A universal sequence of integers generating balanced Steinhaus figures modulo an odd number

In this paper, we partially solve an open problem, due to J.C. Molluzzo in 1976, on the existence of balanced Steinhaus triangles modulo a positive integer $n$, that are Steinhaus triangles containing all the elements of $\mathbb{Z}/n\mathbb{Z}$ with the same multiplicity. For every odd number $n$, we build an orbit in $\mathbb{Z}/n\mathbb{Z}$, by the linear cellular automaton generating the Pascal triangle modulo $n$, which contains infinitely many balanced Steinhaus triangles. This orbit, in $\mathbb{Z}/n\mathbb{Z}$, is obtained from an integer sequence called the universal sequence. We show that there exist balanced Steinhaus triangles for at least $2/3$ of the admissible sizes, in the case where $n$ is an odd prime power. Other balanced Steinhaus figures, such as Steinhaus trapezoids, generalized Pascal triangles, Pascal trapezoids or lozenges, also appear in the orbit of the universal sequence modulo $n$ odd. We prove the existence of balanced generalized Pascal triangles for at least $2/3$ of the admissible sizes, in the case where $n$ is an odd prime power, and the existence of balanced lozenges for all admissible sizes, in the case where $n$ is a square-free odd number.Comment: 30 pages ; 10 figure

Unfolding Concerns about Augmented Reality Technologies: A Qualitative Analysis of User Perceptions

Augmented reality (AR) greatly diffused into the public consciousness in the last years, especially due to the success of mobile applications like Pokémon Go. However, only few people experienced different forms of augmented reality like head-mounted displays (HMDs). Thus, people have only a limited actual experience with AR and form attitudes and perceptions towards this technology only partially based on actual use experiences, but mainly based on hearsay and narratives of others, like the media or friends. Thus, it is highly difficult for developers and product managers of AR solutions to address the needs of potential users. Therefore, we disentangle the perceptions of individuals with a focus on their concerns about AR. Perceived concerns are an important factor for the acceptance of new technologies. We address this research topic based on twelve intensive interviews with laymen as well as AR experts and analyze them with a qualitative research method

Plane four-regular graphs with vertex-to-vertex unit triangles

AbstractFor the smallest number of non-overlapping vertex-to-vertex unit triangles in the plane it is proved ⩽42 in general, and ⩽3800 if additional triangles are not allowed

Augmented Reality in Information Systems Research: A Systematic Literature Review

Augmented Reality (AR) is one of the most prominent emerging technologies recently. This increase in recognition has happened predominantly because of the success of the smartphone game Pokémon Go . But research on AR is not a new strand of literature. Especially computer scientists investigate different technological solutions and areas of application for almost 30 years. This systematic literature review aims at analyzing, synthesizing and categorizing this strand of research in the information systems (IS) domain. We follow an established methodology for conducting the literature review ensuring rigor and replicability. We apply a keyword and backward search resulting in 28 and 118 articles, respectively. Results are categorized with regard to the focus of the research and the domain of the application being investigated. We show that research on user behavior is underrepresented in the current IS literature on AR compared to technical research, especially in the domains gaming and smartphone browsers

Note on a Conjecture of Wegner

The optimal packings of n unit discs in the plane are known for those natural numbers n, which satisfy certain number theoretic conditions. Their geometric realizations are the extremal Groemer packings (or Wegner packings). But an extremal Groemer packing of n unit discs does not exist for all natural numbers n and in this case, the number n is called exceptional. We are interested in number theoretic characterizations of the exceptional numbers. A counterexample is given to a conjecture of Wegner concerning such a characterization. We further give a characterization of the exceptional numbers, whose shape is closely related to that of Wegner's conjecture.Comment: 5 pages; Contributions to Algebra and Geometry, Vol.52 No1 April 201

On h-perfect numbers

Let σ(x) denote the sum of the divisors of x. The diophantine equation σ(x) + σ(y) = 2(x + y) equalizes the abundance and deficiency of x and y. For x = n and y = hn the solutions n are called h-perfect since the classical perfect numbers occur as solutions for h = 1. Some results on h-perfect numbers are determined. Keywords: perfect numbers, amicable number

How Privacy Concerns and Trust and Risk Beliefs Influence Users’ Intentions to Use Privacy-Enhancing Technologies - The Case of Tor

Due to an increasing collection of personal data by internet companies and several data breaches, research related to privacy gained importance in the last years in the information systems domain. Privacy concerns can strongly influence users’ decision to use a service. The Internet Users Information Privacy Concerns (IUIPC) construct is one operationalization to measure the impact of privacy concerns on the use of technologies. However, when applied to a privacy enhancing technology (PET) such as an anonymization service the original rationales do not hold anymore. In particular, an inverted impact of trusting and risk beliefs on behavioral intentions can be expected. We show that the IUIPC model needs to be adapted for the case of PETs. In addition, we extend the original causal model by including trust beliefs in the anonymization service itself. A survey among 124 users of the anonymization service Tor shows that they have a significant effect on the actual use behavior of the PET

Regular Steinhaus graphs of odd degree

A Steinhaus matrix is a binary square matrix of size $n$ which is symmetric, with diagonal of zeros, and whose upper-triangular coefficients satisfy $a_{i,j}=a_{i-1,j-1}+a_{i-1,j}$ for all $2\leq i<j\leq n$. Steinhaus matrices are determined by their first row. A Steinhaus graph is a simple graph whose adjacency matrix is a Steinhaus matrix. We give a short new proof of a theorem, due to Dymacek, which states that even Steinhaus graphs, i.e. those with all vertex degrees even, have doubly-symmetric Steinhaus matrices. In 1979 Dymacek conjectured that the complete graph on two vertices $K_2$ is the only regular Steinhaus graph of odd degree. Using Dymacek's theorem, we prove that if $(a_{i,j})_{1\leq i,j\leq n}$ is a Steinhaus matrix associated with a regular Steinhaus graph of odd degree then its sub-matrix $(a_{i,j})_{2\leq i,j\leq n-1}$ is a multi-symmetric matrix, that is a doubly-symmetric matrix where each row of its upper-triangular part is a symmetric sequence. We prove that the multi-symmetric Steinhaus matrices of size $n$ whose Steinhaus graphs are regular modulo 4, i.e. where all vertex degrees are equal modulo 4, only depend on $\lceil \frac{n}{24}\rceil$ parameters for all even numbers $n$, and on $\lceil \frac{n}{30}\rceil$ parameters in the odd case. This result permits us to verify the Dymacek's conjecture up to 1500 vertices in the odd case.Comment: 16 page

Blocking Coloured Point Sets

This paper studies problems related to visibility among points in the plane. A point $x$ \emph{blocks} two points $v$ and $w$ if $x$ is in the interior of the line segment $\bar{vw}$. A set of points $P$ is \emph{$k$-blocked} if each point in $P$ is assigned one of $k$ colours, such that distinct points $v,w\in P$ are assigned the same colour if and only if some other point in $P$ blocks $v$ and $w$. The focus of this paper is the conjecture that each $k$-blocked set has bounded size (as a function of $k$). Results in the literature imply that every 2-blocked set has at most 3 points, and every 3-blocked set has at most 6 points. We prove that every 4-blocked set has at most 12 points, and that this bound is tight. In fact, we characterise all sets $\{n_1,n_2,n_3,n_4\}$ such that some 4-blocked set has exactly $n_i$ points in the $i$-th colour class. Amongst other results, for infinitely many values of $k$, we construct $k$-blocked sets with $k^{1.79...}$ points