Related papers: Comparing complexities of pairs of modules

Let $R$ be a ring with unity, $\sigma$ an endomorphism of $R$ and $M_R$ a right $R$-module. In this paper, we continue studding $\sigma$-rigid modules that were introduced by Gunner et al. \cite{generalized/rigid}. We give some results on…

Let $R$ be a local ring, and let $M$ and $N$ be finitely generated $R$-modules such that $M$ has finite complete intersection dimension. In this paper we define and study, under certain conditions, a pairing using the modules…

We study the phenomenon that some modules of deep neural networks (DNNs) are more critical than others. Meaning that rewinding their parameter values back to initialization, while keeping other modules fixed at the trained parameters,…

Let $R$ be a commutative Noetherian ring. We give criteria for flatness of $R$-modules in terms of associated primes and torsion-freeness of certain tensor products. This allows us to develop a criterion for regularity if $R$ has…

We study equivariant modules over $GL(V)$ over the polynomial ring $R = Sym V$. We introduce for every partition $\lambda$ the elementary equivariant module $M_{\lambda}$. Then we prove that any finitely generated equivariant module admits…

We present a simple proof of the fact that the base (and independence) polytope of a rank $n$ regular matroid over $m$ elements has an extension complexity $O(mn)$.

Let $M,N$ be finitely generated modules over a local complete intersection $R$. Assume that for each $i>0$, $\mathrm{Tor}^R_i(M,N)=0$. We prove that the cohomological support of $M\otimes_R N$ (in the sense of Avramov-Buchweitz) is equal to…

Let $R$ be a Gorenstein local ring with maximal ideal $\mathfrak{m}$ satisfying $\mathfrak{m}^3=0\ne\mathfrak{m}^2$. Set $k=R/\mathfrak{m}$ and $e=\text{rank}_{k}(\mathfrak{m}/\mathfrak{m}^2)$. If $e>2$ and $M$, $N$ are finitely generated…

Let $M$ be a finitely generated module over a ring $\Lambda$. With certain mild assumptions on $\Lambda$, it is proven that $M$ is a reflexive $\Lambda$-module, once $M \cong M^{**}$ as a $\Lambda$-module.

We introduce the notion of multiplication kernels of birational and $D$-module type and give various examples. We also introduce the notion of a semi-classical multiplication kernel associated with an integrable system and discuss its…

Modular structure is pervasive in many complex networks of interactions observed in natural, social and technological sciences. Its study sheds light on the relation between the structure and function of complex systems. Generally speaking,…

The subword complexity of a finite word $w$ of length $N$ is a function which associates to each $n\le N$ the number of all distinct subwords of $w$ having the length $n$. We define the \emph{maximal complexity} C(w) as the maximum of the…

Depth is a complexity measure for natural systems of the kind studied in statistical physics and is defined in terms of computational complexity. Depth quantifies the length of the shortest parallel computation required to construct a…

Complexity is a multi-faceted phenomenon, involving a variety of features including disorder, nonlinearity, and self-organisation. We use a recently developed rigorous framework for complexity to understand measures of complexity. We…

A $(G,n)$-complex is an $n$-dimensional CW-complex with fundamental group $G$ and whose universal cover is $(n-1)$-connected. If $G$ has periodic cohomology then, for appropriate $n$, we show that there is a one-to-one correspondence…

In the framework of computational complexity and in an effort to define a more natural reduction for problems of equivalence, we investigate the recently introduced kernel reduction, a reduction that operates on each element of a pair…

Neural networks readily learn a subset of the modular arithmetic tasks, while failing to generalize on the rest. This limitation remains unmoved by the choice of architecture and training strategies. On the other hand, an analytical…

Let $(R,\mathfrak{m})$ be a Noetherian local ring and $\widehat{R}$ its $\mathfrak{m}$-adic completion. We study the problem of determining when a finitely generated $\widehat{R}$-module arises from an $R$-module, i.e., when it is…

It is a well known that, for odd $n$, the number of subsets of $\{1,2,\dots,n\}$ the sum of whose elements is divisible by $n$ equals the number of binary necklaces of length $n$. In this paper generalize this result in two directions. On…

Measurement incompatibility is one of the basic aspects of quantum theory. Here we study the structure of the set of compatible -- i.e. jointly measurable -- measurements. We are interested in whether or not there exist compatible…