Dynamics of Oscillators Coupled by a Medium with Adaptive Impact

In this article we study the dynamics of coupled oscillators. We use mechanical metronomes that are placed over a rigid base. The base moves by a motor in a one-dimensional direction and the movements of the base follow some functions of the phases of the metronomes (in other words, it is controlled to move according to a provided function). Because of the motor and the feedback, the phases of the metronomes affect the movements of the base while on the other hand, when the base moves, it affects the phases of the metronomes in return. For a simple function for the base movement (such as $y = \gamma_{x} [r \theta_1 + (1 - r) \theta_2]$ in which $y$ is the velocity of the base, $\gamma_{x}$ is a multiplier, $r$ is a proportion and $\theta_1$ and $\theta_2$ are phases of the metronomes), we show the effects on the dynamics of the oscillators. Then we study how this function changes in time when its parameters adapt by a feedback. By numerical simulations and experimental tests, we show that the dynamic of the set of oscillators and the base tends to evolve towards a certain region. This region is close to a transition in dynamics of the oscillators; where more frequencies start to appear in the frequency spectra of the phases of the metronomes

Semantic spaces

Any natural language can be considered as a tool for producing large databases (consisting of texts, written, or discursive). This tool for its description in turn requires other large databases (dictionaries, grammars etc.). Nowadays, the notion of database is associated with computer processing and computer memory. However, a natural language resides also in human brains and functions in human communication, from interpersonal to intergenerational one. We discuss in this survey/research paper mathematical, in particular geometric, constructions, which help to bridge these two worlds. In particular, in this paper we consider the Vector Space Model of semantics based on frequency matrices, as used in Natural Language Processing. We investigate underlying geometries, formulated in terms of Grassmannians, projective spaces, and flag varieties. We formulate the relation between vector space models and semantic spaces based on semic axes in terms of projectability of subvarieties in Grassmannians and projective spaces. We interpret Latent Semantics as a geometric flow on Grassmannians. We also discuss how to formulate G\"ardenfors' notion of "meeting of minds" in our geometric setting.Comment: 32 pages, TeX, 1 eps figur

The distribution of word matches between Markovian sequences with periodic boundary conditions

Word match counts have traditionally been proposed as an alignment-free measure of similarity for biological sequences. The D2 statistic, which simply counts the number of exact word matches between two sequences, is a useful test bed for developing rigorous mathematical results, which can then be extended to more biologically useful measures. The distributional properties of the D2 statistic under the null hypothesis of identically and independently distributed letters have been studied extensively, but no comprehensive study of the D2 distribution for biologically more realistic higher-order Markovian sequences exists. Here we derive exact formulas for the mean and variance of the D2 statistic for Markovian sequences of any order, and demonstrate through Monte Carlo simulations that the entire distribution is accurately characterized by a Pólya-Aeppli distribution for sequence lengths of biological interest. The approach is novel in that Markovian dependency is defined for sequences with periodic boundary conditions, and this enables exact analytic formulas for the mean and variance to be derived. We also carry out a preliminary comparison between the approximate D2 distribution computed with the theoretical mean and variance under a Markovian hypothesis and an empirical D2 distribution from the human genome

Summer Institute in Biomedical Engineering, 1973

Bioengineering of medical equipment is detailed. Equipment described includes: an environmental control system for a surgical suite; surface potential mapping for an electrode system; the use of speech-modulated-white-noise to differentiate hearers and feelers among the profoundly deaf; the design of an automatic weight scale for an isolette; and an internal tibial torsion correction study. Graphs and charts are included with design specifications of this equipment