Measuring tree balance using symmetry nodes — A new balance index and its extremal properties

AbstractFor a permutation π, and the corresponding permutation matrix, we introduce the notion of discrete derivative, obtained by taking differences of successive entries in π. We characterize the possible derivatives of permutations, and consider questions for permutations with certain properties satisfied by the derivative. For instance, we consider permutations with distinct derivatives, and the relationship to so-called Costas arrays.

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Extremal Properties of an Intermittent Poisson Process Generating 1/f Noise

Fluctuation and Noise Letters ◽

10.1142/s0219477516500280 ◽

2016 ◽

Vol 15 (04) ◽

pp. 1650028 ◽

Cited By ~ 4

Author(s):

Ferdinand Grüneis

Keyword(s):

Stochastic Processes ◽

Poisson Process ◽

Total Power ◽

Spectral Features ◽

Active Period ◽

Quiescent Period ◽

The Family ◽

Random Intervals ◽

Theoretical Results ◽

Extremal Properties

It is well-known that the total power of a signal exhibiting a pure [Formula: see text] shape is divergent. This phenomenon is also called the infrared catastrophe. Mandelbrot claims that the infrared catastrophe can be overcome by stochastic processes which alternate between active and quiescent states. We investigate an intermittent Poisson process (IPP) which belongs to the family of stochastic processes suggested by Mandelbrot. During the intermission [Formula: see text] (quiescent period) the signal is zero. The active period is divided into random intervals of mean length [Formula: see text] consisting of a fluctuating number of events; this is giving rise to so-called clusters. The advantage of our treatment is that the spectral features of the IPP can be derived analytically. Our considerations are focused on the case that intermission is only a small disturbance of the Poisson process, i.e., to the case that [Formula: see text]. This makes it difficult or even impossible to discriminate a spike train of such an IPP from that of a Poisson process. We investigate the conditions under which a [Formula: see text] spectrum can be observed. It is shown that [Formula: see text] noise generated by the IPP is accompanied with extreme variance. In agreement with the considerations of Mandelbrot, the IPP avoids the infrared catastrophe. Spectral analysis of the simulated IPP confirms our theoretical results. The IPP is a model for an almost random walk generating both white and [Formula: see text] noise and can be applied for an interpretation of [Formula: see text] noise in metallic resistors.

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