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Automated annotation of protein function has become a critical task in the post-genomic era. Network-based approaches and homology-based approaches have been widely used and recently tested in large-scale community-wide assessment experiments. It is natural to integrate network data with homology information to further improve the predictive performance. However, integrating these two heterogeneous, high-dimensional and noisy datasets is non-trivial. In this work, we introduce a novel protein function prediction algorithm ProSNet. An integrated heterogeneous network is first built to include molecular networks of multiple species and link together homologous proteins across multiple species. Based on this integrated network, a dimensionality reduction algorithm is introduced to obtain compact low-dimensional vectors to encode proteins in the network. Finally, we develop machine learning classification algorithms that take the vectors as input and make predictions by transferring annotations both within each species and across different species. Extensive experiments on five major species demonstrate that our integration of homology with molecular networks substantially improves the predictive performance over existing approaches.

There is growing use of ontologies for the measurement of cross-species phenotype similarity. Such similarity measurements contribute to diverse applications, such as identifying genetic models for human diseases, transferring knowledge among model organisms, and studying the genetic basis of evolutionary innovations. Two organismal features, whether genes, anatomical parts, or any other inherited feature, are considered to be homologous when they are evolutionarily derived from a single feature in a common ancestor. A classic example is the homology between the paired fins of fishes and vertebrate limbs. Anatomical ontologies that model the structural relations among parts may fail to include some known anatomical homologies unless they are deliberately added as separate axioms. The consequences of neglecting known homologies for applications that rely on such ontologies has not been well studied. Here, we examine how semantic similarity is affected when external homology knowledge is included. We measure phenotypic similarity between orthologous and non-orthologous gene pairs between humans and either mouse or zebrafish, and compare the inclusion of real with faux homology axioms. Semantic similarity was preferentially increased for orthologs when using real homology axioms, but only in the more divergent of the two species comparisons (human to zebrafish, not human to mouse), and the relative increase was less than 1% to non-orthologs. By contrast, inclusion of both real and faux random homology axioms preferentially increased similarities between genes that were initially more dissimilar in the other comparisons. Biologically meaningful increases in semantic similarity were seen for a select subset of gene pairs. Overall, the effect of including homology axioms on cross-species semantic similarity was modest at the levels of divergence examined here, but our results hint that it may be greater for more distant species comparisons.

Ng constructed an invariant of knots in ${ℝ}^3$, a combinatorial knot contact homology. Extending his study, we construct an invariant of surface-knots in ${ℝ}^4$ using diagrams in ${ℝ}^3$.

Feller et al. showed the homology and the intersection form of a closed trisected four-manifold are described in terms of trisection diagram. In this paper, it is confirmed that we are able to calculate those of a trisected four-manifold with boundary in a similar way. Moreover, we describe a representative of the second Stiefel–Whitney class by the relative trisection diagram.

In 2004, Carter, Elhamdadi and Saito defined a homology theory for set-theoretic Yang–Baxter operators (we will call it the “algebraic” version in this paper). In 2012, Przytycki defined another homology theory for pre-Yang–Baxter operators which has a nice graphic visualization (we will call it the “graphic” version in this paper). We show that they are equivalent. The “graphic” homology is also defined for pre-Yang–Baxter operators, and we give some examples of its one-term and two-term homologies. In the two-term case, we have found torsion in homology of Yang–Baxter operator that yields the Jones polynomial.

In the past decade, there has been an extensive advancement in the creation of methods for the design and prediction of protein structures. Expeditious growth in protein structure and sequence databases has charged the development of computational approaches for the prediction of structures. This review focuses on fragment-based strategy, a computational approach for the prediction of the three-dimensional structure of proteins. Fragment assembly has immensely improved protein structure prediction accuracy, especially of the single-domain proteins at the fold level. Fragment libraries are generated using the dihedral angles along with local structural information of known protein structures. This leads to the construction of a full-length polypeptide chain of a query protein using the fragments present in these libraries. The energy function of the proteins is minimized contributing to multiple conformations considering the backbone atoms and “centroid” side-chain pseudo-atoms using conformational sampling. Lastly, Monte Carlo simulation is performed for the sampling of the side-chain rotamers and reduction of energy for more precise and refined model construction. The quality of the fragments determines whether the native-like conformations generated are accurate or not. The future direction as well as tools like ROSETTA, QUARK, FRAGFOLD, M-TASSER, and AlphaFold2 that use fragment assembly for optimal structure prediction have also been described and compared in this review.

Uniform spanning trees on finite graphs and their analogues on infinite graphs are a well-studied area. On a Cayley graph of a group, we show that they are related to the first ℓ²-Betti number of the group. Our main aim, however, is to present the basic elements of a higher-dimensional analogue on finite and infinite CW-complexes, which relate to the higher ℓ²-Betti numbers. One consequence is a uniform isoperimetric inequality extending work of Lyons, Pichot, and Vassout. We also present an enumeration similar to recent work of Duval, Klivans and Martin.

We generalize Kauffman’s famous formula defining the Jones polynomial of an oriented link in $3$-space from his bracket and the writhe of an oriented diagram [L. Kauffman, State models and the Jones polynomial, Topology26(3) (1987) 395–407]. Our generalization is an epimorphism between skein modules of tangles in compact connected oriented $3$-manifolds with markings in the boundary. Besides the usual Jones polynomial of oriented tangles we will consider graded quotients of the bracket skein module and Przytycki’s $q$-analog of the first homology group of a $3$-manifold [J. Przytycki, A $q$-analogue of the first homology group of a $3$-manifold, in Contemporary Mathematics, Vol. 214 (American Mathematical Society, 1998), pp. 135–144]. In certain cases, e.g., for links in submanifolds of rational homology $3$-spheres, we will be able to define an epimorphism from the Jones module onto the Kauffman bracket module. For the general case we define suitably graded quotients of the bracket module, which are graded by homology. The kernels define new skein modules measuring the difference between Jones and bracket skein modules. We also discuss gluing in this setting.

In this paper we study the homology and cohomology groups of the super Schrödinger algebra $S\left(1/1\right)$ in (1 + 1)-dimensional spacetime. We explicitly compute the homology groups of $S\left(1/1\right)$ with coefficients in the trivial module. Then using duality, we finally obtain the dimensions of the cohomology groups of $S\left(1/1\right)$ with coefficients in the trivial module.

We show that for every finite-volume hyperbolic $3$-manifold $M$ and every prime $p$ we have $dim{H}_1(M;F_p)<168.602\cdot \mathrm{\text{vol}}(M)$. There are slightly stronger estimates if $p=2$ or if $M$ is non-compact. This improves on a result proved by Agol, Leininger and Margalit, which gave the same inequality with a coefficient of $334.08$ in place of $168.602$. It also improves on the analogous result with a coefficient of about $260$, which could have been obtained by combining the arguments due to Agol, Leininger and Margalit with a result due to Böröczky. Our inequality involving homology rank is deduced from a result about the rank of the fundamental group: if $M$ is a finite-volume orientable hyperbolic $3$-manifold such that ${\pi }_1(M)$ is $2$-semifree, then $\mathrm{rank}{\pi }_1(M)<1+\lambda 0\cdot \mathrm{\text{vol}}M$, where $\lambda 0$ is a certain constant less than $167.79$.

In the works of Achúcarro and Townsend and also by Witten, a duality between three-dimensional Chern–Simons gauge theories and gravity was established. In all cases, the results made use of the field equations. In a previous work, we were capable of generalizing Witten’s work to the off-shell cases, as well as to the four-dimensional Yang–Mills theory with de Sitter gauge symmetry. The price we paid is that curvature and torsion must obey some constraints under the action of the interior derivative. These constraints implied on the partial breaking of diffeomorphism invariance. In this work, we first formalize our early results in terms of fiber bundle theory by establishing the formal aspects of the map between a principal bundle (gauge theory) and a coframe bundle (gravity) with partial breaking of diffeomorphism invariance. Then, we study the effect of the constraints on the homology defined by the interior derivative. The main result is the emergence of a non-trivial homology in Riemann–Cartan manifolds.

Leech’s (co)homology groups of finite cyclic monoids are computed.

Given a principal $G$-bundle $P\to M$ and two $C^1$ curves in $M$ with coinciding endpoints, we say that the two curves are holonomically equivalent if the parallel transport along them is identical for any smooth connection on $P$. The main result in this paper is that if $G$ is semi-simple, then the two curves are holonomically equivalent if and only if there is a thin, i.e. of rank at most one, $C^1$ homotopy linking them. Additionally, it is also demonstrated that this is equivalent to the factorizability through a tree of the loop formed from the two curves and to the reducibility of a certain transfinite word associated to this loop. The curves are not assumed to be regular.

An anti-associative algebra is a nonassociative algebra whose multiplication satisfies the identity $a(bc)+(ab)c=0.$ Such algebras are nilpotent. We describe the free anti-associativealgebras with a finite number of generators. Other types of nonassociative algebras, obtained either by the polarization process such as Jacobi–Jordan algebras, or obtained by deformation quantization, are associated with this class of algebras. Following Markl-Remm’s work [M. Markl and E. Remm, (Non-)Koszulness of operads for n-ary algebras, galgalim and other curiosities, J. Homotopy Relat. Struct.10 (2015) 939–969], we describe the operads associated with these algebra classes and in particular the cohomology complexes related to deformations.

Starting with a numerically noncritical (at $p$) Hecke eigenclass $f$ in the homology of a congruence subgroup $\mathrm{\Gamma }$ of ${\mathrm{\text{SL}}}_3(\mathbb{Z})$ (where $p$ divides the level of $\mathrm{\Gamma }$) with classical coefficients, we first show how to compute to any desired degree of accuracy a lift of $f$ to a Hecke eigenclass $F$ with coefficients in a module of $p$-adic distributions. Then we show how to find to any desired degree of accuracy the germ of the projection $Z$ to weight space of the eigencurve around the point $z$ corresponding to the system of Hecke eigenvalues of $F$. We do this under the conjecturally mild hypothesis that $Z$ is smooth at $z$.

This paper shows how, in principle, simplicial methods, including the well-known Dold–Kan construction can be applied to convert link homology theories into homotopy theories. The paper studies particularly the case of Khovanov homology and shows how simplicial structures are implicit in the construction of the Khovanov complex from a link diagram and how the homology of the Khovanov category, with coefficients in an appropriate Frobenius algebra, is related to Khovanov homology. This Khovanov category leads to simplicial groups satisfying the Kan condition that are relevant to a homotopy theory for Khovanov homology.

In this paper, relations between topological entropy, volume growth and the growth in topological complexity from homotopical and homological point of view are discussed for random dynamical systems. It is shown that, under certain conditions, the volume growth, the growth in fundamental group and the growth in homological group are all bounded from above by the topological entropy.

Revolutionary in scope and application, the CRISPR Cas9 endonuclease system can be guided by 20-nt single guide RNA (sgRNA) to any complementary loci on the double-stranded DNA. Once the target site is located, Cas9 can then cleave the DNA and introduce mutations. Despite the power of this system, sgRNA is highly susceptible to off-target homologous attachment and can consequently cause Cas9 to cleave DNA at off-target sites. In order to better understand this flaw in the system, the human genome and Streptococcus pyogenes Cas9 (SpCas9) were used in a mathematical and computational study to analyze the probabilities of potential sgRNA off-target homologies. It has been concluded that off-target sites are nearly unavoidable for large-size genomes, such as the human genome. Backed by mathematical analysis, a viable solution is the double-nicking method which has the promise for genome editing specificity. Also applied in this study was a computational algorithm for off-target homology search that was implemented in Java to confirm the mathematical analysis.

Quandles and their homologies are used to construct invariants of oriented links or oriented surface-links in 4-space. On the other hand the knot quandle can still be defined in the case where the links or surface-links are not oriented, but in this case it cannot be used to construct homological invariants. Here we introduce the notion of a quandle with a good involution, and its homology groups. We can use them to construct invariants of unoriented links and unoriented, or non-orientable, surface-links in 4-space.

We give a topological interpretation of the free metabelian group, following the plan described in [11, 12]. Namely, we represent the free metabelian group with d-generators as an extension of the group of the first homology of the d-dimensional lattice (as Cayley graph of the group ℤ^d), with a canonical 2-cocycle. This construction opens a possibility to study metabelian groups from new points of view; in particular to give useful normal forms of the elements of the group, leading to applications to the random walks, and so on. We also describe the satellite groups which correspond to all 2-cocycles of cohomology group associated with the free solvable groups. The homology of the Cayley graph can be used for studying the wide class of groups which include the class of all solvable groups.