Support vector machines (SVMs) are a kind of important machine learning methods generated by the cross interaction of statistical theory and optimization, and have been extensively applied into text categorization, disease diagnosis, face detection and so on. The loss function is the core research content of SVM, and its variational properties play an important role in the analysis of optimality conditions, the design of optimization algorithms, the representation of support vectors and the research of dual problems. This paper summarizes and analyzes the 0-1 loss function and its eighteen popular surrogate loss functions in SVM, and gives three variational properties of these loss functions: subdifferential, proximal operator and Fenchel conjugate, where the nine proximal operators and fifteen Fenchel conjugates are given by this paper.

Based on the modern development of Metrization theorem for context, the main results obtained in recent ten years on point-discrete families are summarized. This paper mainly introduces the theory of the spaces with \sigma-point-discrete bases, the spaces with certain \sigma-point-discrete networks, and the relationship between the above spaces and the spaces with certain \sigma-compact-finite networks.

Let \mathcal{A} be a Banach algebra with unit e and a,b,c\in \mathcal{A},M_c=\left(\begin{array}{cc}a\hfill & c\hfill \\ 0\hfill & b\hfill \end{array}\right)\in M_2\left(\mathcal{A}\right). The concepts of left and right generalized Drazin invertible of elements in a Banach algebra are proposed. A generalized Drazin spectrum of \alpha is defined by {\sigma }_{\mathrm{g}\mathrm{D}}\left(\alpha \right)=\{\lambda \in \mathbb{C}:\alpha -\lambda e\mathrm{i}\mathrm{s}\mathrm{n}\mathrm{o}\mathrm{t}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{z}\mathrm{e}\mathrm{d}\mathrm{D}\mathrm{r}\mathrm{a}\mathrm{z}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{n}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{b}\mathrm{l}\mathrm{e}\}. It is shown that

　　　　　　　　　　 {\sigma }_{\mathrm{g}\mathrm{D}}\left(a\right)\cup {\sigma }_{\mathrm{g}\mathrm{D}}\left(b\right)={\sigma }_{\mathrm{g}\mathrm{D}}\left(M_c\right)\cup W_2,

where W_g is a union of certain holes {\sigma }_{\mathrm{g}\mathrm{D}} and W_g\subseteq {\sigma }_{\mathrm{g}\mathrm{D}}\left(a\right)\cap {\sigma }_{\mathrm{g}\mathrm{D}}\left(b\right), or more finely W_g\subseteq {\sigma }_{\mathrm{r}\mathrm{g}\mathrm{D}}\left(a\right)\cap {\sigma }_{\mathrm{l}\mathrm{g}\mathrm{D}}\left(b\right). In addition, some properties of generalized Drazin spectrum of elements in a Banach algebra are studied.

In this paper, we study the blow-up problem of nonlinear Schrödinger equations

　　　　　　　　 \{\begin{array}{ll}\text{i}{\mathrm{\partial }}_{t}v+\mathrm{\Delta }u+(|u{|}^2+|v{|}^2)u=0,\hfill & (t,x)\in {\mathbb{R}}^{1+n},\hfill \\ \mathrm{i}{\mathrm{\partial }}_{t}v+\mathrm{\Delta }v+(|u{|}^2+|v{|}^2)v=0,\hfill & (t,x)\in {\mathbb{R}}^{1+n},\hfill \\ u(0,x)=u_0\left(x\right),\hfill & v(0,x)=v_0\left(x\right),\hfill \end{array}

and prove that the solution of negative energy blows up in finite or infinite time.