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

Web matrices : structural properties and generating combinatorial identities

In this paper we present new results for the combinatorics of web diagrams and web worlds. These are discrete objects that arise in the physics of calculating scattering amplitudes in non-abelian gauge theories. Web-colouring and web-mixing matrices (collectively known as web matrices) are indexed by ordered pairs of web-diagrams and contain information relating the number of colourings of the first web diagram that will produce the second diagram. We introduce the black diamond product on power series and show how it determines the web-colouring matrix of disjoint web worlds. Furthermore, we show that combining known physical results with the black diamond product gives a new technique for generating combinatorial identities. Due to the complicated action of the product on power series, the resulting identities appear highly non-trivial. We present two results to explain repeated entries that appear in the web matrices. The first of these shows how diagonal web matrix entries will be the same if the comparability graphs of their associated decomposition posets are the same. The second result concerns general repeated entries in conjunction with a flipping operation on web diagrams. We present a combinatorial proof of idempotency of the web-mixing matrices, previously established using physical arguments only. We also show how the entries of the square of the web-colouring matrix can be achieved by a linear transformation that maps the standard basis for formal power series in one variable to a sequence of polynomials. We look at one parameterized web world that is related to indecomposable permutations and show how determining the web-colouring matrix entries in this case is equivalent to a combinatorics on words problem

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

Online algorithms for partially ordered sets

Partially ordered sets (posets) have various applications in computer science ranging from database systems to distributed computing. Content-based routing in publish/subscribe systems is a major poset use case. Content-based routing requires efficient poset online algorithms, including efficient insertion and deletion algorithms. We study the query and total complexities of online operations on posets and poset-like data structures. The main data structures considered are the incidence matrix, Siena poset, ChainMerge, and poset-derived forest. The contributions of this thesis are twofold: First, we present an online adaptation of the ChainMerge data structure as well as several novel poset-derived forest variants. We study the effectiveness of a first-fit-equivalent ChainMerge online insertion algorithm and show that it performs close to optimal query-wise while requiring less CPU processing in a benchmark setting. Second, we present the results of an empirical performance evaluation. In the evaluation we compare the data structures in terms of query complexity and total complexity. The results indicate ChainMerge as the best structure overall. The incidence matrix, although simple, excels in some benchmarks. Poset-derived forest is very fast overall if a 'true' poset data structure is not a requirement. Placing elements in smaller poset-derived forests and then merging them is an efficient way to construct poset-derived forests. Lazy evaluation for poset-derived forests shows some promise as well

Minimal dominating sets enumeration with FPT-delay parameterized by the degeneracy and maximum degree

At STOC 2002, Eiter, Gottlob, and Makino presented a technique called ordered generation that yields an $n^{O(d)}$-delay algorithm listing all minimal transversals of an $n$-vertex hypergraph of degeneracy $d$. Recently at IWOCA 2019, Conte, Kant\'e, Marino, and Uno asked whether this XP-delay algorithm parameterized by $d$ could be made FPT-delay parameterized by $d$ and the maximum degree $\Delta$, i.e., an algorithm with delay $f(d,\Delta)\cdot n^{O(1)}$ for some computable function $f$. Moreover, as a first step toward answering that question, they note that the same delay is open for the intimately related problem of listing all minimal dominating sets in graphs. In this paper, we answer the latter question in the affirmative.Comment: 18 pages, 2 figure

Predicting Proteome-Early Drug Induced Cardiac Toxicity Relationships (Pro-EDICToRs) with Node Overlapping Parameters (NOPs) of a new class of Blood Mass-Spectra graphs

The 11th International Electronic Conference on Synthetic Organic Chemistry session Computational ChemistryBlood Serum Proteome-Mass Spectra (SP-MS) may allow detecting Proteome-Early Drug Induced Cardiac Toxicity Relationships (called here Pro-EDICToRs). However, due to the thousands of proteins in the SP identifying general Pro-EDICToRs patterns instead of a single protein marker may represents a more realistic alternative. In this sense, first we introduced a novel Cartesian 2D spectrum graph for SP-MS. Next, we introduced the graph node-overlapping parameters (nopk) to numerically characterize SP-MS using them as inputs to seek a Quantitative Proteome-Toxicity Relationship (QPTR) classifier for Pro-EDICToRs with accuracy higher than 80%. Principal Component Analysis (PCA) on the nopk values present in the QPTR model explains with one factor (F1) the 82.7% of variance. Next, these nopk values were used to construct by the first time a Pro-EDICToRs Complex Network having nodes (samples) linked by edges (similarity between two samples). We compared the topology of two sub-networks (cardiac toxicity and control samples); finding extreme relative differences for the re-linking (P) and Zagreb (M2) indices (9.5 and 54.2 % respectively) out of 11 parameters. We also compared subnetworks with well known ideal random networks including Barabasi-Albert, Kleinberg Small World, Erdos-Renyi, and Epsstein Power Law models. Finally, we proposed Partial Order (PO) schemes of the 115 samples based on LDA-probabilities, F1-scores and/or network node degrees. PCA-CN and LDA-PCA based POs with Tanimoto’s coefficients equal or higher than 0.75 are promising for the study of Pro-EDICToRs. These results shows that simple QPTRs models based on MS graph numerical parameters are an interesting tool for proteome researchThe authors thank projects funded by the Xunta de Galicia (PXIB20304PR and BTF20302PR) and the Ministerio de Sanidad y Consumo (PI061457). González-Díaz H. acknowledges tenure track research position funded by the Program Isidro Parga Pondal, Xunta de Galici