A treatment of stereochemistry in computer aided organic synthesis

This thesis describes the author’s contributions to a new stereochemical processing module constructed for the ARChem retrosynthesis program. The purpose of the module is to add the ability to perform enantioselective and diastereoselective retrosynthetic disconnections and generate appropriate precursor molecules. The module uses evidence based rules generated from a large database of literature reactions. Chapter 1 provides an introduction and critical review of the published body of work for computer aided synthesis design. The role of computer perception of key structural features (rings, functions groups etc.) and the construction and use of reaction transforms for generating precursors is discussed. Emphasis is also given to the application of strategies in retrosynthetic analysis. The availability of large reaction databases has enabled a new generation of retrosynthesis design programs to be developed that use automatically generated transforms assembled from published reactions. A brief description of the transform generation method employed by ARChem is given. Chapter 2 describes the algorithms devised by the author for handling the computer recognition and representation of the stereochemical features found in molecule and reaction scheme diagrams. The approach is generalised and uses flexible recognition patterns to transform information found in chemical diagrams into concise stereo descriptors for computer processing. An algorithm for efficiently comparing and classifying pairs of stereo descriptors is described. This algorithm is central for solving the stereochemical constraints in a variety of substructure matching problems addressed in chapter 3. The concise representation of reactions and transform rules as hyperstructure graphs is described. Chapter 3 is concerned with the efficient and reliable detection of stereochemical symmetry in both molecules, reactions and rules. A novel symmetry perception algorithm, based on a constraints satisfaction problem (CSP) solver, is described. The use of a CSP solver to implement an isomorph‐free matching algorithm for stereochemical substructure matching is detailed. The prime function of this algorithm is to seek out unique retron locations in target molecules and then to generate precursor molecules without duplications due to symmetry. Novel algorithms for classifying asymmetric, pseudo‐asymmetric and symmetric stereocentres; meso, centro, and C2 symmetric molecules; and the stereotopicity of trigonal (sp2) centres are described. Chapter 4 introduces and formalises the annotated structural language used to create both retrosynthetic rules and the patterns used for functional group recognition. A novel functional group recognition package is described along with its use to detect important electronic features such as electron‐withdrawing or donating groups and leaving groups. The functional groups and electronic features are used as constraints in retron rules to improve transform relevance. Chapter 5 details the approach taken to design detailed stereoselective and substrate controlled transforms from organised hierarchies of rules. The rules employ a rich set of constraints annotations that concisely describe the keying retrons. The application of the transforms for collating evidence based scoring parameters from published reaction examples is described. A survey of available reaction databases and the techniques for mining stereoselective reactions is demonstrated. A data mining tool was developed for finding the best reputable stereoselective reaction types for coding as transforms. For various reasons it was not possible during the research period to fully integrate this work with the ARChem program. Instead, Chapter 6 introduces a novel one‐step retrosynthesis module to test the developed transforms. The retrosynthesis algorithms use the organisation of the transform rule hierarchy to efficiently locate the best retron matches using all applicable stereoselective transforms. This module was tested using a small set of selected target molecules and the generated routes were ranked using a series of measured parameters including: stereocentre clearance and bond cleavage; example reputation; estimated stereoselectivity with reliability; and evidence of tolerated functional groups. In addition a method for detecting regioselectivity issues is presented. This work presents a number of algorithms using common set and graph theory operations and notations. Appendix A lists the set theory symbols and meanings. Appendix B summarises and defines the common graph theory terminology used throughout this thesis

Phase-field modeling of multi-domain evolution in ferromagnetic shape memory alloys and of polycrystalline thin film growth

The phase-field method is a powerful tool in computer-aided materials science as it allows for the analysis of the time-spatial evolution of microstructures on the mesoscale. A multi-phase-field model is adopted to run numerical simulations in two different areas of scientific interest: Polycrystalline thin films growth and the ferromagnetic shape memory effect. FFT-techniques, norm conservative integration and RVE-methods are necessary to make the coupled problems numerically feasible

Algebraic Structures of Neutrosophic Triplets, Neutrosophic Duplets, or Neutrosophic Multisets

Neutrosophy (1995) is a new branch of philosophy that studies triads of the form (, , ), where is an entity {i.e. element, concept, idea, theory, logical proposition, etc.}, is the opposite of , while is the neutral (or indeterminate) between them, i.e., neither nor .Based on neutrosophy, the neutrosophic triplets were founded, which have a similar form (x, neut(x), anti(x)), that satisfy several axioms, for each element x in a given set.This collective book presents original research papers by many neutrosophic researchers from around the world, that report on the state-of-the-art and recent advancements of neutrosophic triplets, neutrosophic duplets, neutrosophic multisets and their algebraic structures – that have been defined recently in 2016 but have gained interest from world researchers. Connections between classical algebraic structures and neutrosophic triplet / duplet / multiset structures are also studied. And numerous neutrosophic applications in various fields, such as: multi-criteria decision making, image segmentation, medical diagnosis, fault diagnosis, clustering data, neutrosophic probability, human resource management, strategic planning, forecasting model, multi-granulation, supplier selection problems, typhoon disaster evaluation, skin lesson detection, mining algorithm for big data analysis, etc

Sphericities of Cycles. What Pólya’s Theorem is Deficient in for Stereoisomer Enumeration

Three methods and their extended versions for enumerating stereoisomers, which have been developed by modifying or simplifying Fujita’s USCI (unit-subduced-cycle-index) approach based on the concept of sphericities of orbits in order not to take account of symmetry itemization, are applied to the enumeration problem of ethane and propane derivatives. The proligand method and its extended version based on the concept of sphericities of cycles are also applied to the same enumeration problems. These results are compared with the results based on Pólya’s theorem (and Pólya’s corona). Thereby, it is shown that Pólya’s theorem enumerates chemical compounds as graphs, not as stereoisomers (3D chemical structures) if all of the permutations corresponding to proper and improper rotations are adopted. Moreover, if the permutations corresponding to proper rotations are adopted, Pólya’s theorem enumerates chemical compounds as chiral ones, where enantiomeric relationship and achiral nature (i.e., self-enantiomeric relationship) are not characterized properly. The two types of applications of Pólya’s theorem do not take account of improper rotations properly. Thereby, what Pólya’s theorem is deficient in is concluded to be the concept of sphericity

Neurogeometry of stereo vision

This work aims to develop a neurogeometric model of stereo vision, based on cortical architectures involved in the problem of 3D perception and neural mechanisms generated by retinal disparities. First, we provide a sub-Riemannian geometry for stereo vision, inspired by the work on the stereo problem by Zucker (2006), and using sub-Riemannian tools introduced by Citti-Sarti (2006) for monocular vision. We present a mathematical interpretation of the neural mechanisms underlying the behavior of binocular cells, that integrate monocular inputs. The natural compatibility between stereo geometry and neurophysiological models shows that these binocular cells are sensitive to position and orientation. Therefore, we model their action in the space R3xS2 equipped with a sub-Riemannian metric. Integral curves of the sub-Riemannian structure model neural connectivity and can be related to the 3D analog of the psychophysical association fields for the 3D process of regular contour formation. Then, we identify 3D perceptual units in the visual scene: they emerge as a consequence of the random cortico-cortical connection of binocular cells. Considering an opportune stochastic version of the integral curves, we generate a family of kernels. These kernels represent the probability of interaction between binocular cells, and they are implemented as facilitation patterns to define the evolution in time of neural population activity at a point. This activity is usually modeled through a mean field equation: steady stable solutions lead to consider the associated eigenvalue problem. We show that three-dimensional perceptual units naturally arise from the discrete version of the eigenvalue problem associated to the integro-differential equation of the population activity

On 4-Dimensional Point Groups and on Realization Spaces of Polytopes

This dissertation consists of two parts. We highlight the main results from each part. Part I. 4-Dimensional Point Groups. (based on a joint work with Günter Rote.) We propose the following classification for the finite groups of orthogonal transformations in 4-space, the so-called 4-dimensional point groups. Theorem A. The 4-dimensional point groups can be classified into * 25 polyhedral groups (Table 5.1), * 21 axial groups (7 pyramidal groups, 7 prismatic groups, and 7 hybrid groups, Table 6.3), * 22 one-parameter families of tubical groups (11 left tubical groups and 11 right tubical groups, Table 3.1), and * 25 infinite families of toroidal groups (2 three-parameter families, 19 two-parameter families, and 4 one-parameter families, Table 4.3.) In contrast to earlier classifications of these groups (notably by Du Val in 1962 and by Conway and Smith in 2003), see Section 1.7), we emphasize a geometric viewpoint, trying to visualize and understand actions of these groups. Besides, we correct some omissions, duplications, and mistakes in these classifications. The 25 polyhedral groups (Chapter 5) are related to the regular polytopes. The symmetries of the regular polytopes are well understood, because they are generated by reflections, and the classification of such groups as Coxeter groups is classic. We will deal with these groups only briefly, dwelling a little on just a few groups that come in enantiomorphic pairs (i.e., groups that are not equal to their own mirror.) The 21 axial groups (Chapter 6) are those that keep one axis fixed. Thus, they essentially operate in the three dimensions perpendicular to this axis (possibly combined with a flip of the axis), and they are easy to handle, based on the well-known classification of the three-dimensional point groups. The tubical groups (Chapter 3) are characterized as those that have (exactly) one Hopf bundle invariant. They come in left and right versions (which are mirrors of each other) depending on the Hopf bundle they keep invariant. They are so named because they arise with a decomposition of the 3-sphere into tube-like structures (discrete Hopf fibrations). The toroidal groups (Chapter 4) are characterized as having an invariant torus. This class of groups is where our main contribution in terms of the completeness of the classification lies. We propose a new, geometric, classification of these groups. Essentially, it boils down to classifying the isometry groups of the two-dimensional square flat torus. We emphasize that, regarding the completeness of the classification, in particular concerning the polyhedral and tubical groups, we rely on the classic approach (see Section 1.6). Only for the toroidal and axial groups, we supplant the classic approach by our geometric approach. We give a self-contained presentation of Hopf fibrations (Chapter 2). In many places in the literature, one particular Hopf map is introduced as “the Hopf map”, either in terms of four real coordinates or two complex coordinates, leading to “the Hopf fibration”. In some sense, this is justified, as all Hopf bundles are (mirror-)congruent. However, for our characterization, we require the full generality of Hopf bundles. As a tool for working with Hopf fibrations, we introduce a parameterization for great circles in S^3 , which might be useful elsewhere. Our main tool to understand tubical groups are polar orbit polytopes. (Chapter 1). In particular, we study the symmetries of a cell of the polar orbit polytope for different starting points. Part II. Realization Spaces of Polytopes (based on a joint work with Rainer Sinn and Günter M. Ziegler.) Robertson in 1988 suggested a model for the realization space of a d-dimensional polytope P, and an approach via the implicit function theorem to prove that the realization space is a smooth manifold of dimension NG(P) := d(f_0 + f_{d−1} ) - f{0,d-1} . We call NG(P) the natural guess for (the dimension of the realization space of) P. We build on Robertson's model and approach to study the realization spaces of higher-dimensional polytopes. We conclude combinatorial criteria (Sections 9.3.3 and 9.4.1) to decide if the realization space of the polytope in consideration is a smooth manifold of dimension given by the natural guess. As another application, we study the realization spaces of the second hypersimplices (Section 9.4.2). We apply these criteria on 4-polytopes with small number of vertices, and along the way, we analyze examples where Robertson’s approach fails, identifying the three smallest examples of 4-polytopes, for which the realization space is still a smooth manifold, but its dimension is not given by the natural guess (Section 9.5). Finally, we investigate the realization space of the 24-cell (Section 9.5.2). We construct families of realizations of the 24-cell, and using them we show that the realization space of the 24-cell has points where it is not a smooth manifold. This provides the first known example of a polytope whose realization space is not a smooth manifold. We conclude by showing that the dimension of the realization space of the 24-cell is at least 48 and at most 52.Diese Dissertation befasst sich mit zwei verschiedenen Themen, von denen jedes seinen eigenen Teil hat. Der erste Teil befasst sich mit 4-dimensionalen Punktgruppen. Wir schlagen eine neue Klassifizierung für diese Gruppen vor (siehe Theorem A), die im Gegensatz zu früheren Klassifizierungen eine geometrische Sichtweise betont und versucht, die Aktionen dieser Gruppen zu visualisieren und zu verstehen. Im Folgenden werden diese Gruppen kurz beschrieben. Die polyedrischen Gruppen (Kapitel 5) sind mit den regelmäßigen Polytopen verwandt. Die axialen Gruppen (Kapitel 6) sind diejenigen, die eine Achse festhalten. Die schlauchartigen Gruppen (Kapitel 3) werden als solche charakterisiert, die genau eine invariantes Hopf-Bündel haben. Sie entstehen bei einer Zerlegung der 3-Sphäre in schlauchartige Strukturen (diskrete Hopf-Faserungen). Die toroidalen Gruppen (Kapitel 4) sind dadurch gekennzeichnet, dass sie einen invarianten Torus haben. Wir schlagen eine neue, geometrische Klassifizierung dieser Gruppen vor. Im Wesentlichen läuft sie darauf hinaus, die Isometriegruppen des zweidimensionalen quadratischen flachen Torus zu klassifizieren. Nebenbei geben wir eine in sich geschlossene Darstellung der Hopf-Faserungen (Kapitel 2). Als Hilfsmittel für die Arbeit mit ihnen führen wir eine Parametrisierung für Großkreise in S 3 ein, die an anderer Stelle nützlich sein könnte. Der zweite Teil befasst sich mit Realisierungsräumen von Polytopen. Wir bauen auf Robertsons Modell und Ansatz auf, um die Realisierungsräume von Polytopen zu untersuchen. Wir stellen kombinatorische Kriterien auf (Abschnitte 9.3.3 und 9.4.1), um zu entscheiden, ob der Realisierungsraum des betrachteten Polytops eine glatte Mannigfaltigkeit der durch die “natürliche Vermutung” gegebenen Dimension ist. Als weitere Anwendung, untersuchen wir die Realisierungsräume der zweiten Hypersimplices (Abschnitt 9.4.2). Nebenbei identifizieren wir die kleinsten Beispiele von 4-Polytopen, für die dieser Ansatz versagt (Abschnitt 9.5). Schließlich untersuchen wir den Realisierungsraum der 24-Zelle (Abschnitt 9.5.2). Wir zeigen, dass es Punkte gibt, an denen sie keine glatte Mannigfaltigkeit ist. Zuletzt zeigen wir, dass seine Dimension mindestens 48 und höchstens 52 beträgt

Structure theorems and extremal problems in incidence geometry

In this thesis, we prove variants and generalisations of the Sylvester-Gallai theorem, which states that a finite non-collinear point set in the plane spans an ordinary line. Green and Tao proved a structure theorem for sufficiently large sets spanning few ordinary lines, and used it to find exact extremal numbers for ordinary and 3-rich lines, solving the Dirac-Motzkin conjecture and the classical orchard problem respectively. We prove structure theorems for sufficiently large sets spanning few ordinary planes, hyperplanes, circles, and hyperspheres, showing that such sets lie mostly on algebraic curves (or on a hyperplane or hypersphere). We then use these structure theorems to solve the corresponding analogues of the Dirac-Motzkin conjecture and the orchard problem. For planes in 3-space and circles in the plane, we are able to find exact extremal numbers for ordinary and 4-rich planes and circles. We also show that there are irreducible rational space quartics such that any n-point subset spans only O(n8=3) coplanar quadruples, answering a question of Raz, Sharir, and De Zeeuw [51]. For hyperplanes in d-space, we are able to find tight asymptotic bounds on the extremal numbers for ordinary and (d + 1)-rich hyperplanes. This also gives a recursive method to compute exact extremal numbers for a fixed dimension d. For hyperspheres in d-space, we are able to find a tight asymptotic bound on the minimum number of ordinary hyperspheres, and an asymptotic bound on the maximum number of (d + 2)-rich hyperspheres that is tight in even dimensions. The recursive method in the hyperplanes case also applies here. Our methods rely on Green and Tao's results on ordinary lines, as well as results from classical algebraic geometry, in particular on projections, inversions, and algebraic curves