In this paper, we study the cofiniteness of generalized local cohomology modules with respect to an ideal of a commutative Noetherian ring and the class of modules in dimension less than a fixed integer. As consequences, we present some results on the ordinary cofiniteness, weakly cofiniteness, and finiteness of the set of associated prime ideals of generalized local cohomology modules.

We determine and classify all finite-dimensional Hopf algebras over an algebraically closed field of characteristic zero whose Hopf coradicals are isomorphic to dual Radford algebras of dimension $4p$ for a prime $p>5$. In particular, we obtain families of new examples of finite-dimensional Hopf algebras without the dual Chevalley property.

Let $k$ be a field. Let $m\in\mathbf{N}_{>0}$ be a positive integer. Let $f\in k[x_1,\ldots,x_m]$ be a polynomial with degree $d\geq 1$ and associated hypersurface $H:=H(f):=\mathrm{Spec}(k[x_1,\ldots,x_m]/\langle f\rangle)$. In this article, we firstly provide a structure property of the weighted-homogeneity of $f$ in terms of the jet schemes ${\mathscr{L}_{H}}$ of $H$. As a by-product, we deduce from this property a new and very \linebreak effective method for the computation of the motivic Poincaré power series $P_H(T):=\sum_{n\geq 0} [\mathscr{L}_{n}(H)]T^n\in K_0(\mathrm{Var}_k)[[T]]$ associated with a homogeneous hypersurface $H$ with a single isolated singularity at the origin $\frak o$ (and more generally with a specific class of isolated quasi-homogeneous hypersurface singularities). With this point of view we obtain various consequences. For the considered class of varieties, our method provides a characteristic-free proof of the rationality of $P_H(T)$ which does not use motivic integration nor the existence of resolutions of singularities; we obtain a precise description of the numerator and the possible poles in the rational expression of $P_H(T)$; when the field is assumed to be of characteristic zero, this allows us to prove the validity of the motivic monodromy conjecture.

The paper continues the study on the distinguished property of the space $C_{p}\left( X\right) $ (of real-valued continuous functions over a Tychonoff space $X$ in the pointwise topology) under the formation of tensor products towards the following research directions: $\left( i\right) $ the injective tensor product $C_{p}\left( X\right) \otimes _{\varepsilon }E$ of $C_{p}\left( X\right) $ and a real locally convex space $E$ (and its completion $C_{p}\left( X\right) \,\widehat{\otimes }_{\varepsilon }\,E$), and $\left( ii\right) $ the space $C_{p}\left( X,E\right) $ of all $E$-valued continuous functions (with a normed space $E$) endowed with the pointwise topology. This work leads also to a new characterization of distinguished Fréchet locally convex spaces $E$. We show, e.g., that if $C_{p}(X)$ is metrizable, then $E$ is distinguished if and only if metrizable $C_{p}\left( X\right) \otimes _{\varepsilon }E$ is distinguished.

We establish the mapping properties of the fractional integral operators on the Herz-Hardy spaces with variable exponents by using extrapolation. In particular, our main results yield the mapping properties for the fractional integral operators on the Herz-Hardy spaces.

In 1994, Mironescu and Panaitopol proved the existence of a triangle with arbitrary prescribed angle bisector lengths. The proof is reduced to a particular fixed point problem. We generalize this problem and propose an alternative mathematical tool. A numerical example is provided. In this frame, we establish some local and uniform stability properties.

Let $G$ be a finite groupoid and $\alpha=(S_g,\alpha_g)_{g\in G}$ a unital partial action of group-type of $G$ on a commutative ring $S=\oplus_{y\in G_0}S_y$. We prove a Galois correspondence between a class of wide subgroupoids of $G$ and a class of subrings of $S$. We recover known results for global groupoid actions and we give several examples to illustrate the correspondence.