The Chow ring for the classifying space of GO(2n)GO(2n)

Let $GO(2n)$ be the general orthogonal group scheme (the group of orthogonal similitudes). In the topological category, Y. Holla and N. Nitsure determined the singular cohomology ring $H^*_{\rm sing}(BGO(2n,\mathbb C),\mathbb F_2)$ of the classifying space $BGO(2n,\mathbb C)$ of the corresponding complex Lie group $GO(2n,\mathbb C)$ in terms of explicit generators and relations. The author of the present note showed that over any algebraically closed field of characteristic not equal to $2$, the smooth-\'etale cohomology ring $H_{\rm sm-\'et}^*(BGO(2n),\mathbb F_2)$ of the classifying algebraic stack $BGO(2n)$ has the same description in terms of generators and relations as the singular cohomology ring $H^*_{\rm sing}(BGO(2n,\mathbb C),\mathbb F_2)$. Totaro defined for any reductive group $G$ over a field, the Chow ring $A^*_G$, which is canonically identified with the ring of characteristic classes in the sense of intersection theory, for principal $G$-bundles, locally trivial in \'etale topology. In this paper, we calculate the Chow group $A^*_{GO(2n)}$ over any field of characteristic different from $2$ in terms of generators and relations.Comment: 11 page

Atomistic aspects of ductile responses of cubic silicon carbide during nanometric cutting

Cubic silicon carbide (SiC) is an extremely hard and brittle material having unique blend of material properties which makes it suitable candidate for microelectromechanical systems and nanoelectromechanical systems applications. Although, SiC can be machined in ductile regime at nanoscale through single-point diamond turning process, the root cause of the ductile response of SiC has not been understood yet which impedes significant exploitation of this ceramic material. In this paper, molecular dynamics simulation has been carried out to investigate the atomistic aspects of ductile response of SiC during nanometric cutting process. Simulation results show that cubic SiC undergoes sp3-sp2 order-disorder transition resulting in the formation of SiC-graphene-like substance with a growth rate dependent on the cutting conditions. The disorder transition of SiC causes the ductile response during its nanometric cutting operations. It was further found out that the continuous abrasive action between the diamond tool and SiC causes simultaneous sp3-sp2 order-disorder transition of diamond tool which results in graphitization of diamond and consequent tool wear

Harder-Narasimhan Filtrations which are not split by the Frobenius maps

Let $X$ be a smooth projective variety over a perfect field $k$ of characteristic $p>0$, and $V$ be a vector bundle over $X$. It is well known that if $X$ is a curve and $V$ is not strongly semistable, then some Frobenius pullback $(F^t)^*V$ is a direct sum of strongly semistable bundles. A natural question to ask is whether this still holds in higher dimension. Indranil Biswas, Yogish I. Holla, A.J. Parameswaran, and S. Subramanian showed that there is always a counterexample to this over any algebraically closed field of positive characteristic which is uncountable. However, we will produce a smooth projective variety over $\mathbb Z$ and a rank 2 vector bundle on it, which, restricted to each prime $p$ in a nonempty open subset of \spec\mathbb Z, constitutes a counterexample over $p$. Indeed, given any split semisimple simply connected algebraic group $G$ of semisimple rank $>1$ over $\mathbb Z$, we will show that there exists a smooth projective homogeneous space $X_Z$ over $\mathbb Z$ and a vector bundle $V$ on $X_Z$ of rank 2 such that for each prime $p$ in a nonempty open subset of \spec\mathbb Z, the restriction $V\otimes\mathbb F_p$ as a vector bundle over $X_Z\otimes\mathbb F_p$ is a counterexample. We only use the Borel-Weil-Bott theorem in characteristic 0 and Frobenius Splitting of $G/B$ in characteristic $p$.Comment: 3 page

Text segmentation on multilabel documents: A distant-supervised approach

Segmenting text into semantically coherent segments is an important task with applications in information retrieval and text summarization. Developing accurate topical segmentation requires the availability of training data with ground truth information at the segment level. However, generating such labeled datasets, especially for applications in which the meaning of the labels is user-defined, is expensive and time-consuming. In this paper, we develop an approach that instead of using segment-level ground truth information, it instead uses the set of labels that are associated with a document and are easier to obtain as the training data essentially corresponds to a multilabel dataset. Our method, which can be thought of as an instance of distant supervision, improves upon the previous approaches by exploiting the fact that consecutive sentences in a document tend to talk about the same topic, and hence, probably belong to the same class. Experiments on the text segmentation task on a variety of datasets show that the segmentation produced by our method beats the competing approaches on four out of five datasets and performs at par on the fifth dataset. On the multilabel text classification task, our method performs at par with the competing approaches, while requiring significantly less time to estimate than the competing approaches.Comment: Accepted in 2018 IEEE International Conference on Data Mining (ICDM