Homology Theory Research Articles

Bio-inspired parallel computing of representative geometrical objects of holes of binary 2D-images

In this paper, we present a bio-inspired parallel implementation of a solution of the problem of looking for the representative geometrical objects of the homology groups in a binary 2D image (extended-HGB2I problem), which is an extended version of a well-known problem in homology theory. In particular, given a binary 2D image, all black connected components and the representative curves of the holes of these components are obtained and labelled. To this aim, a new technique for labelling the connected components of a binary image is presented. In order to compute the solution, the formal framework uses techniques from membrane computing and the implementation has been done in a hardware architecture called compute unified device architecture (CUDA). The computational complexity of the proposed solution is O(m) with respect to the input (image) size m ∼ n². Finally, some examples and applications are also presented.

Braided systems: a unified treatment of algebraic structures with several operations

Bialgebras and Hopf (bi)modules are typical algebraic structures with several interacting operations. Their structural and homological study is therefore quite involved. We develop the machinery of braided systems, tailored for handling such multi-operation situations. Our construction covers the above examples (as well as Poisson algebras, Yetter--Drinfel$'$d modules, and several other structures, treated in separate publications). In spite of this generality, graphical tools allow an efficient study of braided systems, in particular of their representation and homology theories. These latter naturally recover, generalize, and unify standard homology theories for bialgebras and Hopf (bi)modules (due to Gerstenhaber--Schack, Panaite--{\c{S}}tefan, Ospel, Taillefer); and the algebras encoding their representation theories (Heisenberg double, algebras~$\mathscr X$, $\mathscr Y$, $\mathscr Z$ of Cibils--Rosso and Panaite). Our approach yields simplified and conceptual proofs of the properties of these objects.

Bio-inspired parallel computing of representative geometrical objects of holes of binary 2D-images

In this paper, we present a bio-inspired parallel implementation of a solution of the problem of looking for the representative geometrical objects of the homology groups in a binary 2D image (extended-HGB2I problem), which is an extended version of a well-known problem in homology theory. In particular, given a binary 2D image, all black connected components and the representative curves of the holes of these components are obtained and labelled. To this aim, a new technique for labelling the connected components of a binary image is presented. In order to compute the solution, the formal framework uses techniques from membrane computing and the implementation has been done in a hardware architecture called compute unified device architecture (CUDA). The computational complexity of the proposed solution is O(m) with respect to the input (image) size m ∼ n². Finally, some examples and applications are also presented.

Strong homology, derived limits, and set theory

We consider the question of the additivity of strong homology. This entails isolating the set-theoretic content of the higher derived limits of an inverse system indexed by the functions from $\mathbb{N}$ to $\mathbb{N}$. We show that this system governs, at a certain level, the additivity of strong homology over sums of arbitrary cardinality. We show in addition that, under the Proper Forcing Axiom, strong homology is not additive, not even on closed subspaces of $\mathbb{R}^4$.

Bikei homology

We introduce a modified homology and cohomology theory for involutory biquandles (also known as \textit{bikei}). We use bikei 2-cocycles to enhance the bikei counting invariant for unoriented knots and links as well as unoriented and non-orientable knotted surfaces in $\mathbb{R}^4$.

A combinatorial algorithm for computing higher order linking numbers

We develop the intersection theory at relative chain-cochain level, and apply it along with the use of Seifert disks for an oriented link to give a combinatorial algorithm to compute Massey's higher order linking numbers. It is subtle to compute higher-order linking numbers, and it has been a folklore to use the intersection theory in the process, which was first suggested by W. Massey. Massey introduced the higher-order linking as an application of his higher-order cohomology operations defined in terms of suitable cochains, and he calculated the third-order linking numbers for some 3-component links by shifting from cohomology and cup product to homology and intersection theory via duality theorems for manifolds. Later several works along this direction gave methods in various forms for computing the higher-order linking numbers, and others computed Milnor's invariants by using their well-known connection to Massey invariants; but a formal derivation of the general formulae in intersection theory which are used for the computation has not yet been given, and a concrete algorithm for the computation is hence lacking. In this paper we complete Massey's original approach by developing systematically the intersection theory at the relative chain-cochain level in the simplicial category, and use it to derive recursive combinatorial formulae for computing all higher-order linking numbers.

Finite generation and continuity of topological Hochschild and cyclic homology

The goal of this paper is to establish fundamental properties of the Hochschild, topological Hochschild, and topological cyclic homologies of commutative, Noetherian rings, which are assumed only to be F-finite in the majority of our results. This mild hypothesis is satisfied in all cases of interest in finite and mixed characteristic algebraic geometry. We prove firstly that the topological Hochschild homology groups, and the homotopy groups of the fixed point spectra TR, are finitely generated modules. We use this to establish the continuity of these homology theories for any given ideal. A consequence of such continuity results is the pro Hochschild–Kostant–Rosenberg theorem for topological Hochschild and cyclic homology. Finally, we show more generally that the aforementioned finite generation and continuity properties remain true for any proper scheme over such a ring.

Relative Igusa-Todorov Functions and Relative Homological Dimensions

We develope the theory of ${\mathcal {E}}$ -relative Igusa-Todorov functions in an exact I T-context $({\mathcal {C}},{\mathcal {E}})$ (see Definition 2.1). In the case when ${\mathcal {C}}={\text {mod}}\, ({\Lambda })$ is the category of finitely generated left Λ-modules, for an artin algebra Λ, and ${\mathcal {E}}$ is the class of all exact sequences in ${\mathcal {C}},$ we recover the usual Igusa-Todorov functions, Igusa K. and Todorov G. (2005). We use the setting of the exact structures and the Auslander-Solberg relative homological theory to generalise the original Igusa-Todorov’s results. Furthermore, we introduce the ${\mathcal {E}}$ -relative Igusa-Todorov dimension and also we obtain relationships with the relative global and relative finitistic dimensions and the Gorenstein homological dimensions.

On homology of associative shelves

Homology theories for associative algebraic structures are well established and have been studied for a long time. More recently, homology theories for self-distributive algebraic structures motivated by knot theory, such as quandles and their relatives, have been developed and investigated. In this paper, we study associative self-distributive algebraic structures and their one-term and two-term (rack) homology groups.

A new spectral sequence for homology of posets

We develop a new method to compute the homology groups of Alexandroff topological spaces (or equivalently of partially ordered sets) by means of spectral sequences giving a complete and simple description of the corresponding differentials. Applying our tools, we obtain a generalization of a result of Minian in homological Morse theory for posets.

Foliations-Webs-Hessian Geometry-Information Geometry-Entropy and Cohomology

IN MEMORIAM OF ALEXANDER GROTHENDIECK. THE MAN.

Topological diagnostics of the cyclic component of the time series associated with helium

Detection of the deterministic component from noised time series is a common procedure in the solar–terrestrial coupling problem when climate is modeled, solar activity is analyzed, or a signal associated with helium is extracted. Such series are mostly generated by the superposition of different processes for which the concept of a noise component cannot be determined formally. A method based on the combination of time-series topological embedding in Euclidean space and the identification of a persistent cycle by homology theory methods is proposed. The method application is demonstrated based on actual data.

The ℓ∞-semi-norm on uniformly finite homology

Abstract Uniformly finite homology is a coarse homology theory, defined via chains that satisfy a uniform boundedness condition. By construction, uniformly finite homology carries a canonical ℓ ∞ {\ell^{\infty}} -semi-norm. We show that, for uniformly discrete spaces of bounded geometry, this semi-norm on uniformly finite homology in degree 0 with ℤ {\mathbb{Z}} -coefficients allows for a new formulation of Whyte’s rigidity result. In contrast, we prove that this semi-norm is trivial on uniformly finite homology with ℝ {\mathbb{R}} -coefficients in higher degrees.

An approach to intersection theory on singular varieties using motivic complexes

We introduce techniques of Suslin, Voevodsky, and others into the study of singular varieties. Our approach is modeled after Goresky–MacPherson intersection homology. We provide a formulation of perversity cycle spaces leading to perversity homology theory and a companion perversity cohomology theory based on generalized cocycle spaces. These theories lead to conditions on pairs of cycles which can be intersected and a suitable equivalence relation on cocycles/cycles enabling pairings on equivalence classes. We establish suspension and splitting theorems, as well as a localization property. Some examples of intersections on singular varieties are computed.

Virtually compact special hyperbolic groups are conjugacy separable

We prove that any word hyperbolic group which is virtually compact special (in the sense of Haglund and Wise) is conjugacy separable. As a consequence we deduce that all word hyperbolic Coxeter groups and many classical small cancellation groups are conjugacy separable. To get the main result we establish a new criterion for showing that elements of prime order are conjugacy distinguished. This criterion is of independent interest; its proof is based on a combination of discrete and profinite (co)homology theories.

The degenerate distributive complex is degenerate

We prove that the degenerate part of the distributive homology of a multispindle is determined by the normalized homology. In particular, when the multispindle is a quandle Q, the degenerate homology of Q is completely determined by the quandle homology of Q. For this case (and generally for two-term homology of a spindle) we provide an explicit Künneth-type formula for the degenerate part. This solves the mystery in algebraic knot theory of the meaning of the degenerate quandle homology, brought over 15 years ago when the homology theories were defined, and the degenerate part was observed to be nontrivial.

Classification of Degree of Differentiation of Colorectal Neoplasm by Changes in the Betti Number

Introduction/ Background The diagnosis of pathology provides important information for determining the treatment protocol, but other than in developed countries, the number of pathologists is not sufficient. Even where the number is sufficient, they are often unevenly distributed, with most pathologists in large hospitals in metropolitan areas, and adequate services cannot be provided to all patients. For this reason, it is desirable to develop ways to allow for remote diagnosis or to develop automatic diagnostic systems. Digital diagnostic systems based on pattern-recognition techniques have been developed. Because the morphology of cancer tissue is quite complex, a library of all types of cancer morphology would be huge, it would require a high-performance computer system, and the costs in money and time would be very high. This makes it difficult to create an effective system. Aims Recently, a new method based on the homology theory for analyzing histological digital images has been developed. The method evaluates the Betti numbers in a unit area of an image of a colon to determine the region of interest (ROI). The Betti number is an important index for homology theory, and can be used to assess the degree of connectivity in tissue. This method, however, cannot distinguish between different types of tissue. When a tumor forms in the colon, because of the excessive growth of nuclei, the connections increase and become tight. The absolute value of b1 (the one dimensional Betti number) will be high, and the ratio of change will be small. When a tumor becomes more poorly differentiated, it frequently happens that the ties between the cells are weakened, and the intercellular areas are filled with impurities. Because of these impurities, the absolute value of b1 is very high. Its influence, however, will disappear immediately. Namely, b1 decreases very quickly, when we decrease the binarizing threshold. We calculate the decay ratio of the betti number, by changing the binarized thresholds. Our aim in this talk is that using this information, we will classify the degree of differentiation of colorectal neoplasm. Methods In the colon, cancerous regions show stronger color intensity when stained with hematoxylin. This stains blue, and so the distribution of blue in an image indicates the cancerous areas. Thus, for each image, we measured the peak of the blue distribution, and we will refer to it as the reference value. To determine the binarizing threshold, we multiplied the reference value by 0.55 to 0.72, in increments of 0.01. The binarization was carried out in gray scale using these values. We call this interval the reference interval. We may assume that there was no pathological information outside of the reference interval. Results The calculated results can be approximated by quadratic functions. The distribution of the coefficient on the squared term and the x-coordinates of the vertices are shown. We can see a characteristic distribution for each type of cancerous tissue. As the binarizing threshold decreases, the images gradually fade to white, and the structure of the tissue is lost. Under the proposed procedure, in areas where the connections in the tissue are tight and clear, b1 changes slowly. Conversely where the connections are vague, such as in a background area filled with impurities, it changes very quickly. The state of this change can be considered an expression of the strength of the connectivity, and it differs by type of cancerous tissue.

Homology of left non-degenerate set-theoretic solutions to the Yang–Baxter equation

This paper deals with left non-degenerate set-theoretic solutions to the Yang–Baxter equation (= LND solutions), a vast class of algebraic structures encompassing groups, racks, and cycle sets. To each such solution there is associated a shelf (i.e., a self-distributive structure) which captures its major properties. We consider two (co)homology theories for LND solutions, one of which was previously known, in a reduced form, for biracks only. An explicit isomorphism between these theories is described. For groups and racks we recover their classical (co)homology, whereas for cycle sets we get new constructions. For a certain type of LND solutions, including quandles and non-degenerate cycle sets, the (co)homologies split into the degenerate and the normalized parts. We express 2-cocycles of our theories in terms of group cohomology, and, in the case of cycle sets, establish connexions with extensions. This leads to a construction of cycle sets with interesting properties.

The fattened Davis complex and weighted L2–(co)homology of Coxeter groups

Given a Coxeter system $(W,S)$ there is a contractible simplicial complex $\Sigma$ called the Davis complex on which $W$ acts properly and cocompactly. In an article of Dymara, the weighted $L^2$-(co)homology groups of $\Sigma$ were defined, and in an article of Davis-Dymara-Januszkiewicz-Okun, the Singer conjecture for Coxeter groups was appropriately formulated for weighted $L^2$-(co)homology theory. In this article, we prove the weighted version of the Singer conjecture in dimension three under the assumption that the nerve of the Coxeter group is not dual to a hyperbolic simplex, and in dimension four under additional restrictions. We then prove a general version of the conjecture where the nerve of the Coxeter group is assumed to be a flag triangulation of a $3$-manifold.

Fourth order Hamiltonian system with some singular nonlinear term and multiplicity result

We consider a fourth order Hamiltonian system with some singular nonlinear term and multiplicity result. We get two theorems which show the number of weak solutions of this problem. The first theorem is a result which shows that there exists a weak solution for this problem and the second one is an improved result of the first result, which shows that there exist infinitely many weak solutions for this problem. We get the first result by a variational method and critical point theory, and we get the second result by homology theory.