Motivic measures of moduli spaces of 1-dimensional sheaves on rational surfaces

We study the moduli space of rank 0 semistable sheaves on some rational surfaces. We show the irreducibility and stable rationality of them under some conditions. We also compute several (virtual) Betti numbers of those moduli spaces by computing their motivic measures.Comment: arXiv admin note: substantial text overlap with arXiv:1503.0630

Strange duality on rational surfaces

We study Le Potier's strange duality conjecture on a rational surface. We focus on the case involving the moduli space of rank 2 sheaves with trivial first Chern class and second Chern class 2, and the moduli space of 1-dimensional sheaves with determinant $L$ and Euler characteristic 0. We show the conjecture for this case is true under some suitable conditions on $L$, which applies to $L$ ample on any Hirzebruch surface $\Sigma_e:=\mathbb{P}(\mathcal{O}_{\mathbb{P}^1}\oplus\mathcal{O}_{\mathbb{P}^1}(e))$ except for $e=1$. When $e=1$, our result applies to $L=aG+bF$ with $b\geq a+[a/2]$, where $F$ is the fiber class, $G$ is the section class with $G^2=-1$ and $[a/2]$ is the integral part of $a/2$

Affine pavings for moduli spaces of pure sheaves on P2\mathbb{P}^2 with degree ≤5\leq 5

Let $M(d,r)$ be the moduli space of semistable sheaves of rank 0, Euler characteristic $r$ and first Chern class $dH (d>0)$, with $H$ the hyperplane class in $\mathbb{P}^2$. By previous work, we gave an explicit description of the class $[M(d,r)]$ of $M(d,r)$ in the Grothendieck ring of varieties for $d\leq 5$ and $g.c.d(d,r)=1$. In this paper we compute the fixed locus of $M(d,r)$ under some $(\mathbb{C}^{*})^2$-action and show that $M(d,r)$ admits an affine paving for $d\leq 5$ and $g.c.d(d,r)=1$. We also pose a conjecture that for any $d$ and $r$ coprime to $d$, $M(d,r)$ would admit an affine paving.Comment: 22 page

Motivic measures of the moduli spaces of pure sheaves on P2\mathbb{P}^2 with all degrees

Let $\mathcal{M}(d,\chi)$ be the moduli stack of stable sheaves of rank 0, Euler characteristic $\chi$ and first Chern class $dH~(d>0)$, with $H$ the hyperplane class in $\mathbb{P}^2$. We compute the $A$-valued motivic measure $\mu_A(\mathcal{M}(d,\chi))$ of $\mathcal{M}(d,\chi)$ and get explicit formula in codimension $D:=\rho_d-1$, where $\rho_d$ is $d-1$ for $d=p$ or $2p$ with $p$ prime, and $7$ otherwise. As a corollary, we get the last $2(D+1)$ Betti numbers of the moduli scheme $M(d,\chi)$ when $d$ is coprime to $\chi$

Determinant line bundles on Moduli spaces of pure sheaves on rational surfaces and Strange Duality

Let \mhu be the moduli space of semi-stable pure sheaves of class $u$ on a smooth complex projective surface $X$. We specify $u=(0,L,\chi(u)=0),$ i.e. sheaves in $u$ are of dimension $1$. There is a natural morphism $\pi$ from the moduli space \mhu to the linear system \ls. We study a series of determinant line bundles \lcn on \mhu via $\pi.$ Denote $g_L$ the arithmetic genus of curves in \ls. For any $X$ and $g_L\leq0$, we compute the generating function Z^r(t)=\sum_{n}h^0(\mhu,\lcn)t^n. For $X$ being $\mathbb{P}^2$ or \mathbb{P}(\mo_{\pone}\oplus\mo_{\pone}(-e)) with $e=0,1$, we compute $Z^1(t)$ for $g_L>0$ and $Z^r(t)$ for all $r$ and $g_L=1,2$. Our results provide a numerical check to Strange Duality in these specified situations, together with G\"ottsche's computation. And in addition, we get an interesting corollary in the theory of compactified Jacobian of integral curves

Estimates of sections of determinant line bundles on Moduli spaces of pure sheaves on algebraic surfaces

Let $X$ be any smooth simply connected projective surface. We consider some moduli space of pure sheaves of dimension one on $X$, i.e. \mhu with $u=(0,L,\chi(u)=0)$ and $L$ an effective line bundle on $X$, together with a series of determinant line bundles associated to r[\mo_X]-n[\mo_{pt}] in Grothendieck group of $X$. Let $g_L$ denote the arithmetic genus of curves in the linear system \ls. For $g_L\leq2$, we give a upper bound of the dimensions of sections of these line bundles by restricting them to a generic projective line in \ls. Our result gives, together with G\"ottsche's computation, a first step of a check for the strange duality for some cases for $X$ a rational surface

Relativistic interpretation on the nature of nuclear tensor force

The spin-dependent nature of the nuclear tensor force is studied in details within the relativistic Hartree-Fock approach. The relativistic formalism for the tensor force is supplemented with an additional Lorentz-invariant tensor formalism in $\sigma$-scalar channel, so as to take into account almost fully the nature of the tensor force brought about by the Fock diagrams in realistic nuclei. Specifically, the tensor sum rules are tested for the spin and pseudo-spin partners with/without nodes, to further understand the tensor force nature within relativistic model. It is shown that the interference between two components of nucleon spinors brings distinct violations on the tensor sum rules in realistic nuclei , which is mainly due to the opposite sign on $\kappa$ quantities of the upper and lower components as well as the nodal difference. Even though, the sum rules can be precisely reproduced if taking the same radial wave functions for the spin/pseudo-spin partners in addition to neglecting the lower/upper components, revealing clearly the nature of tensor force.Comment: 10 pages, 4 figures, 6 tables, to be published in Chinese Physics

Secrecy Rate Maximization with Outage Constraint in Multihop Relaying Networks

In this paper, we study the secure transmission in multihop wireless networks with randomize-and-forward (RaF) relaying, in the presence of randomly distributed eavesdroppers. By considering adaptive encoder with on-off transmission (OFT) scheme, we investigate the optimal design of the wiretap code and routing strategies to maximize the secrecy rate while satisfying the secrecy outage probability (SOP) constraint. We derive the exact expressions for the optimal rate parameters of the wiretap code. Then the secure routing problem is solved by revising the classical Bellman-Ford algorithm. Simulation results are conducted to verify our analysis.Comment: 4 pages, 2 figures, 1 table, Accepted for publication at the IEEE Communications Lette

Online Learning as Stochastic Approximation of Regularization Paths

In this paper, an online learning algorithm is proposed as sequential stochastic approximation of a regularization path converging to the regression function in reproducing kernel Hilbert spaces (RKHSs). We show that it is possible to produce the best known strong (RKHS norm) convergence rate of batch learning, through a careful choice of the gain or step size sequences, depending on regularity assumptions on the regression function. The corresponding weak (mean square distance) convergence rate is optimal in the sense that it reaches the minimax and individual lower rates in the literature. In both cases we deduce almost sure convergence, using Bernstein-type inequalities for martingales in Hilbert spaces. To achieve this we develop a bias-variance decomposition similar to the batch learning setting; the bias consists in the approximation and drift errors along the regularization path, which display the same rates of convergence, and the variance arises from the sample error analysed as a reverse martingale difference sequence. The rates above are obtained by an optimal trade-off between the bias and the variance.Comment: 37 page

Generating functions for K-theoretic Donaldson invariants and Le Potier's strange duality

K-theoretic Donaldson invariants are holomorphic Euler characteristics of determinant line bundles on moduli spaces of sheaves on surfaces. We compute generating functions of K-theoretic Donaldson invariants on the projective plane and rational ruled surfaces. We apply this result to prove some cases of Le Potier's strange duality.Comment: 50 page