Instant topological relationships hidden in the reality

In most applications of general topology, topology usually is not the first, primary structure, but the information which finally leads to the construction of the certain, for some purpose required topology, is filtered by more or less thick filter of the other mathematical structures. This fact has two main consequences: (1) Most important applied constructions may be done in the primary structure, bypassing the topology. (2) Some topologically important information from the reality may be lost (filtered out by the other, front-end mathematical structures). Thus some natural and direct connection between topology and the reality could be useful. In this contribution we will discuss a pointless topological structure which directly reflects relationship between various locations which are glued together by possible presence of a physical object or a virtual ``observer"

Robustness and Reliability for Virtual Topologies in Wireless Multihop Access Networks

International audienceMobile ad hoc networks (MANet) are a spontaneous collection of mobile terminals. Each node must collaborate in order to structure information exchange. An hybrid network is a MANet connected to Internet via an Access Point (AP). We propose to organize MANet and hybrid networks through a virtual topology. We consider a virtual topology as a hierarchical organization based on the integration of both backbone and clusters. Construction and maintenance procedures of such a virtual topology are detailed and deal with robustness and reliability issues. We present a proactive gratuitous maintenance for our backbone and a new maintenance algorithm for clusters presenting a reduced overhead. Moreover, this improved solution allows to integrate multiple APs in hybrid networks , deleting the previous single point of failure. A method to interconnect backbones is described, which is useful for many applications

Representability of algebraic topology for biomolecules in machine learning based scoring and virtual screening

This work introduces a number of algebraic topology approaches, such as multicomponent persistent homology, multi-level persistent homology and electrostatic persistence for the representation, characterization, and description of small molecules and biomolecular complexes. Multicomponent persistent homology retains critical chemical and biological information during the topological simplification of biomolecular geometric complexity. Multi-level persistent homology enables a tailored topological description of inter- and/or intra-molecular interactions of interest. Electrostatic persistence incorporates partial charge information into topological invariants. These topological methods are paired with Wasserstein distance to characterize similarities between molecules and are further integrated with a variety of machine learning algorithms, including k-nearest neighbors, ensemble of trees, and deep convolutional neural networks, to manifest their descriptive and predictive powers for chemical and biological problems. Extensive numerical experiments involving more than 4,000 protein-ligand complexes from the PDBBind database and near 100,000 ligands and decoys in the DUD database are performed to test respectively the scoring power and the virtual screening power of the proposed topological approaches. It is demonstrated that the present approaches outperform the modern machine learning based methods in protein-ligand binding affinity predictions and ligand-decoy discrimination

Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory

This paper is a concise introduction to virtual knot theory, coupled with a list of research problems in this field.Comment: 65 pages, 24 figures. arXiv admin note: text overlap with arXiv:math/040542

Dynamic virtual private network provisioning from multiple cloud infrastructure service providers

The Cloud infrastructure service providers currently provision basic virtualized computing resources as on demand and dynamic services but there is no common framework in existence that allows the seamless provisioning of even these basic services across multiple cloud service providers, although this is not due to any inherent incompatibility or proprietary nature of the foundation technologies on which these cloud platforms are built. We present a solution idea which aims to provide a dynamic and service oriented provisioning of secure virtual private networks on top of multiple cloud infrastructure service providers. This solution leverages the benefits of peer to peer overlay networks, i.e., the flexibility and scalability to handle the churn of nodes joining and leaving the VPNs and can adapt the topology of the VPN as per the requirements of the applications utilizing its intercloud secure communication framework

Polyfolds: A First and Second Look

Polyfold theory was developed by Hofer-Wysocki-Zehnder by finding commonalities in the analytic framework for a variety of geometric elliptic PDEs, in particular moduli spaces of pseudoholomorphic curves. It aims to systematically address the common difficulties of compactification and transversality with a new notion of smoothness on Banach spaces, new local models for differential geometry, and a nonlinear Fredholm theory in the new context. We shine meta-mathematical light on the bigger picture and core ideas of this theory. In addition, we compiled and condensed the core definitions and theorems of polyfold theory into a streamlined exposition, and outline their application at the example of Morse theory.Comment: 62 pages, 2 figures. Example 2.1.3 has been modified. Final version, to appear in the EMS Surv. Math. Sc

Virtual Knot Theory --Unsolved Problems

This paper is an introduction to the theory of virtual knots and links and it gives a list of unsolved problems in this subject.Comment: 33 pages, 7 figures, LaTeX documen