Homology Theory Research Articles

The effects and underlying mechanisms of medicine and food homologous flowers on the prevention and treatment of related diseases.

The theory of medicine and food homology has a long history in China. Numerous traditional Chinese medicinal could be used as both medicine and food. Many flower medicinal materials also belong to the homology of medicine and food, such as Chrysanthemum morifolium, Lonicera japonica, Crocus sativus, and Lonicera macranthoides. They mainly contain flavonoids, organic acids, terpenoids, and other active ingredients, which have a variety of medicinal values, including anti-inflammatory, anti-tumor, and antioxidant. There are many formulations and functional foods containing these plants in Chinese medicine, which have a variety of nutritional and health effects on the human body. In this review, 10 widely used flowers were selected to review their pharmacological activities, prevention and treatment of related diseases and underlying mechanisms, and discussed the current limitations and future development prospects, hoping to provide references for the research on the development and utilization of natural medical flowers. PRACTICAL APPLICATIONS: The "homology of medicine and food" flowers have a wide range of uses and are of great research value. In this paper, we introduce 10 "homology of medicine and food" flowers. Their active ingredients, pharmacological activities, and treatments for related diseases are reviewed, and the limitations and development prospects of the "homology of medicine and food" flowers are discussed. It is hoped that this will contribute to the development of the food and pharmacological fields.

H0 Tensions in Cosmology and Axion Pseudocycles in the Stringy Universe

The tension between early and late H0 is revised in the context of axion dark matter arising naturally from string theoretical integrations of antisymmetric tensor fields over non-trivial cycles. Certain early universe cycles may appear non-trivial from the perspective of a homology analysis focused on the early universe, while they may become trivial when analysed from the perspective of a homology theory reaching out to lower energies and later times. Such phenomena can introduce variations in the axion potential that would explain the observed H0 tension. The decay of such pseudo-axions when the pseudo-cycles dissipate trigger axion-two-photon (otherwise having an extremely long lifetime) and axion-gravitational processes mediated by Chern–Simons couplings with observable electromagnetic or gravitational wave signals originating in the early universe.

A Lower Bound of the Number of Rewrite Rules Obtained by Homological Methods

It is well-known that some equational theories such as groups or boolean algebras can be defined by fewer equational axioms than the original axioms. However, it is not easy to determine if a given set of axioms is the smallest or not. Malbos and Mimram investigated a general method to find a lower bound of the cardinality of the set of equational axioms (or rewrite rules) that is equivalent to a given equational theory (or term rewriting systems), using homological algebra. Their method is an analog of Squier's homology theory on string rewriting systems. In this paper, we develop the homology theory for term rewriting systems more and provide a better lower bound under a stronger notion of equivalence than their equivalence. The author also implemented a program to compute the lower bounds, and experimented with 64 complete TRSs.

A Shadow Perspective on Equivariant Hochschild Homologies

Abstract Shadows for bicategories, defined by Ponto, provide a useful framework that generalizes classical and topological Hochschild homology. In this paper, we define Hochschild-type invariants for monoids in a symmetric monoidal, simplicial model category $\mathsf V$, as well as for small $\mathsf V$-categories. We show that each of these constructions extends to a shadow on an appropriate bicategory, which implies in particular that they are Morita invariant. We also define a generalized theory of Hochschild homology twisted by an automorphism and show that it is Morita invariant. Hochschild homology of Green functors and $C_n$-twisted topological Hochschild homology fit into this framework, which allows us to conclude that these theories are Morita invariant. We also study linearization maps relating the topological and algebraic theories, proving that the linearization map for topological Hochschild homology arises as a lax shadow functor, and constructing a new linearization map relating topological restriction homology and algebraic restriction homology. Finally, we construct a twisted Dennis trace map from the fixed points of equivariant algebraic $K$-theory to twisted topological Hochschild homology.

Henning Krause: “Homological Theory of Representations”

Henning Krause: “Homological Theory of Representations”

Corrosion characterization of steel wires based on persistent homology theory for magnetostrictive guided wave testing signal

Corrosion of steel wires in cables is a frequent occurrence and is a major factor affecting the durability and safety of cables. Magnetostrictive guided waves have been employed to detect steel wire corrosion. However, due to the complexity of the geometry of the corrosion area, it is difficult to interpret the magnetostrictive guided wave testing signals. Hence, extracting features from testing signals to characterize the corrosion condition is a great challenge. This study proposes a topological feature for corrosion characterization of steel wire, which is extracted from the two-dimensional point cloud with optimal geometry structure of the testing signal based on persistent homology theory. Corrosion experiments were carried out on 30 galvanized steel wires. The experimental results indicate that the topological feature increases roughly linearly with the corrosion depth. Furthermore, compared to other features obtained from the time domain, frequency domain, and joint time–frequency domain, this topological feature can reflect the progression of corrosion more accurately.

On measurings of algebras over operads and homology theories

The notion of a coalgebra measuring, introduced by Sweedler, is a kind of generalized ring map between algebras. We begin by studying maps on Hochschild homology induced by coalgebra measurings. We then introduce a notion of coalgebra measuring between Lie algebras and use it to obtain maps on Lie algebra homology. Further, these measurings between Lie algebras satisfy nice adjoint like properties with respect to universal enveloping algebras. More generally, we introduce and undertake a detailed study of the notion of coalgebra measuring between algebras over any operad $\mathcal O$. In case $\mathcal O$ is a binary and quadratic operad, we show that a measuring of $\mathcal O$-algebras leads to maps on operadic homology. In general, for any operad $\mathcal O$, we construct universal measuring coalgebras to show that the category of $\mathcal O$-algebras is enriched over coalgebras. We develop measuring comodules and universal measuring comodules for this theory. We also relate these to measurings of the universal enveloping algebra $U_{\mathcal O}(\mathscr A)$ of an $\mathcal O$-algebra $\mathscr A$ and the modules over it. Finally, we construct the Sweedler product $C\rhd \mathscr A$ of a coalgebra $C$ and an $\mathcal O$-algebra $\mathscr A$. The object $C\rhd \mathscr A$ is universal among $\mathcal O$-algebras that arise as targets of $C$-measurings starting from $\mathscr A$.

Homology of étale groupoids a graded approach

We introduce a graded homology theory for graded étale groupoids. We prove that the graded homology of a strongly graded ample groupoid is isomorphic to the homology of its ε-th component with ε the identity of the grade group. For Z-graded groupoids, we establish an exact sequence relating the graded zeroth-homology to non-graded one. Specialising to the arbitrary graph groupoids, we prove that the graded zeroth homology group with constant coefficients Z is isomorphic to the graded Grothendieck group of the associated Leavitt path algebra. To do this, we consider the diagonal algebra of the Leavitt path algebra of the covering graph of the original graph and construct the group isomorphism directly. Considering the trivial grading, our result extends Matui's on zeroth homology of finite graphs with no sinks (shifts of finite type) to all arbitrary graphs. We use our results to show that graded zeroth-homology group is a complete invariant for eventual conjugacy of shift of finite types and could be the unifying invariant for the analytic and the algebraic graph algebras.

Templex: A bridge between homologies and templates for chaotic attractors.

The theory of homologies introduces cell complexes to provide an algebraic description of spaces up to topological equivalence. Attractors in state space can be studied using Branched Manifold Analysis through Homologies: this strategy constructs a cell complex from a cloud of points in state space and uses homology groups to characterize its topology. The approach, however, does not consider the action of the flow on the cell complex. The procedure is here extended to take this fundamental property into account, as done with templates. The goal is achieved endowing the cell complex with a directed graph that prescribes the flow direction between its highest-dimensional cells. The tandem of cell complex and directed graph, baptized templex, is shown to allow for a sophisticated characterization of chaotic attractors and for an accurate classification of them. The cases of a few well-known chaotic attractors are investigated-namely, the spiral and funnel Rössler attractors, the Lorenz attractor, the Burke and Shaw attractor, and a four-dimensional system. A link is established with their description in terms of templates.

Vertex F-algebra structures on the complex oriented homology of H-spaces

We give a topological construction of graded vertex F -algebras by generalizing Joyce's vertex algebra construction to complex-oriented homology. Given an H-space X with a B U ( 1 ) -action, a choice of K-theory class, and a complex oriented homology theory E , we build a graded vertex F -algebra structure on E ⁎ ( X ) where F is the formal group law associated with E .

The geometric realization of a normalized set-theoretic Yang–Baxter homology of biquandles

Biracks and biquandles, which are useful for studying the knot theory, are special families of solutions of the set-theoretic Yang–Baxter equation. A homology theory for the set-theoretic Yang–Baxter equation was developed by Carter et al. in order to construct knot invariants. In this paper, we construct a normalized (co)homology theory of a set-theoretic solution of the Yang–Baxter equation. We obtain some concrete examples of nontrivial [Formula: see text]-cocycles for Alexander biquandles. For a biquandle [Formula: see text] its geometric realization [Formula: see text] is discussed, which has the potential to build invariants of links and knotted surfaces. In particular, we demonstrate that the second homotopy group of [Formula: see text] is finitely generated if the biquandle [Formula: see text] is finite.

Shifting chain maps in quandle homology and cocycle invariants

Quandle homology theory has been developed and cocycles have been used to define invariants of oriented classical or surface links. We introduce a shifting chain map σ \sigma on each quandle chain complex that lowers the dimensions by one. By using its pull-back σ ♯ \sigma ^\sharp , each 2 2 -cocycle ϕ \phi gives us the 3 3 -cocycle σ ♯ ϕ \sigma ^\sharp \phi . For oriented classical links in the 3 3 -space, we explore relation between their quandle 2 2 -cocycle invariants associated with ϕ \phi and their shadow 3 3 -cocycle invariants associated with σ ♯ ϕ \sigma ^\sharp \phi . For oriented surface links in the 4 4 -space, we explore how powerful their quandle 3 3 -cocycle invariants associated with σ ♯ ϕ \sigma ^\sharp \phi are. Algebraic behavior of the shifting maps for low-dimensional (co)homology groups is also discussed.

Equivariant Coarse (Co-)Homology Theories

We present an Eilenberg-Steenrod-like axiomatic framework for equivariant coarse homology and cohomology theories. We also discuss a general construction of such coarse theories from topological ones and the associated transgression maps. A large part of this paper is devoted to showing how some well-established coarse (co-)homology theories, whose equivariant versions are either already known or will be introduced in this paper, fit into this setup. Furthermore, a new and more flexible notion of coarse homotopy is given which is more in the spirit of topological homotopies. Some, but not all, coarse (co-)homology theories are even invariant under these new homotopies. They also led us to a meaningful concept of topological actions of locally compact groups on coarse spaces.

Enhanced bivariant homology theory attached to six functor formalism

Bivariant theory is a unified framework for cohomology and Borel–Moore homology theories. In this paper, we extract an ∞ $\infty$ -enhanced bivariant homology theory from Gaitsgory–Rozenblyum's six functor formalism.

4-Manifold topology, Donaldson–Witten theory, Floer homology and higher gauge theory methods in the BV-BFV formalism

We study the behavior of Donaldson’s invariants of 4-manifolds based on the moduli space of anti-self-dual connections (instantons) in the perturbative field theory setting where the underlying source manifold has boundary. It is well-known that these invariants take values in the instanton Floer homology groups of the boundary 3-manifold. Gluing formulae for these constructions lead to a functorial topological field theory description according to a system of axioms developed by Atiyah, which can be also regarded in the setting of perturbative quantum field theory, as it was shown by Witten, using a version of supersymmetric Yang–Mills theory, known today as Donaldson–Witten theory. One can actually formulate an AKSZ model which recovers this theory for a certain gauge-fixing. We consider these constructions in a perturbative quantum gauge formalism for manifolds with boundary that is compatible with cutting and gluing, called the BV-BFV formalism, which was recently developed by Cattaneo, Mnev and Reshetikhin. We prove that this theory satisfies a modified Quantum Master Equation and extend the result to a global picture when perturbing around constant background fields. These methods are expected to extend to higher codimensions and thus might help getting a better understanding for fully extendable [Formula: see text]-dimensional field theories (in the sense of Baez–Dolan and Lurie) in the perturbative setting, especially when [Formula: see text]. Additionally, we relate these constructions to Nekrasov’s partition function by treating an equivariant version of Donaldson–Witten theory in the BV formalism. Moreover, we discuss the extension, as well as the relation, to higher gauge theory and enumerative geometry methods, such as Gromov–Witten and Donaldson–Thomas theory and recall their correspondence conjecture for general Calabi–Yau 3-folds. In particular, we discuss the corresponding (relative) partition functions, defined as the generating function for the given invariants, and gluing phenomena.

Oriented Borel–Moore homologies of toric varieties

We generalize the well known Künneth formula for Chow groups to an arbitrary oriented Borel–Moore homology theory satisfying localization and descent (e.g. algebraic bordism) when taking a product with a toric variety. As a corollary we obtain a universal coefficient theorem for the operational cohomology rings. We also give a new, homological, description for the homology groups of smooth toric varieties, which allows us to compute the algebraic bordism groups of some singular toric varieties.

Research practice and strategy of resource utilization of residues for edible medicinal plants under background of Big Health

In recent years, with the acceleration of population aging in China, the frequent occurrence of chronic diseases, and the increase in the number of sub-health groups, people began to seek ways to maintain health in line with the laws of nature. Healthy China has become a national strategy, and the Big Health industry has ushered in a golden period of development. In this context, the concept of medicine and food homology and edible medicinal substances which have guided Chinese people to pursue health since ancient times have ushered in significant and favorable development opportunities, and the industrial scale has gradually expanded. The development and utilization of edible medicinal plants have also become an important research direction. In the industrialization process of modern health care and edible medicinal products, a large number of drug residues are often abandoned due to ineffective utilization, resulting in a huge waste of resources and increasing the pressure on the ecological environment. Under the guidance of the homology theory of medicine and food, based on the recycling strategy of Chinese medicine resources, this paper analyzed the inherent common characteristics such as physical and chemical properties and biological activity of Chinese medicine residues of edible medicinal plants and put forward some suitable resource utilization strategies and suggestions in combination with the current situation of resource development and utilization of Chinese medicine residues of edible medicinal plants, in order to promote the high-value development and utilization of medicinal and edible resources, and extend the industrial chain of edible medicinal resources, thereby empowering Big Health industry and facilitating Healthy China.

The diagrammatic coaction and cuts of the double box

The diagrammatic coaction encodes the analytic structure of Feynman integrals by mapping any given Feynman diagram into a tensor product of diagrams defined by contractions and cuts of the original diagram. Feynman integrals evaluate to generalized hypergeometric functions in dimensional regularization. Establishing the coaction on this type of functions has helped formulating and checking the diagrammatic coaction of certain two-loop integrals. In this talk we study its application on the fully-massless double-box diagram. We make use of differential equation techniques, which, together with the properties of homology and cohomology theory of the resulting hypergeometric functions, allow us to formulate the coaction on a range of cuts of the double box in closed form.

Some Equivalence Relation between Persistent Homology and Morphological Dynamics

In mathematical morphology, connected filters based on dynamics are used to filter the extrema of an image. Similarly, persistence is a concept coming from persistent homology and Morse theory that represents the stability of the extrema of a Morse function. Since these two concepts seem to be closely related, in this paper we examine their relationship, and we prove that they are equal on n-D Morse functions, \(n\ge 1\). More exactly, pairing a minimum with a 1-saddle by dynamics or pairing the same 1-saddle with a minimum by persistence leads exactly to the same pairing, assuming that the critical values of the studied Morse function are unique. This result is a step further to show how much topological data analysis and mathematical morphology are related, paving the way for a more in-depth study of the relations between these two research fields.

On axiomatic characterization of Alexander-Spanier normal homology theory of general topological spaces

The Alexandroff-Čech normal cohomology theory [27], [3], [1], [2] is the unique continuous extension [36] of the additive cohomology theory [26], [4] from the category of polyhedral pairs KPol2 to the category of closed normally embedded, the so called, P-pairs of general topological spaces KTop2. In this paper we define the Alexander-Spanier normal cohomology theory based on all normal coverings and show that it is isomorphic to the Alexandroff-Čech normal cohomology. Using this fact and methods developed in [6] we construct an exact, the so called, Alexander-Spanier normal homology theory on the category KTop2, which is isomorphic to the Steenrod homology theory on the subcategory of compact pairs KC2. Moreover, we give an axiomatic characterization of the constructed homology theory.