Narrow Results

It was noticed by Harel in [Har86] that "one can define -complete versions of the well-known Post Correspondence Problem". We first give a complete proof of this result, showing that the infinite Post Correspondence Problem in a regular ω-language is -complete, hence located beyond the arithmetical hierarchy and highly undecidable. We infer from this result that it is -complete to determine whether two given infinitary rational relations are disjoint. Then we prove that there is an amazing gap between two decision problems about ω-rational functions realized by finite state Büchi transducers. Indeed Prieur proved in [Pri01, Pri02] that it is decidable whether a given ω-rational function is continuous, while we show here that it is -complete to determine whether a given ω-rational function has at least one point of continuity. Next we prove that it is -complete to determine whether the continuity set of a given ω-rational function is ω-regular. This gives the exact complexity of two problems which were shown to be undecidable in [CFS08].

We prove two new effective properties of rational functions over infinite words which are realized by finite state Büchi transducers. Firstly, for each such function $F:{\mathrm{\Sigma }}^{\omega }\to {\mathrm{\Gamma }}^{\omega }$, one can construct a deterministic Büchi automaton ${𝒜}$ accepting a dense ${\mathrm{\Pi }}_2^0$-subset of ${\mathrm{\Sigma }}^{\omega }$ such that the restriction of $F$ to $L({𝒜})$ is continuous. Secondly, we give a new proof of the decidability of the first Baire class for synchronous $\omega$-rational functions from which we get an extension of this result involving the notion of Wadge classes of regular $\omega$-languages.

Dramatic advances in the field of complex networks have been witnessed in the past few years. This paper reviews some important results in this direction of rapidly evolving research, with emphasis on the relationship between the dynamics and the topology of complex networks. Basic quantities and typical examples of various complex networks are described; and main network models are introduced, including regular, random, small-world and scale-free models. The robustness of connectivity and the epidemic dynamics in complex networks are also evaluated. To that end, synchronization in various dynamical networks are discussed according to their regular, small-world and scale-free connections.

This paper formulates a new approach to the study of chaos in discrete dynamical systems based on the notions of inverse ill-posed problems, set-valued mappings, generalized and multivalued inverses, graphical convergence of a net of functions in an extended multifunction space [Sengupta & Ray, 2000] and the topological theory of convergence. Order, chaos and complexity are described as distinct components of this unified mathematical structure that can be viewed as an application of the theory of convergence in topological spaces to increasingly nonlinear mappings, with the boundary between order and complexity in the topology of graphical convergence being the region in (Multi(X)) that is susceptible to chaos. The paper uses results from the discretized spectral approximation in neutron transport theory [Sengupta, 1988, 1995] and concludes that the numerically exact results obtained by this approximation of the Case singular eigenfunction solution is due to the graphical convergence of the Poisson and conjugate Poisson kernels to the Dirac delta and the principal value multifunctions respectively. In (Multi(X)), the continuous spectrum is shown to reduce to a point spectrum, and we introduce a notion of latent chaotic states to interpret superposition over generalized eigenfunctions. Along with these latent states, spectral theory of nonlinear operators is used to conclude that nature supports complexity to attain efficiently a multiplicity of states that otherwise would remain unavailable to it.

In this paper, we aim to identify a directed complex network with optimal controllability, for which pinning control is to be applied. Since the controllability of a network can be reflected by the smallest nonzero eigenvalue of a matrix related to its topology, an optimal network design problem is formulated based on the maximization of this eigenvalue. To better consider the practical reality, constraints on node degree sequence are specified. Based on the derived bounds of the eigenvalue, an effective rewiring scheme is designed and solutions close to or equal to the upper bound are obtained. Finally, the relationship between network characteristics and controllability is studied. Through complexity analysis, it is concluded that the network with high controllability should possess two properties, i.e. nodes with high out-degree for pinning and other nodes with uniform degree distribution.

This paper develops topological methods for qualitative analysis of the behavior of nonholonomic dynamical systems. Their application is illustrated by considering a new integrable system of nonholonomic mechanics, called a nonholonomic hinge. Although this system is nonholonomic, it can be represented in Hamiltonian form with a Lie–Poisson bracket of rank two. This Lie–Poisson bracket is used to perform stability analysis of fixed points. In addition, all possible types of integral manifolds are found and a classification of trajectories on them is presented.

Characterizing accurately chaotic behaviors is not a trivial problem and must allow to determine the properties that two given chaotic invariant sets share or not. The underlying problem is the classification of chaotic regimes, and their labeling. Addressing these problems corresponds to the development of a dynamical taxonomy, exhibiting the key properties discriminating the variety of chaotic behaviors discussed in the abundant literature. Starting from the hierarchy of chaos initially proposed by one of us, we systematized the description of chaotic regimes observed in three- and four-dimensional spaces, which cover a large variety of known (and less known) examples of chaos. Starting with the spectrum of Lyapunov exponents as the first taxonomic ranks, we extended the description to higher ranks with some concepts inherited from topology (bounding torus, surface of section, first-return map, …).

By treating extensively the Rössler and the Lorenz attractors, we extended the description of branched manifold — the highest known taxonomic rank for classifying chaotic attractor — by a linking matrix (or linker) to multicomponent attractors (bounded by a torus whose genus $g\ge 3$).

We prove that there is a sequence of purely imaginary parameter values ${\lambda }_n$ converging to $0$ such that the Julia set for $z\mapsto {\lambda }_n(z+1/z)$ is homeomorphic to the Sierpiński carpet fractal; however, for any distinct pair of such parameter values, the dynamics of the map restricted to the Julia set are not conjugate.

Essential proteins are important for the survival and development of organisms. Lots of centrality algorithms based on network topology have been proposed to detect essential proteins and achieve good results. However, most of them only focus on the network topology, but ignore the false positive (FP) interactions in protein–protein interaction (PPI) network. In this paper, gene ontology (GO) information is proposed to measure the reliability of the edges in PPI network and we propose a novel algorithm for identifying essential proteins, named EGC algorithm. EGC algorithm integrates topology character of PPI network and GO information. To validate the performance of EGC algorithm, we use EGC and other nine methods (DC, BC, CC, SC, EC, LAC, NC, PEC and CoEWC) to identify the essential proteins in the two different yeast PPI networks: DIP and MIPS. The results show that EGC is better than the other nine methods, which means adding GO information can help in predicting essential proteins.

An algebraic relational theory is being developed in order to represent biological systems. As a result, it is possible to explain, in terms of qualitative relationships, the behaviors of such systems. This paper deals with the periodic continuous responses of a new state derived from the interaction between low energies and matter. This effect was predicted by categoric developments of the algebraic relational theory.

The “reverse engineering” approach to modelling is applied to T-cell vaccination. The novelty here is the representation of the diseased state as a transient.

Electron cryo-microscopy is a fast advancing biophysical technique to derive three-dimensional structures of large protein complexes. Using this technique, many density maps have been generated at intermediate resolution such as 6–10 Å resolution. Although it is challenging to derive the backbone of the protein directly from such density maps, secondary structure elements such as helices and β-sheets can be computationally detected. Our work in this paper provides an approach to enumerate the top-ranked possible topologies instead of enumerating the entire population of the topologies. This approach is particularly practical for large proteins. We developed a directed weighted graph, the topology graph, to represent the secondary structure assignment problem. We prove that the problem of finding the valid topology with the minimum cost is NP hard. We developed an O(N² 2^N) dynamic programming algorithm to identify the topology with the minimum cost. The test of 15 proteins suggests that our dynamic programming approach is feasible to work with proteins of much larger size than we could before. The largest protein in the test contains 18 helical sticks detected from the density map out of 33 helices in the protein.

The determination of the secondary structure topology is a critical step in deriving the atomic structure from the protein density map obtained from electron cryo-microscopy technique. This step often relies on the matching of two sources of information. One source comes from the secondary structures detected from the protein density map at the medium resolution, such as 5–10 Å. The other source comes from the predicted secondary structures from the amino acid sequence. Due to the inaccuracy in either source of information, a pool of possible secondary structure positions needs to be sampled. This paper studies the question, that is, how to reduce the computation of the mapping when the inaccuracy of the secondary structure predictions is considered. We present a method that combines the concept of dynamic graph with our previous work of using constrained shortest path to identify the topology of the secondary structures. We show a reduction of 34.55% of run-time as comparison to the naïve way of handling the inaccuracies. We also show an improved accuracy when the potential secondary structure errors are explicitly sampled verses the use of one consensus prediction. Our framework demonstrated the potential of developing computationally effective exact algorithms to identify the optimal topology of the secondary structures when the inaccuracy of the predicted data is considered.

Bipodal ligand 5,15-bis(4-carboxyphenyl) porphyrin (H${}_{\mathrm{\text{4}}}$DCPP) was designed and synthesized. By adjusting the molar ratio of H${}_{\mathrm{\text{4}}}$DCPP, ancillary ligand 4,4-bipyridine (bpy) and zinc acetate salts, three novel coordination assemblies, namely, zero-dimensional dimeric [Zn${}_{\mathrm{\text{2}}}$(H${}_{\mathrm{\text{2}}}$DCPP)${}_{\mathrm{\text{2}}}$ ·bpy] ·4H${}_{\mathrm{\text{2}}}$O ·4DMF (Zn-D), two-dimensional polymeric {[Zn${}_{\mathrm{\text{2}}}$(DCPP) ·bpy${}_{\mathrm{\text{0.5}}}$ ·H${}_{\mathrm{\text{2}}}$O ·DMF${}_{\mathrm{\text{0.5}}}$] ·solvent}${}_{\mathrm{\text{n}}}$ (Zn-2D), and three-dimensional polymeric [Zn${}_{\mathrm{\text{2}}}$(DCPP) ·bpy${}_{\mathrm{\text{0.5}}}$]${}_{\mathrm{\text{n}}}$ (Zn-3D) were assembled. Due to the delicate integration of multiple chromophores in the coordination space combining bpy, DCPP and MLCT emissions together, photoluminescence (PL) of the three porphyrin-zinc coordination assemblies differ from each other and color tone is tunable from blue to orange with changes of the excitation wavelength. In particular, white light emission (WLE) can be observed by the excitation of 270 to 290 nm, representing the first examples of single component WLE compounds based on porphyrin ligands. Furthermore, temperature-dependent luminescence results in a linear $I$–$T$ relationship in Zn-2D and Zn-3D assemblies, applicable for long wavelength red-emitting thermometers.

Voronoi tessellations and their dual Delaunay tetrahedral tessellations provide versatile representations of three-dimensional structures of proteins, allowing a strict definition of neighborhoods of the different amino acids, considered as rigid entities. They give geometrical and topological information coded in the adjacency matrix (often called contact map in biology). Using this approach, this paper presents a two-dimensional description of internal surfaces, which are tools used to understand the conformation of globular proteins. These surfaces are triangulated surfaces tiled by faces of Delaunay tetrahedra. Two simple types of surfaces are presented, characterized by their topology: disk-like or torus-like. The torus-like surface is relevant in the description of secondary structures. The disk-like surface characterizes the collapse of the chain onto itself.

Researchers hope that establishing a notion of proximity using topology will help to clarify the biological processes underlying the evolution of living organisms. The simple model presented here, using RNA shapes, can carry over to more general and complex genotype–phenotype systems. Proximity is an important component of continuity, in both real-world and topological terms. Consequently, phenotype spaces provide an appropriate setting for modeling and investigating continuous and discontinuous evolutionary change.

The purpose of this work is to construct a new crossover operator using the properties of DNA and RNA by using topological concepts in constructing flexible mathematical models in the field of biomathematics. Also, we investigate and study topological properties of the constructed operators and the associated topological spaces of DNA and RNA. Finally we use the process of exchange for sequence of genotypes structures to construct new types of topological concepts to investigate and discuss several examples and some of their properties.

Mathematical equations are now found not only in the books, but also they help in finding solutions for the biological problems by explaining the technicality of the current biological models and providing predictions that can be validated and complemented to experimental and clinical studies. In this research paper, we use the mset theory to study DNA & RNA mutations to discover the mutation occurrence. Also, we use the link between the concept of the mset and topology to determine the compatibility or similarity between “types”, which may be the strings of bits, vectors, DNA or RNA sequences, etc.

The aim of this paper is to use topological concepts in the construction of flexible mathematical models in the field of biological mathematics. Also, we will build new topographic types to study recombination of deoxyribonucleic acid (DNA) and ribonucleic acid (RNA). Finally, we study the topographical properties of constructed operators and the associated topological spaces of DNA and RNA.

Approximation space can be said to play a critical role in the accuracy of the set’s approximations. The idea of “approximation space” was introduced by Pawlak in 1982 as a core to describe information or knowledge induced from the relationships between objects of the universe. The main objective of this paper is to create new types of rough set models through the use of different neighborhoods generated by a binary relation. New approximations are proposed representing an extension of Pawlak’s rough sets and some of their generalizations, where the precision of these approximations is substantially improved. To elucidate the effectiveness of our approaches, we provide some comparisons between the proposed methods and the previous ones. Finally, we give a medical application of lung cancer disease as well as provide an algorithm which is tested on the basis of hypothetical data in order to compare it with current methods.