Birational geometry of the moduli space of pure sheaves on quadric surface

We study birational geometry of the moduli space of stable sheaves on a quadric surface with Hilbert polynomial $5m+1$ and $c_1=(2,3)$. We describe a birational map between the moduli space and a projective bundle over a Grassmannian as a composition of smooth blow-ups/downs.     

Weighted completion time minimization on a single-machine with a fixed non-availability interval: Differential approximability

This paper is the first successful attempt on differential approximability study for a scheduling problem. Such a study considers the weighted completion time minimization on a single machine with a fixed non-availability interval. The analysis shows that the Weighted Shortest Processing Time (WSPT) rule cannot yield a differential approximation for the problem under consideration in the general case. Nevertheless, a slight modification of this rule provides an approximation with a differential ratio of $\frac{3?\sqrt{5}}{2}\approx 0.38$.     

Counting rational points on smooth cyclic covers

A conjecture of Serre concerns the number of rational points of bounded height on a finite cover of projective space ${\mathbb{P}}^{n?1}$. In this paper, we achieve Serre?s conjecture in the special case of smooth cyclic covers of any degree when $n?10$, and surpass it for covers of degree $r?3$ when $n>10$. This is achieved by a new bound for the number of perfect r-th power values of a polynomial with nonsingular leading form, obtained via a combination of an r-th power sieve and the q-analogue of van der Corput?s method.     

An improved approximation algorithm for maximum edge 2-coloring in simple graphs

We present a polynomial-time approximation algorithm for legally coloring as many edges of a given simple graph as possible using two colors. It achieves an approximation ratio of $\frac{468}{575}$. This improves on the previous best (trivial) ratio of $\frac{4}{5}$.     

New 3-designs and 2-designs having U(3,3) as an automorphism group

In this paper we classify $2$-designs and $3$-designs with $28$ or $36$ points admitting a transitive action of the unitary group $U(3,3)$ on points and blocks. We also construct $2$-designs and $3$-designs with $56$ or $63$ points and strongly regular graphs on 36, 63 or 126 vertices having $U(3,3)$ as a transitive automorphism group. Further, we show that this completes the classification of $3$-designs admitting a transitive action of the group $U(3,3)$, in terms of parameters. A number of the $3$-designs and $2$-designs obtained in this paper have not been known before up to our best knowledge.     

Approximation properties of certain operator-induced norms on Hilbert spaces

We consider a class of operator-induced norms, acting as finite-dimensional surrogates to the $L^2$ norm, and study their approximation properties over Hilbert subspaces of $L^2$. The class includes, as a special case, the usual empirical norm encountered, for example, in the context of nonparametric regression in a reproducing kernel Hilbert space (RKHS). Our results have implications to the analysis of $M$-estimators in models based on finite-dimensional linear approximation of functions, and also to some related packing problems.