensive ab initio calculations using the multi-reference configuration interaction method and then employed to determine the various parameters of a diabatic Hamiltonian of this system. Our previous investigation [T. Mondal and A. J. C. Varandas, J. Chem. Phys. 135, 174304 (2011)], relying on the linear vibronic coupling approach augmented by only a diagonal second-order term of the totally symmetric mode, are extended here by including all possible quadratic coupling constants of JT active e and t(2) modes. Inclusion of these quadratic couplings is found to be important to reproduce correctly the broad vibrational structure and for a better description of dynamical JT effect in the first vibronic band of this radical cation. The impact of large amplitude motions (which are responsible for floppiness of the molecule) on the vibronic structure and dynamics of the first photoelectron band have been examined via readjustment of their linear coupling parameters up to ±10%.

A hybrid nudging-ensemble Kalman filter approach to data assimilation. Part I: application in the Lorenz system

ABSTRACTA hybrid data assimilation approach combining nudging and the ensemble Kalman filter (EnKF) for dynamic analysis and numerical weather prediction is explored here using the non-linear Lorenz three-variable model system with the goal of a smooth, continuous and accurate data assimilation. The hybrid nudging-EnKF (HNEnKF) computes the hybrid nudging coefficients from the flow-dependent, time-varying error covariance matrix from the EnKF's ensemble forecasts. It extends the standard diagonal nudging terms to additional off-diagonal statistical correlation terms for greater inter-variable influence of the innovations in the model's predictive equations to assist in the data assimilation process. The HNEnKF promotes a better fit of an analysis to data compared to that achieved by either nudging or incremental analysis update (IAU). When model error is introduced, it produces similar or better root mean square errors compared to the EnKF while minimising the error spikes/discontinuities created by the intermittent EnKF. It provides a continuous data assimilation with better inter-variable consistency and improved temporal smoothness than that of the EnKF. Data assimilation experiments are also compared to the ensemble Kalman smoother (EnKS). The HNEnKF has similar or better temporal smoothness than that of the EnKS, and with much smaller central processing unit (CPU) time and data storage requirements.

A Multiscale Framework for EHL and Micro-EHL

In this article, a heterogeneous multiscale method is introduced to analyze the microelastohydrodynamic lubrication (micro-EHL) of bearings with topological features. Two scales are adopted in the analysis: the large-scale simulations describe the entire bearing domain, and the small-scale simulations describe the fluid–structure interaction (FSI) at the small-scale features. Conservation of mass and momentum of the lubricant and the bearing's elastic deformation are solved for. The relationship between the pressure gradient and mass flow is obtained from homogenized small-scale FSI simulations and applied on a global scale via a scattered data interpolation method. When the micro structure is periodic the exact model at micro scale is replaced by an effective derived equation, i.e., homogenized model. The elastic deformation of the textured bearing surface is addressed at both the large and small scales, by decomposing the displacement influence matrix into the diagonal terms and nondiagonal terms (sorted at the small scale and large scale, respectively). The multiscale method was demonstrated as being capable of modeling the global pressure and film thickness for a bearing with surface texture while maintaining the accuracy of the small-scale modeling features. The illustrative geometry was that of a linear converging pad bearing in two dimensions. The solutions were compared with those obtained using lubrication theory for the smooth surface case, and good agreement was obtained. The method was then demonstrated for geometries incorporating topographical features.

On convex relaxations for quadratically constrained quadratic programming

We consider convex relaxations for the problem of minimizing a (possibly nonconvex) quadratic objective subject to linear and (possibly nonconvex) quadratic constraints. Let $$\mathcal{F }$$ denote the feasible region for the linear constraints. We first show that replacing the quadratic objective and constraint functions with their convex lower envelopes on $$\mathcal{F }$$ is dominated by an alternative methodology based on convexifying the range of the quadratic form $$\genfrac(){0.0pt}{}{1}{x}\genfrac(){0.0pt}{}{1}{x}^T$$ for $$x\in \mathcal{F }$$. We next show that the use of "$$\alpha $$BB" underestimators as computable estimates of convex lower envelopes is dominated by a relaxation of the convex hull of the quadratic form that imposes semidefiniteness and linear constraints on diagonal terms. Finally, we show that the use of a large class of D.C. ("difference of convex") underestimators is dominated by a relaxation that combines semidefiniteness with RLT constraints.

Solvated dissipative electro-elastic network model of hydrated proteins

Elastic network models coarse grain proteins into a network of residue beads connected by springs. We add dissipative dynamics to this mechanical system by applying overdamped Langevin equations of motion to normal-mode vibrations of the network. In addition, the network is made heterogeneous and softened at the protein surface by accounting for hydration of the ionized residues. Solvation changes the network Hessian in two ways. Diagonal solvation terms soften the spring constants and off-diagonal dipole-dipole terms correlate displacements of the ionized residues. The model is used to formulate the response functions of the electrostatic potential and electric field appearing in theories of redox reactions and spectroscopy. We also formulate the dielectric response of the protein and find that solvation of the surface ionized residues leads to a slow relaxation peak in the dielectric loss spectrum, about two orders of magnitude slower than the main peak of protein relaxation. Finally, the solvated network is used to formulate the allosteric response of the protein to ion binding. The global thermodynamics of ion binding is not strongly affected by the network solvation, but it dramatically enhances conformational changes in response to placing a charge at the active site of the protein.

On Cantor identities

In this paper, some special varieties which generalize Jónsson–Tarski algebras are considered. We prove that every nontrivial algebra from such a variety is term infinite and contains infinitely many distinct proper diagonal term operations of every arity.

A terrain-following grid transform and preconditioner for parallel, large-scale, integrated hydrologic modeling

A terrain-following grid formulation (TFG) is presented for simulation of coupled variably-saturated subsurface and surface water flow. The TFG is introduced into the integrated hydrologic model, ParFlow, which uses an implicit, Newton Krylov solution technique. The analytical Jacobian is also formulated and presented and both the diagonal and non-symmetric terms are used to precondition the Krylov linear system. The new formulation is verified against an orthogonal stencil and is shown to provide increased accuracy at lower lateral spatial discretization for hillslope simulations. Using TFG, efficient scaling to a large number of processors (16,384) and a large domain size (8.1 Billion unknowns) is shown. This demonstrates the applicability of this formulation to high-resolution, large-spatial extent hydrology applications where topographic effects are important. Furthermore, cases where the analytical Jacobian is used for the Newton iteration and as a non-symmetric preconditioner for the linear system are shown to have faster computation times and better scaling. This demonstrates the importance of solver efficiency in parallel scaling through the use of an appropriate preconditioner.

Reply to the Discussion by H. Jiang and B. Fu on “Experimental study of concrete masonry infills bounded by steel frames”1Appears in the Canadian Journal of Civil Engineering, 39(10): 1168–1169 [doi: 10.1139/l2012-082

The authors appreciate comments and input provided by the discussers on the discussed paper. The additional information presented in the discussion is in most part in line with the conclusions of the paper. However, there are a couple of points that we feel are worthy of clarification. Equation [3] in the discussion for diagonal strut width calculation is misquoted. The diagonal strut width equation given in the Canadian masonry design standard CSA S304.1-04 (CSA 2004) is expressed as w ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 2 þ a 2 p , rather than w = ah + al as suggested in the discussion. However, the discussion following this equation is still valid. While the diagonal strut width may be considered as the source of the large difference in design stiffness between the Canadian and American Standards (MSJC 2011), it is not correct to say that it also counts for the large difference in design strength between the two standards. In the Canadian standard, the corner crushing failure is evaluated based on the compressive strength of the diagonal strut, thus the width of diagonal strut plays an important role in the strength calculation. The discussed paper showed a mean test-to-design strength ratio of 2.32 based on the Canadian standard. The authors believe that this underestimate of strength is more attributed to the fact that the compressive strength of masonry is assumed to be 0:5f 0 m in the diagonal strut compression calculation. It is not a result of diagonal width value. On the other hand, the corner crushing strength in the American standard is calibrated using test results from experimental work conducted by Flanagan and Bennett (1999) on clay tiles infilled steel frames. The equation is simple to apply but does not contain an explicit diagonal width term. The diagonal width seems to be replaced by a constant term. While it provided strength values closer to those test results presented in the discussed paper than the CSA S304 equation, it is premature to conclude that this equation is equally applicable to infills of different material and geometry. More testing data covering a wide range of variables on infills should be studied to have an overall performance assessment. References

How covariant is the galaxy luminosity function?

We investigate the error properties of certain galaxy luminosity function (GLF) estimators. Using a cluster expansion of the density field, we show how, for both volume and flux limited samples, the GLF estimates are covariant. The covariance matrix can be decomposed into three pieces: a diagonal term arising from Poisson noise; a sample variance term arising from large-scale structure in the survey volume; an occupancy covariance term arising due to galaxies of different luminosities inhabiting the same cluster. To evaluate the theory one needs: the mass function and bias of clusters, and the conditional luminosity function (CLF). We use a semi-analytic model (SAM) galaxy catalogue from the Millennium run N-body simulation and the CLF of Yang et al. (2003) to explore these effects. The GLF estimates from the SAM and the CLF qualitatively reproduce results from the 2dFGRS. We also measure the luminosity dependence of clustering in the SAM and find reasonable agreement with 2dFGRS results for bright galaxies. However, for fainter galaxies, L<L*, the SAM overpredicts the relative bias by ~10-20%. We use the SAM data to estimate the errors in the GLF estimates for a volume limited survey of volume V~0.13 [Gpc/h]^3. We find that different luminosity bins are highly correlated: for L<L* the correlation coefficient is r>0.5. Our theory is in good agreement with these measurements. These strong correlations can be attributed to sample variance. For a flux-limited survey of similar volume, the estimates are only slightly less correlated. We explore the importance of these effects for GLF model parameter estimation. We show that neglecting to take into account the bin-to-bin covariances can lead to significant systematic errors in best-fit parameters.

Dispersion analysis of non-normal reflection spectra from monoclinic crystals

Results of the first dispersion analyses of polarized reflection spectra from a monoclinic a–c crystal plane taking properly into account the monoclinic symmetry and non-normal incidence on the crystal surface are reported. Spectra were recorded for the incidence angles α=8°, 16°, 40° and 60°, and for each at four different polarization angles. For the first two incidence angles the spectra were additionally analyzed under the assumption of normal incidence and the results were compared to those which resulted from the exact treatment. The comparison focused on two different aspects. The first aspect concerns the validity of the oscillator parameters. Accordingly, oscillator parameters of IR active modes obtained by dispersion analysis under the assumption of normal incidence do not show systematic errors despite the simplified treatment. This is a consequence of the fact that errors due to the incorrect assumption of normal incidence are compensated mainly by increases of the diagonal terms of the high frequency dielectric tensor. The corresponding values obtained under the assumption of normal incidence show averaged standard deviations of about 2% at 8° and about 9% at 16°. The second aspect investigated focuses on the employment of the oscillator parameters to generate the dielectric function tensor for optical calculations. Here we found that, as a direct consequence of the increased high frequency dielectric tensor, errors substantially increase if the resulting dielectric tensor functions are employed to calculate the reflectance at higher angles of incidence. Therefore, and since the increase of computational effort due to the exact treatment is small, we strongly suggest and encourage its use instead of the simplified treatment.

저차 크로스모드를 고려한 단순지지 사각 접수 평판의 방사효율해석

In this paper, radiation efficiency of the plate surround by an infinite rigid baffle is studied. The plate is simply supported and one side is in contact with air, while other side with water. The pressure and normal velocity over the plate surface are assumed as modal summations, from which a set of linear equations is obtained for fluid-structure coupled problem. It is shown that neglect of the cross modes results in overestimation of the radiation efficiency specifically for mid-frequency ranges. Based on the fact that the responses are mainly determined from the first few cross modes in addition to the diagonal terms, a new algorithm is proposed, where banded matrix is iteratively solved in computing radiation efficiency. In numerical examples, it is found that radiation efficiency obtained from banded matrix is in excellent agreement with the one from the full matrix, while computing time is significantly reduced. It is also found that as frequency grows larger, radiation efficiency considering only diagonal terms is a good approximation.

Geometrically non-linear dynamics of composite four-bar mechanisms

This work intends to demonstrate the importance of a geometrically nonlinear cross-sectional analysis of certain composite beam-based four-bar mechanisms in predicting system dynamic characteristics. All component bars of the mechanism are made of fiber reinforced laminates and have thin rectangular cross-sections. They could, in general, be pre-twisted and/or possess initial curvature, either by design or by defect. They are linked to each other by means of revolute joints. We restrict ourselves to linear materials with small strains within each elastic body (beam). Each component of the mechanism is modeled as a beam based on geometrically non-linear 3-D elasticity theory. The component problems are thus split into 2-D analyses of reference beam cross-sections and non-linear 1-D analyses along the three beam reference curves. For the thin rectangular cross-sections considered here, the 2-D cross-sectional non-linearity is also overwhelming. This can be perceived from the fact that such sections constitute a limiting case between thin-walled open and closed sections, thus inviting the non-linear phenomena observed in both. The strong elastic couplings of anisotropic composite laminates complicate the model further. However, a powerful mathematical tool called the Variational Asymptotic Method (VAM) not only enables such a dimensional reduction, but also provides asymptotically correct analytical solutions to the non-linear cross-sectional analysis. Such closed-form solutions are used here in conjunction with numerical techniques for the rest of the problem to predict multi-body dynamic responses more quickly and accurately than would otherwise be possible. The analysis methodology can be viewed as a three-step procedure: First, the cross-sectional properties of each bar of the mechanism is determined analytically based on an asymptotic procedure, starting from Classical Laminated Shell Theory (CLST) and taking advantage of its thin strip geometry. Second, the dynamic response of the non-linear, flexible four-bar mechanism is simulated by treating each bar as a 1-D beam, discretized using finite elements, and employing energy-preserving and -decaying time integration schemes for unconditional stability. Finally, local 3-D deformations and stresses in the entire system are recovered, based on the 1-D responses predicted in the previous step. With the model, tools and procedure in place, we identify and investigate a few four-bar mechanism problems where the cross-sectional non-linearities are significant in predicting better and critical system dynamic characteristics. This is carried out by varying stacking sequences (i.e. the arrangement of ply orientations within a laminate) and material properties, and speculating on the dominating diagonal and coupling terms in the closed-form non-linear beam stiffness matrix. A numerical example is presented which illustrates the importance of 2-D cross-sectional non-linearities and the behavior of the system is also observed by using commercial software (I-DEAS + NASTRAN + ADAMS).