Coloring intersection graphs of x-monotone curves in the plane

A class of graphs G is χ-bounded if the chromatic number of the graphs in G is bounded by some function of their clique number. We show that the class of intersection graphs of simple families of x-monotone curves in the plane intersecting a vertical line is χ-bounded. As a corollary, we show that the class of intersection graphs of rays in the plane is χ-bounded, and the class of intersection graphs of unit segments in the plane is χ-bounded.     

Non-homogeneous random walks on a semi-infinite strip

We study the asymptotic behaviour of Markov chains (_Xn,_ηn)$(X_n,{\eta }_n)$ on _Z+×S${\mathbb{Z}}_{+}\times S$, where _Z+${\mathbb{Z}}_{+}$ is the non-negative integers and S$S$ is a finite set. Neither coordinate is assumed to be Markov. We assume a moments bound on the jumps of _Xn$X_n$, and that, roughly speaking, _ηn${\eta }_n$ is close to being Markov when _Xn$X_n$ is large. This departure from much of the literature, which assumes that _ηn${\eta }_n$ is itself a Markov chain, enables us to probe precisely the recurrence phase transitions by assuming asymptotically zero drift for _Xn$X_n$ given _ηn${\eta }_n$. We give a recurrence classification in terms of increment moment parameters for _Xn$X_n$ and the stationary distribution for the large- X$X$ limit of _ηn${\eta }_n$. In the null case we also provide a weak convergence result, which demonstrates a form of asymptotic independence between _Xn$X_n$ (rescaled) and _ηn${\eta }_n$. Our results can be seen as generalizations of Lamperti’s results for non-homogeneous random walks on _Z+${\mathbb{Z}}_{+}$ (the case where S$S$ is a singleton). Motivation arises from modulated queues or processes with hidden variables where _ηn${\eta }_n$ tracks an internal state of the system.     

Pullback attractors of non-autonomous stochastic degenerate parabolic equations on unbounded domains

This paper is concerned with pullback attractors of the stochastic p  -Laplace equation defined on the entire space Rⁿ${\mathbb{R}}^n$. We first establish the asymptotic compactness of the equation in L²(Rⁿ)$L^2({\mathbb{R}}^n)$ and then prove the existence and uniqueness of non-autonomous random attractors. This attractor is pathwise periodic if the non-autonomous deterministic forcing is time periodic. The difficulty of non-compactness of Sobolev embeddings on Rⁿ${\mathbb{R}}^n$ is overcome by the uniform smallness of solutions outside a bounded domain.     

A Biologically Inspired Optimization Algorithm for Solving Fuzzy Shortest Path Problems with Mixed Fuzzy Arc Lengths

The shortest path problem is among fundamental problems of network optimization. Majority of the optimization algorithms assume that weights of data graph’s edges are pre-determined real numbers. However, in real-world situations, the parameters (costs, capacities, demands, time) are not well defined. The fuzzy set has been widely used as it is very flexible and cost less time when compared with the stochastic approaches. We design a bio-inspired algorithm for computing a shortest path in a network with various types of fuzzy arc lengths by defining a distance function for fuzzy edge weights using \(\alpha \) cuts. We illustrate effectiveness and adaptability of the proposed method with numerical examples, and compare our algorithm with existing approaches.     

Orthogonal Pfister involutions in characteristic two

We show that over a field of characteristic 2 a central simple algebra with orthogonal involution that decomposes into a product of quaternion algebras with involution is either anisotropic or metabolic. We use this to define an invariant of such orthogonal involutions that completely determines the isotropy behaviour of the involution. We also give an example of a non-totally decomposable algebra with orthogonal involution that becomes totally decomposable over every splitting field of the algebra.     

Boosting the feasibility pump

The feasibility pump (FP) has proved to be an effective method for finding feasible solutions to mixed integer programming problems. FP iterates between a rounding procedure and a projection procedure, which together provide a sequence of points alternating between LP feasible but fractional solutions, and integer but LP infeasible solutions. The process attempts to minimize the distance between consecutive iterates, producing an integer feasible solution when closing the distance between them. We investigate the benefits of enhancing the rounding procedure with a clever integer line search that efficiently explores a large set of integer points. An extensive computational study on benchmark instances demonstrates the efficacy of the proposed approach.     

Topological duality and lattice expansions,II: Lattice expansions with quasioperators

The main objective of this paper (the second of two parts) is to show that quasioperators can be dealt with smoothly in the topological duality established in Part I. A quasioperator is an operation on a lattice that either is join preserving and meet reversing in each argument or is meet preserving and join reversing in each argument. The paper discusses several common examples, including orthocomplementation on the closed subspaces of a fixed Hilbert space (sending meets to joins), modal operators ? and □ on a bounded modal lattice (preserving joins, resp. meets), residuation on a bounded residuated lattice (sending joins to meets in the first argument and meets to meets in the second). This paper introduces a refinement of the topological duality of Part I that makes explicit the topological distinction between the duals of meet homomorphisms and of join homomorphisms. As a result, quasioperators can be represented by certain continuous maps on the topological duals.     

Highly efficient on-DNA amide couplings promoted by micelle forming surfactants for the synthesis of DNA encoded libraries

DNA encoded libraries (DELs) represent powerful new technology for finding small molecule ligands for proteins and are increasingly being applied to hit finding in medicinal chemistry. Crucial to the synthesis of high quality DELs is the identification of chemical reactions for their assembly that proceed with very high conversion across a range of different substrates, under conditions compatible with DNA-tagged substrates. Many current chemistries used in DEL synthesis do not meet this requirement, resulting in libraries of low fidelity. Amide couplings are the most commonly used reaction in synthesis of screening libraries and also in DELs. The ability to carry out highly efficient, widely applicable amide couplings in DEL synthesis would therefore be highly desirable. We report a method for amide coupling using micelle forming surfactants, promoted by a modified linker, that is broadly applicable across a wide range of substrates. Most significantly, this works exceptionally well for coupling of DNA-conjugated carboxylic acids (N-to-C) with amines in solution, a procedure that is currently very inefficient. The optimisation of separate procedures for coupling of DNA-conjugated acids and amines by reagent screening and statistically driven optimisation is described. The generality of the method is illustrated by the application to a wide range of examples with unprecedented levels of conversion. The utility of the (N-to-C) coupling of DNA-conjugated acids in DEL synthesis is illustrated by the three cycle synthesis of a fully DNA-encoded compound by two cycles of coupling of an aminoester, with intermediate ester hydrolysis, followed by capping with an amine. This methodology will be of great utility in the synthesis of high fidelity DELs.

Highly efficient forward and reverse on-DNA amide couplings were developed exploiting hydrophobic linkers in combination with the micelle forming surfactant TPGS-750M. The method is highly effective for a wide range of substrates in the synthesis of DNA-encoded libraries.     

A facile synthesis of novel 2-amino-6-arylmethyl-7-carboxamido-7,8-dihydropyrimido[5,4-f][1,4]thiazepin-5-ones

A facile synthesis of novel 2-amino-6-arylmethyl-7-carboxamido-7,8-dihydropyrimido[5,4-f][1,4]thiazepin-5-ones is described. The synthesis was developed on solid phase and was applied to provide a series of analogs in good yield. The key reactions are acylation of a cysteine derivative with 2,4-dichloropyrimidine-5-carbonyl chloride followed by cyclization to generate a 6-arylmethyl-7-carboxamido-2-chloro-7,8-dihydropyrimido[5,4-f][1,4]thiazepin-5-one, which is further derivatized with an amine to give the desired 2-amino-6-arylmethyl-7-carboxamido-7,8-dihydropyrimido[5,4-f][1,4]thiazepin-5-one.