465 research outputs found
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 , we show
that , the Waldschmidt constant of , can be expressed as
the optimal solution to a linear program constructed from the primary
decomposition of . By applying results from fractional graph theory, we can
then express in terms of the fractional chromatic number of
a hypergraph also constructed from the primary decomposition of . Moreover,
expressing as the solution to a linear program enables us
to prove a Chudnovsky-like lower bound on , 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 with few
components compared to , 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
The Waldschmidt constant of a standard k-configuration in P^2
A k-configuration of type (d(1), ..., d(s)), where 1 <= d < ... < d(s) are integers, is a set of points in P2 that has a number of algebraic and geometric properties. For example, the graded Betti numbers and Hilbert functions of all k-configurations in P-2 are determined by the type (d(1),........., ds). However the Waldschmidt constant of a k-configuration in P-2 of the same type may vary. In this paper, we find that the Waldschmidt constant of a k-configuration in P-2 of type (d(1),..., d(s)) with d(1) >= s >= 1 is s. Then we deal with the Waldschmidt constants of standard k-configurations in P2 of type (a), (a, b), and (a, b, c) with a >= 1. In particular, we prove that the Waldschmidt constant of a standard k-configuration in P-2 of type (1, b, c) with c >= 2b+2 does not depend on c
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
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