scholarly journals Second Order Superintegrable Systems in Three Dimensions

2005 ◽  
Author(s):  
Willard Miller
Keyword(s):  
Second Order ◽  
Three Dimensions ◽  
Superintegrable Systems

scholarly journals Can We Compare Solidarity Across Europe? What, Why, When, and How to Assess Exact and Approximate Equivalence of First- and Second-Order Factor Models

2021 ◽  
Vol 3 ◽  
Author(s):  
Vera Lomazzi
Keyword(s):  
Factor Model ◽  
Second Order ◽  
Three Dimensions ◽  
Alignment Method ◽  
Cross Sectional ◽  
Political Values ◽  
First Order ◽  
Measurement Models ◽  
European Values ◽  
Order Factor

Although measurement invariance is widely considered a precondition for meaningful cross-sectional comparisons, substantive studies have often neglected evaluating this assumption, thereby risking drawing conclusions and making theoretical generalizations based on misleading results. This study offers a theoretical overview of the key issues concerning the measurement and the comparison of socio-political values and aims to answer the questions of what must be evaluated, why, when, and how to assess measurement equivalence. This paper discusses the implications of formative and reflective approaches to the measurement of socio-political values and introduces challenges in their comparison across different countries. From this perspective, exact and approximate approaches to equivalence are described as well as their empirical translation in statistical techniques, such as the multigroup confirmatory factor analysis (MGCFA) and the frequentist alignment method. To illustrate the application of these methods, the study investigates the construct of solidarity as measured by European Values Study (EVS) and using data collected in 34 countries in the last wave of the EVS (2017–2020). The concept is captured through a battery of nine items reflecting three dimensions of solidarity: social, local, and global. Two measurement models are hypothesized: a first-order factor model, in which the three independent dimensions of solidarity are correlated, and a second-order factor model, in which solidarity is conceived according to a hierarchical principle, and the construct of solidarity is reflected in the three sub-factors. In testing the equivalence of the first-order factor model, the results of the MGCFA indicated that metric invariance was achieved. The alignment method supported approximate equivalence only when the model was reduced to two factors, excluding global solidarity. The second-order factor model fit the data of only seven countries, in which this model could be used to study solidarity as a second-order concept. However, the comparison across countries resulted not appropriate at any level of invariance. Finally, the implications of these results for further substantive research are discussed.

scholarly journals Modeling of rose coco beans using twenty four points optimum second order rotatable design

2017 ◽  
Vol 5 (2) ◽  
pp. 96
Author(s):  
Isaac Tum ◽  
John Mutiso ◽  
Joseph Koske
Keyword(s):  
Response Surface ◽  
Maximum Yield ◽  
Second Order ◽  
Three Dimensions ◽  
Coefficient Of Determination ◽  
Yield Response ◽  
Efficient Production ◽  
Phosphorus And Potassium ◽  
Nitrogen Phosphorus ◽  
Rotatable Design

The response surface methodology (RSM) is a collection of mathematical and statistical techniques useful for the modeling and analysis of problems in which a response of interest is influenced by several variables, and the objective is to optimize the response. The objective of the study was to model the rose coco beans (Phaseolus vulgaris) through an existing A-optimum and D-efficient second order rotatable design of twenty four points in three dimensions in a greenhouse setting using three inorganic fertilizers, namely, nitrogen, phosphorus and potassium. Thus, the objective of the study was accomplished using the calculus optimum value of the free/letter parameter f=1.1072569. This was done by estimating the parameters via least square's techniques, by making available for the yield response of rose coco beans at calculus optimum value design for the first time. The results showed that, the three factors: nitrogen, phosphorus, and potassium contributed significantly on the yield of rose coco beans (p<0.05). In GP3G, the second-order model was adequate for 1% level of significance with p value of 0.0034. The analysis of variance (ANOVA) of response surface for rose coco yield showed that this design was adequate due to satisfactory level of a coefficient of determination, R2, 0.8066 and coefficient variation, CV was 10.30. This study demonstrated the importance of statistical methods in the optimal and efficient production of rose coco beans. We do recommend a randomize screening of all the fertilizer components with which it has influence on rose coco beans be done to ascertain the right initial amount of each fertilizer that could achieve maximum yield than this study realized.

scholarly journals Development of the high-order decoupled direct method in three dimensions for particulate matter: enabling advanced sensitivity analysis in air quality models

2012 ◽  
Vol 5 (2) ◽  
pp. 355-368 ◽  
Author(s):  
W. Zhang ◽  
S. L. Capps ◽  
Y. Hu ◽  
A. Nenes ◽  
S. L. Napelenok ◽  
...  
Keyword(s):  
Sensitivity Analysis ◽  
Particulate Matter ◽  
Air Quality ◽  
Direct Method ◽  
Statistical Significance ◽  
Second Order ◽  
High Order ◽  
Three Dimensions ◽  
So2 Emissions ◽  
Aerosol Module

Abstract. The high-order decoupled direct method in three dimensions for particulate matter (HDDM-3D/PM) has been implemented in the Community Multiscale Air Quality (CMAQ) model to enable advanced sensitivity analysis. The major effort of this work is to develop high-order DDM sensitivity analysis of ISORROPIA, the inorganic aerosol module of CMAQ. A case-specific approach has been applied, and the sensitivities of activity coefficients and water content are explicitly computed. Stand-alone tests are performed for ISORROPIA by comparing the sensitivities (first- and second-order) computed by HDDM and the brute force (BF) approximations. Similar comparison has also been carried out for CMAQ sensitivities simulated using a week-long winter episode for a continental US domain. Second-order sensitivities of aerosol species (e.g., sulfate, nitrate, and ammonium) with respect to domain-wide SO2, NOx, and NH3 emissions show agreement with BF results, yet exhibit less noise in locations where BF results are demonstrably inaccurate. Second-order sensitivity analysis elucidates poorly understood nonlinear responses of secondary inorganic aerosols to their precursors and competing species. Adding second-order sensitivity terms to the Taylor series projection of the nitrate concentrations with a 50% reduction in domain-wide NOx or SO2 emissions rates improves the prediction with statistical significance.

HIGH-ORDER GALERKIN APPROXIMATIONS FOR PARAMETRIC SECOND-ORDER ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

2013 ◽  
Vol 23 (09) ◽  
pp. 1729-1760 ◽  
Author(s):  
VICTOR NISTOR ◽  
CHRISTOPH SCHWAB
Keyword(s):  
Finite Element ◽  
Differential Operators ◽  
Weighted Sobolev Spaces ◽  
Regularity Result ◽  
Second Order ◽  
Three Dimensions ◽  
Open Unit ◽  
Partial Differential ◽  
Open Unit Ball ◽  
Galerkin Approximations

Let D ⊂ ℝd, d = 2, 3, be a bounded domain with piecewise smooth boundary, Y = ℓ∞(ℕ) and U = B1(Y), the open unit ball of Y. We consider a parametric family (Py)y∈U of uniformly strongly elliptic, second-order partial differential operators Py on D. Under suitable assumptions on the coefficients, we establish a regularity result for the solution u of the parametric boundary value problem Py u(x, y) = f(x, y), x ∈ D, y ∈ U, with mixed Dirichlet–Neumann boundary conditions on ∂d D and, respectively, on ∂n D. Our regularity and well-posedness results are formulated in a scale of weighted Sobolev spaces [Formula: see text] of Kondrat'ev type. We prove that the (Py)y ∈ U admit a shift theorem that is uniform in the parameter y ∈ U. Specifically, if the coefficients of P satisfy [Formula: see text], y = (yk)k≥1 ∈ U and if the sequences [Formula: see text] are p-summable in k, for 0 < p< 1, then the parametric solution u admits an expansion into tensorized Legendre polynomials Lν(y) such that the corresponding sequence [Formula: see text], where [Formula: see text]. We also show optimal algebraic orders of convergence for the Galerkin approximations uℓ of the solution u using suitable Finite Element spaces in two and three dimensions. Namely, let t = m/d and s = 1/p-1/2, where [Formula: see text], 0 < p < 1. We show that, for each m ∈ ℕ, there exists a sequence {Sℓ}ℓ≥0 of nested, finite-dimensional spaces Sℓ ⊂ L2(U;V) such that the Galerkin projections uℓ ∈ Sℓ of u satisfy ‖u - uℓ‖L2(U;V) ≤ C dim (Sℓ)- min {s, t} ‖f‖Hm-1(D), dim (Sℓ) → ∞. The sequence Sℓ is constructed using a sequence Vμ⊂V of Finite Element spaces in D with graded mesh refinements toward the singularities. Each subspace Sℓ is defined by a finite subset [Formula: see text] of "active polynomial chaos" coefficients uν ∈ V, ν ∈ Λℓ in the Legendre chaos expansion of u which are approximated by vν ∈ Vμ(ℓ, ν), for each ν ∈ Λℓ, with a suitable choice of μ(ℓ, ν).

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