Path homomorphisms,graph colorings,and boolean matrices

We investigate bounds on the chromatic number of a graph G derived from the nonexistence of homomorphisms from some path \begin{eqnarray*}\vec{P}\end{eqnarray*} into some orientation \begin{eqnarray*}\vec{G}\end{eqnarray*} of G. The condition is often efficiently verifiable using boolean matrix multiplications. However, the bound associated to a path \begin{eqnarray*}\vec{P}\end{eqnarray*} depends on the relation between the “algebraic length” and “derived algebraic length” of \begin{eqnarray*}\vec{P}\end{eqnarray*}. This suggests that paths yielding efficient bounds may be exponentially large with respect to G, and the corresponding heuristic may not be constructive. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 198–209, 2010     

Pythagoras numbers of fields

A field of characteristic is said to have finite Pythagoras number if there exists an integer such that each nonzero sum of squares in can be written as a sum of squares, in which case the Pythagoras number of is defined to be the least such integer. As a consequence of Pfister's results on the level of fields, of a nonformally real field is always of the form or , and all integers of such type can be realized as Pythagoras numbers of nonformally real fields. Prestel showed that values of the form , , and can always be realized as Pythagoras numbers of formally real fields. We will show that in fact to every integer there exists a formally real field with . As a refinement, we will show that if and are integers such that , then there exists a uniquely ordered field with and (resp. ), where (resp. ) denotes the supremum of the dimensions of anisotropic forms over which are torsion in the Witt ring of (resp. which are indefinite with respect to each ordering on ).

     

Novel characterization of the folding of proteins

In this paper, we consider a characterization of the folding of 3D model proteins. In this characterization, we describe the different kinds of 3‐step path conformation, then construct an augmented Hasse matrix which reflects some properties of the protein folding. This method can characterize the folding degree of 3D model proteins and evaluate the similarity/diversity of these five model proteins. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

On universally decodable matrices for space–time coding

The notion of universally decodable matrices (UDMs) was recently introduced by Tavildar and Viswanath while studying slow-fading channels. It turns out that the problem of constructing UDMs is tightly connected to the problem of constructing maximum-distance separable codes. In this paper, we first study the properties of UDMs in general and then we discuss an explicit construction of a class of UDMs, a construction which can be seen as an extension of Reed–Solomon codes. In fact, we show that this extension is, in a sense to be made more precise later on, unique. Moreover, the structure of this class of UDMs allows us to answer some open conjectures by Tavildar, Viswanath, and Doshi in the positive, and it also allows us to formulate an efficient decoding algorithm for this class of UDMs. It turns out that our construction yields a coding scheme that is essentially equivalent to a class of codes that was proposed by Rosenbloom and Tsfasman. Moreover, we point out connections to so-called repeated-root cyclic codes.     

A Multilevel Search Algorithm for the Maximization of Submodular Functions Applied to the Quadratic Cost Partition Problem

Maximization of submodular functions on a ground set is a NP-hard combinatorial optimization problem. Data correcting algorithms are among the several algorithms suggested for solving this problem exactly and approximately. From the point of view of Hasse diagrams data correcting algorithms use information belonging to only one level in the Hasse diagram adjacent to the level of the solution at hand. In this paper, we propose a data correcting algorithm that looks at multiple levels of the Hasse diagram and hence makes the data correcting algorithm more efficient. Our computations with quadratic cost partition problems show that this multilevel search effects a 8- to 10-fold reduction in computation times, so that some of the dense quadratic partition problem instances of size 500, currently considered as some of the most difficult problems and far beyond the capabilities of current exact methods, are solvable on a personal computer working at 300 MHz within 10 min.     

Congruences between the coefficients of the Tate curve via formal groups

Let be the Tate curve with canonical differential, . If the characteristic is , then the Hasse invariant, , of the pair should equal one. If , then calculation of leads to a nontrivial separable relation between the coefficients and . If or , Thakur related and via elementary methods and an identity of Ramanujan. Here, we treat uniformly all characteristics via explicit calculation of the formal group law of . Our analysis was motivated by the study of the invariant which is an infinite Witt vector generalizing the Hasse invariant.

     

Phase Diagrams and Synthesis of Diamond

For reasons of phase equilibria, the lowest temperatures T min , above which at high pressures the diamond crystallization from melt solutions is allowable in terms of thermodynamics, have been found for a number of metal-carbon systems. In the Ta-C and Nb-C systems, the diamond synthesis is possible at temperatures below T min , while to synthesize diamond in the Mg-Zn-C system, the temperatures much higher than T min , are required because of the necessity to overcome the kinetic difficulties.     

QSARs for identifying and prioritizing substances with persistence and bioconcentration potential

From the 8511 chemicals with 1998 production volumes reported to the U.S. Environmental Protection Agency (U.S. EPA), the TSCA Interagency Testing Committee's (ITC's) Degradation Effects Bioconcentration Information Testing Strategies (DEBITS) was used to identify 56 chemicals. The DEBITS Quantitative Structure-Activity Relationships (QSARs) and the U.S. EPA's PBT profiler QSARs were used to predict the persistence and bioconcentration factors of these 56 chemicals. Partial order ranking was used to prioritise the chemicals based on persistence and bioconcentration potential.