On positivity of time-frequency distributions.

Consideration is given to the problem of how to regard the fundamental impossibility with time-frequency energy distributions of Cohen's class always to be nonnegative and, at the same time, to have correct marginal distributions. It is shown that the Wigner distribution is the only member of a large class of bilinear time-frequency distributions that becomes nonnegative after smoothing in the time-frequency plane by means of Gaussian weight functions with BT product equal to unity

An Analysis of Minimum Entropy Time-Frequency Distributions

The subject area of time-frequency analysis is concerned with creating meaningful representations of signals in the time-frequency domain that exhibit certain properties. Different applications require different characteristics in the representation. Some of the properties that are often desired include satisfying the time and frequency marginals, positivity, high localization, and strong finite support. Proper time-frequency distributions, which are defined as distributions that are manifestly positive and satisfy both the time and frequency marginals, are of particular interest since they can be viewed as a joint time-frequency density function and ensure strong finite support. Since an infinite number of proper time-frequency distributions exist, it is often necessary to impose additional constraints on the distribution in order to create a meaningful representation of the signal. A significant amount of research has been spent attempting to find constraints that produce meaningful representations.Recently, the idea was proposed of using the concept of minimum entropy to create time-frequency distributions that are highly localized and contain a large number of zero-points. The proposed method starts with an initial distribution that is proper and iteratively reduces the total entropy of the distribution while maintaining the positivity and marginal properties. The result of this method is a highly localized, proper TFD.This thesis will further explore and analyze the proposed minimum entropy algorithm. First, the minimum entropy algorithm and the concepts behind the algorithm will be introduced and discussed. After the introduction, a simple example of the method will be examined to help gain a basic understanding of the algorithm. Next, we will explore different rectangle selection methods which define the order in which the entropy of the distribution is minimized. We will then evaluate the effect of using different initial distributions with the minimum entropy algorithm. Afterwards, the results of the different rectangle selection methods and initial distributions will be analyzed and some more advanced concepts will be explored. Finally, we will draw conclusions and consider the overall effectiveness of the algorithm

Living on the edge of chaos: minimally nonlinear models of genetic regulatory dynamics

Linearized catalytic reaction equations modeling e.g. the dynamics of genetic regulatory networks under the constraint that expression levels, i.e. molecular concentrations of nucleic material are positive, exhibit nontrivial dynamical properties, which depend on the average connectivity of the reaction network. In these systems the inflation of the edge of chaos and multi-stability have been demonstrated to exist. The positivity constraint introduces a nonlinearity which makes chaotic dynamics possible. Despite the simplicity of such minimally nonlinear systems, their basic properties allow to understand fundamental dynamical properties of complex biological reaction networks. We analyze the Lyapunov spectrum, determine the probability to find stationary oscillating solutions, demonstrate the effect of the nonlinearity on the effective in- and out-degree of the active interaction network and study how the frequency distributions of oscillatory modes of such system depend on the average connectivity.Comment: 11 pages, 5 figure

Positivity of the English language

Over the last million years, human language has emerged and evolved as a fundamental instrument of social communication and semiotic representation. People use language in part to convey emotional information, leading to the central and contingent questions: (1) What is the emotional spectrum of natural language? and (2) Are natural languages neutrally, positively, or negatively biased? Here, we report that the human-perceived positivity of over 10,000 of the most frequently used English words exhibits a clear positive bias. More deeply, we characterize and quantify distributions of word positivity for four large and distinct corpora, demonstrating that their form is broadly invariant with respect to frequency of word use.Comment: Manuscript: 9 pages, 3 tables, 5 figures; Supplementary Information: 12 pages, 3 tables, 8 figure

Speed-of-light limitations in passive linear media

We prove that well-known speed of light restrictions on electromagnetic energy velocity can be extended to a new level of generality, encompassing even nonlocal chiral media in periodic geometries, while at the same time weakening the underlying assumptions to only passivity and linearity of the medium (either with a transparency window or with dissipation). As was also shown by other authors under more limiting assumptions, passivity alone is sufficient to guarantee causality and positivity of the energy density (with no thermodynamic assumptions). Our proof is general enough to include a very broad range of material properties, including anisotropy, bianisotropy (chirality), nonlocality, dispersion, periodicity, and even delta functions or similar generalized functions. We also show that the "dynamical energy density" used by some previous authors in dissipative media reduces to the standard Brillouin formula for dispersive energy density in a transparency window. The results in this paper are proved by exploiting deep results from linear-response theory, harmonic analysis, and functional analysis that had previously not been brought together in the context of electrodynamics.Comment: 19 pages, 1 figur

An application of multivariate total positivity to peacocks

We use multivariate total positivity theory to exhibit new families of peacocks. As the authors of \cite{HPRY}, our guiding example is the result of Carr-Ewald-Xiao \cite{CEX}. We shall introduce the notion of strong conditional monotonicity. This concept is strictly more restrictive than the conditional monotonicity as defined in \cite{HPRY} (see also \cite{Be}, \cite{BPR1} and \cite{ShS1}). There are many random vectors which are strongly conditionally monotone (SCM). Indeed, we shall prove that multivariate totally positive of order 2 (MTP$_2$) random vectors are SCM. As a consequence, stochastic processes with MTP$_2$ finite-dimensional marginals are SCM. This family includes processes with independent and log-concave increments, and one-dimensional diffusions which have absolutely continuous transition kernels.Comment: 29 page