26,881 research outputs found

### 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 $a/2$

### Motivic measures of the moduli spaces of pure sheaves on $\mathbb{P}^2$ 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 $\chi$

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

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

### 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

### 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

### 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

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