Carlsson's rank conjecture and a conjecture on square-zero upper triangular matrices

Let k be an algebraically closed field and A the polynomial algebra in r variables with coefficients in k. In case the characteristic of k is 2, Carlsson [9] conjectured that for any DG-A-module M of dimension N as a free A-module, if the homology of M is nontrivial and finite dimensional as a k-vector space, then $2^r\le N$. Here we state a stronger conjecture about varieties of square-zero upper triangular $N\times N$ matrices with entries in A. Using stratifications of these varieties via Borel orbits, we show that the stronger conjecture holds when $N<8$ or $r<3$ without any restriction on the characteristic of k. As a consequence, we obtain a new proof for many of the known cases of Carlsson's conjecture and give new results when $N>4$ and $r=2$.     

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.     

Measure and integration: The basic extension and representation theorems in terms of new inner and outer envelopes

The work of the author in measure and integration is based on new inner and outer envelope formations, which replace the traditional Carathéodory outer measure and certain simple suprema and infima. The new formations lead to essential improvements in both the extent and the adequacy of the basic results. However, they did not find an entrance into the recent textbook literature. The present paper seeks to demonstrate their power with the examples of the basic inner and outer extension and representation theorems for set functions and functionals.     

A conformal mapping approach for investigating the free-boundary seepage from symmetric channels

We present a new method to investigate the two-dimensional free-boundary groundwater seepage from symmetric soil channels into a homogeneous isotropic porous medium. We use Levi–Civitá’s function to construct an integral representation for the conformal mapping of the complex potential domain onto the physical flow domain. A genetic algorithm (GA) is used to calculate the coefficients of the Maclaurin series expansion of Levi–Civitá’s function. The coordinates of the points from the channel contour, calculated by means of the integral representation, must satisfy the analytic equation of the contour. We use this condition to define the objective function of the genetic algorithm. Levi–Civitá’s function is afterwards used to calculate the seepage loss, the free lines, the streamlines, the equipotential lines, the isobars and the velocity field. Some examples illustrate the method.     

Spectral representation of reduced protein models

We consider a spectrum-like two-dimensional graphical representation of proteins based on a reduced protein model in which 20 amino acids are grouped into five classes. This particular grouping of amino acids was suggested by Riddle and co-workers in 1997. The graphical representation is based on depicting sequentially the amino acids on five horizontal lines at equal separations. One-letter codes, B, O, U, X and Y, to which numerical values 1 to 5 have been assigned, are suggested as labels for the fictional amino acids that represent all the amino acids within each group. The approach is illustrated on ND6 proteins of eight species having from 168 to 175 amino acids. While visual inspection of the novel spectral graphical representations of proteins may reveal local similarities and dissimilarities of protein sequences, arithmetic manipulations of spectra offer an elegant route to graphic visualization of the degree of similarity for selected pairs of proteins.     

TargetATPsite: A template‐free method for ATP‐binding sites prediction with residue evolution image sparse representation and classifier ensemble

Understanding the interactions between proteins and ligands is critical for protein function annotations and drug discovery. We report a new sequence‐based template‐free predictor (TargetATPsite) to identify the Adenosine‐5′‐triphosphate (ATP) binding sites with machine‐learning approaches. Two steps are implemented in TargetATPsite: binding residues and pockets predictions, respectively. To predict the binding residues, a novel image sparse representation technique is proposed to encode residue evolution information treated as the input features. An ensemble classifier constructed based on support vector machines (SVM) from multiple random under‐samplings is used as the prediction model, which is effective for dealing with imbalance phenomenon between the positive and negative training samples. Compared with the existing ATP‐specific sequence‐based predictors, TargetATPsite is featured by the second step of possessing the capability of further identifying the binding pockets from the predicted binding residues through a spatial clustering algorithm. Experimental results on three benchmark datasets demonstrate the efficacy of TargetATPsite. © 2013 Wiley Periodicals, Inc.