The necessary and sufficient conditions under which a ring satisfies regular power-substitution are investigated. It is shown that a ring R satisfies regular power-substitution if and only if a≂b in R implies that there exist n ∈ ℕ and a U ∈ GL _n(R) such that aU = Ub if and only if for any regular x ∈ R there exist m, n ∈ ℕ and U ∈ GL _n(R) such that x ^m I _n = x ^m Ux ^m, where a≂b means that there exists x, y, z ∈ R such that a = ybx, b = xaz and x = xyx = xzx. It is proved that every directly finite simple ring satisfies regular power-substitution. Some applications for stably free R-modules are also obtained.

The relative transpose via Gorenstein projective modules is introduced, and some corresponding results on the Auslander-Reiten sequences and the Auslander-Reiten formula to this relative version are generalized.

The canard phenomenon occurring in planar fast-slow systems under nongeneric conditions is investigated. When the critical manifold has a non-generic fold point, by using the method of asymptotic analysis combined with the recently developed blow-up technique, the existence of a canard is established and the asymptotic expansion of the parameter for which a canard exists is obtained.

Let π and π′ be automorphic irreducible cuspidal representations of GL _m(ℚ$\mathbb{Q}_\mathbb{A} $) and GL _m′ (ℚ$\mathbb{Q}_\mathbb{A} $), respectively, and L(s, π × $\tilde \pi '$) be the Rankin-Selberg L-function attached to π and π′. Without assuming the Generalized Ramanujan Conjecture (GRC), the author gives the generalized prime number theorem for L(s, π × $\tilde \pi '$) when π ≅ π′. The result generalizes the corresponding result of Liu and Ye in 2007.

This paper focuses on a part of the presentation given by the third author at the Shanghai Forum on Industrial and Applied Mathematics (Shanghai 2006). It is related to the existence of a periodic solution of evolution variational inequalities. The approach is based on the method of guiding functions.

The aim of this note is to improve the regularity results obtained by H. Beirão da Veiga in 2008 for a class of p-fluid flows in a cubic domain. The key idea is exploiting the better regularity of solutions in the tangential directions with respect to the normal one, by appealing to anisotropic Sobolev embeddings.

Let $\tilde S$ be a Riemann surface of analytically finite type (p, n) with 3p − 3 + n > 0. Let a ∈ $\tilde S$ and S = $\tilde S$ − {a}. In this article, the author studies those pseudo-Anosov maps on S that are isotopic to the identity on $\tilde S$ and can be represented by products of Dehn twists. It is also proved that for any pseudo-Anosov map f of S isotopic to the identity on $\tilde S$, there are infinitely many pseudo-Anosov maps F on S − {b} = $\tilde S$ − {a, b}, where b is a point on S, such that F is isotopic to f on S as b is filled in.

The authors give some new necessary conditions for the boundedness of Toeplitz products $T_f^\alpha T_{\bar g}^\alpha $ on the weighted Bergman space A _α ² of the unit ball, where f and g are analytic on the unit ball. Hankel products H _f H _g ^* on the weighted Bergman space of the unit ball are studied, and the results analogous to those Stroethoff and Zheng obtained in the setting of unit disk are proved.

Let G be a locally compact Abelian group with Haar measure. The authors discuss some basic properties of $L_{w_1 }^r $(G) ∩ L(p,q,w ₂dµ)(G) spaces. Then the necessary conditions for compact embeddings of the spaces $L_{w_1 }^r $(R ^d)∩L(p,q,w ₂dµ)(R ^d) are showed.

The existence and uniqueness of solutions to the multivalued stochastic differential equations with non-Lipschitz coefficients are proved, and bicontinuous modifications of the solutions are obtained.