Covering Partial Cubes with Zones

A partial cube is a graph having an isometric embedding in a hypercube. Partial cubes are characterized by a natural equivalence relation on the edges, whose classes are called zones. The number of zones determines the minimal dimension of a hypercube in which the graph can be embedded. We consider the problem of covering the vertices of a partial cube with the minimum number of zones. The problem admits several special cases, among which are the problem of covering the cells of a line arrangement with a minimum number of lines, and the problem of finding a minimum-size fibre in a bipartite poset. For several such special cases, we give upper and lower bounds on the minimum size of a covering by zones. We also consider the computational complexity of those problems, and establish some hardness results

On infinite-finite duality pairs of directed graphs

The (A,D) duality pairs play crucial role in the theory of general relational structures and in the Constraint Satisfaction Problem. The case where both classes are finite is fully characterized. The case when both side are infinite seems to be very complex. It is also known that no finite-infinite duality pair is possible if we make the additional restriction that both classes are antichains. In this paper (which is the first one of a series) we start the detailed study of the infinite-finite case. Here we concentrate on directed graphs. We prove some elementary properties of the infinite-finite duality pairs, including lower and upper bounds on the size of D, and show that the elements of A must be equivalent to forests if A is an antichain. Then we construct instructive examples, where the elements of A are paths or trees. Note that the existence of infinite-finite antichain dualities was not previously known

Hitting all Maximal Independent Sets of a Bipartite Graph

We prove that given a bipartite graph G with vertex set V and an integer k, deciding whether there exists a subset of V of size k hitting all maximal independent sets of G is complete for the class Sigma_2^P.Comment: v3: minor chang

On 1-factorizations of Bipartite Kneser Graphs

It is a challenging open problem to construct an explicit 1-factorization of the bipartite Kneser graph $H(v,t)$, which contains as vertices all $t$-element and $(v-t)$-element subsets of $[v]:=\{1,\ldots,v\}$ and an edge between any two vertices when one is a subset of the other. In this paper, we propose a new framework for designing such 1-factorizations, by which we solve a nontrivial case where $t=2$ and $v$ is an odd prime power. We also revisit two classic constructions for the case $v=2t+1$ --- the \emph{lexical factorization} and \emph{modular factorization}. We provide their simplified definitions and study their inner structures. As a result, an optimal algorithm is designed for computing the lexical factorizations. (An analogous algorithm for the modular factorization is trivial.)Comment: We design the first explicit 1-factorization of H(2,q), where q is a odd prime powe

Gene Expression Pattern Analysis of Anterior Hox Genes during Zebrafish (Danio rerio) Embryonic Development Reveals Divergent Expression Patterns from Other Teleosts

The regional identity of organs and organ systems along the anterior-posterior axis during embryonic development is patterned, in part, by Hox genes, which encode transcription factor proteins that activate or repress the expression of downstream target genes. Divergent nested Hox gene expression patterns may have had a role in facilitating morphological divergence of structures, such as the pharyngeal jaw apparatus, among evolutionarily divergent teleost fishes. Recent studies from several evolutionarily divergent teleosts, such as the Japanese Medaka (Oryzias latipes) and the Nile Tilapia (Oreochromis niloticus), have shown the presence of divergent expression patterns of several Hox genes within paralog groups 2–5 between these species. Specifically, these expression patterns were documented in the pharyngeal arches, which give rise to the pharyngeal jaw apparatus. While the expression patterns of several Zebrafish (Danio rerio) Hox genes that are orthologous to those of Medaka and Tilapia have been documented within the developing hindbrain and pharyngeal arches, many still have yet to be documented, especially within the pharyngeal arches during the postmigratory cranial neural crest cell stages. Here, we present the expression patterns of six Zebrafish Hox genes, hoxc3a, d3a, a4a, d4a, b5a, and c5a, within the pharyngeal arches during a postmigratory cranial neural crest cell stage and compare them to their orthologous genes of Medaka and Tilapia at similar stages. We show that while hoxc3a, d3a, and c5a of Zebrafish are absent from the pharyngeal arches, hoxa4a, d4a, and b5a show divergent expression patterns from their orthologs in Medaka and Tilapia. These observed divergences may be, in part, responsible for the divergent pharyngeal jaw apparatus structures exhibited by these fishes

Solving order constraints in logarithmic space.

We combine methods of order theory, finite model theory, and universal algebra to study, within the constraint satisfaction framework, the complexity of some well-known combinatorial problems connected with a finite poset. We identify some conditions on a poset which guarantee solvability of the problems in (deterministic, symmetric, or non-deterministic) logarithmic space. On the example of order constraints we study how a certain algebraic invariance property is related to solvability of a constraint satisfaction problem in non-deterministic logarithmic space

Control over phase separation and nucleation using a laser-tweezing potential

Control over the nucleation of new phases is highly desirable but elusive. Even though there is a long history of crystallization engineering by varying physicochemical parameters, controlling which polymorph crystallizes or whether a molecule crystallizes or forms an amorphous precipitate is still a poorly understood practice. Although there are now numerous examples of control using laser-induced nucleation, the absence of physical understanding is preventing progress. Here we show that the proximity of a liquid–liquid critical point or the corresponding binodal line can be used by a laser-tweezing potential to induce concentration gradients. A simple theoretical model shows that the stored electromagnetic energy of the laser beam produces a free-energy potential that forces phase separation or triggers the nucleation of a new phase. Experiments in a liquid mixture using a low-power laser diode confirm the effect. Phase separation and nucleation using a laser-tweezing potential explains the physics behind non-photochemical laser-induced nucleation and suggests new ways of manipulating matter