Steiner systems and configurations of points

The aim of this paper is to make a connection between design theory and algebraic geometry/commutative algebra. In particular, given any Steiner System S(t, n, v) we associate two ideals, in a suitable polynomial ring, defining a Steiner configuration of points and its Complement. We focus on the latter, studying its homological invariants, such as Hilbert Function and Betti numbers. We also study symbolic and regular powers associated to the ideal defining a Complement of a Steiner configuration of points, finding its Waldschmidt constant, regularity, bounds on its resurgence and asymptotic resurgence. We also compute the parameters of linear codes associated to any Steiner configuration of points and its Complement

The Waldschmidt constant for squarefree monomial ideals

Given a squarefree monomial ideal $I \subseteq R =k[x_1,\ldots,x_n]$, we show that $\widehat\alpha(I)$, the Waldschmidt constant of $I$, can be expressed as the optimal solution to a linear program constructed from the primary decomposition of $I$. By applying results from fractional graph theory, we can then express $\widehat\alpha(I)$ in terms of the fractional chromatic number of a hypergraph also constructed from the primary decomposition of $I$. Moreover, expressing $\widehat\alpha(I)$ as the solution to a linear program enables us to prove a Chudnovsky-like lower bound on $\widehat\alpha(I)$, thus verifying a conjecture of Cooper-Embree-H\`a-Hoefel for monomial ideals in the squarefree case. As an application, we compute the Waldschmidt constant and the resurgence for some families of squarefree monomial ideals. For example, we determine both constants for unions of general linear subspaces of $\mathbb{P}^n$ with few components compared to $n$, and we find the Waldschmidt constant for the Stanley-Reisner ideal of a uniform matroid.Comment: 26 pages. This project was started at the Mathematisches Forschungsinstitut Oberwolfach (MFO) as part of the mini-workshop "Ideals of Linear Subspaces, Their Symbolic Powers and Waring Problems" held in February 2015. Comments are welcome. Revised version corrects some typos, updates the references, and clarifies some hypotheses. To appear in the Journal of Algebraic Combinatoric

Hadamard Products of Symbolic Powers and Hadamard Fat Grids

In this paper we address the question if, for points P, Q ? P-2, I(P)I-m(*)(Q)(n) = I(P(*)Q)(m+n-1 )and we obtain different results according to the number of zero coordinates in P and Q. Successively, we use our results to define the so called Hadamard fat grids, which are the result of the Hadamard product of two sets of collinear points with given multiplicities. The most important invariants of Hadamard fat grids, as minimal resolution, Waldschmidt constant and resurgence, are then computed

Hilbert functions of schemes of double and reduced points

It remains an open problem to classify the Hilbert functions of double points in P2. Given a valid Hilbert function Hof a zero-dimensional scheme in P2, we show how to construct a set of fat points Z⊆P2of double and reduced points such that HZ, the Hilbert function of Z, is the same as H. In other words, we show that any valid Hilbert function Hof a zero-dimensional scheme is the Hilbert function of a set a positive number of double points and some reduced points. Fo r some families of valid Hilbert functions, we are also able to show that His the Hilbert function of only double points. In addition, we give necessary and sufficient conditions for the Hilbert function of a scheme of a double points, or double points plus one additional reduced point, to be the Hilbert function of points with support on a star configuration of lines

Steiner configurations ideals: Containment and colouring

Given a homogeneous ideal I ⊆ k[x0, …, xn ], the Containment problem studies the relation between symbolic and regular powers of I, that is, it asks for which pairs m, r ∈ N, I(m) ⊆ Ir holds. In the last years, several conjectures have been posed on this problem, creating an active area of current interests and ongoing investigations. In this paper, we investigated the Stable Harbourne Conjecture and the Stable Harbourne–Huneke Conjecture, and we show that they hold for the defining ideal of a Complement of a Steiner configuration of points in Pnk. We can also show that the ideal of a Complement of a Steiner Configuration of points has expected resurgence, that is, its resurgence is strictly less than its big height, and it also satisfies Chudnovsky and Demailly’s Conjectures. Moreover, given a hypergraph H, we also study the relation between its colourability and the failure of the containment problem for the cover ideal associated to H. We apply these results in the case that H is a Steiner System

Sound absorbing and insulating low-cost panels from end-of-life household materials for the development of vulnerable contexts in circular economy perspective

From a construction point of view, neighborhoods with residents living at or below the poverty threshold are characterized by low energy efficiency buildings, in which people live in acoustic discomfort with no viable options for home improvements, as they usually can not afford the materials and labor costs associated. An alternative to this is to use low-cost insulating elements made of non-conventional materials with acceptable acoustic properties. Given that household materials at their end-of-life (EoLHM) are free of costs and available also to the more disadvantaged population, they can be used to build acoustic panels for such contexts. This approach embraces several benefits since it reduces the amount of waste produced, the footprint deriving from the extraction of new raw materials and, by highlighting the potential of the EoLHM, discourages the abandonment of waste. In this paper, the acoustic properties of EoLHM, such as cardboard, egg-cartons, clothes, metal elements and combinations of them, are investigated by means of the impedance tube technique. The measured sound absorption coefficient and transmission loss have shown that EoLHM can be used for the realization of acoustic panels. However, since none of the analyzed materials shows absorbing and insulating properties at the same time, EoLHM must be wisely selected. This innovative approach supports the circular economy and the improvement for the living condition of low-income households

Conversion of end-of-life household materials into building insulating low-cost solutions for the development of vulnerable contexts: Review and outlook towards a circular and sustainable economy

In a world increasingly aware of the environmental cost of the current production/ consumption model, the use of sustainable practices to reduce our environmental impact as a society becomes imperative. One way to reduce this impact is to increase the reuse of materials that are considered, by current definitions of ”waste”, at their end of life. End-of-Life Household Materials (EoLHM) can be defined as household waste materials that still possess exploitable properties, thus making them suitable for reuse. There are several studies in the literature that address the recycling of these materials. When it comes to their reuse, unfortunately, only a limited number of studies are available. This paper aims to fill this gap by investigating the possibility to convert EoLHM, such as clothes or packaging, into low-cost thermal insulating materials for the improvement of the indoor thermal comfort in buildings, especially for households at risk of suffering from energy poverty. For this purpose, a comprehensive literature review and a qualitative analysis of both commercial and EoLHM are proposed. Commercial thermal insulating materials analysis is used as a reference to measure the performance of EoLHM. Important aspects to be considered when choosing suitable EoLHM for a smart conversion and reuse are also investigated. The most important outcome of this investigation is the comprehension that the conversion of EoLHM into insulating material is possible, and it implies a direct reduction in waste production, with environmental benefits and positive social implications. However, some aspects such as adaptability, life expectancy, collection and storage are, at present, in need of further thinking and development to make the EoLHM reuse and re-conversion processes viable on a large (neighborhood/city) scale

Sustainable and low-cost solutions for thermal and acoustic refurbishment of old buildings

This paper investigates the possibility to realize solutions for buildings thermal and acoustic refurbishment by using end-of-life household materials, such as cardboard, clothes, and egg-boxes. These solutions can be installed to improve the indoor quality in neighborhoods populated by people below the poverty threshold. The considered end-of-life household materials and their combination have been analyzed from the acoustic and thermal points of view. First of all, the sound insulation and the sound absorption properties have been determined by means of an impedance tube. Then, the summer and the winter thermal performances, when coupled to different wall systems, have been investigated analitically. The results suggest that good thermal and acoustic characteristics can be achieved in a contained thickness by coupling end-of-life household materials