Betti number
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The topology of the cosmic web in terms of persistent Betti numbers Open
We introduce a multiscale topological description of the Megaparsec weblike\ncosmic matter distribution. Betti numbers and topological persistence offer a\npowerful means of describing the rich connectivity structure of the cosmic web\nand…
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Topology-Preserving Deep Image Segmentation Open
Segmentation algorithms are prone to make topological errors on fine-scale structures, e.g., broken connections. We propose a novel method that learns to segment with correct topology. In particular, we design a continuous-valued loss func…
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Simplicial complexes: Spectrum, homology and random walks Open
This paper studies the dynamical and asymptotic aspects of high‐dimensional expanders. We define a stochastic process on simplicial complexes of arbitrary dimension, which detects the existence of homology in the same way that a random wal…
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Beyond the clustering coefficient: A topological analysis of node neighbourhoods in complex networks Open
In Network Science, node neighbourhoods, also called ego-centered networks, have attracted significant attention. In particular the clustering coefficient has been extensively used to measure their local cohesiveness. In this paper, we sho…
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Exact topological inference of the resting-state brain networks in twins Open
A cycle in a brain network is a subset of a connected component with redundant additional connections. If there are many cycles in a connected component, the connected component is more densely connected. Whereas the number of connected co…
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Topology and geometry of Gaussian random fields I: on Betti numbers, Euler characteristic, and Minkowski functionals Open
This study presents a numerical analysis of the topology of a set of\ncosmologically interesting three-dimensional Gaussian random fields in terms of\ntheir Betti numbers $\\beta_0$, $\\beta_1$ and $\\beta_2$. We show that Betti\nnumbers e…
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Recent advances in the synthesis and synthetic applications of Betti base (aminoalkylnaphthol) and bis-Betti base derivatives Open
The multicomponent reaction between 2-naphthol, arylaldehydes and ammonia yields aminobenzylnaphthols in a process known as the Betti reaction, which was first uncovered at the beginning of the 20th century.
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Persistent Laplacians: Properties, Algorithms and Implications Open
We present a thorough study of the theoretical properties and devise\nefficient algorithms for the \\emph{persistent Laplacian}, an extension of the\nstandard combinatorial Laplacian to the setting of pairs (or, in more\ngenerality, sequen…
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Topology-Preserving Deep Image Segmentation Open
Segmentation algorithms are prone to make topological errors on fine-scale struc- tures, e.g., broken connections. We propose a novel method that learns to segment with correct topology. In particular, we design a continuous-valued loss fu…
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Evolutionary de Rham-Hodge method Open
The de Rham-Hodge theory is a landmark of the 20th Century's mathematics and has had a great impact on mathematics, physics, computer science, and engineering. This work introduces an evolutionary de Rham-Hodge method to provide a unified …
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Measuring the topology of reionization with Betti numbers Open
The distribution of ionized hydrogen during the epoch of reionization (EoR) has a complex morphology. We propose to measure the 3D topology of ionized regions using the Betti numbers. These quantify the topology using the number of compone…
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On the growth of $L^2$-invariants for sequences of lattices in Lie groups Open
We study the asymptotic behaviour of Betti numbers, twisted torsion and other spectral invariants of sequences of locally symmetric spaces. Our main results are uniform versions of the DeGeorge--Wallach Theorem, of a theorem of Delorme and…
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DAHA and iterated torus knots Open
The theory of DAHA-Jones polynomials is extended from torus knots to their arbitrary iterations (for any reduced root systems and weights), which includes the polynomiality, duality and other properties of the DAHA superpolynomials. Presum…
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The topology of the directed clique complex as a network invariant Open
We introduce new algebro-topological invariants of directed networks, based on the topological construction of the directed clique complex. The shape of the underlying directed graph is encoded in a way that can be studied mathematically t…
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HERMES: Persistent spectral graph software Open
Persistent homology (PH) is one of the most popular tools in topological data analysis (TDA), while graph theory has had a significant impact on data science. Our earlier work introduced the persistent spectral graph (PSG) theory as a unif…
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Using Topological Data Analysis (TDA) and Persistent Homology to Analyze the Stock Markets in Singapore and Taiwan Open
In recent years, persistent homology (PH) and topological data analysis (TDA) have gained increasing attention in the fields of shape recognition, image analysis, data analysis, machine learning, computer vision, computational biology, bra…
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Combinatorics and topology of proper toric maps Open
We study the topology of toric maps. We show that if f : X → Y {f\colon X\to Y} is a proper toric morphism, with X simplicial, then the cohomology of every fiber of f is pure and of Hodge–Tate type. When the map is a fibration, we give an …
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New curvature conditions for the Bochner Technique Open
We prove a vanishing and estimation theorem for the $p^{\\text{th}}$-Betti\nnumber of closed $n$-dimensional Riemannian manifolds with a lower bound on the\naverage of the lowest $n-p$ eigenvalues of the curvature operator. This\ngeneraliz…
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Mixing in High-Dimensional Expanders Open
We establish a generalization of the Expander Mixing Lemma for arbitrary (finite) simplicial complexes. The original lemma states that concentration of the Laplace spectrum of a graph implies combinatorial expansion (which is also referred…
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Betti numbers of configuration spaces of surfaces Open
We give explicit formulas for the Betti numbers, both stable and unstable, of\nthe unordered configuration spaces of an arbitrary surface of finite type.\n
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Computing the Homology of Basic Semialgebraic Sets in Weak Exponential Time Open
We describe and analyze an algorithm for computing the homology (Betti numbers and torsion coefficients) of basic semialgebraic sets that works in weak exponential time. That is, of a set of exponentially small measure in the space of data…
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Structure of force networks in tapped particulate systems of disks and pentagons. II. Persistence analysis Open
In the companion paper [Pugnaloni et al., Phys. Rev. E 93, 062902 (2016)10.1103/PhysRevE.93.062902], we use classical measures based on force probability density functions (PDFs), as well as Betti numbers (quantifying the number of compone…
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Profinite rigidity and surface bundles over the circle Open
If $M$ is a compact 3-manifold whose first betti number is 1, and $N$ is a compact 3-manifold such that $π_1N$ and $π_1M$ have the same finite quotients, then $M$ fibres over the circle if and only if $N$ does. We prove that groups of the …
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Topological View of Flows Inside the BOLD Spontaneous Activity of the Human Brain Open
Spatio-temporal brain activities with variable delay detectable in resting-state functional magnetic resonance imaging (rs-fMRI) give rise to highly reproducible structures, termed cortical lag threads, that propagate from one brain region…
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C2-equivariant stable homotopy from realmotivic stable homotopy Open
We give a method for computing the [math] -equivariant homotopy groups of the Betti realization of a [math] -complete cellular motivic spectrum over [math] in terms of its motivic homotopy groups. More generally, we show that Betti realiza…
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Simplicial generation of Chow rings of matroids Open
We introduce a presentation of the Chow ring of a matroid by a new set of generators, called “simplicial generators.” These generators are analogous to nef divisors on projective toric varieties, and admit a combinatorial interpretation vi…
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Categorified duality in Boij–Söderberg theory and invariants of free complexes Open
We present a robust categorical foundation for the duality theory introduced by Eisenbud and Schreyer to prove the Boij–Söderberg conjectures describing numerical invariants of syzygies. The new foundation allows us to extend the reach of …
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Naphthol-derived Betti bases as potential SLC6A14 blockers Open
Betti bases (aminobenzylnaphthols) have not been studied extensively to explore their possible pharmacological applications. Our group prepared a small and focused library of twenty-three Betti bases from the multicomponent reaction of 2-n…
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Associahedra for finite‐type cluster algebras and minimal relations between <i>g</i>‐vectors Open
We show that the mesh mutations are the minimal relations among the ‐vectors with respect to any initial seed in any finite‐type cluster algebra. We then use this algebraic result to derive geometric properties of the ‐vector fan: we show …
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Persistent homology in two-dimensional atomic networks Open
The topology of two-dimensional network materials is investigated by persistent homology analysis. The constraint of two dimensions allows for a direct comparison of key persistent homology metrics (persistence diagrams, cycles, and Betti …