A variant smoothing Newton method for P0-NCP based on a new smoothing function

In this paper, we present a new one-step smoothing Newton method proposed for solving the non-linear complementarity problem with _P0$P_0$-function based on a new smoothing NCP$NCP$-function. We adopt a variant merit function. Our algorithm needs only to solve one linear system of equations and perform one line search per iteration. It shows that any accumulation point of the iteration sequence generated by our algorithm is a solution of _P0-NCP$P_0\text{-}NCP$. Furthermore, under the assumption that the solution set is non-empty and bounded, we can guarantee at least one accumulation point of the generated sequence. Numerical experiments show the feasibility and efficiency of the algorithm.     

The approximation characteristic of diagonal matrix in probabilistic setting

In this paper, the approximation characteristic of a diagonal matrix in probabilistic and average case settings is investigated. And the asymptotic degree of the probabilistic linear (n,δ)$(n,\delta )$-width and p$p$-average linear n$n$-width of diagonal matrix M$M$ are determined.     

Stable categories of higher preprojective algebras

We introduce (n+1)$(n+1)$-preprojective algebras of algebras of global dimension n$n$. We show that if an algebra is n$n$-representation-finite then its (n+1)$(n+1)$-preprojective algebra is self-injective. In this situation, we show that the stable module category of the (n+1)$(n+1)$-preprojective algebra is (n+1)$(n+1)$-Calabi–Yau, and, more precisely, it is the (n+1)$(n+1)$-Amiot cluster category of the stable n$n$-Auslander algebra of the original algebra. In particular this stable category contains an (n+1)$(n+1)$-cluster tilting object. We show that even if the (n+1)$(n+1)$-preprojective algebra is not self-injective, under certain assumptions (which are always satisfied for n∈{1,2}$n\in \{1,2\}$) the results above still hold for the stable category of Cohen–Macaulay modules.     

Necessary and sufficient condition for uniqueness of solution to the first boundary value problem for the diffusion equation in unbounded domains

This paper introduces a notion of regularity of t=-∞$t=-\infty$ for the diffusion (or heat) equation and establishes a necessary and sufficient condition for the existence of a unique bounded solution to the first boundary value problem for the diffusion equation in a general domain Ω⊂R^N+1$\Omega \subset {\mathbb{R}}^{N+1}$ which extends up to t=-∞$t=-\infty$.     

Compact covers and function spaces

For a Tychonoff space X  , we denote by _Cp(X)$C_p(X)$ and _Cc(X)$C_c(X)$ the space of continuous real-valued functions on X equipped with the topology of pointwise convergence and the compact-open topology respectively. Providing a characterization of the Lindelöf Σ-property of X   in terms of _Cp(X)$C_p(X)$, we extend Okunev?s results by showing that if there exists a surjection from _Cp(X)$C_p(X)$ onto _Cp(Y)$C_p(Y)$ (resp. from _Lp(X)$L_p(X)$ onto _Lp(Y)$L_p(Y)$) that takes bounded sequences to bounded sequences, then υY is a Lindelöf Σ-space (respectively K-analytic) if υX has this property. In the second part, applying Christensen?s theorem, we extend Pelant?s result by proving that if X is a separable completely metrizable space and Y   is first countable, and there is a quotient linear map from _Cc(X)$C_c(X)$ onto _Cc(Y)$C_c(Y)$, then Y   is a separable completely metrizable space. We study also a non-separable case, and consider a different approach to the result of J. Baars, J. de Groot, J. Pelant and V. Valov, which is based on the combination of two facts: Complete metrizability is preserved by _?p${?}_p$-equivalence in the class of metric spaces (J. Baars, J. de Groot, J. Pelant). If X   is completely metrizable and _?p${?}_p$-equivalent to a first-countable Y, then Y is metrizable (V. Valov). Some additional results are presented.