30 research outputs found

    ์ƒํƒœ๊ณ„์—์„œ์˜ ๊ฒฝ์Ÿ ๊ด€์ ์œผ๋กœ ๊ทธ๋ž˜ํ”„์™€ ์œ ํ–ฅ๊ทธ๋ž˜ํ”„์˜ ๊ตฌ์กฐ ์—ฐ๊ตฌ

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    ํ•™์œ„๋…ผ๋ฌธ(๋ฐ•์‚ฌ) -- ์„œ์šธ๋Œ€ํ•™๊ต๋Œ€ํ•™์› : ์‚ฌ๋ฒ”๋Œ€ํ•™ ์ˆ˜ํ•™๊ต์œก๊ณผ, 2023. 2. ๊น€์„œ๋ น.In this thesis, we study m-step competition graphs, (1, 2)-step competition graphs, phylogeny graphs, and competition-common enemy graphs (CCE graphs), which are primary variants of competition graphs. Cohen [11] introduced the notion of competition graph while studying predator-prey concepts in ecological food webs.An ecosystem is a biological community of interacting species and their physical environment. For each species in an ecosystem, there can be m conditions of the good environment by regarding lower and upper bounds on numerous dimensions such as soil, climate, temperature, etc, which may be represented by an m-dimensional rectangle, so-called an ecological niche. An elemental ecological truth is that two species compete if and only if their ecological niches overlap. Biologists often describe competitive relations among species cohabiting in a community by a food web that is a digraph whose vertices are the species and an arc goes from a predator to a prey. In this context, Cohen [11] defined the competition graph of a digraph as follows. The competition graph C(D) of a digraph D is defined to be a simple graph whose vertex set is the same as V (D) and which has an edge joining two distinct vertices u and v if and only if there are arcs (u, w) and (v, w) for some vertex w in D. Since Cohen introduced this definition, its variants such as m-step competition graphs, (i, j)-step competition graphs, phylogeny graphs, CCE graphs, p-competition graphs, and niche graphs have been introduced and studied. As part of these studies, we show that the connected triangle-free m-step competition graph on n vertices is a tree and completely characterize the digraphs of order n whose m-step competition graphs are star graphs for positive integers 2 โ‰ค m < n. We completely identify (1,2)-step competition graphs C_{1,2}(D) of orientations D of a complete k-partite graph for some k โ‰ฅ 3 when each partite set of D forms a clique in C_{1,2}(D). In addition, we show that the diameter of each component of C_{1,2}(D) is at most three and provide a sharp upper bound on the domination number of C_{1,2}(D) and give a sufficient condition for C_{1,2}(D) being an interval graph. On the other hand, we study on phylogeny graphs and CCE graphs of degreebounded acyclic digraphs. An acyclic digraph in which every vertex has indegree at most i and outdegree at most j is called an (i, j) digraph for some positive integers i and j. If each vertex of a (not necessarily acyclic) digraph D has indegree at most i and outdegree at most j, then D is called an hi, ji digraph. We give a sufficient condition on the size of hole of an underlying graph of an (i, 2) digraph D for the phylogeny graph of D being a chordal graph where D is an (i, 2) digraph. Moreover, we go further to completely characterize phylogeny graphs of (i, j) digraphs by listing the forbidden induced subgraphs. We completely identify the graphs with the least components among the CCE graphs of (2, 2) digraphs containing at most one cycle and exactly two isolated vertices, and their digraphs. Finally, we gives a sufficient condition for CCE graphs being interval graphs.์ด ๋…ผ๋ฌธ์—์„œ ๊ฒฝ์Ÿ๊ทธ๋ž˜ํ”„์˜ ์ฃผ์š” ๋ณ€์ด๋“ค ์ค‘ m-step ๊ฒฝ์Ÿ๊ทธ๋ž˜ํ”„, (1, 2)-step ๊ฒฝ์Ÿ ๊ทธ๋ž˜ํ”„, ๊ณ„ํ†ต ๊ทธ๋ž˜ํ”„, ๊ฒฝ์Ÿ๊ณต์ ๊ทธ๋ž˜ํ”„์— ๋Œ€ํ•œ ์—ฐ๊ตฌ ๊ฒฐ๊ณผ๋ฅผ ์ข…ํ•ฉํ–ˆ๋‹ค. Cohen [11]์€ ๋จน์ด์‚ฌ์Šฌ์—์„œ ํฌ์‹์ž-ํ”ผ์‹์ž ๊ฐœ๋…์„ ์—ฐ๊ตฌํ•˜๋ฉด์„œ ๊ฒฝ์Ÿ๊ทธ๋ž˜ํ”„ ๊ฐœ๋…์„ ๊ณ ์•ˆํ–ˆ๋‹ค. ์ƒํƒœ๊ณ„๋Š” ์ƒํ˜ธ์ž‘์šฉํ•˜๋Š” ์ข…๋“ค๊ณผ ๊ทธ๋“ค์˜ ๋ฌผ๋ฆฌ์  ํ™˜๊ฒฝ์˜ ์ƒ๋ฌผํ•™์  ์ฒด๊ณ„์ด๋‹ค. ์ƒํƒœ๊ณ„์˜ ๊ฐ ์ข…์— ๋Œ€ํ•ด์„œ, ํ† ์–‘, ๊ธฐํ›„, ์˜จ๋„ ๋“ฑ๊ณผ ๊ฐ™์€ ๋‹ค์–‘ํ•œ ์ฐจ์›์˜ ํ•˜๊ณ„ ๋ฐ ์ƒ๊ณ„๋ฅผ ๊ณ ๋ คํ•˜์—ฌ ์ข‹์€ ํ™˜๊ฒฝ์„ m๊ฐœ์˜ ์กฐ๊ฑด๋“ค๋กœ ๋‚˜ํƒ€๋‚ผ ์ˆ˜ ์žˆ๋Š”๋ฐ ์ด๋ฅผ ์ƒํƒœ์  ์ง€์œ„(ecological niche)๋ผ๊ณ  ํ•œ๋‹ค. ์ƒํƒœํ•™์  ๊ธฐ๋ณธ๊ฐ€์ •์€ ๋‘ ์ข…์ด ์ƒํƒœ์  ์ง€์œ„๊ฐ€ ๊ฒน์น˜๋ฉด ๊ฒฝ์Ÿํ•˜๊ณ (compete), ๊ฒฝ์Ÿํ•˜๋Š” ๋‘ ์ข…์€ ์ƒํƒœ์  ์ง€์œ„๊ฐ€ ๊ฒน์นœ๋‹ค๋Š” ๊ฒƒ์ด๋‹ค. ํ”ํžˆ ์ƒ๋ฌผํ•™์ž๋“ค์€ ํ•œ ์ฒด์ œ์—์„œ ์„œ์‹ํ•˜๋Š” ์ข…๋“ค์˜ ๊ฒฝ์Ÿ์  ๊ด€๊ณ„๋ฅผ ๊ฐ ์ข…์€ ๊ผญ์ง“์ ์œผ๋กœ, ํฌ์‹์ž์—์„œ ํ”ผ์‹์ž์—๊ฒŒ๋Š” ์œ ํ–ฅ๋ณ€(arc)์„ ๊ทธ์–ด์„œ ๋จน์ด์‚ฌ์Šฌ๋กœ ํ‘œํ˜„ํ•œ๋‹ค. ์ด๋Ÿฌํ•œ ๋งฅ๋ฝ์—์„œ Cohen [11]์€ ๋‹ค์Œ๊ณผ ๊ฐ™์ด ์œ ํ–ฅ๊ทธ๋ž˜ํ”„์˜ ๊ฒฝ์Ÿ ๊ทธ๋ž˜ํ”„๋ฅผ ์ •์˜ํ–ˆ๋‹ค. ์œ ํ–ฅ๊ทธ๋ž˜ํ”„(digraph) D์˜ ๊ฒฝ์Ÿ๊ทธ๋ž˜ํ”„(competition graph) C(D) ๋ž€ V (D)๋ฅผ ๊ผญ์ง“์  ์ง‘ํ•ฉ์œผ๋กœ ํ•˜๊ณ  ๋‘ ๊ผญ์ง“์  u, v๋ฅผ ์–‘ ๋์ ์œผ๋กœ ๊ฐ–๋Š” ๋ณ€์ด ์กด์žฌํ•œ๋‹ค๋Š” ๊ฒƒ๊ณผ ๊ผญ์ง“์  w๊ฐ€ ์กด์žฌํ•˜์—ฌ (u, w),(v, w)๊ฐ€ ๋ชจ๋‘ D์—์„œ ์œ ํ–ฅ๋ณ€์ด ๋˜๋Š” ๊ฒƒ์ด ๋™์น˜์ธ ๊ทธ๋ž˜ํ”„๋ฅผ ์˜๋ฏธํ•œ๋‹ค. Cohen์ด ๊ฒฝ์Ÿ๊ทธ๋ž˜ํ”„์˜ ์ •์˜๋ฅผ ๋„์ž…ํ•œ ์ดํ›„๋กœ ๊ทธ ๋ณ€์ด๋“ค๋กœ m-step ๊ฒฝ์Ÿ๊ทธ๋ž˜ํ”„(m-step competition graph), (i, j)-step ๊ฒฝ์Ÿ๊ทธ๋ž˜ํ”„((i, j)-step competition graph), ๊ณ„ํ†ต๊ทธ๋ž˜ํ”„(phylogeny graph), ๊ฒฝ์Ÿ๊ณต์ ๊ทธ๋ž˜ํ”„(competition-common enemy graph), p-๊ฒฝ์Ÿ๊ทธ๋ž˜ํ”„(p-competition graph), ๊ทธ๋ฆฌ๊ณ  ์ง€์œ„๊ทธ๋ž˜ํ”„(niche graph)๊ฐ€ ๋„์ž…๋˜์—ˆ๊ณ  ์—ฐ๊ตฌ๋˜๊ณ  ์žˆ๋‹ค. ์ด ๋…ผ๋ฌธ์˜ ์—ฐ๊ตฌ ๊ฒฐ๊ณผ๋“ค์˜ ์ผ๋ถ€๋Š” ๋‹ค์Œ๊ณผ ๊ฐ™๋‹ค. ์‚ผ๊ฐํ˜•์ด ์—†์ด ์—ฐ๊ฒฐ๋œ m-step ๊ฒฝ์Ÿ ๊ทธ๋ž˜ํ”„๋Š” ํŠธ๋ฆฌ(tree)์ž„์„ ๋ณด์˜€์œผ๋ฉฐ 2 โ‰ค m < n์„ ๋งŒ์กฑํ•˜๋Š” ์ •์ˆ˜ m, n์— ๋Œ€ํ•˜์—ฌ ๊ผญ์ง“์ ์˜ ๊ฐœ์ˆ˜๊ฐ€ n๊ฐœ์ด๊ณ  m-step ๊ฒฝ์Ÿ๊ทธ๋ž˜ํ”„๊ฐ€ ๋ณ„๊ทธ๋ž˜ํ”„(star graph)๊ฐ€ ๋˜๋Š” ์œ ํ–ฅ๊ทธ๋ž˜ํ”„๋ฅผ ์™„๋ฒฝํ•˜๊ฒŒ ํŠน์ง•ํ™” ํ•˜์˜€๋‹ค. k โ‰ฅ 3์ด๊ณ  ๋ฐฉํ–ฅ์ง€์–ด์ง„ ์™„์ „ k-๋ถ„ํ•  ๊ทธ๋ž˜ํ”„(oriented complete k-partite graph)์˜ (1, 2)-step ๊ฒฝ์Ÿ๊ทธ๋ž˜ํ”„ C_{1,2}(D)์—์„œ ๊ฐ ๋ถ„ํ• ์ด ์™„์ „ ๋ถ€๋ถ„ ๊ทธ๋ž˜ํ”„๋ฅผ ์ด๋ฃฐ ๋•Œ, C_{1,2}(D)์„ ๋ชจ๋‘ ํŠน์ง•ํ™” ํ•˜์˜€๋‹ค. ๋˜ํ•œ, C_{1,2}(D)์˜ ๊ฐ ์„ฑ๋ถ„(component)์˜ ์ง€๋ฆ„(diameter)์˜ ๊ธธ์ด๊ฐ€ ์ตœ๋Œ€ 3์ด๋ฉฐ C_{1,2}(D)์˜ ์ง€๋ฐฐ์ˆ˜(domination number)์— ๋Œ€ํ•œ ์ƒ๊ณ„์™€ ์ตœ๋Œ“๊ฐ’์„ ๊ตฌํ•˜๊ณ  ๊ตฌ๊ฐ„๊ทธ๋ž˜ํ”„(interval graph)๊ฐ€ ๋˜๊ธฐ ์œ„ํ•œ ์ถฉ๋ถ„ ์กฐ๊ฑด์„ ๊ตฌํ•˜์˜€๋‹ค. ์ฐจ์ˆ˜๊ฐ€ ์ œํ•œ๋œ ์œ ํ–ฅํšŒ๋กœ๋ฅผ ๊ฐ–์ง€ ์•Š๋Š” ์œ ํ–ฅ๊ทธ๋ž˜ํ”„(degree-bounded acyclic digraph)์˜ ๊ณ„ํ†ต๊ทธ๋ž˜ํ”„์™€ ๊ฒฝ์Ÿ๊ณต์ ๊ทธ๋ž˜ํ”„์— ๋Œ€ํ•ด์„œ๋„ ์—ฐ๊ตฌํ•˜์˜€๋‹ค. ์–‘์˜ ์ •์ˆ˜๋“ค i, j์— ๋Œ€ํ•˜์—ฌ (i, j) ์œ ํ–ฅ๊ทธ๋ž˜ํ”„๋ž€ ๊ฐ ๊ผญ์ง“์ ์˜ ๋‚ด์ฐจ์ˆ˜๋Š” ์ตœ๋Œ€ i, ์™ธ์ฐจ์ˆ˜๋Š” ์ตœ๋Œ€ j์ธ ์œ ํ–ฅํšŒ๋กœ ๊ฐ–์ง€ ์•Š๋Š” ์œ ํ–ฅ๊ทธ๋ž˜ํ”„์ด๋‹ค. ๋งŒ์•ฝ ์œ ํ–ฅ๊ทธ๋ž˜ํ”„ D์— ๊ฐ ๊ผญ์ง“์ ์ด ๋‚ด์ฐจ์ˆ˜๊ฐ€ ์ตœ๋Œ€ i, ์™ธ์ฐจ์ˆ˜๊ฐ€ ์ตœ๋Œ€ j ์ธ ๊ฒฝ์šฐ์— D๋ฅผ hi, ji ์œ ํ–ฅ๊ทธ๋ž˜ํ”„๋ผ ํ•œ๋‹ค. D๊ฐ€ (i, 2) ์œ ํ–ฅ๊ทธ๋ž˜ํ”„์ผ ๋•Œ, D์˜ ๊ณ„ํ†ต๊ทธ๋ž˜ํ”„๊ฐ€ ํ˜„๊ทธ๋ž˜ํ”„(chordal graph)๊ฐ€ ๋˜๊ธฐ ์œ„ํ•œ D์˜ ๋ฐฉํ–ฅ์„ ๊ณ ๋ คํ•˜์ง€ ์•Š๊ณ  ์–ป์–ด์ง€๋Š” ๊ทธ๋ž˜ํ”„(underlying graph)์—์„œ ๊ธธ์ด๊ฐ€ 4์ด์ƒ์ธ ํšŒ๋กœ(hole)์˜ ๊ธธ์ด์— ๋Œ€ํ•œ ์ถฉ๋ถ„์กฐ๊ฑด์„ ๊ตฌํ•˜์˜€๋‹ค. ๊ฒŒ๋‹ค๊ฐ€ (i, j) ์œ ํ–ฅ๊ทธ๋ž˜ํ”„์˜ ๊ณ„ํ†ต๊ทธ๋ž˜ํ”„์—์„œ ๋‚˜์˜ฌ ์ˆ˜ ์—†๋Š” ์ƒ์„ฑ ๋ถ€๋ถ„ ๊ทธ๋ž˜ํ”„(forbidden induced subgraph)๋ฅผ ํŠน์ง•ํ™” ํ•˜์˜€๋‹ค. (2, 2) ์œ ํ–ฅ๊ทธ๋ž˜ํ”„ D์˜ ๊ฒฝ์Ÿ๊ณต์ ๊ทธ๋ž˜ํ”„ CCE(D)๊ฐ€ 2๊ฐœ์˜ ๊ณ ๋ฆฝ์ (isolated vertex)๊ณผ ์ตœ๋Œ€ 1๊ฐœ์˜ ํšŒ๋กœ๋ฅผ ๊ฐ–์œผ๋ฉด์„œ ๊ฐ€์žฅ ์ ์€ ์„ฑ๋ถ„์„ ๊ฐ–๋Š” ๊ฒฝ์šฐ์ผ ๋•Œ์˜ ๊ตฌ์กฐ๋ฅผ ๊ทœ๋ช…ํ–ˆ๋‹ค. ๋งˆ์ง€๋ง‰์œผ๋กœ, CCE(D)๊ฐ€ ๊ตฌ๊ฐ„๊ทธ๋ž˜ํ”„๊ฐ€ ๋˜๊ธฐ ์œ„ํ•œ ์„ฑ๋ถ„์˜ ๊ฐœ์ˆ˜์— ๋Œ€ํ•œ ์ถฉ๋ถ„์กฐ๊ฑด์„ ๊ตฌํ•˜์˜€๋‹ค.1 Introduction 1 1.1 Graph theory terminology and basic concepts 1 1.2 Competition graphs and its variants 6 1.2.1 A brief background of competition graphs 6 1.2.2 Variants of competition graphs 8 1.2.3 m-step competition graphs 10 1.2.4 (1, 2)-step competition graphs 13 1.2.5 Phylogeny graphs 14 1.2.6 CCE graphs 16 1.3 A preview of the thesis 17 2 Digraphs whose m-step competition graphs are trees 19 2.1 The triangle-free m-step competition graphs 23 2.2 Digraphs whose m-step competition graphs are trees 29 2.3 The digraphs whose m-step competition graphs are star graphs 38 3 On (1, 2)-step competition graphs of multipartite tournaments 47 3.1 Preliminaries 48 3.2 C1,2(D) with a non-clique partite set of D 51 3.3 C1,2(D) without a non-clique partite set of D 66 3.4 C1,2(D) as a complete graph 74 3.5 Diameters and domination numbers of C1,2(D) 79 3.6 Disconnected (1, 2)-step competition graphs 82 3.7 Interval (1, 2)-step competition graphs 84 4 The forbidden induced subgraphs of (i, j) phylogeny graphs 90 4.1 A necessary condition for an (i, 2) phylogeny graph being chordal 91 4.2 Forbidden subgraphs for phylogeny graphs of degree bounded digraphs 99 5 On CCE graphs of (2, 2) digraphs 122 5.1 CCE graphs of h2, 2i digraphs 128 5.2 CCE graphs of (2, 2) digraphs 134 Abstract (in Korean) 168 Acknowledgement (in Korean) 170๋ฐ•

    Probing Structure and Dynamics of Amorphous Ice with Small-Molecule Nanoprobes

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    Water (H2O) is omnipresent on the surface of the Earth, the atmosphere, in nature, and on various celestial bodies.1 The phase diagram of ice exhibits enormous complexity with a plethora of structures and at least two amorphous ices. 2, 3 One of these, low-density amorphous ice, is the most abundant solid in the Universe. Despite H2Oโ€™s significance, a full understanding of its role in physical processes remains elusive.4 H2O is capable of building complex hydrogen-bonded networks, and solvates hydrophobic/hydrophilic species.2 Carbon and H2O often coexist, forming interfaces in highly diverse environments.5, 6 This thesis focuses on the structure of H2O in the hydration shells of hydrophobes, tracking the structure and dynamics of vapour deposited amorphous ice with finely dispersed small-molecule nanoprobes. Detailed insights into the morphology of amorphous solid water (ASW)7 and evidence for the presence of three discernible desorption processes present in macroscopic films of amorphous ice have been demonstrated. They are attributed to gas desorption from open cracks, from the collapse of internal voids, and from matrix-isolated gas induced by the irreversible crystallisation of H2O to stacking disordered ice. 7 Due to adamantane (C10H16) being expelled from the amorphous ice matrix upon heating, a number of important insights were gained โ€“ the uncharted regime of small hydrophobes surrounded by a H2O network were detected. Neutron diffraction studies of C10H16/ASW employing structure refinement modelling identified a new type of cage structure, with 28 H2O molecules constructed from distorted five- and six-membered rings, named the 566 4 polyhedron. Beyond this phenomenon, unusual, yet strong orientation correlations of the H2O molecules were detected. Intriguingly, the closest Oโ€“ H bonds were found to point towards the centre of mass of C10H16 โ€“ it is quite striking that such a non-polar solute induces intense orientation correlations in its hydration shells. H2O has been at the forefront of many breakthroughs and will continue to push boundaries, probing the chemistry and physics of ice research. 3 References 1. L. del Rosso, M. Celli, F. Grazzi, M. Catti, T. C. Hansen, A. D. Fortes and L. Ulivi, Nat. Mater., 2020, 19, 663-668. 2. C. G. Salzmann, P. G. Radaelli, B. Slater and J. L. Finney, Phys. Chem. Chem. Phys., 2011, 13, 18468-18480. 3. C. G. Salzmann, J. Chem. Phys., 2019, 150, 060901. Abstract 4 4. T. Loerting, K. Winkel, M. Seidl, M. Bauer, C. Mitterdorfer, P. H. Handle, C. G. Salzmann, E. Mayer, J. L. Finney and D. T. Bowron, Phys. Chem. Chem. Phys., 2011, 13, 8783-8794. 5. M. C. De Sanctis, F. Capaccioni, M. Ciarniello, G. Filacchione, M. Formisano, S. Mottola, A. Raponi, F. Tosi, D. Bockelรฉe-Morvan, S. Erard, C. Leyrat, B. Schmitt, E. Ammannito, G. Arnold, M. A. Barucci, M. Combi, M. T. Capria, P. Cerroni, W. H. Ip, E. Kuehrt, T. B. McCord, E. Palomba, P. Beck, E. Quirico, V. T. The, G. Piccioni, G. Bellucci, M. Fulchignoni, R. Jaumann, K. Stephan, A. Longobardo, V. Mennella, A. Migliorini, J. Benkhoff, J. P. Bibring, A. Blanco, M. Blecka, R. Carlson, U. Carsenty, L. Colangeli, M. Combes, J. Crovisier, P. Drossart, T. Encrenaz, C. Federico, U. Fink, S. Fonti, P. Irwin, Y. Langevin, G. Magni, L. Moroz, V. Orofino, U. Schade, F. Taylor, D. Tiphene, G. P. Tozzi, N. Biver, L. Bonal, J. P. Combe, D. Despan, E. Flamini, S. Fornasier, A. Frigeri, D. Grassi, M. S. Gudipati, F. Mancarella, K. Markus, F. Merlin, R. Orosei, G. Rinaldi, M. Cartacci, A. Cicchetti, S. Giuppi, Y. Hello, F. Henry, S. Jacquinod, J. M. Reess, R. Noschese, R. Politi and G. Peter, Nature, 2015, 525, 500. 6. B. A. Buffett, Annu. Rev. Earth Planet Sci., 2000, 28, 477-507. 7. S. K. Talewar, S. O. Halukeerthi, R. Riedlaicher, J. J. Shephard, A. E. Clout, A. Rosu-Finsen, G. R. Williams, A. Langhoff, D. Johannsmann and C. G. Salzmann, J. Chem. Phys., 2019, 151, 134505

    Quantum Computation with Superconducting Parametric Cavity

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    Multimode superconducting parametric cavity is a flexible platform that has been used to study a variety of topics in microwave quantum optics ranging from parametric amplification, entanglement generation to higher order spontaneous parametric downconversion (SPDC). Leveraging the extensive toolbox of interactions available in this system, we can look to explore exciting applications in quantum computation and simulation. In this thesis, we study the use of the parametric cavity to realize continuous variable (CV) quantum computation. We propose and examine in detail the scheme to compute with the microwave photons in the orthogonal frequency modes of the cavity via successive application of parametric pump pulses or cavity drives. The family of all Gaussian transformations can be accomplished easily with interactions already demonstrated in this system. From recent results and proposals involving higher order SPDC, there are also clear pathways towards realizing the non-Gaussian resources necessary for universal computation. Common measurements on the system are accomplished with standard measurement techniques on the output state of the cavity and additional useful measurements may be implemented using available parametric interactions or new device designs involving a qubit as a nonlinear probe. Using the parametric cavity, we experimentally implemented a hybrid quantum-classical machine learning algorithm called the Quantum Kitchen Sinks (QKS) as the first step towards developing this platform for quantum computation. The algorithm is studied over two sets of experiments starting from partial experimental implementation of the quantum variational circuits to fully experimental implementation using multiple simultaneous continuous wave (CW) pumps. In both cases, we find that the quantum part of the algorithm implemented in the parametric cavity improved the classification accuracy on a difficult synthetic data set up to 90.1% and 99.5% respectively when compared to a classical linear machine learning algorithm

    GPU fast multipole method with lambda-dynamics features

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    A significant and computationally most demanding part of molecular dynamics simulations is the calculation of long-range electrostatic interactions. Such interactions can be evaluated directly by the naรฏve pairwise summation algorithm, which is a ubiquitous showcase example for the compute power of graphics processing units (GPUS). However, the pairwise summation has O(N^2) computational complexity for N interacting particles; thus, an approximation method with a better scaling is required. Today, the prevalent method for such approximation in the field is particle mesh Ewald (PME). PME takes advantage of fast Fourier transforms (FFTS) to approximate the solution efficiently. However, as the underlying FFTS require all-to-all communication between ranks, PME runs into a communication bottleneck. Such communication overhead is negligible only for a moderate parallelization. With increased parallelization, as needed for high-performance applications, the usage of PME becomes unprofitable. Another PME drawback is its inability to perform constant pH simulations efficiently. In such simulations, the protonation states of a protein are allowed to change dynamically during the simulation. The description of this process requires a separate evaluation of the energies for each protonation state. This can not be calculated efficiently with PME as the algorithm requires a repeated FFT for each state, which leads to a linear overhead with respect to the number of states. For a fast approximation of pairwise Coulombic interactions, which does not suffer from PME drawbacks, the Fast Multipole Method (FMM) has been implemented and fully parallelized with CUDA. To assure the optimal FMM performance for diverse MD systems multiple parallelization strategies have been developed. The algorithm has been efficiently incorporated into GROMACS and subsequently tested to determine the optimal FMM parameter set for MD simulations. Finally, the FMM has been incorporated into GROMACS to allow for out-of-the-box electrostatic calculations. The performance of the single-GPU FMM implementation, tested in GROMACS 2019, achieves about a third of highly optimized CUDA PME performance when simulating systems with uniform particle distributions. However, the FMM is expected to outperform PME at high parallelization because the FMM global communication overhead is minimal compared to that of PME. Further, the FMM has been enhanced to provide the energies of an arbitrary number of titratable sites as needed in the constant-pH method. The extension is not fully optimized yet, but the first results show the strength of the FMM for constant pH simulations. For a relatively large system with half a million particles and more than a hundred titratable sites, a straightforward approach to compute alternative energies requires the repetition of a simulation for each state of the sites. The FMM calculates all energy terms only a factor 1.5 slower than a single simulation step. Further improvements of the GPU implementation are expected to yield even more speedup compared to the actual implementation.2021-11-1

    New insights into water's phase diagram using ammonium fluoride

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    Ice is a complex, yet highly relevant material and has been a rife area of research since the beginning of the 20th century.1-5 Understanding ice is expected to have consequences not just for furthering our appreciation of the different states of water, but also general chemistry, physics and geology.2, 6, 7 It has often been found that properties first observed in ice (e.g. stacking disorder) are also present in other materials.2, 6, 8, 9 This thesis largely builds on work performed by Shephard et al. which explored the effect of 2.5 mol% NH4F in ice, and astoundingly fully prevented ice II formation.10 Initially, the thesis focuses on the effect of adding NH4F to ice at ambient pressure, which is demonstrated to produce a denser material than pure ice. At 0.5 GPa, NH4F-ice solid solutions (โ‰ฅ 12 mol%) surprisingly produce stable ice XII-type structures. Additionally, upon the mapping of the 2.5 mol% NH4F phase diagram to 1.7 GPa, it was found that phase-pure ice XII could be quenched at 1.1 GPa. Both ice XIItype structures did not require an amorphous precursor. The influence of NH4F in ice is explored in mixtures that are subjected to the compression conditions that yield high-density amorphous ice โ€˜pressure-induced amorphisedโ€™ upon their compression to 1.4 GPa at 77 K. Unexpectedly, the crossover of PIA to recrystallisation is determined as beginning on the water-rich side (35 mol% NH4F) of the solid solutions. Stacking disorder from the heating of NH4F II and III at ambient pressure is quantified. The materials reach a maximum cubicity of 77%, yet the stacking disorder obtained from each material is unique. Remarkably NH4F III did not transform to an amorphous phase upon heating. The final standalone chapter focuses on the ordering of ices V/XIII and IX with 0.01 M HCl doping

    Local Boxicity, Local Dimension, and Maximum Degree

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    In this paper, we focus on two recently introduced parameters in the literature, namely `local boxicity' (a parameter on graphs) and `local dimension' (a parameter on partially ordered sets). We give an `almost linear' upper bound for both the parameters in terms of the maximum degree of a graph (for local dimension we consider the comparability graph of a poset). Further, we give an O(nฮ”2)O(n\Delta^2) time deterministic algorithm to compute a local box representation of dimension at most 3ฮ”3\Delta for a claw-free graph, where nn and ฮ”\Delta denote the number of vertices and the maximum degree, respectively, of the graph under consideration. We also prove two other upper bounds for the local boxicity of a graph, one in terms of the number of vertices and the other in terms of the number of edges. Finally, we show that the local boxicity of a graph is upper bounded by its `product dimension'.Comment: 11 page

    Merging Data-Driven And Computational Methods to Understand Ice Nucleation

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    Heterogeneous ice nucleation (IN) is one of the most ubiquitous phase transitions on earth and impacts a plethora of fields in industry (e.g. air transport, food freezing and harsh-weather operations) and science (e.g. freeze avoidance of animals, cryobiology, cloud research). Still to date, we are lacking reliable answers to the question: What is it at the molecular scale that causes an impurity to facilitate the freezing process of supercooled liquid water? In this thesis we make headway towards identifying such microscopic principles by performing computational studies combined with data-driven approaches. In chapter 3 we screen a range of model substrates to disentangle the contributions of lattice match and hydrophobicity and find that there is a complex interplay and an enormous sensitivity to the atomistic details of the interface. In chapter 4 we show that the heterogeneous setting can alter the polymorph of ice that forms and introduce the concept of pre-critical fluctuations, yielding new ideas to design polymorph-targeting substrates. Chapter 5 deals with the liquid dynamics before and during the nucleation event, an aspect of nucleation that mostly goes unrecognized. We show that the homogeneous nucleation event happens in relatively immobile regions of the supercooled liquid, a finding that opens new avenues to understand and influence heterogeneous nucleation by targeting dynamics rather than structure. Finally, Chapter 6 builds on the large amount of data created during this project in that we combine all previously simulated systems and devise a machine-learning approach to find the most important descriptors for their ice nucleation activity (INA). With this we identify new microscopic guidelines and demonstrate that the quantitative prediction of heterogeneous INA is in reach. The unveiling of a computational artifact that potentially affects many computational interface studies is also part of this thesis

    On local structures of cubicity 2 graphs

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    A 2-stab unit interval graph (2SUIG) is an axes-parallel unit square intersection graph where the unit squares intersect either of the two fixed lines parallel to the XX-axis, distance 1+ฯต1 + \epsilon (0<ฯต<10 < \epsilon < 1) apart. This family of graphs allow us to study local structures of unit square intersection graphs, that is, graphs with cubicity 2. The complexity of determining whether a tree has cubicity 2 is unknown while the graph recognition problem for unit square intersection graph is known to be NP-hard. We present a polynomial time algorithm for recognizing trees that admit a 2SUIG representation
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