Families of determinantal schemes

Given integers a_0 \le a_1 \le ... \le a_{t+c-2} and b_1 \le ... \le b_t, we denote by W(b;a) \subset Hilb^p(\PP^{n}) the locus of good determinantal schemes X \subset \PP^{n} of codimension c defined by the maximal minors of a t x (t+c-1) homogeneous matrix with entries homogeneous polynomials of degree a_j-b_i. The goal of this short note is to extend and complete the results given by the authors in [10] and determine under weakened numerical assumptions the dimension of W(b;a), as well as whether the closure of W(b;a) is a generically smooth irreducible component of the Hilbert scheme Hilb^p(\PP^{n}).Comment: The non-emptiness of W(b;a) is restated as (2.2) in this version; the codimension c=2 case in (2.5)-(2.6) is reconsidered, and c > 2 (c > 3) is now an assumption in (2.16)-(2.17). 13 page

Ideals generated by submaximal minors

The goal of this paper is to study irreducible families W(b;a) of codimension 4, arithmetically Gorenstein schemes X of P^n defined by the submaximal minors of a t x t matrix A with entries homogeneous forms of degree a_j-b_i. Under some numerical assumption on a_j and b_i we prove that the closure of W(b;a) is an irreducible component of Hilb^{p(x)}(P^n), we show that Hilb^{p(x)}(P^n) is generically smooth along W(b;a) and we compute the dimension of W(b;a) in terms of a_j and b_i. To achieve these results we first prove that X is determined by a regular section of the twisted conormal sheaf I_Y/I^2_Y(s) where s=deg(det(A)) and Y is a codimension 2, arithmetically Cohen-Macaulay scheme of P^n defined by the maximal minors of the matrix obtained deleting a suitable row of A.Comment: 22 page

The representation type of determinantal varieties

This work is entirely devoted to construct huge families of indecomposable arithmetically Cohen-Macaulay (resp. Ulrich) sheaves $\mathcal{E}$ of arbitrary high rank on a general standard (resp. linear) determinantal scheme $X \subset \mathbb{P}^n$ of codimension $c \geq 1, n-c \geq 1$ and defined by the maximal minors of a $t \times(t+c-1)$ homogeneous matrix $\mathcal{A}$. The sheaves $\mathcal{E}$ are constructed as iterated extensions of sheaves of lower rank. As applications: (1) we prove that any general standard determinantal scheme $X \subset \mathbb{P}^n$ is of wild representation type provided the degrees of the entries of the matrix $\mathcal{A}$ satisfy some weak numerical assumptions; and (2) we determine values of $t, n$ and $n-c$ for which a linear standard determinantal scheme $X \subset \mathbb{P}^n$ is of wild representation type with respect to the much more restrictive category of its indecomposable Ulrich sheaves, i.e. $X$ is of Ulrich wild representation type

Threading of Unconcatenated Ring Polymers at High Concentrations: Double-Folded vs Time-Equilibrated Structures

Unconcatenated ring polymers in concentrated solutions and melt are remarkably well described as double-folded conformations on randomly branched primitive trees. This picture though contrasts recent evidence for extensive intermingling between close-by rings in the form of long-lived topological constraints or threadings. Here, we employ the concept of ring minimal surface to quantify the extent of threadings in polymer solutions of the double-folded rings vs rings in equilibrated molecular dynamics computer simulations. Our results show that the double-folded ring polymers are significantly less threaded compared to their counterparts at equilibrium. Second, threadings form through a slow process whose characteristic time-scale is of the same order of magnitude as that of the diffusion of the rings in solution. These findings are robust, being based on universal (model-independent) observables as the average fraction of threaded length or the total penetrations between close-by rings and the corresponding distribution functions

ToyArchitecture: Unsupervised Learning of Interpretable Models of the World

Research in Artificial Intelligence (AI) has focused mostly on two extremes: either on small improvements in narrow AI domains, or on universal theoretical frameworks which are usually uncomputable, incompatible with theories of biological intelligence, or lack practical implementations. The goal of this work is to combine the main advantages of the two: to follow a big picture view, while providing a particular theory and its implementation. In contrast with purely theoretical approaches, the resulting architecture should be usable in realistic settings, but also form the core of a framework containing all the basic mechanisms, into which it should be easier to integrate additional required functionality. In this paper, we present a novel, purposely simple, and interpretable hierarchical architecture which combines multiple different mechanisms into one system: unsupervised learning of a model of the world, learning the influence of one's own actions on the world, model-based reinforcement learning, hierarchical planning and plan execution, and symbolic/sub-symbolic integration in general. The learned model is stored in the form of hierarchical representations with the following properties: 1) they are increasingly more abstract, but can retain details when needed, and 2) they are easy to manipulate in their local and symbolic-like form, thus also allowing one to observe the learning process at each level of abstraction. On all levels of the system, the representation of the data can be interpreted in both a symbolic and a sub-symbolic manner. This enables the architecture to learn efficiently using sub-symbolic methods and to employ symbolic inference.Comment: Revision: changed the pdftitl

AI-discovery of a new charging protocol in a micromaser quantum battery

We propose a general computational framework for optimizing model-dependent parameters in quantum batteries (QB). We apply this method to two different charging scenarios in the micromaser QB and we discover a new charging protocol for stabilizing the battery in upper-laying Hilbert space chambers in a controlled and automatic way. This protocol is found to be stable and robust, and it leads to an improved charging efficiency in micromaser QBs. Moreover, our optimization framework is highly versatile and efficient, holding great promise for the advancement of QB technologies at all scales.Comment: 15 pages, 7 figure

On the normal sheaf of determinantal varieties

Let X be a standard determinantal scheme X of P^n of codimension c, i.e. a scheme defined by the maximal minors of a tx(t+c-1)homogeneous polynomial matrix A. In this paper, we study the main features of its normal sheaf \shN_X. We prove that under some mild restrictions: (1) there exists a line bundle \shL on X-Sing(X) such that \shN_X \otimes \shL is arithmetically Cohen–Macaulay and, even more, it is Ulrich whenever the entries of A are linear forms, (2) \shN_X is simple (hence, indecomposable) and, finally, (3) \shN_X is \mu-(semi)stable provided the entries of A are linear form