Solving And Applications Of Multi-Facility Location Problems

This thesis is devoted towards the study and solving of a new class of multi-facility location problems. This class is of a great theoretical interest both in variational analysis and optimization while being of high importance to a variety of practical applications. Optimization problems of this type cannot be reduced to convex programming like, the much more investigated facility location problems with only one center. In contrast, such classes of multi-facility location problems can be described by using DC (difference of convex) programming, which are significantly more involved from both theoretical and numerical viewpoints.In this thesis, we present a new approach to solve multi-facility location problems, which is based on mixed integer programming and algorithms for minimizing differences of convex (DC) functions. We then computationally implement the proposed algorithm on both artificial and real data sets and provide many numerical examples. Finally, some directions and insights for future work are detailed

Validation of the German Classification of Diverticular Disease (VADIS)—a prospective bicentric observational study

Purpose: The German Classification of Diverticular Disease was introduced a few years ago. The aim of this study was to determine whether Classification of Diverticular Disease enables an exact stratification of different types of diverticular disease in terms of course and treatment. Methods: This was a prospective, bicentric observational trial. Patients aged ≥ 18 years with diverticular disease were prospectively included. The primary endpoint was the rate of recurrence within 2 year follow-up. Secondary outcome measures were Gastrointestinal Quality of Life Index, Quality of life measured by SF-36, frequency of gastrointestinal complaints, and postoperative complications. Results: A total of 172 patients were included. After conservative management, 40% of patients required surgery for recurrence in type 1b vs. 80% in type 2a/b (p = 0.04). Sixty percent of patients with type 2a (micro-abscess) were in need of surgery for recurrence vs. 100% of patients with type 2b (macro-abscess) (p = 0.11). Patients with type 2a reached 123 ± 15 points in the Gastrointestinal Quality of Life Index compared with 111 ± 14 in type 2b (p = 0.05) and higher scores in the “Mental Component Summary” scale of SF-36 (52 ± 10 vs. 43 ± 13; p = 0.04). Patients with recurrent diverticulitis without complications (type 3b) had less often painful constipation (30% vs. 73%; p = 0.006) when they were operated compared with conservative treatment. Conclusion: Differentiation into type 2a and 2b based on abscess size seems reasonable as patients with type 2b required surgery while patients with type 2a may be treated conservatively. Sigmoid colectomy in patients with type 3b seems to have gastrointestinal complaints during long-term follow-up. Trial registration: https://www.drks.de ID: DRKS0000557

FPT Constant-Approximations for Capacitated Clustering to Minimize the Sum of Cluster Radii

Clustering with capacity constraints is a fundamental problem that attracted significant attention throughout the years. In this paper, we give the first FPT constant-factor approximation algorithm for the problem of clustering points in a general metric into $k$ clusters to minimize the sum of cluster radii, subject to non-uniform hard capacity constraints. In particular, we give a $(15+\epsilon)$-approximation algorithm that runs in $2^{0(k^2\log k)}\cdot n^3$ time. When capacities are uniform, we obtain the following improved approximation bounds: A (4 + $\epsilon$)-approximation with running time $2^{O(k\log(k/\epsilon))}n^3$, which significantly improves over the FPT 28-approximation of Inamdar and Varadarajan [ESA 2020]; a (2 + $\epsilon$)-approximation with running time $2^{O(k/\epsilon^2 \cdot\log(k/\epsilon))}dn^3$ and a $(1+\epsilon)$-approximation with running time $2^{O(kd\log ((k/\epsilon)))}n^{3}$ in the Euclidean space; and a (1 + $\epsilon$)-approximation in the Euclidean space with running time $2^{O(k/\epsilon^2 \cdot\log(k/\epsilon))}dn^3$ if we are allowed to violate the capacities by (1 + $\epsilon$)-factor. We complement this result by showing that there is no (1 + $\epsilon$)-approximation algorithm running in time $f(k)\cdot n^{O(1)}$, if any capacity violation is not allowed.Comment: Full version of a paper accepted to SoCG 202

Rainbow Turan Methods for Trees

The rainbow Turan number, a natural extension of the well-studied traditionalTuran number, was introduced in 2007 by Keevash, Mubayi, Sudakov and Verstraete. The rainbow Tur ́an number of a graph F , ex*(n, F ), is the largest number of edges for an n vertex graph G that can be properly edge colored with no rainbow F subgraph. Chapter 1 of this dissertation gives relevant definitions and a brief history of extremal graph theory. Chapter 2 defines k-unique colorings and the related k-unique Turan number and provides preliminary results on this new variant. In Chapter 3, we explore the reduction method for finding upper bounds on rainbow Turan numbers and use this to inform results for the rainbow Turan numbers of specific families of trees. These results are used in Chapter 4 to prove that the rainbow Turan numbers of all trees are linear in n, which correlates to a well-known property of the traditional Turan numbers of trees. We discuss improvements to the constant term in Chapters 4 and 5, and conclude with a discussion on avenues for future work

Chapter 8: Selective Stoichiometric and Catalytic Reactivity in the Confines of a Chiral Supramolecular Assembly

Nature uses enzymes to activate otherwise unreactive compounds in remarkable ways. For example, DNases are capable of hydrolyzing phosphate diester bonds in DNA within seconds,[1-3]--a reaction with an estimated half-life of 200 million years without an enzyme.[4] The fundamental features of enzyme catalysis have been much discussed over the last sixty years in an effort to explain the dramatic rate increases and high selectivities of enzymes. As early as 1946, Linus Pauling suggested that enzymes must preferentially recognize and stabilize the transition state over the ground state of a substrate.[5] Despite the intense study of enzymatic selectivity and ability to catalyze chemical reactions, the entire nature of enzyme-based catalysis is still poorly understood. For example, Houk and co-workers recently reported a survey of binding affinities in a wide variety of enzyme-ligand, enzyme-transition-state, and synthetic host-guest complexes and found that the average binding affinities were insufficient to generate many of the rate accelerations observed in biological systems.[6] Therefore, transition-state stabilization cannot be the sole contributor to the high reactivity and selectivity of enzymes, but rather, other forces must contribute to the activation of substrate molecules. Inspired by the efficiency and selectivity of Nature, synthetic chemists have admired the ability of enzymes to activate otherwise unreactive molecules in the confines of an active site. Although much less complex than the evolved active sites of enzymes, synthetic host molecules have been developed that can carry out complex reactions with their cavities. While progress has been made toward highly efficient and selective reactivity inside of synthetic hosts, the lofty goal of duplicating enzymes specificity remains.[7-9] Pioneered by Lehn, Cram, Pedersen, and Breslow, supramolecular chemistry has evolved well beyond the crown ethers and cryptands originally studied.[10-12] Despite the increased complexity of synthetic host molecules, most assembly conditions utilize self-assembly to form complex highly-symmetric structures from relatively simple subunits. For supramolecular assemblies able to encapsulate guest molecules, the chemical environment in each assembly--defined by the size, shape, charge, and functional group availability--greatly influences the guest-binding characteristics.[6, 13-17

Energetic materials based on isocyanuric acid and 1,2,4-oxadiazole derivatives

The synthesis and properties of explosive urea and triazine derivatives is investigated on behalf of the explosive parameters and the full characterization of the molecules. (Chapter I-III) The class of oxadiazole derivatives is enhanced from the known explosive 1,2,5 oxadiazole (furazane) derivatives to the 1,2,4 oxadiazole derivatives. This molecule class is thoroughly investigated by all terms of chemical and explosive material matter and especially the 1,2,4-oxadiazol-5-one derivatives are compared to the corresponding tetrazole derivatives which were by far the most investigated molecule moiety of Prof. Dr. T.M. Klapoetke et al. for more than the last ten years. The 1,2,4 oxadiazol-5-one derivatives do only value as comparable model molecule to the tetrazole but were found to be good explosives themselves. So the triaminoguanidinium 1,2,4-oxadiazol-5-onate is suitable as low temperature propellant, the potassium and cesium 1,2,4-oxadiazol-5-onate are found to be good additions for NIR-flares and last but not least the best performing molecule was found to be the 3,5-diamino-1,2,4-oxadiazolium 5-aminotetrazolate, which combines the stability of the oxadiazole moiety with the very exothermic properties of a tetrazole in its best way. (Chapter IV-V) The 3-amino-1,2,4(4H)-oxadiazol-5-one is investigated thoroughly and detected to be a chemically and thermodynamically more stable system which can be functionalized according to methods known prior in the working group. The 3-dinitromethyl-1,2,4(4H)-oxadiazol-5-one is found a promising explosive class which can be combined as anion with a wide range of cations to tailor the stability and performance. The overall conclusion is that the 1,2,4-oxadiazole are chemical suitable as well as secondary explosives, propellants and pyrotechnics