Search results
Results from the WOW.Com Content Network
Thus, for example, a real function could be odd or even (or neither), as could a complex-valued function of a vector variable, and so on. The given examples are real functions, to illustrate the symmetry of their graphs .
Geometrically, the graph of an odd function has rotational symmetry with respect to the origin, meaning that its graph remains unchanged after rotation of 180 degrees about the origin. Examples of odd functions are x, x 3, sin(x), sinh(x), and erf(x).
Examples include even and odd functions in calculus, symmetric groups in abstract algebra, symmetric matrices in linear algebra, and Galois groups in Galois theory. In statistics , symmetry also manifests as symmetric probability distributions , and as skewness —the asymmetry of distributions.
For n odd, the dihedral group is centerless, so any element defines a non-trivial inner automorphism; for n even, the rotation by 180° (reflection through the origin) is the non-trivial element of the center. Thus for n odd, the inner automorphism group has order 2n, and for n even (other than n = 2) the inner automorphism group has order n.
Since the Fourier transform of a real signal is guaranteed to be Hermitian, it can be compressed using the Hermitian even/odd symmetry. This, for example, allows the discrete Fourier transform of a signal (which is in general complex) to be stored in the same space as the original real signal.
For example. a square has four axes of symmetry, because there are four different ways to fold it and have the edges match each other. Another example would be that of a circle, which has infinitely many axes of symmetry passing through its center for the same reason. [10] If the letter T is reflected along a vertical axis, it appears the same.
In the Schoenflies notation the symbol S n (German, Spiegel, for mirror), where n must be even, denotes the symmetry group generated by an n-fold improper rotation. For example, the symmetry operation S 6 is the combination of a rotation of (360°/6)=60° and a mirror plane
Every odd graph is 3-arc-transitive: every directed three-edge path in an odd graph can be transformed into every other such path by a symmetry of the graph. [12] Odd graphs are distance transitive, hence distance regular. [2]