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Opis: When manufacturing complicated products where both material and design play a role, especially thin and curved components, it is difcult to maintain accurate dimensions in Selective Laser Melting. Considering these difculties, this article presents the dimensional errors in the fabrication of Ti-6Al-4V discs and their thermomechanics during manufacturing. Various combinations of laser processing parameters were used to fabricate the 2.00 mm thick discs with a diameter of 5.70 mm. It was found that the thickness shortened and the round shape changed to an oval shape for most of the discs. The thickness decreased along the build-up direction from the bottom to the top and formed a taper that increased with increasing energy density (ED). The horizontal diameter of the discs changed slightly, while the vertical diameters changed remarkably with increasing ED. On the other hand, reducing the laser power resulted in a reduction of the roundness error, while it caused a reduction of the thickness. The hatch spacing signifcantly afected the volume of the melt pool and caused a change in the vertical diameter. The central part of the curved surface of the discs became concave and the concavity increased due to the increasing ED.
Ključne besede: dimension, Ti-6Al-4V, lase power, scanning speed, hatch spacing, selective laser melting
Objavljeno v DKUM: 29.03.2024; Ogledov: 219; Prenosov: 9
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Opis: This doctoral dissertation is concerned with aspects on distance related topics in graphs. We study three main topics, namely a recently introduced measure called the Hausdorff distance of graphs and two new graph invariants - the edge metric dimension and the mixed metric dimension of graphs. All three topics are part of the metric graph theory since they are tightly connected with the basic concept of distance between two vertices of a graph. The Hausdorff distance is a relatively new measure of the similarity of graphs. The notion of the Hausdorff distance considers a special kind of common subgraph of the compared graphs and depends on the structural properties outside of the common subgraph. We study the Hausdorff distance between certain families of graphs that often appear in chemical graph theory. Next to a few results for general graphs, we determine formulae for the distance between paths and cycles. Previously, there was no known efficient algorithm for the problem of determining the Hausdorff distance between two trees, and in this dissertation we present a polynomial-time algorithm for it. The algorithm is recursive and it utilizes the divide and conquer technique. As a subtask it also uses a procedure that is based on the well-known graph algorithm for finding a maximum bipartite matching. The edge metric dimension is a graph invariant that deals with distinguishing the edges of a graph. Let $G=(V(G),E(G))$ be a connected graph, let $w \in V(G)$ be a vertex, and let $e=uv \in E(G)$ be an edge. The distance between the vertex $w$ and the edge $e$ is given by $d_G(e,w)=\min\{d_G(u,w),d_G(v,w)\}$. A vertex $w \in V(G)$ distinguishes two edges $e_1,e_2 \in E(G)$ if $d_G(w,e_1) \ne d_G(w,e_2)$. A set $S$ of vertices in a connected graph $G$ is an edge metric generator of $G$ if every two distinct edges of $G$ are distinguished by some vertex of $S$. The smallest cardinality of an edge metric generator of $G$ is called the edge metric dimension and is denoted by $dim_e(G)$. The concept of the edge metric dimension is new. We study its mathematical properties. We make a comparison between the edge metric dimension and the standard metric dimension of graphs while presenting some realization results concerning the two. We prove that computing the edge metric dimension of connected graphs is NP-hard and give some approximation results. Moreover, we present bounds and closed formulae for the edge metric dimension of several classes of graphs. The mixed metric dimension is a graph invariant similar to the edge metric dimension that deals with distinguishing the elements (vertices and edges) of a graph. A vertex $w \in V(G)$ distinguishes two elements of a graph $x,y \in E(G)\cup V(G)$ if $d_G(w,x) \ne d_G(w,y)$. A set $S$ of vertices in a connected graph $G$ is a mixed metric generator of $G$ if every two elements $x,y \in E(G) \cup V(G)$ of $G$, where $x \neq y$, are distinguished by some vertex of $S$. The smallest cardinality of a mixed metric generator of $G$ is called the mixed metric dimension and is denoted by $dim_m(G)$. In this dissertation, we consider the structure of mixed metric generators and characterize graphs for which the mixed metric dimension equals the trivial lower and upper bounds. We also give results on the mixed metric dimension of certain families of graphs and present an upper bound with respect to the girth of a graph. Finally, we prove that the problem of determining the mixed metric dimension of a graph is NP-hard in the general case.
Ključne besede: Hausdorff distance, distance between graphs, graph algorithms, trees, graph similarity, edge metric dimension, edge metric generator, mixed metric dimension, metric dimension
Objavljeno v DKUM: 03.08.2020; Ogledov: 1587; Prenosov: 129
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Opis: The Fibonacci dimension ▫$text{fdim}(G)$▫ of a graph ▫$G$▫ was introduced [in S. Cabello, D. Eppstein, S. Klavžar, The Fibonacci dimension of a graph Electron. J. Combin., 18 (2011) P 55, 23 pp] as the smallest integer ▫$d$▫ such that ▫$G$▫ admits an isometric embedding into ▫$Gamma_d$▫, the ▫$d$▫-dimensional Fibonacci cube. The Fibonacci dimension of the resonance graphs of catacondensed benzenoid systems is studied. This study is inspired by the fact, that the Fibonacci cubes are precisely the resonance graphs of a subclass of the catacondensed benzenoid systems. Our results show that the Fibonacci dimension of the resonance graph of a catacondensed benzenoid system ▫$G$▫ depends on the inner dual of ▫$G$▫. Moreover, we show that computing the Fibonacci dimension can be done in linear time for a graph of this class.
Ključne besede: Fibonaccijeva dimenzija, benzenoidni sistemi, resonančni grafi, algoritem, Fibonacci dimension, benzenoid systems, resonance graphs, algorithm
Objavljeno v DKUM: 10.07.2015; Ogledov: 1171; Prenosov: 87
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Ključne besede: matematika, topologija, dimenzija pokrivanja, posplošena krivulja Sierpińskega, univerzalni prostor, Lipscombov univerzalni prostor, vložitev, razširitev, poln metrični prostor, zaprta vložitev, mathematics, topology, covering dimension, embedding, closed embedding, generalized Sierpiński curve, universal space, Lipscomb universal space, complete metric space, extension
Objavljeno v DKUM: 10.07.2015; Ogledov: 1401; Prenosov: 29
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Opis: Uvedemo teorijo Cantorjevega prostor-časa. V tej teoriji je vsak delec možno interpretirati kot razcep nekega drugega. Nekateri delci so razcepni s protonom in so izraženi s ▫$phioverline{alpha_0}$▫. Če sledimo idejam El Naschieja so limitne množice Kleinove grupe Cantorjeve množice, s Haussdorffovo dimenzijo ▫$phi$▫ ali ▫$frac{1}{phi}, frac{1}{phi^2}, frac{1}{phi^3}...$▫ Z uporabo E-neskončne teorije je masni spekter elementarnih delcev, kot funkcija zlatega reza, v limitni množici Mobius-Kleinove geometrije kvantnega prostor-časa, kot je bilo obravnavano pri Dattu.
Ključne besede: E-neskončna teorija, Hausdorffova dimenzija, Cantorjeva množica, Mobius-Kleinova transformacija, E-infinity theory, Haussdorff dimension, Cantor set, Mobius-Klein transformation
Objavljeno v DKUM: 10.07.2015; Ogledov: 1101; Prenosov: 91
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