Topological Problems Research Articles

Topological rigidity problems

We survey the recent results and current issues on the topological rigidity problem for closed aspherical manifolds, i.e., connected closed manifolds whose universal coverings are contractible. A number of open problems and conjectures are presented during the course of the discussion. We also review the status and applications of the Farrell-Jones Conjecture for algebraic \(K\)-and \(L\)-theory for a group ring $RG$ and coefficients in an additive category. These conjectures imply many other well-known and important conjectures. Examples are the Borel Conjecture about the topological rigidity of closed aspherical manifolds, the Novikov Conjecture about the homotopy invariance of higher signatures and the Conjecture for vanishing of the Whitehead group. We here present the status of the Borel, Novikov and vanishing of the Whitehead group Conjectures.

Minimal disconnected cuts in planar graphs

The problem of finding a disconnected cut in a graph is NP‐hard in general but polynomial‐time solvable on planar graphs. The problem of finding a minimal disconnected cut is also NP‐hard but its computational complexity was not known for planar graphs. We show that it is polynomial‐time solvable on 3‐connected planar graphs but NP‐hard for 2‐connected planar graphs. Our technique for the first result is based on a structural characterization of minimal disconnected cuts in 3‐connected ‐free‐minor graphs and on solving a topological minor problem in the dual. In addition we show that the problem of finding a minimal connected cut of size at least 3 is NP‐hard for 2‐connected apex graphs. Finally, we relax the notion of minimality and prove that the problem of finding a so‐called semi‐minimal disconnected cut is still polynomial‐time solvable on planar graphs. © 2016 Wiley Periodicals, Inc. NETWORKS, Vol. 68(4), 250–259 2016

DNA Topoisomerase IIα contributes to the early steps of adipogenesis in 3T3-L1 cells

DNA topoisomerases (Topo) are multifunctional enzymes resolving DNA topological problems such as those arising during DNA replication, transcription and mitosis. Mammalian cells express 2 class II isoforms, Topoisomerases IIα (Topo IIα) and IIβ (Topo IIβ), which have similar enzymatic properties but are differently expressed, in dividing and pluripotent cells, and in post-mitotic and differentiated cells respectively. Pre-adipocytes re-enter the cell cycle prior to committing to their differentiation and we hypothesised that Topo II could contribute to these processes. We show that Topo IIα expression in 3T3-L1 cells is induced within 16h after the initiation of the differentiation programme, peaks at 24h and rapidly declines thereafter. In contrast Topo IIβ was present both in pre-adipocytes and throughout differentiation. Inhibition of PI3K with LY294002, known to prevent adipocyte differentiation, consistently reduced the expression of Topo IIα, whereas a clear effect on Topo IIβ was not apparent. In addition, inhibition of mTOR with rapamycin also reduced the protein levels of Topo IIα. Using specific class IA PI3K catalytic subunit inhibitors, we show that p110α inhibition with A66 has the greatest reduction of Topo IIα expression and of differentiation, as measured by triglyceride storage. The timing of Topo IIα expression coincides with the mitotic clonal expansion (MCE) phase of differentiation and inhibition of Topo II with ICRF-187 during this stage decreased PPARγ1 and 2 protein levels and triglyceride storage, whereas inhibition later on has little impact. Moreover, the addition of ICRF-187 had no effect on the incorporation of EdU during S-phase at day 1 but lowered the relative cell numbers on day 2. ICRF-187 also induced an increase in the centri/pericentromeric heterochromatin localisation of Topo IIα, indicating a role for Topo IIα at these locations during MCE. In summary, we present evidence that Topo IIα plays an important role in adipogenesis during MCE and in a PI3K/mTOR-dependent manner. Considering that Topoisomerases II are targets in cancer chemotherapy, our results highlight that treatment of cancer with Topo II inhibitors may alter metabolic processes in the adipose tissue.

On the nonexistence of elements of Kervaire invariant one

We show that the Kervaire invariant one elements θj ∈ π2j+1−2S exist only for j ≤ 6. By Browder’s Theorem, this means that smooth framed manifolds of Kervaire invariant one exist only in dimensions 2, 6, 14, 30, 62, and possibly 126. Except for dimension 126 this resolves a longstanding problem in algebraic topology.

Computational Geometric and Algebraic Topology

Computational topology is a young, emerging field of mathematics that seeks out practical algorithmic methods for solving complex and fundamental problems in geometry and topology. It draws on a wide variety of techniques from across pure mathematics (including topology, differential geometry, combinatorics, algebra, and discrete geometry), as well as applied mathematics and theoretical computer science. In turn, solutions to these problems have a wide-ranging impact: already they have enabled significant progress in the core area of geometric topology, introduced new methods in applied mathematics, and yielded new insights into the role that topology has to play in fundamental problems surrounding computational complexity. At least three significant branches have emerged in computational topology: algorithmic 3-manifold and knot theory, persistent homology and surfaces and graph embeddings. These branches have emerged largely independently. However, it is clear that they have much to offer each other. The goal of this workshop was to be the first significant step to bring these three areas together , to share ideas in depth, and to pool our expertise in approaching some of the major open problems in the field.

Targeting of topoisomerases for prognosis and drug resistance in ovarian cancer.

BackgroudAs magicians of the DNA world, topoisomerases resolve all of the topological problems in relation to DNA during a variety of genetic processes. While the prognostic value of topoisomerase isoenzymes in epithelial ovarian carcinoma (EOC) is still elusive. In current study, we investigated the prognostic value of topoisomerase isoenzymes in the EOC patients. Kaplan Meier plotter (KM plotter) database were used to assess the relevance of individual topoisomerase isoenzyme mRNA expression to EOC patients overall survival (OS), in which updated survival information and gene expression data were from a total of 1,648 EOC patients.ResultsHigh expression of TOP1 and TOP2A were found to be correlated to worse OS in all patients and serous patients, but not in endometrioid patients. Contrary to TOP1 and TOP2A, TOP3A and TOP3B expression were associated with better OS in all patients and serous patients, but not in endometrioid patients. While TOP2B were not found any significant prognostic value for EOC patients. From the Oncomine database, we also found widespread upregulation in the expression of TOP1 and TOP2A genes in primary tumor tissues. Albeit limited in number, all datasets exhibiting differential expression showed TOP3A and TOP3B under-regulated.ConclusionThese results strongly supported that TOP1 and TOP2A were potential biomarkers for predicting poor survival of EOC patients, while TOP3A and TOP3B were expected to be further exploited as tumor suppressors. Comprehensive understanding of the topoisomerase isoforms may have guiding significance for the diagnosis treatment and prognosis in EOC patients.Electronic supplementary materialThe online version of this article (doi:10.1186/s13048-016-0244-9) contains supplementary material, which is available to authorized users.

RNA topoisomerase is prevalent in all domains of life and associates with polyribosomes in animals

DNA Topoisomerases are essential to resolve topological problems during DNA metabolism in all species. However, the prevalence and function of RNA topoisomerases remain uncertain. Here, we show that RNA topoisomerase activity is prevalent in Type IA topoisomerases from bacteria, archaea, and eukarya. Moreover, this activity always requires the conserved Type IA core domains and the same catalytic residue used in DNA topoisomerase reaction; however, it does not absolutely require the non-conserved carboxyl-terminal domain (CTD), which is necessary for relaxation reactions of supercoiled DNA. The RNA topoisomerase activity of human Top3β differs from that of Escherichia coli topoisomerase I in that the former but not the latter requires the CTD, indicating that topoisomerases have developed distinct mechanisms during evolution to catalyze RNA topoisomerase reactions. Notably, Top3β proteins from several animals associate with polyribosomes, which are units of mRNA translation, whereas the Top3 homologs from E. coli and yeast lack the association. The Top3β-polyribosome association requires TDRD3, which directly interacts with Top3β and is present in animals but not bacteria or yeast. We propose that RNA topoisomerases arose in the early RNA world, and that they are retained through all domains of DNA-based life, where they mediate mRNA translation as part of polyribosomes in animals.

Products of Admissible Monomials in the Polynomial Algebra as a Module over the Steenrod Algebra

Let ${\P}(n) ={\F}[x_1,\ldots,x_n]$ be the polynomial algebra in $n$ variables $x_i$, of degree one, over the field $\F$ of two elements. The mod-2 Steenrod algebra $\A$ acts on ${\P }(n)$ according to well known rules. A major problem in algebraic topology is that of determining $\A^+{\P}(n)$, the image of the action of the positively graded part of $\A$. We are interested in the related problem of determining a basis for the quotient vector space ${\Q}(n) = {\P}(n)/\A^{+}\P(n)$. Both ${\P }(n) =\bigoplus_{d \geq 0} {\P}^{d}(n)$ and ${\Q}(n)$ are graded, where ${\P}^{d}(n)$ denotes the set of homogeneous polynomials of degree $d$. ${\Q}(n)$ has been explicitly calculated for $n=1,2,3,4$ but problems remain for $n \geq 5.$ In this note we show that if $u = x_{1}^{m_1} \cdots x_{k}^{m_{k}} \in {\P}^{d}(k)$ and $v = x_{1}^{e_1} \cdots x_{r}^{e_{r}} \in {\P}^{d'}(r)$ are an admissible monomials, (that is, $u$ and $v$ meet a criterion to be in a certain basis for ${\Q}(k)$ and ${\Q}(r)$ respectively), then for each permutation $\sigma \in S_{k+r}$ for which $\sigma(i)<\sigma(j),$ $i<j\leq k$ and $\sigma(s)<\sigma(t),$ $k<s<t\leq k+r,$ the monomial $x_{\sigma(1)}^{m_1} \cdots x_{\sigma(k)}^{m_{k}} x_{\sigma(k+1)}^{e_1} \cdots x_{\sigma(k+r)}^{e_r} \in {\P}^{d+d'}(k+r)$ is admissible. As an application we consider a few cases when $n=5.$

Relay Placement in Wireless Networks: Minimizing Communication Cost

Given $n$ source nodes and $k$ relay nodes, we model the optimal relay topology problem allowing for simultaneous optimization of the relay node locations and traffic through the network, so that the overall number of packet retransmissions is minimized. We argue that state-of-the-art models and algorithms for relay placement in wireless networks do not reflect salient characteristics of the optimal relays topology and lead to suboptimal solutions. We do not constrain the position of relays to a finite set of discrete points, as the latter may not be feasible in practical networks. In this case, we show that just listing a set of feasible sites for the relays is already at least $APX$ -hard. Exploiting convexity in a special case of the network communication cost function, we give an optimal algorithm for the relay placement problem. However, the algorithm is exponential on the number of nodes in the network. We suggest a practical heuristic algorithm for relay placement: RePlace . We compare RePlace numerically to the optimal algorithm and show that RePlace achieves the optimal or almost optimal solutions. We implement RePlace in the full network stack simulator JiST/SWANS. The relay topologies generated by RePlace eliminate overhead communication cost almost entirely.

A wireless sensor network-based monitoring system with dynamic convergecast tree algorithm for precision cultivation management in orchid greenhouses

The orchid industry in Taiwan has established large-scale orchid greenhouses to achieve high-precision cultivation of orchids, especially for Phalaenopsis. The wireless sensor network (WSN) technology has been shown to be able to play an important and useful role for effectively acquiring environmental parameters in real-time. However, the mobile benches equipped with different sensors used in an orchid greenhouse create a problem of susceptible dynamic network topology. To meet the requirements of reliable data acquisition in the monitoring of orchid growth, a novel dynamic convergecast tree algorithm (DCTA) based on a tree-like topology was designed and implemented in the WSN-based monitoring system. The proposed WSN algorithm uses the information of the received signal strength indication and hop count to dynamically adjust the routing path of each sensor node. The proposed algorithm includes a flexible scheduling-based design for the medium access control protocol to guarantee higher transmission reliability of the sensor data. An extensive series of experiments, including tests in the lab and an orchid greenhouse, were conducted to examine the performance of the proposed DCTA. The experimental results show that the proposed algorithm can reliably collect environmental data; average successful data delivery rates up to 92.5 % of the entire tested networks with multiple mobile nodes in the greenhouse can be achieved. The WSN-based monitoring system equipped with the proposed DCTA provides environmental measurements with better spatio-temporal resolution to achieve precision cultivation management for orchids.

Certified computation of planar Morse–Smale complexes

The Morse–Smale complex is an important tool for global topological analysis in various problems of computational geometry and topology. Algorithms for Morse–Smale complexes have been presented in case of piecewise linear manifolds (Edelsbrunner et al., 2003a). However, previous research in this field is incomplete in the case of smooth functions. In the current paper we address the following question: Given an arbitrarily complex Morse–Smale system on a planar domain, is it possible to compute its certified (topologically correct) Morse–Smale complex? Towards this, we develop an algorithm using interval arithmetic to compute certified critical points and separatrices forming the Morse–Smale complexes of smooth functions on bounded planar domain. Our algorithm can also compute geometrically close Morse–Smale complexes.

Similarity of binary relations based on complete Boolean lattices and related results

Rough set theory is concerned with the lower and upper approximations through a binary relation on the universe. Binary relations play an important role in this theory. This paper investigates similarity of binary relations based on complete Boolean lattices. First, rough approximations based on complete Boolean lattices are proposed through the predecessor neighborhood. Second, L-fuzzy topologies induced by binary relations are researched where L is a complete Boolean lattice. Thirdly, similarity of binary relations is introduced by using L-fuzzy topology and the fact that every binary relation is solely similar to some preorder relation is showed. Finally, as its theoretical expansion, a topological problem "when can the given L-fuzzy topology be coincided with the L-fuzzy topology induced by some binary relation where L is a complete Boolean lattice?" is considered and this problem is answered by introducing L-fuzzy approximating spaces.

Coordinated Dynamics Orchestrating the DNA Re-Ligation by De-Poisoned Topoisomerase II

Topoisomerase II is a crucial enzyme that tightly controls the topology of DNA to facilitate the critical sub-cellular processes involving these genetic materials. It generates transient double-strand breaks (DSB) and transports another DNA duplex through to resolve their topological problems. Although topoisomerase poisons, the drugs arresting the subsequent DNA re-ligation, have been used in anticancer therapy and the recent structural characterization of the enzyme, our understanding of the re-ligation mechanism was still distressingly restricted by the transitory nature of this process. Captivated by the fluent re-ligation following the characteristic DSB by the enzyme, we selected a high-resolution crystal structure of the drug-entrapped intermediate and exploited atomistic molecular dynamics simulations of the very large systems for up to a microsecond to investigate the dynamics of this process [1]. We observed a nearly vectorial transition of the de-poisoned complex toward the resealing-compliant configuration, and the statistical analysis revealed the non-concerted maneuvers of the two different DNA strands mediated by the enzyme. Using a neighborhood-based metric, we captured the important dynamical transitions during the re-ligation from the vast amount of correlated motions and provided the visible links between previously unsettled experimental observations. With the simulations of the drug-bound complexes, we also presented the conformational population shifts of an important residue in response to the binding of different drug molecules [2]. Furthermore, by comparing the coordinated dynamics revealed by different isoforms, we are able to dissect the differential dynamics of the critical DNA re-ligation conducted by the enzymes. We believe these are useful insights for the development of new anticancer drugs with lower cardiotoxicity.

A Primary Way of Solving Sampling Bias Problem in Complex Internet Topology

Measuring complex Internet topology is primary research activities in Internet-related research fields and has become a focus recently. And due to the complexity of Internet topology, further processing is required and is thoroughly studied in this paper. First, Internet topology measurement is performed by CAIDA measurement approach with more than twenty monitors. Then the problem of sampling bias is quantitatively studied. Mathematical models are introduced into both the node bias problem and link bias problem, and final option is gained. Final results are that more than 156 CAIDA monitors are necessary in order to have a full Internet topology measurement result.

Optimum design of phononic crystal perforated plate structures for widest bandgap of fundamental guided wave modes and maximized in-plane stiffness

This paper presents a topology optimization of single material phononic crystal plate (PhP) to be produced by perforation of a uniform background plate. The primary objective of this optimization study is to explore widest exclusive bandgaps of fundamental (first order) symmetric or asymmetric guided wave modes as well as widest complete bandgap of mixed wave modes (symmetric and asymmetric). However, in the case of single material porous phononic crystals the bandgap width essentially depends on the resultant structural integration introduced by achieved unitcell topology. Thinner connections of scattering segments (i.e. lower effective stiffness) generally lead to (i) wider bandgap due to enhanced interfacial reflections, and (ii) lower bandgap frequency range due to lower wave speed. In other words higher relative bandgap width (RBW) is produced by topology with lower effective stiffness. Hence in order to study the bandgap efficiency of PhP unitcell with respect to its structural worthiness, the in-plane stiffness is incorporated in optimization algorithm as an opposing objective to be maximized. Thick and relatively thin Polysilicon PhP unitcells with square symmetry are studied. Non-dominated sorting genetic algorithm NSGA-II is employed for this multi-objective optimization problem and modal band analysis of individual topologies is performed through finite element method. Specialized topology initiation, evaluation and filtering are applied to achieve refined feasible topologies without penalizing the randomness of genetic algorithm (GA) and diversity of search space. Selected Pareto topologies are presented and gradient of RBW and elastic properties in between the two Pareto front extremes are investigated. Chosen intermediate Pareto topology, even not extreme topology with widest bandgap, show superior bandgap efficiency compared with the results reported in other works on widest bandgap topology of asymmetric guided waves, available in the literature. Finally, steady state and transient frequency response of finite thin PhP structures of selected Pareto topologies are studied and validity of obtained bandgaps is confirmed.

Infinite games and chain conditions

We apply the theory of infinite two-person games to two well-known problems in topology: Suslin's Problem and Arhangel'skii's problem on $G_\delta$ covers of compact spaces. More specifically, we prove results of which the following two are special cases: 1) every linearly ordered topological space satisfying the game-theoretic version of the countable chain condition is separable and 2) in every compact space satisfying the game-theoretic version of the weak Lindel\of property, every cover by $G_\delta$ sets has a continuum-sized subcollection whose union is $G_\delta$-dense.

Optimal topology for consensus of heterogeneous multi-agent systems

Consensus can be achieved under various topologies for multi-agent systems. Then, which one is the optimal? In this paper, we consider the problem of optimal topology for consensus of heterogeneous multi-agent systems. We assume that the system consists of a leader and several followers with heterogeneous dynamics. Firstly, we define a quadratic cost function composed of consensus error and control effort for the heterogeneous multi-agent system. Secondly, by using linear-quadratic regulator theory and matrix theory, we prove that the optimal position topology and the optimal velocity topology are star graphs. Finally, simulations are carried out to illustrate the effectiveness of the theoretical results.

Mary Ellen's conjectures

For a special issue like this, it would seem to be very appropriate for us, the editors, to write an article summarizing Mary Ellen’s contributions to the field, and giving some account of her mathematical career. But there is an excellent article by Frank Tall [65] in the book The Work of Mary Ellen Rudin that does exactly this, covering her work and giving a sketch of her mathematical life up until shortly after her retirement in 1991. The article also includes a list of her students and their theses titles, her postdocs, and her publications. While she published a couple of papers on other topics after that time, she spent most of her post-retirement working on Nikiel’s conjecture, that the images of compact ordered spaces are exactly the compact monotonically normal spaces. She eventually solved it (positively) in 2001 [63]; it took one of the deepest and most difficult arguments of her career to do so. Instead of copying Frank’s article, we would like to do something a bit different. In the book Open Problems in Topology, published in 1990, Mary Ellen has an article [62] in which she states 17 “conjectures”. This is the way she chose to phrase some unsolved problems, most of them due originally to others, that she was particularly interested in. We quote from her article to give a sense of her reasons for choosing these particular problems: “The problems I list here have mostly been listed in Rudin [1988]; I have worked on all of them long enough to have some real respect for them; they have all stood the test of time and labor by various people; and I feel any solution to any of them would have applications at least to other problems exhibiting similar pathologies.” Here we go over Mary Ellen’s list of conjectures, giving a brief summary of her contributions and where each conjecture stands today. Many of these problems had or could be expected to have consistent answers. But Mary Ellen clearly had a bias for ZFC results, so instead of asking “Is it consistent that P holds?” she would ask “Is it true (in ZFC) that “not P” holds?”, or, as in this article, state “Conjecture: Not P holds.”.

On some types of covering rough sets from topological points of view

The concept of coverings is one of the fundamental concepts in topological spaces and plays a big part in the study of topological problems. This motivates the research of covering rough sets from topological points of view. From topological points of view, we can get a good insight into the essence of covering rough sets and make our discussions concise and profound. In this paper, we first construct a type of topology called the topology induced by the covering on a covering approximation space. This notion is indeed in the core of this paper. Then we use it to define the concepts of neighborhoods, closures, connected spaces, and components. Drawing on these concepts, we define several pairs of approximation operators. We not only investigate the relationships among them, but also give clear explanations of the concepts discussed in this paper. For a given covering approximation space, we can use the topology induced by the covering to investigate the topological properties of the space such as separation, connectedness, etc. Finally, a diagram is presented to show that the collection of all the lower and upper approximations considered in this paper constructs a lattice in terms of the inclusion relation ⊆.

Semi-topological Galois Theory

We introduce splitting coverings to enhance the well known analogy between field extensions and covering spaces. Semi-topological Galois groups are defined for Weierstrass polynomials and a Galois correspondence is proved. Combining results from braid groups, we solve the topological inverse Galois problem. As an application, symmetric and cyclic groups are realized over ℚ.