A function is called strictly concave if
A continuous function on C is concave if and only if
A differentiable function f is concave on an interval if its derivative function f ′ is monotonically decreasing on that interval: a concave function has a decreasing slope. ("Decreasing" here means "non-increasing", rather than "strictly decreasing", and thus allows zero slopes.)
For a twice-differentiable function f, if the second derivative, f ′′(x), is positive (or, if the acceleration is positive), then the graph is convex; if f ′′(x) is negative, then the graph is concave. Points where concavity changes are inflection points.
If a convex (i.e., concave upward) function has a "bottom", any point at the bottom is a minimal extremum. If a concave (i.e., concave downward) function has an "apex", any point at the apex is a maximal extremum.
If f(x) is twice-differentiable, then f(x) is concave if and only if f ′′(x) is non-positive. If its second derivative is negative then it is strictly concave, but the opposite is not true, as shown by f(x) = -x4.
A function is called quasiconcave if and only if there is an such that for all
US Patent Issued to NEC Laboratories America on Aug. 17 for "Method and Apparatus for Transductive Support Vector Machines" (New Jersey, New York Inventors)
Aug 18, 2010; ALEXANDRIA, Va., Aug. 24 -- United States Patent no. 7,778,949, issued on Aug. 17, was assigned to NEC Laboratories America Inc....