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Journal articles on the topic 'Linear dependence (Mathematics)'

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Published: 4 June 2021
Last updated: 27 January 2023

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1

Bleicher, M. N., and G. B. Preston. "Abstract linear dependence relations." Publicationes Mathematicae Debrecen 8, no. 1-2 (2022): 55–63. http://dx.doi.org/10.5486/pmd.1961.8.1-2.03.

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2

Halas, Zdeněk. "Continuous dependence of solutions of generalized linear ordinary differential equations on a parameter." Mathematica Bohemica 132, no. 2 (2007): 205–18. http://dx.doi.org/10.21136/mb.2007.134190.

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3

Johnson, K. G. "A theorem on abstract linear dependence relations." Publicationes Mathematicae Debrecen 8, no. 1-2 (2022): 64–67. http://dx.doi.org/10.5486/pmd.1961.8.1-2.04.

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4

Liu, Dayan, Xiankun Du, and Xiaosong Sun. "The linear dependence problem for power linear maps." Linear Algebra and its Applications 426, no. 2-3 (2007): 706–15. http://dx.doi.org/10.1016/j.laa.2007.05.047.

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5

Kumar, K. Senthil. "Linear dependence of quasi-periods over the rationals." Comptes Rendus. Mathématique 359, no. 4 (2021): 409–14. http://dx.doi.org/10.5802/crmath.171.

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6

CAI, GUANG-HUI. "ALMOST SURE CONVERGENCE FOR LINEAR PROCESS GENERATED BY ASYMPTOTICALLY LINEAR NEGATIVE QUADRANT DEPENDENCE PROCESSES." Communications of the Korean Mathematical Society 20, no. 1 (2005): 161–68. http://dx.doi.org/10.4134/ckms.2005.20.1.161.

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7

Okhrimenko, Viacheslav, and Maiia Zbіtnieva. "Mathematical Model of Tubular Linear Induction Motor." Mathematical Modelling of Engineering Problems 8, no. 1 (2021): 103–9. http://dx.doi.org/10.18280/mmep.080113.

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Problem of calculation of distribution of magnetic field induction in clearance of tubular linear induction motor (TLIM) is considered. Mathematical model is represented by Fredholm integral equations of second kind for complexes of electric field strength and density of coupled magnetization currents at interface of environments. Algorithm of calculation of distribution of magnetic field induction in TLIM clearance has been developed. Dependence of magnetic field induction in motor clearance on value of pole division is investigated. There is area of optimum pole pitch. Reliability of results of calculations on mathematical model is confirmed by their comparison with results obtained on physical model. Calculated dependence of induction on thickness of runner's iron circuit also has extreme character. Given model can be used at design stage of TLIM. Model allows calculating its optimal geometric dimensions based on criterion of maximum induction in motor clearance, taking into account physical properties of applied materials.
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8

Xie, Yuan-tao, Juan Yang, Chong-guang Jiang, Zi-yu Cai, and Joshua Adagblenya. "Incidence, Dependence Structure of Disease, and Rate Making for Health Insurance." Mathematical Problems in Engineering 2018 (August 12, 2018): 1–13. http://dx.doi.org/10.1155/2018/4265801.

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In order to analyze the two goals under the national strategy of “Healthy China”, this paper attempts to solve the problem of coverage rate and guarantee level of health insurance, as well as the rational allocation of full life cycle health insurance resources. This paper uses pair copula to model the dependence of different disease incidence and proposes an actuarial model for rate making in health insurance based on the dependence captured by pair copula. These are far more accurate than any other model and more proper for covering a basket of several different diseases. The data for the paper was drawn from the experience incidence table of major diseases (malignant tumor, acute myocardial infarction, and stroke sequelae) from the ages 0-65 years in the Chinese life insurance industry. Extending the hypothesis of independence in actuarial modeling, the authors comprehensively use a hierarchical copula theory to extract the dependence structure of risk variable in insurance. The classification rate making technology and survival analysis method in traditional actuarial pricing were also considered. This paper applied the generalized linear model, which is commonly used in nonlife insurance pricing for empirical study of health insurance rate making. The authors discovered that the incidence of major diseases and the single premium rate calculated by the generalized linear model under HAC dependence structure were both significantly different from that calculated by the Manchester United method without dependency. The authors also stated that the rate based on the generalized linear model under HAC dependence structure was a bit different from that without dependency but both were generally the same as that of Care Expert in PICC Health. The underestimation or overestimation of systematic risks and the distortion of the rate system can be eliminated if we combine risk dependence into modeling.
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9

Knops, R. J., and L. E. Payne. "Improved estimates for continuous data dependence in linear elastodynamics." Mathematical Proceedings of the Cambridge Philosophical Society 103, no. 3 (1988): 535–59. http://dx.doi.org/10.1017/s0305004100065142.

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In a previous paper [6], the present authors established estimates for the continuous dependence of the solution on various data in the initial boundary value problem of linear elastodynamics on a bounded region of space. The main conclusion concerned continuous dependence on the body-force, but also it was shown how this result could be used to derive continuous dependence on the initial data, elasticities, boundary data and initial geometry. The method adopted was based upon logarithmic convexity arguments and hence led naturally to continuity in the sense of Hölder on compact sub-intervals of time. A special feature of the study entailed the lack of any sign-definiteness conditions on the elasticities which, of course, in the absence of any a priori constraint on the solution always gives rise to an ill-posed problem. (See, for instance, the comprehensive survey by Payne [10].)
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10

Barańczuk, Stefan. "Hasse principle for linear dependence in Mordell-Weil groups." Functiones et Approximatio Commentarii Mathematici 62, no. 1 (2020): 81–85. http://dx.doi.org/10.7169/facm/1792.

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11

Dall'Aglio, Marco, and Marco Scarsini. "Zonoids, linear dependence, and size-biased distributions on the simplex." Advances in Applied Probability 35, no. 4 (2003): 871–84. http://dx.doi.org/10.1239/aap/1067436324.

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The zonoid of a d-dimensional random vector is used as a tool for measuring linear dependence among its components. A preorder of linear dependence is defined through inclusion of the zonoids. The zonoid of a random vector does not characterize its distribution, but it does characterize the size-biased distribution of its compositional variables. This fact will allow a characterization of our linear dependence order in terms of a linear-convex order for the size-biased compositional variables. In dimension 2 the linear dependence preorder will be shown to be weaker than the concordance order. Some examples related to the Marshall-Olkin distribution and to a copula model will be presented, and a class of measures of linear dependence will be proposed.
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12

Jossen, Peter. "Detecting linear dependence on an abelian variety via reduction maps." Commentarii Mathematici Helvetici 88, no. 2 (2013): 323–52. http://dx.doi.org/10.4171/cmh/287.

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13

Brešar, Matej, and Igor Klep. "A local-global principle for linear dependence of noncommutative polynomials." Israel Journal of Mathematics 193, no. 1 (2012): 71–82. http://dx.doi.org/10.1007/s11856-012-0066-4.

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14

Kocsis, Imre. "On the Linear Dependence of a Finite Set of Additive Functions." Results in Mathematics 62, no. 1-2 (2011): 67–71. http://dx.doi.org/10.1007/s00025-011-0130-0.

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15

Aneiros-Pérez, G., and J. M. Vilar-Fernández. "Local polynomial estimation in partial linear regression models under dependence." Computational Statistics & Data Analysis 52, no. 5 (2008): 2757–77. http://dx.doi.org/10.1016/j.csda.2007.10.009.

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16

Brock, Friedemann, Leonelo Iturriaga, and Pedro Ubilla. "Semi-linear singular elliptic equations with dependence on the gradient." Nonlinear Analysis: Theory, Methods & Applications 65, no. 3 (2006): 601–14. http://dx.doi.org/10.1016/j.na.2005.09.034.

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17

Andreani, Roberto, Gabriel Haeser, María Laura Schuverdt, and Paulo J. S. Silva. "A relaxed constant positive linear dependence constraint qualification and applications." Mathematical Programming 135, no. 1-2 (2011): 255–73. http://dx.doi.org/10.1007/s10107-011-0456-0.

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18

Hirukawa, Junichi. "On the Causality between Multiple Locally Stationary Processes." Advances in Decision Sciences 2012 (May 30, 2012): 1–15. http://dx.doi.org/10.1155/2012/261707.

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When one would like to describe the relations between multivariate time series, the concepts of dependence and causality are of importance. These concepts also appear to be useful when one is describing the properties of an engineering or econometric model. Although the measures of dependence and causality under stationary assumption are well established, empirical studies show that these measures are not constant in time. Recently one of the most important classes of nonstationary processes has been formulated in a rigorous asymptotic framework by Dahlhaus in (1996), (1997), and (2000), called locally stationary processes. Locally stationary processes have time-varying spectral densities whose spectral structures smoothly change in time. Here, we generalize measures of linear dependence and causality to multiple locally stationary processes. We give the measures of linear dependence, linear causality from one series to the other, and instantaneous linear feedback, at time t and frequency λ.
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19

Galliani, Pietro, and Jouko Väänänen. "Diversity, dependence and independence." Annals of Mathematics and Artificial Intelligence 90, no. 2-3 (2021): 211–33. http://dx.doi.org/10.1007/s10472-021-09778-8.

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AbstractWe propose a very general, unifying framework for the concepts of dependence and independence. For this purpose, we introduce the notion of diversity rank. By means of this diversity rank we identify total determination with the inability to create more diversity, and independence with the presence of maximum diversity. We show that our theory of dependence and independence covers a variety of dependence concepts, for example the seemingly unrelated concepts of linear dependence in algebra and dependence of variables in logic.
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20

Astrauskas, A., J. B. L�vy, and M. S. Taqqu. "The asymptotic dependence structure of the linear fractional L�vy motion." Lithuanian Mathematical Journal 31, no. 1 (1992): 1–19. http://dx.doi.org/10.1007/bf00972312.

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21

Dahmen, Wolfgang, Rong Qing Jia, and Charles A. Micchelli. "On Linear Dependence Relations for Integer Translates of Compactly Supported Distributions." Mathematische Nachrichten 151, no. 1 (1991): 303–10. http://dx.doi.org/10.1002/mana.19911510119.

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22

Cooley, D., and E. Thibaud. "Decompositions of dependence for high-dimensional extremes." Biometrika 106, no. 3 (2019): 587–604. http://dx.doi.org/10.1093/biomet/asz028.

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Summary We propose two decompositions that help to summarize and describe high-dimensional tail dependence within the framework of regular variation. We use a transformation to define a vector space on the positive orthant and show that transformed-linear operations applied to regularly-varying random vectors preserve regular variation. We summarize tail dependence via a matrix of pairwise tail dependence metrics that is positive semidefinite; eigendecomposition allows one to interpret tail dependence in terms of the resulting eigenbasis. This matrix is completely positive, and one can easily construct regularly-varying random vectors that share the same pairwise tail dependencies. We illustrate our methods with Swiss rainfall and financial returns data.
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23

Rosenblatt, Joseph. "Linear independence of translations." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 59, no. 1 (1995): 131–33. http://dx.doi.org/10.1017/s1446788700038519.

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AbstractIt was shown by Edgar and Rosenblatt that f ∈ Lp (ℝn), 1 ≤ p < 2n/ (n-1), and f ≠0, then f has linearly independent translates. Using a result of Hömander, it is shown here that the same theorem holds if p = 2n / (n−1). This gives a sharp result because for n ≥2, there exists f ∈C0 (ℝn), f ≠0, which is simultaneously in all Lp (ℝn), p > 2n/(n−1), that has a linear dependence relation among its translates. References and some discussion are included.
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24

BARREIRA, LUIS, and CLAUDIA VALLS. "PARAMETER DEPENDENCE OF STABLE MANIFOLDS FOR IMPULSIVE EQUATIONS." Analysis and Applications 12, no. 02 (2014): 131–60. http://dx.doi.org/10.1142/s0219530514500092.

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We establish the existence of stable manifolds under sufficiently small perturbations of a linear impulsive equation. Our results are optimal, in the sense that for vector fields of class C1 outside the jumping times, the invariant manifolds are also of class C1 outside these times. We also consider the case of C1 parameter-dependent perturbations and we establish the C1 dependence of the stable manifolds on the parameter. The proof uses the fiber contraction principle. We emphasize that we consider the general case of nonautonomous equations for which the linear part has a nonuniform exponential dichotomy.
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25

Sivakumar, N. "Concerning the linear dependence of integer translates of exponential box splines." Journal of Approximation Theory 64, no. 1 (1991): 95–118. http://dx.doi.org/10.1016/0021-9045(91)90088-r.

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26

Hladík, Milan. "Solution Set Characterization of Linear Interval Systems with a Specific Dependence Structure." Reliable Computing 13, no. 4 (2007): 361–74. http://dx.doi.org/10.1007/s11155-007-9033-x.

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27

Karbauskaitė, Rasa, Gintautas Dzemyda, and Virginijus Marcinkevičius. "Dependence of locally linear embedding on the regularization parameter." TOP 18, no. 2 (2010): 354–76. http://dx.doi.org/10.1007/s11750-010-0151-y.

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28

Aouadi, Moncef. "Temperature dependence of an elastic modulus in generalized linear micropolar thermoelasticity." Zeitschrift für angewandte Mathematik und Physik 57, no. 6 (2006): 1057–74. http://dx.doi.org/10.1007/s00033-005-0055-0.

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29

BARANOVSKI, ALEXANDER L., and ANTHONY J. LAWRANCE. "SENSITIVE PARAMETER DEPENDENCE OF AUTOCORRELATION FUNCTION IN PIECEWISE LINEAR MAPS." International Journal of Bifurcation and Chaos 17, no. 04 (2007): 1185–97. http://dx.doi.org/10.1142/s0218127407017768.

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This paper is principally concerned with the effect of splitting and permuting, or shifting, the branches of an onto piecewise linear map, with particular regard to the effect on the autocorrelation of the associated chaotic sequence. This is shown to be a chaotic function of the shifting parameter of the map, and its sensitivity with respect to minute changes in this parameter is termed autocorrelation chaos. This paper presents both analytical and computational studies of the phenomenon.
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30

Faraci, F., and D. Puglisi. "Nodal solutions of semi-linear elliptic equations with dependence on the gradient." Bulletin des Sciences Mathématiques 175 (March 2022): 103101. http://dx.doi.org/10.1016/j.bulsci.2022.103101.

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31

Motreanu, Dumitru, and Elisabetta Tornatore. "Location of solutions for quasi-linear elliptic equations with general gradient dependence." Electronic Journal of Qualitative Theory of Differential Equations, no. 87 (2017): 1–10. http://dx.doi.org/10.14232/ejqtde.2017.1.87.

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32

Borodina, E. B. "An algorithm for constructing multidimensional continued fractions and linear dependence of numbers." Mathematical Notes 99, no. 1-2 (2016): 37–45. http://dx.doi.org/10.1134/s0001434616010041.

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33

Giraitis, L., and D. Surgailis. "Limit theorem for polynomials of a linear process with long-range dependence." Lithuanian Mathematical Journal 29, no. 2 (1990): 128–45. http://dx.doi.org/10.1007/bf00966075.

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34

Tam, K. K. "Initiation of thermal explosion by intense light: criticality dependence on data and parameters." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 28, no. 3 (1987): 287–95. http://dx.doi.org/10.1017/s0334270000005403.

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AbstractA model for thermal ignition by intense light is studied. The governing non-linear parabolic equation is linearized in a two-step manner with the aid of a non-linear ordinary differential equation which captures the salient features of the non-linear parabolic equation. The critical parameters are computed from the steady-state solution of the ordinary differential equation, which can be obtained without actually solving the equation. Comparison with available data shows that the present method yields good results.
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35

Lu, Fangyan. "Linear dependence of operators characterized by trace functionals." Linear Algebra and its Applications 434, no. 1 (2011): 343–55. http://dx.doi.org/10.1016/j.laa.2010.08.017.

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36

Galdi, G. P., R. J. Knops, and S. Rionero. "Uniqueness and continuous dependence in the linear elastodynamic exterior and half-space problems." Mathematical Proceedings of the Cambridge Philosophical Society 99, no. 2 (1986): 357–66. http://dx.doi.org/10.1017/s0305004100064276.

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A method based upon the Lagrange identity has been used by Brun [2] in the linear theories of thermoelasticity and viscoelasticity to establish uniqueness of the solution to the initial boundary value problem on bounded three-dimensional regions. A major feature of Brun's analysis is that it does not require any sign-definiteness assumptions on, for instance, the elasticities. The technique was extended by Knops and Payne [14] to derive certain continuous dependence results in linear elastodynamics, again for a bounded region. These authors had earlier recovered Brun's uniqueness result for linear elasticity [11] and derived other continuous dependence results based upon logarithmic convexity arguments [12, 14] (see also [13] for a similar treatment of thermoelasticity). Levine [18] later treated an abstract version of the Brun approach and applied it to a family of abstract linear operator equations. Among his results is a simplified proof that equipartition of the kinetic and potential energies occurs. Other applications of the Lagrange identity in proofs of uniqueness for bounded regions include those by Naghdi and Trapp [19] for a Cosserat surface, and by Green [9] for a theory of linear thermoelasticity that allows second sound.
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37

Loperfido, Nicola Maria Rinaldo. "Canonical Correlations and Nonlinear Dependencies." Symmetry 13, no. 7 (2021): 1308. http://dx.doi.org/10.3390/sym13071308.

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Canonical correlation analysis (CCA) is the default method for investigating the linear dependence structure between two random vectors, but it might not detect nonlinear dependencies. This paper models the nonlinear dependencies between two random vectors by the perturbed independence distribution, a multivariate semiparametric model where CCA provides an insight into their nonlinear dependence structure. The paper also investigates some of its probabilistic and inferential properties, including marginal and conditional distributions, nonlinear transformations, maximum likelihood estimation and independence testing. Perturbed independence distributions are closely related to skew-symmetric ones.
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38

Sonkin, T. N. "Prediction of confidence limits for reliability indexes with linear dependence on parameters." Journal of Soviet Mathematics 39, no. 4 (1987): 2909–14. http://dx.doi.org/10.1007/bf01092344.

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39

Tazaki, Asahi, Shintaro Takeuchi, Suguru Miyauchi, Lucy T. Zhang, Ryo Onishi, and Takeo Kajishima. "Fluid Permeation Through A Membrane With Infinitesimal Permeability Under Reynolds Lubrication." Journal of Mechanics 36, no. 5 (2020): 637–48. http://dx.doi.org/10.1017/jmech.2020.38.

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ABSTRACTTo understand the lubrication-dominated permeation through a membrane, numerical simulations of permeation through a moving corrugated permeable membrane is carried out with a fully validated numerical method. Through comparisons between the numerical results and the results of an asymptotic analysis of permeate flux (under an infinitesimal permeability condition) using Reynolds lubrication equation, the effect of permeation on lubrication and its inverse effect (i.e., the dependence of permeation on lubrication) are discussed. The linear and non-linear dependences of the relaxation of the lubrication pressure due to membrane permeation are identified. The effect of the tangential component of the permeate flux is evaluated by a linear analysis, and the limitation of Reynolds-type lubrication is discussed.
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40

Lomadze, Vakhtang. "Continuous dependence of linear differential systems on polynomial modules." Mathematics of Control, Signals, and Systems 32, no. 3 (2020): 385–409. http://dx.doi.org/10.1007/s00498-020-00263-x.

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41

Jossen, Peter, and Antonella Perucca. "A counterexample to the local–global principle of linear dependence for Abelian varieties." Comptes Rendus Mathematique 348, no. 1-2 (2010): 9–10. http://dx.doi.org/10.1016/j.crma.2009.11.019.

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42

Saavedra, Angeles, Javier Taboada, María Araújo, and Eduardo Giráldez. "Generalized Linear Spatial Models to Predict Slate Exploitability." Journal of Applied Mathematics 2013 (2013): 1–7. http://dx.doi.org/10.1155/2013/531062.

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The aim of this research was to determine the variables that characterize slate exploitability and to model spatial distribution. A generalized linear spatial model (GLSMs) was fitted in order to explore relationship between exploitability and different explanatory variables that characterize slate quality. Modelling the influence of these variables and analysing the spatial distribution of the model residuals yielded a GLSM that allows slate exploitability to be predicted more effectively than when using generalized linear models (GLM), which do not take spatial dependence into account. Studying the residuals and comparing the prediction capacities of the two models lead us to conclude that the GLSM is more appropriate when the response variable presents spatial distribution.
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43

Shan, Qingsong, and Qianning Liu. "Nonparametric estimation of the measure of functional dependence." AIMS Mathematics 6, no. 12 (2021): 13488–502. http://dx.doi.org/10.3934/math.2021782.

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<abstract><p>In this paper, we propose a beta kernel estimator to measure functional dependence (MFD). The MFD not only can measure the strength of linear or monotonic relationships, but it is also suitable for more complicated functional dependence. We derive the asymptotic distribution of the proposed estimator and then use several simulated examples to compare our estimator with the traditional measures. Our simulation results demonstrate that beta kernel provides high accuracy in estimation. A real data example is also given to illustrate one possible application of the new estimator.</p></abstract>
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44

Akbar, Saddam Sultan, and Meraj Mustafa. "Application of Exponential Temperature Dependent Viscosity Model for Fluid Flow over a Moving or Stationary Slender Surface." Mathematics 10, no. 18 (2022): 3269. http://dx.doi.org/10.3390/math10183269.

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The problem of laminar flow around a moving thin needle or slender surface with free stream velocity is analyzed when viscosity is supposed to have an exponential temperature dependency. Additionally, the temperature dependence in thermal conductivity is retained. Consideration of variable viscosity and thermal conductivity makes the governing equations coupled and non-linear. A self-similar solution of the problem is achieved, which depends on a parameter , which is the quotient of wall and ambient temperatures. A comparison of present findings is made with those of inversely linear temperature-dependent viscosity and constant viscosity cases. The size of the needle plays an important part in enhancing thermal boundary layer thickness. The expressions of skin friction coefficient and local Nusselt number in case of exponential temperature dependent viscosity are just derived in this study. An important observation is that computational results are qualitatively like those noticed for the case of inversely linear temperature dependency.
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45

Walker, Ronald A. "Regarding an adaptive algorithm for testing multivariate linear dependence." Linear Algebra and its Applications 434, no. 2 (2011): 605–13. http://dx.doi.org/10.1016/j.laa.2010.09.024.

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46

Sergeev, I. N. "Study of the linear dependence of scalar-valued functions using the Wronskian determinant." Journal of Mathematical Sciences 164, no. 6 (2010): 945–47. http://dx.doi.org/10.1007/s10958-010-9775-6.

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47

FOKAS, A. S., and B. PELLONI. "Two-point boundary value problems for linear evolution equations." Mathematical Proceedings of the Cambridge Philosophical Society 131, no. 3 (2001): 521–43. http://dx.doi.org/10.1017/s0305004101005436.

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We study boundary value problems for a linear evolution equation with spatial derivatives of arbitrary order, on the domain 0 < x < L, 0 < t < T, with L and T positive finite constants. We present a general method for identifying well-posed problems, as well as for constructing an explicit representation of the solution of such problems. This representation has explicit x and t dependence, and it consists of an integral in the k-complex plane and of a discrete sum. As illustrative examples we solve some two-point boundary value problems for the equations iqt + qxx = 0 and qt + qxxx = 0.
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48

Leszczyński, H. "Finite Difference Schemes for a Non-linear Heat Equation with Functional Dependence." ZAMM 79, no. 1 (1999): 53–64. http://dx.doi.org/10.1002/(sici)1521-4001(199901)79:1<53::aid-zamm53>3.0.co;2-a.

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49

Lorenzi, Alfredo, Luca Lorenzi, and Masahiro Yamamoto. "Continuous dependence and uniqueness for lateral Cauchy problems for linear integro-differential parabolic equations." Journal of Inverse and Ill-posed Problems 25, no. 5 (2017): 617–31. http://dx.doi.org/10.1515/jiip-2016-0066.

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AbstractVia Carleman estimates we prove uniqueness and continuous dependence results for lateral Cauchy problems for linear integro-differential parabolic equations without initial conditions. The additional information supplied prescribes the conormal derivative of the temperature on a relatively open subset of the lateral boundary of the space-time domain.
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50

Bulgariu, Emilian. "On the Uniqueness and Continuous Dependence in the Linear Theory of Thermo-Microstretch Elasticity Backward in Time." Annals of the Alexandru Ioan Cuza University - Mathematics 59, no. 2 (2013): 339–55. http://dx.doi.org/10.2478/v10157-012-0042-7.

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Abstract We study the uniqueness and the continuous dependence problems for the thermo-microstretch elastic processes backward in time. The data are given for the final time t = 0 and we want to study the solution at the previous moments. We transform the problem in a boundary-initial value problem by an appropriate change of variables. The uniqueness theorems presented in this article extend in a particular case the uniqueness theorem of Passarella and Tibullo (2010) and we also discuss a different class of problems than the one considered by them. We find some estimates that prove the continuous dependence of solution with respect to the final data.
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