Starting with a commutative ring $R$ and an ideal $I$ , it is possible to define a family of rings $R(I)_{a,b}$ , with $a,b \in R$ , as quotients of the Rees algebra $\oplus_{n \geq0} I^{n}t^{n}$ ; among the rings appearing in this family we find Nagata’s idealization and amalgamated duplication. Many properties of these rings depend only on $R$ and $I$ and not on $a$ , $b$ ; in this paper we show that the Gorenstein and the almost Gorenstein properties are independent of $a$ , $b$ . More precisely, we characterize when the rings in the family are Gorenstein, complete intersection, or almost Gorenstein and we find a formula for the type.

We extend recent results by Pisier on$K$-subcouples, i.e. subcouples of an interpolation couple that preserve the$K$-functional (up to constants) and corresponding notions for quotient couples. Examples include interpolation (in the pointwise sense) and a reinterpretation of the Adamyan-Arov-Krein theorem for Hankel operators.

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We study the cluster algebras arising from cluster tubes with rank bigger than 1. Cluster tubes are 2−Calabi-Yau triangulated categories which contain no cluster tilting objects, but maximal rigid objects. Fix a certain maximal rigid object T in the cluster tube C_n of rank n. For any indecomposable rigid object M in C_n, we define an analogous X_M of Caldero-Chapton's formula (or Palu's cluster character formula) by using the geometric information of M. We show that X_M,X_{M′} satisfy the mutation formula when M,M′ form an exchange pair, and that X_? : M↦X_M gives a bijection from the set of indecomposable rigid objects in C_n to the set of cluster variables of cluster algebra of type C_{n−1}, which induces a bijection between the set of basic maximal rigid objects in C_n and the set of clusters. This strengths a surprising result proved recently by Buan-Marsh-Vatne that the combinatorics of maximal rigid objects in the cluster tube Cn encode the combinatorics of the cluster algebra of type B_{n−1} since the combinatorics of cluster algebras of type B_{n−1} or of type C_{n−1} are the same by a result of Fomin and Zelevinsky. As a consequence, we give a categorification of cluster algebras of type C.

In its simplest form, convolution neural networks (CNNs) consist of a fully connected layer g composed with a sequence of convolution layers T . Although g is known to have the universal approximation property, it is not known if CNNs, which has the form g ◦ T inherits this property, especially when the kernel size in T is small. In this paper, we show that under suitable conditions, CNNs does inherit the universal approximation property and its sample complexity can be characterized. In addition, we discuss concretely how the nonlinearity of T can improve the approximation power. Finally, we show that when the target function class has a certain compositional form, convolutional networks are far more advantageous compared with fully connected networks, in terms of number of parameters needed to achieve a desired accuracy.