MT-MAG: Accurate and interpretable machine learning based taxonomic assignment of metagenome-assembled genomes, with a partial classification option

We propose MT-MAG, a novel machine learning-based taxonomic assignment tool for hierarchically-structured local classification of metagenome-assembled genomes (MAGs). MT-MAG is capable of classifying large and diverse real metagenomic datasets, having analyzed for this study a total of 240 Gbp of data in the training set, and 7 Gbp of data in the test set. MT-MAG is, to the best of our knowledge, the first machine learning method for taxonomic assignment of metagenomic data that offers a "partial classification" option. MT-MAG outputs complete or a partial classification paths, and interpretable numerical classification confidences of its classifications, at all taxonomic ranks. MT-MAG is able to completely classify 48% more sequences than DeepMicrobes to the Species level (the only comparable taxonomic rank for DeepMicrobes), and it outperforms DeepMicrobes by an average of 33% in weighted accuracy, and by 89% in constrained accuracy.

Geometry and Automorphisms of Non-Kähler Holomorphic Symplectic Manifolds

We consider the only one known class of non-Kähler irreducible holomorphic symplectic manifolds, described in the works by D. Guan and the 1st author. Any such manifold $Q$ of dimension $2n-2$ is obtained as a finite degree $n^2$ cover of some non-Kähler manifold $W_F$, which we call the base of $Q$. We show that the algebraic reduction of $Q$ and its base is the projective space of dimension $n-1$. Besides, we give a partial classification of submanifolds in $Q$, describe the degeneracy locus of its algebraic reduction and prove that the automorphism group of $Q$ satisfies the Jordan property.

Wronskian indices and rational conformal field theories

The classification scheme for rational conformal field theories, given by the Mathur-Mukhi-Sen (MMS) program, identifies a rational conformal field theory by two numbers: (n, l). n is the number of characters of the rational conformal field theory. The characters form linearly independent solutions to a modular linear differential equation (which is also labelled by (n, l)); the Wronskian index l is a non-negative integer associated to the structure of zeroes of the Wronskian.In this paper, we compute the (n, l) values for three classes of well-known CFTs viz. the WZW CFTs, the Virasoro minimal models and the $$ \mathcal{N} $$ N = 1 super-Virasoro minimal models. For the latter two, we obtain exact formulae for the Wronskian indices. For WZW CFTs, we get exact formulae for small ranks (upto 2) and all levels and for all ranks and small levels (upto 2) and for the rest we compute using a computer program. We find that any WZW CFT at level 1 has a vanishing Wronskian index as does the $$ {\hat{\mathbf{A}}}_{\mathbf{1}} $$ A ̂ 1 CFT at all levels. We find intriguing coincidences such as: (i) for the same level CFTs with $$ {\hat{\mathbf{A}}}_{\mathbf{2}} $$ A ̂ 2 and $$ {\hat{\mathbf{G}}}_{\mathbf{2}} $$ G ̂ 2 have the same (n, l) values, (ii) for the same level CFTs with $$ {\hat{\mathbf{B}}}_{\mathbf{r}} $$ B ̂ r and $$ {\hat{\mathbf{D}}}_{\mathbf{r}} $$ D ̂ r have the same (n, l) values for all r ≥ 5.Classifying all rational conformal field theories for a given (n, l) is one of the aims of the MMS program. We can use our computations to provide partial classifications. For the famous (2, 0) case, our partial classification turns out to be the full classification (achieved by MMS three decades ago). For the (3, 0) case, our partial classification includes two infinite series of CFTs as well as fifteen “discrete” CFTs; except three all others have Kac-Moody symmetry.

Classification of $ \mathbf{(3 \!\mod 5)} $ arcs in $ \mathbf{ \operatorname{PG}(3,5)} $

<p style='text-indent:20px;'>The proof of the non-existence of Griesmer <inline-formula><tex-math id="M3">\begin{document}$ [104, 4, 82]_5 $\end{document}</tex-math></inline-formula>-codes is just one of many examples where extendability results are used. In a series of papers Landjev and Rousseva have introduced the concept of <inline-formula><tex-math id="M4">\begin{document}$ (t\mod q) $\end{document}</tex-math></inline-formula>-arcs as a general framework for extendability results for codes and arcs. Here we complete the known partial classification of <inline-formula><tex-math id="M5">\begin{document}$ (3 \mod 5) $\end{document}</tex-math></inline-formula>-arcs in <inline-formula><tex-math id="M6">\begin{document}$ \operatorname{PG}(3,5) $\end{document}</tex-math></inline-formula> and uncover two missing, rather exceptional, examples disproving a conjecture of Landjev and Rousseva. As also the original non-existence proof of Griesmer <inline-formula><tex-math id="M7">\begin{document}$ [104, 4, 82]_5 $\end{document}</tex-math></inline-formula>-codes is affected, we present an extended proof to fill this gap.</p>

On classification of super-modular categories of rank 8

We develop categorical and number-theoretical tools for the classification of super-modular categories. We apply these tools to obtain a partial classification of super-modular categories of rank [Formula: see text]. In particular we find three distinct families of prime categories in rank [Formula: see text] in contrast to the lower rank cases for which there is only one such family.

Origin Preserving Path Formulation for Multiparameter ℤ2-Equivariant Corank 2 Bifurcation Problems

A singularity theory, in the form of path formulation, is developed to analyze and organize the qualitative behavior of multiparameter [Formula: see text]-equivariant bifurcation problems of corank 2 and their deformations when the trivial solution is preserved as parameters vary. Path formulation allows for an efficient discussion of different parameter structures with a minimal modification of the algebra between cases. We give a partial classification of one-parameter problems. With a couple of parameter hierarchies, we show that the generic bifurcation problems are 2-determined and of topological codimension-0. We also show that the preservation of the trivial solutions is an important hypotheses for multiparameter bifurcation problems. We apply our results to the bifurcation of a cylindrical panel under axial compression.

Antiautomorphisms and Biantiautomorphisms of Some Finite Abelian Groups

We extend the concepts of antimorphism and antiautomorphism of the additive group of integers modulo n, given by Gaitanas Konstantinos, to abelian groups. We give a lower bound for the number of antiautomorphisms of cyclic groups of odd order and give an exact formula for the number of linear antiautomorphisms of cyclic groups of odd order. Finally, we give a partial classification of the finite abelian groups which admit antiautomorphisms and state some open questions.

Second-order PDEs in four dimensions with half-flat conformal structure

We study second-order partial differential equations (PDEs) in four dimensions for which the conformal structure defined by the characteristic variety of the equation is half-flat (self-dual or anti-self-dual) on every solution. We prove that this requirement implies the Monge–Ampère property. Since half-flatness of the conformal structure is equivalent to the existence of a non-trivial dispersionless Lax pair, our result explains the observation that all known scalar second-order integrable dispersionless PDEs in dimensions four and higher are of Monge–Ampère type. Some partial classification results of Monge–Ampère equations in four dimensions with half-flat conformal structure are also obtained.