Breaking Instance-Independent Symmetries In Exact Graph Coloring

Code optimization and high level synthesis can be posed as constraint satisfaction and optimization problems, such as graph coloring used in register allocation. Graph coloring is also used to model more traditional CSPs relevant to AI, such as planning, time-tabling and scheduling. Provably optimal solutions may be desirable for commercial and defense applications. Additionally, for applications such as register allocation and code optimization, naturally-occurring instances of graph coloring are often small and can be solved optimally. A recent wave of improvements in algorithms for Boolean satisfiability (SAT) and 0-1 Integer Linear Programming (ILP) suggests generic problem-reduction methods, rather than problem-specific heuristics, because (1) heuristics may be upset by new constraints, (2) heuristics tend to ignore structure, and (3) many relevant problems are provably inapproximable. Problem reductions often lead to highly symmetric SAT instances, and symmetries are known to slow down SAT solvers. In this work, we compare several avenues for symmetry breaking, in particular when certain kinds of symmetry are present in all generated instances. Our focus on reducing CSPs to SAT allows us to leverage recent dramatic improvement in SAT solvers and automatically benefit from future progress. We can use a variety of black-box SAT solvers without modifying their source code because our symmetry-breaking techniques are static, i.e., we detect symmetries and add symmetry breaking predicates (SBPs) during pre-processing. An important result of our work is that among the types of instance-independent SBPs we studied and their combinations, the simplest and least complete constructions are the most effective. Our experiments also clearly indicate that instance-independent symmetries should mostly be processed together with instance-specific symmetries rather than at the specification level, contrary to what has been suggested in the literature

Bilinear Discrete Painleve-II and its Particular Solutions

By analogy to the continuous Painlev\'e II equation, we present particular solutions of the discrete Painlev\'e II (d-P$\rm_{II}$) equation. These solutions are of rational and special function (Airy) type. Our analysis is based on the bilinear formalism that allows us to obtain the $\tau$ function for d-P$\rm_{II}$. Two different forms of bilinear d-P$\rm_{II}$ are obtained and we show that they can be related by a simple gauge transformation.Comment: 9 pages in plain Te

On a q-difference Painlev\'e III equation: I. Derivation, symmetry and Riccati type solutions

A q-difference analogue of the Painlev\'e III equation is considered. Its derivations, affine Weyl group symmetry, and two kinds of special function type solutions are discussed.Comment: arxiv version is already officia

Third-order integrable difference equations generated by a pair of second-order equations

We show that the third-order difference equations proposed by Hirota, Kimura and Yahagi are generated by a pair of second-order difference equations. In some cases, the pair of the second-order equations are equivalent to the Quispel-Robert-Thomson(QRT) system, but in the other cases, they are irrelevant to the QRT system. We also discuss an ultradiscretization of the equations.Comment: 15 pages, 3 figures; Accepted for Publication in J. Phys.

Singularity confinement and algebraic integrability

Two important notions of integrability for discrete mappings are algebraic integrability and singularity confinement, have been used for discrete mappings. Algebraic integrability is related to the existence of sufficiently many conserved quantities whereas singularity confinement is associated with the local analysis of singularities. In this paper, the relationship between these two notions is explored for birational autonomous mappings. Two types of results are obtained: first, algebraically integrable mappings are shown to have the singularity confinement property. Second, a proof of the non-existence of algebraic conserved quantities of discrete systems based on the lack of confinement property is given.Comment: 18 pages, no figur

A qq-anaolg of the sixth Painlev\'e equation

A $q$-difference analog of the sixth Painlev\'e equation is presented. It arises as the condition for preserving the connection matrix of linear $q$-difference equations, in close analogy with the monodromy preserving deformation of linear differential equations. The continuous limit and special solutions in terms of $q$-hypergeometric functions are also discussed.Comment: 8 pages, LaTeX file (Two misprints corrected

Comparative RNAi Screens in C. elegans and C. briggsae Reveal the Impact of Developmental System Drift on Gene Function

Although two related species may have extremely similar phenotypes, the genetic networks underpinning this conserved biology may have diverged substantially since they last shared a common ancestor. This is termed Developmental System Drift (DSD) and reflects the plasticity of genetic networks. One consequence of DSD is that some orthologous genes will have evolved different in vivo functions in two such phenotypically similar, related species and will therefore have different loss of function phenotypes. Here we report an RNAi screen in C. elegans and C. briggsae to identify such cases. We screened 1333 genes in both species and identified 91 orthologues that have different RNAi phenotypes. Intriguingly, we find that recently evolved genes of unknown function have the fastest evolving in vivo functions and, in several cases, we identify the molecular events driving these changes. We thus find that DSD has a major impact on the evolution of gene function and we anticipate that the C. briggsae RNAi library reported here will drive future studies on comparative functional genomics screens in these nematodes

Enhancing through thickness thermal conductivity of ultra-thin composite laminates. Final report

The materials used in electronic applications have specific requirements for stiffness, thermal conductivity, and electromagnetic shielding making the choice of materials used very important. Electronic components are very sensitive to heat, hence the heat dissipation or cooling of the various components is necessary to prevent failure. Thus, any material used in the electronic industry must have a high thermal conductivity in addition to a specified thermal expansion, stiffness and strength properties. The purpose of this project was to design and manufacture composite panels which would conduct heat from an electronic chip attached to the top surface to a cooling liquid flowing at its lower surface. To maximize the heat conducted from the chip to the cooling liquid, the composite must have a high through thickness thermal conductivity. Further, design restrictions on the thickness of the composite panel had to be taken into account. It was found that the presence of excess resin adversely affects the conductivity of a woven fabric composite due to which the through thickness conductivity of the 400 {micro}m thick panel was better than the 500 {micro}m thick panel. The through thickness conductivity of the panel with short fibers alone was better than that of the woven cloth panel. The finite element model developed for a priori prediction of the through thickness thermal conductivity of the composite panels is a very powerful tool that can save enormous prototyping times an associates coats