Related papers: Quantifying Homology Classes
A generalization of the formula of Fine and Rao for the ranks of the intersection homology groups of a complex algebraic variety is given. The proof uses geometric properties of intersection homology and mixed Hodge theory.
Persistent homology has been studied to better understand the structural properties and topology features of weighted networks. It can reveal hidden layers of information about the higher-order structures formed by non-pairwise interactions…
We consider three types of entities for quantum measurements. In order of generality, these types are: observables, instruments and measurement models. If $\alpha$ and $\beta$ are entities, we define what it means for $\alpha$ to be a part…
Motivated by recent best case analyses for some sorting algorithms and based on the type of complexity we partition the algorithms into two classes: homogeneous and non homogeneous algorithms. Although both classes contain algorithms with…
We compute numerically the homology of several graph complexes in low loop orders, extending previous results.
The homography matrix is a key component in various vision-based robotic tasks. Traditionally, homography estimation algorithms are classified into feature- or intensity-based. The main advantages of the latter are their versatility,…
We study the homology groups of the complement of a complexified real line arrangement with coefficients in complex rank-one local systems. Using Borel--Moore homology, we establish an algorithm computing their dimensions via the real…
In this paper we survey three approaches to computing the homology of a finite dimensional compact smooth closed manifold using a Morse-Bott function and discuss relationships among the three approaches. The first approach is to perturb the…
We define symmetry classes and commutation symmetries in the Hilbert space H of the 1D spin-1/2 Heisenberg magnetic ring with N sites and investigate them by means of tools from the representation theory of symmetric groups S_N such as…
Like (co)homology group theory of formal Hamiltonian vector fields on symplectic vector spaces, we try studying homology group theory on symplecit tori introducing the notion of weight.
Persistent homology is a powerful mathematical tool that summarizes useful information about the shape of data allowing one to detect persistent topological features while one adjusts the resolution. However, the computation of such…
Many first-order equational theories, such as the theory of groups or boolean algebras, can be presented by a smaller set of axioms than the original one. Recent studies showed that a homological approach to equational theories gives us…
A qualgebra $G$ is a set having two binary operations that satisfy compatibility conditions which are modeled upon a group under conjugation and multiplication. We develop a homology theory for qualgebras and describe a classifying space…
We provide upper bounds for logarithmic torsion homology growth and Betti number growth of groups, phrased in the language of measured group theory.
Suppose $S$ is a closed, oriented surface of genus at least two. This paper investigates the geometry of the homology multicurve complex, $\mathcal{HC}(S,\alpha)$, of $S$; a complex closely related to complexes studied by…
Such modern applications of topology as data analysis and digital image analysis have to deal with noise and other uncertainty. In this environment, topological spaces often appear equipped with a real valued function. Persistence is a…
Assessing how adequate clusters fit a dataset and finding an optimum number of clusters is a difficult process. A membership matrix and the degree of membership matrix is suggested to determine the homogeneity of a cluster fit. Maximisation…
Attempts to incorporate topological information in supervised learning tasks have resulted in the creation of several techniques for vectorizing persistent homology barcodes. In this paper, we study thirteen such methods. Besides describing…
Let $X$ be a $2n$-manifold with a locally standard action of a compact torus $T^n$. If the free part of action is trivial and proper faces of the orbit space $Q$ are acyclic, then there are three types of homology classes in $X$: (1)…
Let $\mathcal{M}_{0,n+1}$ be the moduli space of genus zero Riemann surfaces with $n+1$ marked points. In this paper we compute $H_*^{\Sigma_n}(\mathcal{M}_{0,n+1};\mathbb{F}_p)$ and $H_*^{\Sigma_n}(\mathcal{M}_{0,n+1};\mathbb{F}_p(\pm 1))$…