One of our main results is the following: Let$X$be a compact connected subset of the Euclidean space$R$^{$n$}and$r(X, d$_{2}) the rendezvous number of$X$, where$d$_{2}denotes the Euclidean distance in$R$^{$n$}. (The rendezvous number$r(X, d$_{2}) is the unique positive real number with the property that for each positive integer$n$and for all (not necessarily distinct)$x$_{1},$x$_{2},...,$x$_{$n$}in$X$, there exists some$x$in$X$such that $$(1/n)\sum\nolimits_{i = 1}^n {d_2 (x_i ,x)} = r(X,d_2 )$$ .) Then there exists some regular Borel probability measure μ_{0}on$X$such that the value of ∫_{$X$}$d$_{2}($x, y$)$d$μ_{0}($y$) is independent of the choice$x$in$X$, if and only if$r$($X, d$_{2}) = sup_{μ}∫_{$X$}∫_{$X$}$d$_{2}($x, y$)$d$μ($x$)$d$μ($y$), where the supremum is taken over all regular Borel probability measures μ on$X$.

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It is well-known that sparse grid algorithm has been widely accepted as an efficient tool to overcome the "curse of dimensionality" in some degree. In this note, we first give the error estimate of hyperbolic cross (HC) approximations with generalized Hermite functions. The exponential convergence in both regular and optimized hyperbolic cross approximations has been shown. Moreover, the error estimate of Hermite spectral method to high-dimensional linear parabolic PDEs with HC approximations has been investigated in the properly weighted Korobov spaces. The numerical result verifies the exponential convergence of this approach.

In this paper we consider several homological dimensions of crossed products $A _{\alpha} ^{\sigma} G$, where $A$ is a left Noetherian ring and $G$ is a finite group. We revisit the induction and restriction functors in derived categories, generalizing a few classical results for separable extensions. The global dimension and finitistic dimension of $A ^{\sigma} _{\alpha} G$ are classified: global dimension of $A ^{\sigma} _{\alpha} G$ is either infinity or equal to that of $A$, and finitistic dimension of $A ^{\sigma} _{\alpha} G$ coincides with that of $A$. A criterion for skew group rings to have finite global dimensions is deduced. Under the hypothesis that $A$ is a semiprimary algebra containing a complete set of primitive orthogonal idempotents closed under the action of a Sylow $p$-subgroup $S \leqslant G$, we show that $A$ and $A _{\alpha} ^{\sigma} G$ share the same homological dimensions under extra assumptions.

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