Calculus in ℝn

Grossman, Stanley I.

doi:10.1016/b978-0-12-304380-1.50014-2

Cited by 3 publications

(7 citation statements)

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“…With a linear change m in thickness, the cross-sectional area of flow at a given radius is the surface area of a cylinder with a sloping bottom. This surface area can be determined by integrating along a parametric surface (adapted from Grossman [1992]):…”

Section: Nonstationary Casementioning

confidence: 99%

Flow dimensions corresponding to hydrogeologic conditions

Walker

Roberts

2003

Water Resources Research

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[1] The generalized radial flow approach to hydraulic test interpretation uses the flow dimension to describe the change in flow area versus radial distance from the borehole. The flow dimension of a hydraulic test may reflect several characteristics of the hydrogeologic system, including heterogeneity, boundaries, and leakage. We show that a radial flow system with a linear, constant-head boundary asymptotically reaches a flow dimension of four, while the flow dimension of a leaky aquifer is a function of time and the leakage factor. We use numerical techniques to show that a stationary transmissivity field with a modest level of heterogeneity has a flow dimension that stabilizes at two. We also show that the flow dimension for a nonstationary transmissivity field depends on the form and magnitude of the nonstationarity. The flow dimension observed during hydraulic tests helps identify admissible conceptual models for the tested system, and places hydraulic testing in its full hydrogeologic context.

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Section: Nonstationary Casementioning

confidence: 99%

Flow dimensions corresponding to hydrogeologic conditions

Walker

Roberts

2003

Water Resources Research

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“…4.2 and 5. Since it will turn out to be the case that the likelihood ratio curves of interest in those two cases are closed loops, it will also be useful to consider the curvature [21] of such curves, which is similar in spirit to the second derivative of a function but defined more generally for parametric curves as…”

Section: General Theorymentioning

confidence: 99%

“…By the implicit function theorem, we can again obtain v n as a function of ( v 1 , …, v n −1 ) as in Eq. 47 [21]; however, we are no longer assuming that this function has n − 1 linearly independent tangent vectors at every point (take, e. g., the case of a curve in a three-dimensional space). We will from here on refer to the number of linearly independent tangent vectors to a surface at a point on that surface as the number of degrees of freedom of the surface at that point.…”

Section: Parametric Surfaces and Jacobiansmentioning

confidence: 99%

“…But this matrix in turn is just the Jacobian of the coordinate transformation determined by the first n − 1 equations of Eq. 48 [21]. …”

Section: Parametric Surfaces and Jacobiansmentioning

confidence: 99%

“…61, namely J x → P , has only 4 columns; while the second matrix, namely J m → x , is a 4 × 5 matrix. The rank of each is thus necessarily four or less [21]. It follows that the rank of the product, J m → P , is also four or less.…”

Section: Degrees Of Freedom Of Roc Surfacesmentioning

confidence: 99%

See 2 more Smart Citations

The three-class ideal observer for univariate normal data: Decision variable and ROC surface properties

Edwards¹,

Metz²

2012

Journal of Mathematical Psychology

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Although a fully general extension of ROC analysis to classification tasks with more than two classes has yet to be developed, the potential benefits to be gained from a practical performance evaluation methodology for classification tasks with three classes have motivated a number of research groups to propose methods based on constrained or simplified observer or data models. Here we consider an ideal observer in a task with underlying data drawn from three univariate normal distributions. We investigate the behavior of the resulting ideal observer's decision variables and ROC surface. In particular, we show that the pair of ideal observer decision variables is constrained to a parametric curve in two-dimensional likelihood ratio space, and that the decision boundary line segments used by the ideal observer can intersect this curve in at most six places. From this, we further show that the resulting ROC surface has at most four degrees of freedom at any point, and not the five that would be required, in general, for a surface in a sixdimensional space to be non-degenerate. In light of the difficulties we have previously pointed out in generalizing the well-known area under the ROC curve performance metric to tasks with three or more classes, the problem of developing a suitable and fully general performance metric for classification tasks with three or more classes remains unsolved.

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Response Time of Thin‐Layer Electrolytic Cells to Potential‐Step Signals

Yamada¹,

Kitamura²,

Ohsaka³

et al. 1996

J. Electrochem. Soc.

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A theoretical approach is presented to the problem of ohmic potential drops in response to potential-step excitation in circular thin-layer cells such as those used for in situ infrared reflection-absorption measurements of electrode surfaces. Both the faradaic current due to the reversible electrode reaction of adsorbed redox species on the electrode surface and the double-layer charging current have been taken into account. The problem formulated as an initial and boundary value problem was transformed into an integral equation, and then it was solved numerically. An approximate equation is presented which allows us to estimate the response time of the circular thin-layer cell from the relevant cell geometries and cell parameters.

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Calculus in ℝn

Cited by 3 publications

References 0 publications

Flow dimensions corresponding to hydrogeologic conditions

Flow dimensions corresponding to hydrogeologic conditions

The three-class ideal observer for univariate normal data: Decision variable and ROC surface properties

Response Time of Thin‐Layer Electrolytic Cells to Potential‐Step Signals

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