Three‐unit semicircles curve: A compact 3D graphical representation of DNA sequences based on classifications of nucleotides

A new three‐dimensional graphical representation of DNA sequences, three‐unit semicircles (TUS)‐curve, which maps a given sequence into a dot sequences embedded in three‐unit semicircles, is proposed based on three biclassifications of nucleotides. TUS‐curve has the merit of compactness and could avoid the degeneracy and loss of information. The geometrical center of the curve, which indicates the distribution of base frequencies of the corresponding DNA sequence, is extracted and applied to analyze the similarity of various species. Phylogenetic tree of 11 species based on their first exons of β‐globin genes showed that the TUS‐curve is a powerful tool to get valuable biological information. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Classroom note: The area under x α by geometry

By stretching the area under the curve x ^α it is shown to be of the form x ^{α + 1} p(α). Geometry is then used to prove p(α) = 1/(α + 1).     

Functional ergodic limits for occupation time processes of site-dependent branching Brownian motions in R

We consider a kind of site-dependent branching Brownian motions whose branching laws depend on the site-branching factorσ(·).We focus on the functional ergodic limits for the occupation time processes of the models in R.It is proved that the limiting process has the form ofλξ(·),where A is the Lebesgue measure on IE andξ(·)is a real-valued process which is non-degenerate if and only ifσis integrable.Whenξ(·)is non-degenerate,it is strictly positive for t0.Moreover,ξconverges to O in finite-dimensional distributions if the integral ofσtends to infinity.     

Integral representation in the theory of continuous trading

This paper deals with predictable representation in continuous trading. First of all we give a positive answer to a conjecture of Harrison and Pliska [2]. Then we prove the completeness of a model of mixed type (see[2]). A slight modificationof this example shows that the continuous and purely discontinuous parts of a local martingale depend on the filtration.     

Code2vect: An efficient heterogenous data classifier and nonlinear regression technique

The aim of this paper is to present a new classification and regression algorithm based on Artificial Intelligence. The main feature of this algorithm, which will be called Code2Vect, is the nature of the data to treat: qualitative or quantitative and continuous or discrete. Contrary to other artificial intelligence techniques based on the “Big-Data,” this new approach will enable working with a reduced amount of data, within the so-called “Smart Data” paradigm. Moreover, the main purpose of this algorithm is to enable the representation of high-dimensional data and more specifically grouping and visualizing this data according to a given target. For that purpose, the data will be projected into a vectorial space equipped with an appropriate metric, able to group data according to their affinity (with respect to a given output of interest). Furthermore, another application of this algorithm lies on its prediction capability. As it occurs with most common data-mining techniques such as regression trees, by giving an input the output will be inferred, in this case considering the nature of the data formerly described. In order to illustrate its potentialities, two different applications will be addressed, one concerning the representation of high-dimensional and categorical data and another featuring the prediction capabilities of the algorithm.     

Asyptotics for linear difference equations I:basic theory

We present here a unified treatment of asymptotic theory of linear difference equations. This is based on anadapted theory of discrete dichotomy. The obtained results narrow the gap between Poincarés Theorem and (the discrete analoue of) Levinson's Theorem.