Topological Ranks Research Articles

Topological ranks reveal functional knowledge encoded in biological networks: a comparative analysis.

Biological networks topology yields important insights into biological function, occurrence of diseases and drug design. In the last few years, different types of topological measures have been introduced and applied to infer the biological relevance of network components/interactions, according to their position within the network structure. Although comparisons of such measures have been previously proposed, to what extent the topology per se may lead to the extraction of novel biological knowledge has never been critically examined nor formalized in the literature. We present a comparative analysis of nine outstanding topological measures, based on compact views obtained from the rank they induce on a given input biological network. The goal is to understand their ability in correctly positioning nodes/edges in the rank, according to the functional knowledge implicitly encoded in biological networks. To this aim, both internal and external (gold standard) validation criteria are taken into account, and six networks involving three different organisms (yeast, worm and human) are included in the comparison. The results show that a distinct handful of best-performing measures can be identified for each of the considered organisms, independently from the reference gold standard. Input files and code for the computation of the considered topological measures and K-haus distance are available at https://gitlab.com/MaryBonomo/ranking. simona.rombo@unipa.it. Supplementary data are available at Briefings in Bioinformatics online.

The profinite topology of free groups and weakly generic tuples of automorphisms

AbstractLet be a countable first order structure and endow the universe of with the discrete topology. Then the automorphism group of becomes a topological group. A tuple of automorphisms is defined to be weakly generic iff its diagonal conjugacy class (in the algebraic sense) is dense (in the topological sense) and the ‐orbit of each is finite. Existence of tuples of weakly generic automorphisms are interesting from the point of view of model theory as well as from the point of view of finite combinatorics.The main results of the present work are as follows. In Theorem 2.6 we characterize the existence of tuples of weakly generic automorphisms with the aid of the profinite topology of free groups. In Corollary 2.12 we will show that if has finite topological rank r (and satisfies a further, mild technical condition) then the existence of a weakly generic tuple in implies the existence of weakly generic tuples in for all natural number . Finally, in Theorem 3.2 we show that if is a countable model of an ℵ0‐categorical, simple theory in which all types over the empty set are stationary and has a pair of weakly generic automorphisms then it has tuples of weakly generic automorphisms of arbitrary finite length.At the technical level we will combine elementary investigations about the profinite topology of free groups with the results of [11] about topological ranks of the automorphism groups of some structures.

The Bass and topological stable ranks for algebras of almost periodic functions on the real line, II

Let Λ be either a subgroup of the integers Z, a semigroup in N, or Λ=Q (resp., Q+). We determine the Bass and topological stable ranks of the algebras APΛ={f∈AP:σ(f)⊆Λ} of almost periodic functions on the real line and with Bohr spectrum in Λ. This answers a question in the first part of this series of articles under the same heading, where it was shown that, in contrast to the present situation, these ranks were infinite for each semigroup Λ of real numbers for which the Q-vector space generated by Λ had infinite dimension.

Comparing the topological rank of journals in Web of Science and Mendeley

Recently, there has been a surge of interest in new data emerged due to the rapid development of the information technologies in scholarly communication. Since the 2010s, altmetrics has become a common trend in scientometric research. However, researchers have not treated in much detail the question of the probability distributions underlying these new data. The principal objective of this study was to investigate one of the classic problems of scientometrics—the problem of citation and readership distributions. The study is based on the data obtained from two information systems: Web of Science and Mendeley. Here we based on the concept of the cumulative empirical distribution function to explore the differences and similarities between citations and readership counts of biological journals indexed in Web of Science and Mendeley. The basic idea was to determine, for any journal, a “size” (it is said to be the topological rank) of citation and readership empirical cumulative distributions, and then to compare distributions of the topological ranks of Web of Science and Mendeley. In order to verify our model, we employ it to the bibliometric and altmetric research of 305 biological journals indexed in Journal Citation Reports 2015. The findings show that both distributions of the topological rank of biological journals are statistically close to the Wakeby distribution. The findings presented in this study add to our understanding of information processes of the scholarly communication in the new digital environment.

Pathways of topological rank analysis (PoTRA): a novel method to detect pathways involved in hepatocellular carcinoma.

Complex diseases such as cancer are usually the result of a combination of environmental factors and one or several biological pathways consisting of sets of genes. Each biological pathway exerts its function by delivering signaling through the gene network. Theoretically, a pathway is supposed to have a robust topological structure under normal physiological conditions. However, the pathway’s topological structure could be altered under some pathological condition. It is well known that a normal biological network includes a small number of well-connected hub nodes and a large number of nodes that are non-hubs. In addition, it is reported that the loss of connectivity is a common topological trait of cancer networks, which is an assumption of our method. Hence, from normal to cancer, the process of the network losing connectivity might be the process of disrupting the structure of the network, namely, the number of hub genes might be altered in cancer compared to that in normal or the distribution of topological ranks of genes might be altered. Based on this, we propose a new PageRank-based method called Pathways of Topological Rank Analysis (PoTRA) to detect pathways involved in cancer. We use PageRank to measure the relative topological ranks of genes in each biological pathway, then select hub genes for each pathway, and use Fisher’s exact test to test if the number of hub genes in each pathway is altered from normal to cancer. Alternatively, if the distribution of topological ranks of gene in a pathway is altered between normal and cancer, this pathway might also be involved in cancer. Hence, we use the Kolmogorov–Smirnov test to detect pathways that have an altered distribution of topological ranks of genes between two phenotypes. We apply PoTRA to study hepatocellular carcinoma (HCC) and several subtypes of HCC. Very interestingly, we discover that all significant pathways in HCC are cancer-associated generally, while several significant pathways in subtypes of HCC are HCC subtype-associated specifically. In conclusion, PoTRA is a new approach to explore and discover pathways involved in cancer. PoTRA can be used as a complement to other existing methods to broaden our understanding of the biological mechanisms behind cancer at the system-level.

Rank 2 proximal Cantor systems are residually scrambled

ABSTRACTDownarowicz and Maass [7] proposed topological ranks for all homeomorphic Cantor minimal dynamical systems using properly ordered Bratteli diagrams. In this study, we adopt this definition to the case of the essentially minimal zero-dimensional systems. We consider the cases in which topological ranks are 2 and unique minimal sets are fixed points. Akin and Kolyada [2], had shown that if the unique minimal set of an essentially minimal system is a fixed point, then the system must be proximal. The finite topological rank implies expansiveness; furthermore, in the case of proximal Cantor systems with topological rank 2, the expansiveness is always from the lowest degree. Rank 2 proximal Cantor systems are residually scrambled. We present a necessary and sufficient condition for the unique ergodicity of these systems. In addition, we show that the number of ergodic measures of the systems that are topologically mixing can be 1 and 2. Moreover, we present examples that are topologically weakly mixing, not topologically mixing, and uniquely ergodic. Finally, we show that the number of ergodic measures of the systems that are not weakly mixing can be 1 and 2.

TOPOLOGICAL RANK DOES NOT INCREASE BY NATURAL EXTENSION OF CANTOR MINIMALS

Downarowicz and Maass (2008) have defined the topological rank for all Cantor minimal homeomorphisms. On the other hand, Gambaudo and Martens (2006) have expressed all Cantor minimal continuous surjections as the inverse limits of certain graph coverings. Using the aforementioned results, we previously extended the notion of topological rank to all Cantor minimal continuous surjections. In this paper, we show that taking natural extensions of Cantor minimal continuous surjections does not increase their topological ranks. Further, we apply the result to the minimal symbolic case.

The Bass and topological stable ranks for algebras of almost periodic functions on the real line

Let $\Lambda$ be a sub-semigroup of the reals. We show that the Bass and topological stable ranks of the algebras $\textrm {AP}_\Lambda =\{f\in \textrm {AP}: \sigma (f)\subseteq \Lambda \}$ of almost periodic functions on the real line and with Bohr spectrum in $\Lambda$ are infinite whenever the algebraic dimension of the $\mathbb {Q}$-vector space generated by $\Lambda$ is infinite. This extends Suárez’s result for $\textrm {AP}_\mathbb {R}=\textrm {AP}$. Also considered are general subalgebras of AP.

The topological decomposition of inverse limits of iterated wreath products of finite Abelian groups

Abstract. Let be an arbitrary infinite sequence of nontrivial finite Abelian transitive groups such that the topological rank ρ of the infinite Cartesian product of these groups is finite. We consider the corresponding inverse limit of iterated permutational wreath products. By using the geometric language of automorphisms of a certain rooted tree we show that the group is topologically generated by the union of two sets each containing ρ elements and such that both the group generated by S and the group generated by are free Abelian groups of rank ρ. Moreover, the group G generated by the above union decomposes as a semidirect product of the group generated by one of these sets and the normal closure of the other set, and the semigroup generated by this union is a free product of semigroups generated by S and , respectively. We derive other algebraic properties of the group G. In particular, the condition for the groups to be finitely generated as profinite groups and some nontrivial results concerning topological ranks of their subgroups are given in this way. For example, for every our construction gives an explicit and naturally defined finitely generated subgroup of an inverse limit of iterated wreath products of finite Abelian groups such that the rank of this subgroup and the topological rank of its topological closure differ by n.

The covering dimension of a distinguished subset of the spectrum [formula omitted] of [formula omitted] and the algebra of real-symmetric and continuous functions on [formula omitted

We show that the covering dimension, dim E, of the closure E of the interval ] − 1 , 1 [ in the spectrum of H ∞ equals one. Using Suárezʼs result that dim M ( H ∞ ) = 2 , we then compute the Bass and topological stable ranks of the algebra C ( M ( H ∞ ) ) sym of real-symmetric continuous functions on M ( H ∞ ) .

Real-Symmetric Extensions of Invertible Tuples of Multivariable Continuous Functions

We give necessary and sufficient conditions on pairs of real-symmetric compact sets (K, L) in \({{\mathbb C}^n, K\, {\subseteq}\, L}\), such that every invertible n-tuple of real-symmetric continuous functions on K can be extended to such an invertible tuple on L. We also determine the Bass and topological stable ranks of \({C(K)_{{\rm sym}}}\).

Stable ranks for the real function algebra $C(X,\tau)$

We determine the Bass and topological stable ranks of the real algebra C(X, ) of all complex-valued continuous functions on the compact Hausdor space X that satisfy f = f, where is a topological involution on X.

Topological stable rank of H ∞(Ω) for circular domains Ω

Let Ω be a circular domain, that is, an open disk with finitely many closed disjoint disks removed. Denote by H ∞(Ω) the Banach algebra of all bounded holomorphic functions on Ω, with pointwise operations and the supremum norm. We show that the topological stable rank of H ∞(Ω) is equal to 2. The proof is based on Suárez’s theorem that the topological stable rank of H ∞(\( \mathbb{D} \)) is equal to 2, where \( \mathbb{D} \) is the unit disk. We also show that for circular domains symmetric to the real axis, the Bass and topological stable ranks of the real-symmetric algebra H ∞ℝ (Ω) are 2.

Matricial topological ranks for two algebras of bounded holomorphic functions

Let N and D be two matrices over the algebra H ∞ of bounded analytic functions in the disk, or its real counterpart . Suppose that N and D have the same number n of columns. In a generalization of the notion of topological stable rank 2, it is shown that N and D can be approximated (in the operator norm) by two matrices Ñ and , so that the Aryabhatta–Bezout equation admits a solution. This has particular interesting consequences in systems theory. Moreover, in case that N is a square matrix, X can be chosen to be invertible in the case of the algebra H ∞, but not always in the case of .

Spectral characteristics and stable ranks for the Sarason algebra $H^\inf + C$

We prove a Corona type theorem with bounds for the Sarason algebra $H^\infty+C$ and determine its spectral characteristics. We also determine the Bass, the dense, and the topological stable ranks of $H^\infty+C$.

On the stable rank and reducibility in algebras of real symmetric functions

Let Aℝ(𝔻) denote the set of functions belonging to the disc algebra having real Fourier coefficients. We show that Aℝ(𝔻) has Bass and topological stable ranks equal to 2, which settles the conjecture made by Brett Wick in [18]. We also give a necessary and sufficient condition for reducibility in some real algebras of functions on symmetric domains with holes, which is a generalization of the main theorem in [18]. A sufficient topological condition on the symmetric open set 𝔻 is given for the corresponding real algebra Aℝ(𝔻) to have Bass stable rank equal to 1 (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Ideal Structure and Stable Rank of $${\mathbb {C} e+ \ell^2(I)}$$ with the Hadamard Product

Let I be any index set. We consider the Banach algebra \({\mathbb {C} e+ \ell^2(I)}\) with the Hadamard product, and prove that its Bass and topological stable ranks are both equal to 1. We also characterize divisors, maximal ideals, closed ideals and closed principal ideals. For \({I=\mathbb {N}}\) we also characterize all prime z-ideals in this Banach algebra.

Bass and Topological Stable Ranks of Complex and Real Algebras of Measures, Functions and Sequences

We compute the Bass stable rank and the topological stable rank of several convolution Banach algebras of complex measures on (-∞,∞) or on [0,∞) consisting of a discrete measure and/or of an absolutely continuous measure. We also compute the stable ranks of the convolution algebras , , l1(S) and , where S is an arbitrary subgroup of , of the almost periodic algebra AP and of , etc. We answer affirmatively the question posed by Mortini (Studia Mathematica 103(3):275–281, 1992). For the above algebras, the polydisc algebra , the algebra of continuous functions, and others, we also study their subsets (real Banach algebras) of real-valued measures, real-valued sequences or real-symmetric functions, and of corresponding exponentially stable algebras (for example, the Callier–Desoer algebra of causal exponentially decaying measures and L1 functions), and we compute their stable ranks. Finally, we show that in some of these real algebras a variant of the parity interlacing property is equivalent to reducibility of a unimodular (or coprime) pair. Also corona theorems are presented and the existence of coprime fractions is studied; in particular, we list which of these algebras are Bezout domains.

The Bass and topological stable ranks of and

In this note we prove that the Bass stable rank of is two. This establishes the validity of a conjecture by S. Treil. We accomplish this in two different ways, one by giving a direct proof, and the other, by first showing that the topological stable rank of is two. We apply these results to give new proofs of results by R. Rupp and A. Sasane stating that the Bass stable rank of is two and the topological stable rank of is two, settling a conjecture by the second author. We also present a -free proof of the second author's characterization of the reducible pairs in .

On the topological stable rank of non-selfadjoint operator algebras

We provide a negative solution to a question of M. Rieffel who asked if the right and left topological stable ranks of a Banach algebra must always agree. Our example is found amongst a class of nest algebras. We show that for many other nest algebras, both the left and right topological stable ranks are infinite. We extend this latter result to Popescu’s non-commutative disc algebras and to free semigroup algebras as well.