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Transitive points in CR-dynamical systems
Iztok Banič, Goran Erceg, Sina Greenwood, Judy A. Kennedy, 2023, original scientific article

Abstract: We study different types of transitive points in CR-dynamical systems with closed relations G on compact metric spaces X. We also introduce transitive and dense orbit transitive CR-dynamical systems and discuss their properties and the relations between them. This generalizes the notion of transitive topological dynamical systems.
Keywords: dynamical systems, transitivity, closed relatio, Mahavier product
Published in DKUM: 17.06.2024; Views: 56; Downloads: 2
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Distance formula for direct-co-direct product in the case of disconnected factors
Aleksander Kelenc, Iztok Peterin, 2023, original scientific article

Abstract: Direktni-ko-direktni produkt ▫$G\circledast H$▫ grafov ▫$G$▫ in ▫$H$▫ je graf na množizi vozlišč ▫$V(G)\times V(H)$▫. Vozlišči ▫$(g,h)$▫ in ▫$(g',h')$▫ sta sosednji, če je ▫$gg'\in E(G)$▫ in ▫$hh'\in E(H)$▫ ali ▫$gg'\notin E(G)$▫ in ▫$hh'\notin E(H)$▫. Naj bo največ eden izmed faktorjev ▫$G$▫ in ▫$H$▫ povezan. Pokažemo da je razdalja med dvema vozliščema v ▫$G\circledast H$▫ omejena s tri, razen v majhnem številu izjem. Vse izjeme so natančno popisane, kar prinese razdaljno formulo za ▫$G\circledast H$▫.
Keywords: direktni-ko-direktni produkt, razdalja, ekscentričnost, nepovezan graf, direct-co-direct product, distance, eccentricity, disconnected graphs
Published in DKUM: 21.05.2024; Views: 48; Downloads: 1
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The essential facilities doctrine, intellectual property rights, and access to big data
Rok Dacar, 2023, original scientific article

Abstract: This paper analyzes the criteria for applying the essential facilities doctrine to intellectual property rights and the possibility of applying it in cases where Big Data is the alleged essential facility. It aims to answer the research question: ‘‘What are the specifics of the intellectual property criteria in essential facilities cases and are these criteria applicable to Big Data?’’ It points to the semantic openness of the ‘‘new product’’ and ‘‘technical progress’’ conditions that have been developed for assessing whether an intellectual property right constitutes an essential facility. The paper argues that the intellectual property criteria are not applicable in all access to Big Data cases because Big Data is not necessarily protected by copyright. While a set of Big Data could be protected by copyright if certain conditions are met, even in such cases the lack of intrinsic value of Big Data significantly limits the applicability of the intellectual property criteria.
Keywords: essential facilities doctrine, intellectual property rights, big data, new product condition, technical progress condition
Published in DKUM: 11.04.2024; Views: 129; Downloads: 10
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General Position Sets in Two Families of Cartesian Product Graphs
Danilo Korže, Aleksander Vesel, 2023, original scientific article

Abstract: For a given graph G, the general position problem asks for the largest set of vertices S⊆V(G) , such that no three distinct vertices of S belong to a common shortest path of G. The general position problem for Cartesian products of two cycles as well as for hypercubes is considered. The problem is completely solved for the first family of graphs, while for the hypercubes, some partial results based on reduction to SAT are given.
Keywords: general position set, cartesian product, hypercube, SAT
Published in DKUM: 02.04.2024; Views: 181; Downloads: 20
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The Lelek fan as the inverse limit of intervals with a single set-valued bonding function whose graph is an arc
Iztok Banič, Goran Erceg, Judy A. Kennedy, 2023, original scientific article

Abstract: We consider a family of inverse limits of inverse sequences of closed unit intervals with a single upper semi-continuous set-valued bonding function whose graph is an arc; the graph is the union of two line segments in [0,1]2, both of which contain the origin (0, 0) and have positive slope. One of the segments extends to the top-boundary and the other to the right side boundary of [0,1] x [0,1]. We show that there is a large subfamily F of these bonding functions such that for each fEF, the inverse limit of the inverse sequence of closed unit intervals using f as a single bonding function is homeomorphic to the Lelek fan.
Keywords: fan, cantor fan, Lelek fan, closed relation, Mahavier product, inverse limit, set-valued function
Published in DKUM: 14.03.2024; Views: 487; Downloads: 530
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A new alternative to Szeged, Mostar, and PI polynomials : the SMP polynomials
Martin Knor, Niko Tratnik, 2023, original scientific article

Abstract: Szeged-like topological indices are well-studied distance-based molecular descriptors, which include, for example, the (edge-)Szeged index, the (edge-)Mostar index, and the (vertex-)PI index. For these indices, the corresponding polynomials were also defined, i.e., the (edge-)Szeged polynomial, the Mostar polynomial, the PI polynomial, etc. It is well known that, by evaluating the first derivative of such a polynomial at x = 1, we obtain the related topological index. The aim of this paper is to introduce and investigate a new graph polynomial of two variables, which is called the SMP polynomial, such that all three vertex versions of the above-mentioned indices can be easily calculated using this polynomial. Moreover, we also define the edge-SMP polynomial, which is the edge version of the SMP polynomial. Various properties of the new polynomials are studied on some basic families of graphs, extremal problems are considered, and several open problems are stated. Then, we focus on the Cartesian product, and we show how the (edge-)SMP polynomial of the Cartesian product of n graphs can be calculated using the (weighted) SMP polynomials of its factors.
Keywords: SMP polynomial, edge-SMP polynomial, Cartesian product, Szeged index, Mostar index, PI index
Published in DKUM: 09.02.2024; Views: 202; Downloads: 15
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Hamiltonicity of certain cartesian products of graphs
Tjaša Paj Erker, 2022, original scientific article

Abstract: A graph is Hamiltonian if it contains a spanning cycle. In this paper, we examine the hamiltonicity of the Cartesian product of a tree with a path. We offer sufficient conditions for the Cartesian product of a tree with a path to be Hamiltonian.
Keywords: hamiltonicity, cartesian product, path factor
Published in DKUM: 26.10.2023; Views: 341; Downloads: 9
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Jordan maps and zero Lie product determined algebras
Matej Brešar, 2022, original scientific article

Abstract: Let ▫$A$▫ be an algebra over a field ▫$F$▫ with ▫$\mathrm{char} (F) \ne 2$▫. If ▫$A$▫ is generated as an algebra by ▫$[[A,A],[A,A]]$▫, then for every skew-symmetric bilinear map ▫$\Phi:A \times A \to X$▫, where ▫$X$▫ is an arbitrary vector space over ▫$F$▫, the condition that ▫$\Phi(x^2,x)=0$▫ for all ▫$x \in A$▫ implies that ▫$\Phi(xy,z) +\Phi(zx,y) + \Phi(yz,x)=0$▫ for all ▫$x,y,z \in A$▫. This is applicable to the question of whether ▫$A$▫ is zero Lie product determined, and is also used in proving that a Jordan homomorphism from ▫$A$▫ onto a semiprime algebra ▫$B$▫ is the sum of a homomorphism and an antihomomorphism.
Keywords: bilinear map, zero Lie product determined algebra, derivation, Jordan derivation, Jordan homomorphism, functional identity
Published in DKUM: 18.08.2023; Views: 337; Downloads: 29
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Orientable domination in product-like graphs
Sarah Anderson, Boštjan Brešar, Sandi Klavžar, Kirsti Kuenzel, Douglas F. Rall, 2023, original scientific article

Abstract: The orientable domination number, ▫${\rm DOM}(G)$▫, of a graph ▫$G$▫ is the largest domination number over all orientations of ▫$G$▫. In this paper, ▫${\rm DOM}$▫ is studied on different product graphs and related graph operations. The orientable domination number of arbitrary corona products is determined, while sharp lower and upper bounds are proved for Cartesian and lexicographic products. A result of Chartrand et al. from 1996 is extended by establishing the values of ▫${\rm DOM}(K_{n_1,n_2,n_3})$▫ for arbitrary positive integers ▫$n_1,n_2$▫ and ▫$n_3$▫. While considering the orientable domination number of lexicographic product graphs, we answer in the negative a question concerning domination and packing numbers in acyclic digraphs posed in [Domination in digraphs and their direct and Cartesian products, J. Graph Theory 99 (2022) 359-377].
Keywords: digraph, domination, orientable domination number, packing, graph product, corona graph
Published in DKUM: 09.08.2023; Views: 406; Downloads: 29
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Product assembly assistance system based on pick-to-light and computer vision technology
Darko Hercog, Primož Bencak, Uroš Vincetič, Tone Lerher, 2022, original scientific article

Abstract: Product assembly is often one of the last steps in the production process. Product assembly is often carried out by workers (assemblers) rather than robots, as it is generally challenging to adapt automation to any product. When assembling complex products, it can take a long time before the assembler masters all the steps and can assemble the product independently. Training time has no added value; therefore, it should be reduced as much as possible. This paper presents a custom-developed system that enables the guided assembly of complex and diverse products using modern technologies. The system is based on pick-to-light (PTL) modules, used primarily in logistics as an additional aid in the order picking process, and Computer Vision technology. The designed system includes a personal computer (PC), several custom-developed PTL modules and a USB camera. The PC with a touchscreen visualizes the assembly process and allows the assembler to interact with the system. The developed PC application guides the operator through the assembly process by showing all the necessary assembly steps and parts. Two-step verification is used to ensure that the correct part is picked out of the bin, first by checking that the correct pushbutton on the PTL module has been pressed and second by using a camera with a Computer Vision algorithm. The paper is supported by a use case demonstrating that the proposed system reduces the assembly time of the used product. The presented solution is scalable and flexible as it can be easily adapted to show the assembly steps of another product.
Keywords: product assembly, PTL, pick-to-light, computer vision, LabVIEW, performance analysis
Published in DKUM: 31.07.2023; Views: 417; Downloads: 23
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