Abstract:
This paper establishes a dynamic portfolio insurance model under the condition of continuous time, based on Meton's optimal investment-consumption model, which combined the method of replicating dynamic synthetic put option using risk-free and risk assets. And it transfers the problem of investor's individual intertemporal dynamic portfolio insurance decision into a problem of static utility maximization under condition of continuous time, and give the optimal capital combination strategies corresponding to the optimal wealth level of the portfolio insurers, and compares the difference of strategies between this model and Merton model. The conclusions show that investors' optimal strategies of portfolio insurance are not dependent on their wealth, but market risk. That is to say, the higher the risk is, the more the demand of portfolio insurance is.

Abstract:
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.

Abstract:
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.

Abstract:
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.

Abstract:
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.

Abstract:
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 $\chi$.

Abstract:
We study Le Potier's strange duality conjecture on $\mathbb{P}^2$. We show the conjecture is true for the pair ($M(2,0,2),~M(d,0)$) with $d>0$, where $M(2,0,2)$ is the moduli space of semistable sheaves of rank 2, zero first Chern class and second Chern class 2, and $M(d,0)$ is the moduli space of 1-dimensional semistable sheaves of first Chern class $dH$ and Euler characteristic 0.

Abstract:
Let $M(d,\chi)$ be the moduli space of semistable sheaves of rank 0, Euler characteristic $\chi$ and first Chern class $dH (d>0)$, with $H$ the hyperplane class in $\mathbb{P}^2$. We give a description of $M(d,\chi)$, viewing each sheaf as a class of matrices with entries in $\bigoplus_{i\geq0}H^0(\mathcal{O}_{\mathbb{P}^2}(i))$. We show that there is a big open subset of $M(d,1)$ isomorphic to a projective bundle over an open subset of a Hilbert scheme of points on $\mathbb{P}^2.$ Finally we compute the classes of M(4,1), M(5,1) and M(5,2) in the Grothendieck group of varieties, especially we conclude that M(5,1) and M(5,2) are of the same class.

Acquired
immunodeficiency syndrome (AIDS) is a fatal disease which highly threatens the
health of human being. Human immunodeficiency virus (HIV) is the pathogeny for
this disease. Investigating HIV-1 protease cleavage sites can help researchers
find or develop protease inhibitors which can restrain the replication of HIV-1,
thus resisting AIDS. Feature selection is a new approach for solving the HIV-1
protease cleavage site prediction task and it’s a key point in our research.
Comparing with the previous work, there are several advantages in our work.
First, a filter method is used to eliminate the redundant features. Second,
besides traditional orthogonal encoding (OE), two kinds of newly proposed
features extracted by conducting principal component analysis (PCA) and
non-linear Fisher transformation (NLF) on AAindex database are used. The two
new features are proven to perform better
than OE. Third, the data set used here is largely expanded to 1922 samples.
Also to improve prediction performance, we conduct parameter optimization for
SVM, thus the classifier can obtain better prediction capability. We also fuse
the three kinds of features to make sure comprehensive feature representation
and improve prediction performance. To effectively evaluate the prediction
performance of our method, five parameters, which are much more than previous
work, are used to conduct complete comparison. The experimental results of our
method show that our method gain better performance than the state of art
method. This means that the feature selection combined with feature fusion and
classifier parameter optimization can effectively improve HIV-1 cleavage site
prediction. Moreover, our work can provide useful help for HIV-1 protease inhibitor
developing in the future.

Abstract:
Studies of past accidents have revealed that various elements such as failure to identify hazards, crowd behaviors out of controlling, deficiency of the egress signage system, inconsistency between process behavior and process plan, and environmental constraints, etc. affected crowd evacuation. Above all, the human factor is the key issue in safety and disaster management, although it is bound to other factors inextricably. This paper explores crowd behaviors that may influence an urgent situation, and discusses the technique applied to the crowd prediction. Based on risk rating relative to crowd density, risk plans for different levels are proposed to dispose the potential threats. Also practical crowd management measures at different risk levels are illustrated in a case of a metro station in China. Finally, the strategies for crowd security management are advised that all stakeholders are amenable to form risk consciousness and implement safety procedures consistent with risk plans professionally and scientifically.