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The main purpose of this paper is to provide formulas for the Hilbert-Kunz multiplicity of fiber products and idealization rings. We calculate the Hilbert-Kunz multiplicity of a fiber product $R \times_T S$, with $R$, $S$ and $T$ being Noetherian local rings with common characteristic and residue field. Inspired by a work of Strazzanti and Armengou we calculate the Hilbert-Kunz multiplicity of an idealization ring $R \ltimes M$, where $M$ is a finite $R$-module. We explore certain examples and structural consequences and obtain some bounds for the Hilbert-Kunz multiplicity., Comment: 16 pages - Comments are welcome!

The Internet revolution in 1990, followed by the data-driven and information revolution, has transformed the world as we know it. Nowadays, what seam to be 10 to 20 years ago, a science fiction idea (i.e., machines dominating the world) is seen as possible. This revolution also brought a need for new regulatory practices where user trust and artificial Intelligence (AI) discourse has a central role. This work aims to clarify some misconceptions about user trust in AI discourse and fight the tendency to design vulnerable interactions that lead to further breaches of trust, both real and perceived. Findings illustrate the lack of clarity in understanding user trust and its effects on computer science, especially in measuring user trust characteristics. It argues for clarifying those notions to avoid possible trust gaps and misinterpretations in AI adoption and appropriation.

The rationale of this work is based on the current user trust discourse of Artificial Intelligence (AI). We aim to produce novel HCI approaches that use trust as a facilitator for the uptake (or appropriation) of current technologies. We propose a framework (HCTFrame) to guide non-experts to unlock the full potential of user trust in AI design. Results derived from a data triangulation of findings from three literature reviews demystify some misconceptions of user trust in computer science and AI discourse, and three case studies are conducted to assess the effectiveness of a psychometric scale in mapping potential users' trust breakdowns and concerns. This work primarily contributes to the fight against the tendency to design technical-centered vulnerable interactions, which can eventually lead to additional real and perceived breaches of trust. The proposed framework can be used to guide system designers on how to map and define user trust and the socioethical and organisational needs and characteristics of AI system design. It can also guide AI system designers on how to develop a prototype and operationalise a solution that meets user trust requirements. The article ends by providing some user research tools that can be employed to measure users' trust intentions and behaviours towards a proposed solution.

We consider two-species random sequential adsorption (RSA) in which species A and B adsorb randomly on a lattice with the restriction that opposite species cannot occupy nearest-neighbor sites. When the probability $x_A$ of choosing an A particle for an adsorption trial reaches a critical value $0.626441(1)$, the A species percolates and/or the blocked sites X (those with at least one A and one B nearest neighbor) percolate. Analysis of the size-distribution exponent $\tau$, the wrapping probabilities, and the excess cluster number shows that the percolation transition is consistent with that of ordinary percolation. We obtain an exact result for the low $x_B = 1 - x_A$ jamming behavior: $\theta_A = 1 - x_B +b_2 x_B^2+\mathcal{O}(x_B^3)$, $\theta_B = x_B/(z+1)+\mathcal{O}(x_B^2)$ for a $z$-coordinated lattice, where $\theta_A$ and $\theta_B$ are respectively the saturation coverages of species A and B. We also show how differences between wrapping probabilities of A and X clusters, as well as differences in the number of A and X clusters, can be used to find the transition point accurately. For the one-dimensional case a three-site approximation appears to provide exact results for the coverages.

Relational Keyword Search (R-KwS) systems enable naive/informal users to explore and retrieve information from relational databases without knowing schema details or query languages. These systems take the keywords from the input query, locate the elements of the target database that correspond to these keywords, and look for ways to "connect" these elements using information on referential integrity constraints, i.e., key/foreign key pairs. Although several such systems have been proposed in the literature, most of them only support queries whose keywords refer to the contents of the target database and just very few support queries in which keywords refer to elements of the database schema. This paper proposes LATHE, a novel R-KwS designed to support such queries. To this end, in our work, we first generalize the well-known concepts of Query Matches (QMs) and Candidate Joining Networks (CJNs) to handle keywords referring to schema elements and propose new algorithms to generate them. Then, we introduce an approach to automatically select the CJNs that are more likely to represent the user intent when issuing a keyword query. This approach includes two major innovations: a ranking algorithm for selecting better QMs, yielding the generation of fewer but better CJNs, and an eager evaluation strategy for pruning void useless CJNs. We present a comprehensive set of experiments performed with query sets and datasets previously used in experiments with state-of-the-art R-KwS systems and methods. Our results indicate that LATHE can handle a wider variety of keyword queries while remaining highly effective, even for large databases with intricate schemas.