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Similarly, f is strictly logarithmically convex if and only if, in the above two expressions, strict inequality holds for all t ∈ (0, 1). The above definition permits f to be zero, but if f is logarithmically convex and vanishes anywhere in X, then it vanishes everywhere in the interior of X.
A convex mirror diagram showing the focus, focal length, centre of curvature, principal axis, etc. A convex mirror or diverging mirror is a curved mirror in which the reflective surface bulges towards the light source. [1] Convex mirrors reflect light outwards, therefore they are not used to focus light.
Every concave function that is nonnegative on its domain is log-concave. However, the reverse does not necessarily hold. An example is the Gaussian function f(x) = exp(−x 2 /2) which is log-concave since log f(x) = −x 2 /2 is a concave function of x. But f is not concave since the second derivative is positive for | x | > 1:
as the only positive function f , with domain on the interval x > 0, that simultaneously has the following three properties: f (1) = 1, and f (x + 1) = x f (x) for x > 0 and f is logarithmically convex. A treatment of this theorem is in Artin's book The Gamma Function, [4] which has been reprinted by the AMS in a collection of Artin's writings.
The focal point F and focal length f of a positive (convex) lens, a negative (concave) lens, a concave mirror, and a convex mirror.. The focal length of an optical system is a measure of how strongly the system converges or diverges light; it is the inverse of the system's optical power.
In other words, a real image is an image which is located in the plane of convergence for the light rays that originate from a given object. Examples of real images include the image produced on a detector in the rear of a camera, and the image produced on an eyeball retina (the camera and eye focus light through an internal convex lens).
Instead, the angular aperture of a lens (or an imaging mirror) is expressed by the f-number, written f /N, where N is the f-number given by the ratio of the focal length f to the diameter of the entrance pupil D: =. This ratio is related to the image-space numerical aperture when the lens is focused at infinity. [3]
A convex hyperbolic mirror will reflect rays emanating from the focal point in front of the mirror as if they were emanating from the focal point behind the mirror. Conversely, it can focus rays directed at the focal point that is behind the mirror towards the focal point that is in front of the mirror as in a Cassegrain telescope.