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For example, x ∗ is a strict global maximum point if for all x in X with x ≠ x ∗, we have f(x ∗) > f(x), and x ∗ is a strict local maximum point if there exists some ε > 0 such that, for all x in X within distance ε of x ∗ with x ≠ x ∗, we have f(x ∗) > f(x). Note that a point is a strict global maximum point if and only if ...
The four-vertex theorem was first proved for convex curves (i.e. curves with strictly positive curvature) in 1909 by Syamadas Mukhopadhyaya. [8] His proof utilizes the fact that a point on the curve is an extremum of the curvature function if and only if the osculating circle at that point has fourth-order contact with the curve; in general the osculating circle has only third-order contact ...
Low-order polynomials tend to be smooth and high order polynomial curves tend to be "lumpy". To define this more precisely, the maximum number of inflection points possible in a polynomial curve is n-2, where n is the order of the polynomial equation. An inflection point is a location on the curve where it switches from a positive radius to ...
Water retention curve is the relationship between the water content, θ, and the soil water potential, ψ. The soil moisture curve is characteristic for different types of soil, and is also called the soil moisture characteristic. It is used to predict the soil water storage, water supply to the plants (field capacity) and soil aggregate stability.
The consolidation curve for a saturated clay showing the procedure for finding the preconsolidation pressure. Using a consolidation curve:(Casagrande 1936) [6] Choose by eye the point of maximum curvature on the consolidation curve. Draw a horizontal line from this point. Draw a line tangent to the curve at the point found in part 1.
From that point on, the Larsons cut their spending in half. Ken and Kathy Larson. They downsized from the big house on the river to a smaller one on one-third of an acre a few blocks away.
Figure 1: The red curve shows the constraint g(x, y) = c. The blue curves are contours of f(x, y). The point where the red constraint tangentially touches a blue contour is the maximum of f(x, y) along the constraint, since d 1 > d 2.
On Thursday, Nov. 21, Beyoncé attended the opening night of Broadway musical Death Becomes Her in New York City, accompanied by her mother, Tina Knowles, and Kelly Rowland.Rowland was joined by ...