Narrow Results

In the present paper, a modeling in the complex space is combined with complex-valued fractional Brownian motion to get some new results in biological systems. The rational of this approach is as follows. Biological dynamics which evolve continuously in time but are not time differentiable, necessarily exhibit random properties. These random features appear also as a result of the randomness of the proper time of biological systems. Usually, this is taken into account by using white noises that is to say fractals of order two. Fractals of order n larger than two are more suitable for increments with large amplitudes, and they may be introduced by using either real-valued fractal noises with long range memory or Brownian motions with independent increments, which are necessarily complex-valued. In the later case, we are then led to describe biological systems in the complex plane. After some background on the complex-valued fractional Brownian motion, we shall deal successively with population growth, information thermodynamics of order n, nonequilibrium phase transition via fractal noises and complexity of Markovian processes via the concept of informational divergence.

The paper analyzes Information Geometry (IG) generated by the evolutionary Information Macrodynamics (IMD) which are memorized by IG renovated geometrical structures. The IG encloses a cellular structure which carries a genetic code for the IMD restoration. The paper focuses on the natural enfolding of the universal optimal genetic code into the IG, obtained from the preceding dynamics, with a forthcoming creation of initial genetic information by the IG cells. The field of information geometry consists of the cell's sets that enable the discrete production of a quantum of information at the cells' boundaries. A particular spatial–time dynamic trajectory, located on a curved information space, enfolds and encodes a specific triplet's code, accumulated by the IG cellular information structure. The paper explores a potential conversion of imaginary into real information, associated with the transformation of mind into matter, and evaluates the dissimilar directional "difficulty" of crossing the boundary between imaginary, virtual and physical real information. The attractive information forces, as attributes of the curved IG, assemble the elementary quantums of time-space virtual dynamics into the cooperative macrostructures and information networks. The results analytically prove for the first time that a macrostructure's genetic information is enfolded into the IG geometrical structures. This means that the morphology of any biological macrostructure, as well as its external surface and inner structures, encloses the code of the macrostructure's genetic information. The complexity of encoded macrostructure depends not just on the total number of its corresponding code's symbols, but essentially on the complexity of the coding structure presented by a hierarchy of the code's symbols. This information systemic property, applied to the encoding of biogenetic information, demonstrates that the same gene's number (for different bio-species) can encode proteins with a variety of distinctive complexities, characterizing the huge diversity of biosystems from plants to humans.

This article introduces a systemic theory of phyllotaxis (study of primordial patterns on plants) and updates a mathematical model which is central in the theory. The theory deals with the descriptive and the functional aspects of phyllotaxis, and studies the origins of patterns as well. The article concentrates on the formal aspects of the model and on its explanatory values. The model possesses biological foundations which will not be recalled here. It supposes a principle of optimal design and the representation of phyllotactic patterns with control hierarchies. These hierarchies can be generated with irreducible matrices and L-systems. In the hierarchies, parameters can be identified representing important characteristics of growth that is complexity, stability and rhythm. A formula linking those parameters allows us to calculate the numerical cost of each one of the phyllotactic patterns and to order the costs. The various types of patterns come out, including whorled patterns which are seen as special cases of spiral patterns. The model proposes predictions which can be compared to observations. It predicts the existence of improbable patterns which have been later identified and it possesses explanatory values which have been interestingly put to contribution in difficult problems of pattern recognition in botany. It also possesses mathematical by-products in the theory of growth functions of L-systems, thus related to Perron-Frobenius spectral theory.

This article is a survey of recent advances on multilayer neural networks. The first section is a short summary on multilayer neural networks, their history, their architecture and their learning rule, the well-known back-propagation. In the following section, several theorems are cited, which present one-hidden-layer neural networks as universal approximators. The next section points out that two hidden layers are often required for exactly realizing d-dimensional dichotomies. Defining the frontier between one-hidden-layer and two-hidden-layer networks is still an open problem.

Several bounds on the size of a multilayer network which learns from examples are presented and we enhance the fact that, even if all can be done with only one hidden layer, more often, things can be done better with two or more hidden layers. Finally, this assertion 'is supported by the behaviour of multilayer neural networks in two applications: prediction of pollution and odor recognition modelling.