Homology Groups Research Articles

The homology groups $H_{n+1} \left( \mathbb{C}\Omega_n \right)$

The topic of the paper is the investigation of the homology groups of the $(2n+1)$-dimensional CW-complex $\mathbb{C}\Omega_n$. The spaces $\mathbb{C}\Omega_n$ consist of complex-valued functions and are the analogue of the spaces $\Omega_n$, widely known in the approximation theory. The spaces $\mathbb{C}\Omega_n$ have been introduced in 2015 by A.M. Pasko who has built the CW-structure of the spaces $\mathbb{C}\Omega_n$ and using this CW-structure established that the spaces $\mathbb{C}\Omega_n$ are simply connected. Note that the mentioned CW-structure of the spaces $\mathbb{C}\Omega_n$ is the analogue of the CW-structure of the spaces $\Omega_n$ constructed by V.I. Ruban. Further A.M. Pasko found the homology groups of the space $\mathbb{C}\Omega_n$ in the dimensionalities $0, 1, \ldots, n, 2n-1, 2n, 2n+1$. The goal of the present paper is to find the homology group $H_{n+1}\left ( \mathbb{C}\Omega_n \right )$. It is proved that $H_{n+1} \left ( \mathbb{C}\Omega_n \right )=\mathbb{Z}^\frac{n+1}{2}$ if $n$ is odd and $H_{n+1} \left ( \mathbb{C}\Omega_n \right )=\mathbb{Z}^\frac{n+2}{2}$ if $n$ is even.

Torsions in cohomology of [formula omitted] and congruence of modular forms

We describe torsion classes in the first cohomology group of SL2(Z). In particular, we obtain generalized Dickson's invariants for p-power polynomial rings. Secondly, we describe torsion classes in the zero-th homology group of SL2(Z) as a module over the torsion invariants. As application, we obtain various congruences between cuspidal forms of level one and torsion Eisenstein classes.

Short- and medium-chain chlorinated paraffins in urban road dust of Shanghai, China: concentrations, source apportionment and human exposure assessment.

Chlorinated paraffins (CPs) are ubiquitous anthropogenic contaminants that have been found in various environmental media. The objective of this study was to determine concentrations, spatial distribution, possible sources and potential health risk of SCCPs and MCCPs in urban road dust collected from Shanghai, China. The concentrations ranged from 9.74 to 11,400ngg-1 for ΣSCCPs, 44.1 to 49,900ngg-1 for ΣMCCPs and 53.9 to 61,400ngg-1 for total CPs, respectively. MCCPs were the dominant component in all road dust, averagely accounting for 82.8% of total CPs. The concentrations of CPs in dust collected from traffic and commercial areas were significantly higher than those from campus, industrial, park and residential areas (p < 0.01), which could be attributed to tire wear in heavy traffic. All dust samples were divided into two groups by hierarchical cluster analysis for both SCCPs and MCCPs, and the most abundant homologue groups in most samples were C10Cl7-10 and C13Cl7-9 for SCCPs, and C14Cl7-9 and C15Cl8-9 for MCCPs. Correlation analysis showed that all carbon homologues in road dusts were highly correlated each other, suggesting SCCPs and MCCPs in dust maybe came from similar sources. Three sources for CPs in dust samples were apportioned by the PMF model; their relative contributions to the total CPs burden in dust were 25.6% for factor 1 (commercial CP mixture), 13.7% for factor 2 (long-distance transport) and 60.7% for factor 3 (commercial CP mixture). The median estimated daily intakes of total CPs via road dust were 1.78 × 10-5 for children and 3.0 × 10-6mgkg-1day-1 for adults, respectively. Quantitative risk assessment using non-cancer hazard index and total margin of exposure of total CPs indicated that total CPs at the present level in road dust pose no significant risk for both children and adults in Shanghai.

Almost finiteness and homology of certain non-free actions

We show that Cantor minimal \mathbb{Z}\rtimes\mathbb{Z}_2 -systems and essentially free amenable odometers are almost finite. We also compute the homology groups of Cantor minimal \mathbb{Z}\rtimes\mathbb{Z}_2 -systems and show that the associated transformation groupoids satisfy the HK conjecture if and only if the action is free.

On the symmetric group action on rigid disks in a strip

In this paper we decompose the rational homology of the ordered configuration space of n open unit-diameter disks in the infinite strip of width 2 as a direct sum of induced \(S_{n}\)-representations. In (Alpert in Generalized representation stability for disks in a strip and no-k-equal spaces, arXiv:2006.01240, 2020), Alpert proves that the \(k^{\text {th}}\)-integral homology of the ordered configuration space of n open unit-diameter disks in the infinite strip of width 2 is an FI\(_{k+1}\)-module by studying certain operations on homology called “high-insertion maps.” The integral homology groups \(H_{k}\big (\text {Conf}(n,2)\big )\) are free abelian, and Alpert computes a basis for \(H_{k}\big (\text {Conf}(n,2)\big )\) as an abelian group. In this paper, we study the rational homology groups as \(S_{n}\)-representations. We find a new basis for \(H_{1}\big (\text {Conf}(n,2);\mathbb {Q}\big ),\) and use this, along with results of Ramos (Proc Am Math Soc 145(11):4647–4660, 2017), to give an explicit description of \(H_{k}\big (\text {Conf}(n,2);\mathbb {Q}\big )\) as a direct sum of induced \(S_{n}\)-representations arising from free FI\(_{d}\)-modules. We use this decomposition to calculate the rank of the rational homology of the unordered configuration space of n open unit-diameter disks in the infinite strip of width 2.

Identification of MYB Transcription Factors Involving in Fruit Quality Regulation of Fragaria × ananassa Duch.

The cultivated strawberry (Fragaria × ananassa Duch.) is an important horticultural crop. The economic values of strawberry cultivars are decided by their fruit qualities including taste, color and aroma. The important role of MYB transcription factors in fruit quality regulation is recognized increasingly with the identification of MYB genes involved in metabolism. A total of 407 MYB genes of F. × ananassa (FaMYBs) were identified in the genome-wide scale and named according to subgenome locations. The 407 FaMYBs were clustered into 36 groups based on phylogenetic analysis. According to synteny analysis, whole genome duplication and segmental duplication contributed over 90% of the expansion of the FaMYBs family. A total of 101 FaMYB loci with 1-6 alleles were identified by the homologous gene groups on homologous chromosomes. The differentially expressed FaMYB profiles of three cultivars with different fruit quality and fruit ripe processes provided the 8 candidate loci involved in fruit quality regulation. In this experiment, 7, 5, and 4 FaMYBs were screeded as candidate genes involved in the regulation of metabolism/transportation of anthocyanins, sugars or organic acids and 4-hydroxy-2, 5-dimethyl-3(2H)-furanone, respectively. These results pointed out the key FaMYBs for further functional analysis of gene regulation of strawberry fruit quality and would be helpful in the clarification on ofe roles of MYBs in the metabolism of fruit crops.

The Free Factor Complex and the Dualizing Module for the Automorphism Group of a Free Group

Abstract Answering a question of Hatcher and Vogtmann, we prove that the top homology group of the free factor complex is not the dualizing module for $\operatorname {Aut}(F_{n})$, at least for $n = 5$.

Comparison between the homologous BNT162b2 and the heterologous Gam-COVID-Vac/BNT162b2 vaccine regimen in republic of North Macedonia

The medical and socio-economic consequences that stemmed from the COVID-19 pandemic, forced the healthcare policymakers in Republic of North Macedonia to rely on five different vaccines against the SARS-CoV-2 virus, in order to reach a satisfactory level of herd immunity. It is here where we got the idea to compare the heterologous Gam-COVID-Vac/BNT162b2 regimen to the homologous BNT162b2 regimen, with our main focus being the immunogenicity differences between the two of them. Additionally, we researched the variation in humoral immune response relative to age strata; the reactogenicity differences; and discrepancies in SARS-CoV-2 infection incidence between the two regimens. To achieve this, antibody titers in sera samples from fifty-three (53) healthcare workers, divided in heterologous and homologous group, were analysed at six different time checkpoints. Our results showed robust immunogenic response after the administration of the booster dose (4. 2-fold increase in antibody titers), followed by a slower-waning humoral immune protection in the heterologous regimen, compared to the homologous BNT162b2 schedule, furthermore confirmed by non-inferiority testing (Geometric Mean Ratio=0,98) at the final checkpoint. That, coupled with the similar reactogenicity (p=0,767) of both regimens, imply that the Gam-COVID-Vac/BNT162b2 combination might be a feasible approach in the effort to contain the COVID-19 pandemic.

On a Runge theorem over $${\mathbb {R}}_3$$

In this paper, we investigate a topological characterization of the Runge theorem in the Clifford algebra \( {\mathbb {R}}_3\) via the description of the homology groups of axially symmetric open subsets of the quadratic cone in \({\mathbb {R}}_3\).

Homology groups of embedded fractional Brownian motion.

A well-known class of nonstationary self-similar time series is the fractional Brownian motion (fBm) considered to model ubiquitous stochastic processes in nature. Due to noise and trends superimposed on data and even sample size and irregularity impacts, the well-known computational algorithm to compute the Hurst exponent (H) has encountered superior results. Motivated by this discrepancy, we examine the homology groups of high-dimensional point cloud data (PCD), a subset of the unit D-dimensional cube, constructed from synthetic fBm data as a pipeline to compute the H exponent. We compute topological measures for embedded PCD as a function of the associated Hurst exponent for different embedding dimensions, time delays, and amount of irregularity existing in the dataset in various scales. Our results show that for a regular synthetic fBm, the higher value of the embedding dimension leads to increasing the H dependency of topological measures based on zeroth and first homology groups. To achieve a reliable classification of fBm, we should consider the small value of time delay irrespective of the irregularity presented in the data. More interestingly, the value of the scale for which the PCD to be path connected and the postloopless regime scale are more robust concerning irregularity for distinguishing the fBm signal. Such robustness becomes less for the higher value of the embedding dimension. Finally, the associated Hurst exponents for our topological feature vector for the S&P500 were computed, and the results are consistent with the detrended fluctuation analysis method.

Immunogenicity, Immune Dynamics, and Subsequent Response to the Booster Dose of Heterologous versus Homologous Prime-Boost Regimens with Adenoviral Vector and mRNA SARS-CoV-2 Vaccine among Liver Transplant Recipients: A Prospective Study.

Background: Heterologous prime-boost vaccination potentially augments the immune response against SARS-CoV-2 in liver transplant (LT) recipients. We investigated immunogenicity induced by different primary prime-boost vaccination protocols and the subsequent response to the booster vaccine among LT recipients. Methods: LT recipients, who received primary immunisation with ChAdOx1/ChAdOx1 or ChAdOx1/BNT162b2, were administered the third dose of mRNA-1273 three months following the primary vaccination. Blood samples were collected before and after primary vaccination and post-booster. The levels of receptor binding domain antibody (anti-RBD) and neutralising antibody (sVNT) and spike-specific T-cell responses were assessed. Results: Among the 89 LT recipients, patients receiving ChAdOx1/BNT162b2 had significantly higher anti-RBD titres, sVNT, and cellular response after primary vaccination than those receiving ChAdOx1/ChAdOx1 (p < 0.05). The antibody response decreased 12 weeks after the primary vaccination. After the booster, humoral and cellular responses significantly improved, with comparable seroconversion rates between the heterologous and homologous groups. Positive sVNT against the wild type occurred in >90% of LT patients, with only 12.3% positive against the Omicron variant. Conclusions: ChAdOx1/BNT162b2 evoked a significantly higher immunological response than ChAdOx1/ChAdOx1 in LT recipients. The booster strategy substantially induced robust immunity against wild type in most patients but was less effective against the Omicron strain.

The embedded homology of hypergraph pairs

In this paper, we generalize the embedded homology groups of hypergraphs initially given in [S. Bressan, J. Li, S. Ren and J. Wu, The embedded homology of hypergraphs and applications, Asian J. Math. 23(3) (2019) 479–500] and study the relative embedded homology groups of hypergraph pairs. We prove some long exact sequences as well as a Mayer–Vietoris sequence for the relative embedded homology groups of hypergraph pairs. Moreover, we briefly discuss the two-dimensional persistence for the relative embedded homology groups of hypergraph pairs.

Lagrangian traces for the Johnson filtration of the handlebody group

In this paper, we define trace-like operators on a subspace of the space of derivations of the free Lie algebra generated by the first homology group H of a surface Σ. This definition depends on the choice of a Lagrangian of H, and we call these operators the Lagrangian traces. We suppose that Σ is the boundary of a handlebody with first homology group H′, and we show that, in degree greater than 2, the Lagrangian traces corresponding to the Lagrangian Ker(H→H′) vanish on the image by the Johnson homomorphisms of the elements of the Johnson filtration that extend to the handlebody.

An Efficient Algorithm for 1-Dimensional (Persistent) Path Homology

This paper focuses on developing an efficient algorithm for analyzing a directed network (graph) from a topological viewpoint. A prevalent technique for such topological analysis involves computation of homology groups and their persistence. These concepts are well suited for spaces that are not directed. As a result, one needs a concept of homology that accommodates orientations in input space. Path-homology developed for directed graphs by Grigoryan et al. (2012) has been effectively adapted for this purpose recently by Chowdhury and Mémoli (2018). They also give an algorithm to compute the path-homology and the persistent path-homology. Our main contribution in this paper is an algorithm that computes this path-homology and its persistence more efficiently for the 1-dimensional (\(H_1\)) case. In developing such an algorithm, we discover various structures and their efficient computations that aid computing the 1-dimensional path-homology. We implement our algorithm and present some preliminary experimental results.

MM-CCNB: Essential protein prediction using MAX-MIN strategies and compartment of common neighboring approach

Background and objectiveProteins are indispensable for the flow of the life of living organisms. Protein pairs in interaction exhibit more functional activities than individuals. These activities have been considered an essential measure in predicting their essentiality. Neighborhood approaches have been used frequently in the prediction of essentiality scores. All paired neighbors of the essential proteins are nominated for the suitable candidate seeds for prediction. Still now Jaccard's coefficient is limited to predicting functions, homologous groups, sequence analysis, etc. It really motivate us to predict essential proteins efficiently using different computational approaches. MethodsIn our work, we proposed modified Jaccard's coefficient to predict essential proteins. We have proposed a novel methodology for predicting essential proteins using MAX-MIN strategies and modified Jaccard's coefficient approach. ResultsThe performance of our proposed methodology has been analyzed for Saccharomyces cerevisiae datasets with an accuracy of more than 80%. It has been observed that the proposed algorithm is outperforms with an accuracy of 0.78, 0.74, 0.79, and 0.862 for YDIP, YMIPS, YHQ, and YMBD datasets respectivly. ConclusionsThere are several computational approaches in the existing state-of-art model of essential protein prediction. It has been noted that our predicted methodology outperforms other existing models viz. different centralities, local interaction density combined with protein complexes, modified monkey algorithm and ortho_sim_loc methods.

Haupt’s theorem for strata of abelian differentials

Let S be a closed topological surface. Haupt’s theorem provides necessary and sufficient conditions for a complex-valued character of the first integer homology group of S to be realized by integration against a complex-valued 1-form that is holomorphic with respect to some complex structure on S. We prove a refinement of this theorem that takes into account the divisor data of the 1-form.

Haplotype-resolved Genome of Sika Deer Reveals Allele-specific Gene Expression and Chromosome Evolution

Despite the scientific and medicinal importance of diploid sika deer (Cervus nippon), its genome resources are limited and haplotype-resolved chromosome-scale assembly is urgently needed. To explore mechanisms underlying the expression patterns of the allele-specific genes in antlers and the chromosome evolution in Cervidae, we report, for the first time, a high-quality haplotype-resolved chromosome-scale genome of sika deer by integrating multiple sequencing strategies, which was anchored to 32 homologous groups with a pair of sex chromosomes (XY). Several expanded genes (RET, PPP2R1A, PPP2R1B, YWHAB, YWHAZ, and RPS6) and positively selected genes (eIF4E, Wnt8A, Wnt9B, BMP4, and TP53) were identified, which could contribute to rapid antler growth without carcinogenesis. A comprehensive and systematic genome-wide analysis of allele expression patterns revealed that most alleles were functionally equivalent in regulating rapid antler growth and inhibiting oncogenesis. Comparative genomic analysis revealed that chromosome fission might occur during the divergence of sika deer and red deer (Cervus elaphus), and the olfactory sensation of sika deer might be more powerful than that of red deer. Obvious inversion regions containing olfactory receptor genes were also identified, which arose since the divergence. In conclusion, the high-quality allele-aware reference genome provides valuable resources for further illustration of the unique biological characteristics of antler, chromosome evolution, and multi-omics research of cervid animals.

Group-contribution SAFT equations of state: A review

We present a review and thorough assessment of recent advances and applications of group-contribution methods (GCMs) implemented in the Statistical Associating Fluid Theory (SAFT) equations of state (EoSs). Modern SAFT EoSs are some of the most advanced and effective thermodynamic models for industrial applications, because they are predictive and have a strong molecular foundation. GCMs are simple and useful approaches that can enhance the predictive ability and applicability of EoSs, for successful evaluation of thermodynamic properties of complex systems. A major advantage of using a GCM is the transferability of functional group parameters, i.e., a single set of parameters can be used for a functional group found in more than one homologous group. Because such a GCM requires less effort in system-specific model fitting and regression, it has also been incorporated into the SAFT-framework, yielding various incarnations of GCM SAFT EoSs. These versatile EoSs have been applied by many researchers to a wide variety of fluids and mixtures, including electrolytes, ionic liquids, polymers, petroleum fluids, biofuels, solvents for CO2 capture, and computer-aided molecular design.

Disoriented homology and double branched covers

AbstractThis paper provides a convenient and practical method to compute the homology and intersection pairing of a branched double cover of the 4-ball.To projections of links in the 3-ball, and to projections of surfaces in the 4-ball into the boundary sphere, we associate a sequence of homology groups, called the disoriented homology. We show that the disoriented homology is isomorphic to the homology of the double branched cover of the link or surface. We define a pairing on the first disoriented homology group of a surface and show that this is equal to the intersection pairing of the branched cover. These results generalize work of Gordon and Litherland, for embedded surfaces in the 3-sphere, to arbitrary surfaces in the 4-ball. We also give a generalization of the signature formula of Gordon–Litherland to the general setting.Our results are underpinned by a theorem describing a handle decomposition of the branched double cover of a codimension-2 submanifold in the n-ball, which generalizes previous results of Akbulut–Kirby and others.

The Complex World of Emaraviruses—Challenges, Insights, and Prospects

Emaravirus (Order Bunyavirales; Family Fimoviridae) is a genus comprising over 20 emerging plant viruses with a worldwide distribution and economic impact. Emaraviruses infect a variety of host plants and have especially become prevalent in important long-living woody plants. These viruses are enveloped, with a segmented, single-stranded, negative-sense RNA genome and are transmitted by eriophyid mites or mechanical transmission. Emaraviruses have four core genome segments encoding an RNA-dependent RNA polymerase, a glycoprotein precursor, a nucleocapsid protein, and a movement protein. They also have additional genome segments, whose number varies widely. We report here that the proteins encoded by these segments form three main homology groups: a homolog of the sadwavirus Glu2 Pro glutamic protease; a protein involved in pathogenicity, which we named “ABC”; and a protein of unknown function, which we named “P55”. The distribution of these proteins parallels the emaravirus phylogeny and suggests, with other analyses, that emaraviruses should be split into at least two genera. Reliable diagnosis systems are urgently needed to detect emaraviruses, assess their economic and ecological importance, and take appropriate measures to prevent their spread (such as routine testing, hygiene measures, and control of mite vectors). Additional research needs include understanding the function of emaravirus proteins, breeding resistant plants, and clarifying transmission modes.