Nonisomorphic Ordered Sets with Arbitrarily Many Ranks That Produce Equal Decks

We prove that for any $n$ there is a pair $(P_1 ^n , P_2 ^n )$ of nonisomorphic ordered sets such that $P_1 ^n$ and $P_2 ^n$ have equal maximal and minimal decks, equal neighborhood decks, and there are $n+1$ ranks $k_0 , \ldots , k_n$ such that for each $i$ the decks obtained by removing the points of rank $k_i$ are equal. The ranks $k_1 , \ldots , k_n$ do not contain extremal elements and at each of the other ranks there are elements whose removal will produce isomorphic cards. Moreover, we show that such sets can be constructed such that only for ranks $1$ and $2$, both without extremal elements, the decks obtained by removing the points of rank $r_i$ are not equal.Comment: 30 pages, 6 figures, straight LaTe

Isomorphic pastings and the two possible structures for a pair of graphs having the same deck

When G denotes a graph, the unlabeled subgraph obtained by deleting a vertex from G is called a card of G and the collection of all cards of G is the deck of G. A graph having the same deck as G is called a hypomorph of G. A graph is called reconstructible if it is isomorphic to all its hypomorphs. Reconstruction Conjecture claims that all graphs are reconstructible and it is open. A representation of a hypomorph of G in terms of two of its cards, called pasting, is introduced. Isomorphic pastings of two cards is defined. In the case of a digraph, a card with which the degree triple of the deleted vertex is also given is called a degree associated card or dacard. Dadeck, dareconstructible digraphs, dapastings and isomorphic dapastings based on dacards are defined analogously. DARC claims that all digraphs are dareconstructible and it is also open. Results: Two hypomorphs G and H of a graph are isomorphic if and only if a pair of cards in their common deck is pasted isomorphically in both G and H. Either every pair of cards in their common deck is pasted isomorphically in both G and H, or no pair of cards is pasted isomorphically in both G and H. Results analogous to the above hold for dapastings in dahypomorphs of a digraph. Some results on pastings are proved and two graph parameters are reconstructed. The neighborhood degree quintuple of a vertex and a new family of digraphs are dareconstructible. New approaches for proving the reconstruction conjecture and DARC by the method of contradiction arise.Comment: 20 pages, 8 figure

Integrals and Potentials of Differential 1-forms on the Sierpinski Gasket

We provide a definition of integral, along paths in the Sierpinski gasket K, for differential smooth 1-forms associated to the standard Dirichlet form K. We show how this tool can be used to study the potential theory on K. In particular, we prove: i) a de Rham reconstruction of a 1-form from its periods around lacunas in K; ii) a Hodge decomposition of 1-forms with respect to the Hilbertian energy norm; iii) the existence of potentials of smooth 1-forms on a suitable covering space of K. We finally show that this framework provides versions of the de Rham duality theorem for the fractal K.Comment: Some proofs have been clarified, reference to previous literature is now more accurate, 33 pages, 6 figure

Reconstruction of Finite Truncated Semi-Modular Lattices

AbstractIn this paper, we prove reconstruction results for truncated lattices. The main results are that truncated lattices that contain a 4-crown and truncated semi-modular lattices are reconstructible. Reconstruction of the truncated lattices not covered by this work appears challenging. Indeed, the remaining truncated lattices possess very little lattice-typical structure. This seems to indicate that further progress on the reconstruction of truncated lattices is closely correlated with progress on reconstructing ordered sets in general

A framework for hull form reverse engineering and geometry integration into numerical simulations

The thesis presents a ship hull form specific reverse engineering and CAD integration framework. The reverse engineering part proposes three alternative suitable reconstruction approaches namely curves network, direct surface fitting, and triangulated surface reconstruction. The CAD integration part includes surface healing, region identification, and domain preparation strategies which used to adapt the CAD model to downstream application requirements. In general, the developed framework bridges a point cloud and a CAD model obtained from IGES and STL file into downstream applications