Two essays in computational optimization: computing the clar number in fullerene graphs and distributing the errors in iterative interior point methods

Fullerene are cage-like hollow carbon molecules graph of pseudospherical sym- metry consisting of only pentagons and hexagons faces. It has been the object of interest for chemists and mathematicians due to its widespread application in various fields, namely including electronic and optic engineering, medical sci- ence and biotechnology. A Fullerene molecular, Γ n of n atoms has a multiplicity of isomers which increases as N iso ∼ O(n 9 ). For instance, Γ 180 has 79,538,751 isomers. The Fries and Clar numbers are stability predictors of a Fullerene molecule. These number can be computed by solving a (possibly N P -hard) combinatorial optimization problem. We propose several ILP formulation of such a problem each yielding a solution algorithm that provides the exact value of the Fries and Clar numbers. We compare the performances of the algorithm derived from the proposed ILP formulations. One of this algorithm is used to find the Clar isomers, i.e., those for which the Clar number is maximum among all isomers having a given size. We repeated this computational experiment for all sizes up to 204 atoms. In the course of the study a total of 2 649 413 774 isomers were analyzed.The second essay concerns developing an iterative primal dual infeasible path following (PDIPF) interior point (IP) algorithm for separable convex quadratic minimum cost flow network problem. In each iteration of PDIPF algorithm, the main computational effort is solving the underlying Newton search direction system. We concentrated on finding the solution of the corresponding linear system iteratively and inexactly. We assumed that all the involved inequalities can be solved inexactly and to this purpose, we focused on different approaches for distributing the error generated by iterative linear solvers such that the convergences of the PDIPF algorithm are guaranteed. As a result, we achieved theoretical bases that open the path to further interesting practical investiga- tion

The topology of fullerenes

Fullerenes are carbon molecules that form polyhedral cages. Their bond structures are exactly the planar cubic graphs that have only pentagon and hexagon faces. Strikingly, a number of chemical properties of a fullerene can be derived from its graph structure. A rich mathematics of cubic planar graphs and fullerene graphs has grown since they were studied by Goldberg, Coxeter, and others in the early 20th century, and many mathematical properties of fullerenes have found simple and beautiful solutions. Yet many interesting chemical and mathematical problems in the field remain open. In this paper, we present a general overview of recent topological and graph theoretical developments in fullerene research over the past two decades, describing both solved and open problems. WIREs Comput Mol Sci 2015, 5:96–145. doi: 10.1002/wcms.1207 Conflict of interest: The authors have declared no conflicts of interest for this article. For further resources related to this article, please visit the WIREs website

The Polytope Formalism: isomerism and associated unimolecular isomerisation

This thesis concerns the ontology of isomerism, this encompassing the conceptual frameworks and relationships that comprise the subject matter; the necessary formal definitions, nomenclature, and representations that have impacts reaching into unexpected areas such as drug registration and patent specifications; the requisite controlled and precise vocabulary that facilitates nuanced communication; and the digital/computational formalisms that underpin the chemistry software and database tools that empower chemists to perform much of their work. Using conceptual tools taken from Combinatorics, and Graph Theory, means are presented to provide a unified description of isomerism and associated unimolecular isomerisation spanning both constitutional isomerism and stereoisomerism called the Polytope Formalism. This includes unification of the varying approaches historically taken to describe and understand stereoisomerism in organic and inorganic compounds. Work for this Thesis began with the synthesis, isolation, and characterisation of compounds not adequately describable using existing IUPAC recommendations. Generalisation of the polytopal-rearrangements model of stereoisomerisation used for inorganic chemistry led to the prescriptions that could deal with the synthesised compounds, revealing an unrecognised fundamental form of isomerism called akamptisomerism. Following on, this Thesis describes how in attempting to place akamptisomerism within the context of existing stereoisomerism reveals significant systematic deficiencies in the IUPAC recommendations. These shortcomings have limited the conceptualisation of broad classes of compounds and hindered development of molecules for medicinal and technological applications. It is shown how the Polytope Formalism can be applied to the description of constitutional isomerism in a practical manner. Finally, a radically different medicinal chemistry design strategy with broad application, based upon the principles, is describe

Polymer side-chains as arms for molecular recognition

This thesis describes research based on synthetic protocols, methodologies, and applications of polymers containing side-chain molecular recognition elements. The use of molecular recognition, in lieu of covalent chemistry, potentially presents a path through the current limits of polymer science. The work described in this thesis is, at least in part, a testament to this proposal. The first two chapters presen a basic introduction of noncovalent interactions that are ubiquitous in the research of supramolecular polymers. Chapter 2 lays the foundation for the remaining chapters of this thesis by presenting several examples of prior work related specifically to the use of molecular recognition on the side-chains of polymers. The next two chapters present research focused on advancing the functionalization of polymers through molecular recognition. These chapters demonstrate that both architecturally controlled block copolymers and random terpolymers can accept a full load of different substrates without interference among distinct molecular recognition elements along the polymer backbone. Chapters 5 and 6 present a unique application of polymers containing molecular recognition elements, templated synthesis. Chapter 5 first discusses lessons learned from small molecule based templated synthesis in which a template and a substrate are held together by metal coordination and a subsequent bond forming reaction occurs. Chapter 6 discusses template polymerizations, in which a polymeric template was used, and a daughter monomer was polymerized while attached to the template. Another application of polymers containing molecular recognition elements is presented in Chapters 7 and 8. In these chapters, metal coordination is used to assemble polymer multilayer thin films that are both responsive to external stimuli, stable, and erasable. Finally, Chapter 9 summarizes the main conclusions of each chapter and presents a potential view of new projects that might result from the research presented in this thesis.Ph.D.Committee Chair: Weck, Marcus; Committee Member: Beckham, Haskell; Committee Member: Bunz, Uwe; Committee Member: Jones, Christopher; Committee Member: Marder, Set

Carbon in the Galaxy: Studies from Earth and Space

Presented here is the text of the invited papers presented during a meeting entitled, Carbon in the Galaxy: Studies from Earth and Space, that was held at NASA Ames Research Center on November 5 and 6, 1987. For completeness, abstracts from all of the poster papers and the text of a paper summarizing what was learned during the course of the meeting are also included. The underlying premise for the meeting was that there is much to be gained by bringing together scientists from very different disciplines, all of whom study carbon in different ways for different reasons. The interchanges took place during the meeting and the contents of the enclosed papers validate that premise

Non-covalent interactions in organotin(IV) derivatives of 5,7-ditertbutyl- and 5,7-diphenyl-1,2,4-triazolo[1,5-a]pyrimidine as recognition motifs in crystalline self- assembly and their in vitro antistaphylococcal activity

Non-covalent interactions are known to play a key role in biological compounds due to their stabilization of the tertiary and quaternary structure of proteins [1]. Ligands similar to purine rings, such as triazolo pyrimidine ones, are very versatile in their interactions with metals and can act as model systems for natural bio-inorganic compounds [2]. A considerable series (twelve novel compounds are reported) of 5,7-ditertbutyl-1,2,4-triazolo[1,5-a]pyrimidine (dbtp) and 5,7-diphenyl- 1,2,4-triazolo[1,5-a]pyrimidine (dptp) were synthesized and investigated by FT-IR and 119Sn M\uf6ssbauer in the solid state and by 1H and 13C NMR spectroscopy, in solution [3]. The X-ray crystal and molecular structures of Et2SnCl2(dbtp)2 and Ph2SnCl2(EtOH)2(dptp)2 were described, in this latter pyrimidine molecules are not directly bound to the metal center but strictly H-bonded, through N(3), to the -OH group of the ethanol moieties. The network of hydrogen bonding and aromatic interactions involving pyrimidine and phenyl rings in both complexes drives their self-assembly. Noncovalent interactions involving aromatic rings are key processes in both chemical and biological recognition, contributing to overall complex stability and forming recognition motifs. It is noteworthy that in Ph2SnCl2(EtOH)2(dptp)2 \u3c0\u2013\u3c0 stacking interactions between pairs of antiparallel triazolopyrimidine rings mimick basepair interactions physiologically occurring in DNA (Fig.1). M\uf6ssbauer spectra suggest for Et2SnCl2(dbtp)2 a distorted octahedral structure, with C-Sn-C bond angles lower than 180\ub0. The estimated angle for Et2SnCl2(dbtp)2 is virtually identical to that determined by X-ray diffraction. Ph2SnCl2(EtOH)2(dptp)2 is characterized by an essentially linear C-Sn-C fragment according to the X-ray all-trans structure. The compounds were screened for their in vitro antibacterial activity on a group of reference staphylococcal strains susceptible or resistant to methicillin and against two reference Gramnegative pathogens [4] . We tested the biological activity of all the specimen against a group of staphylococcal reference strains (S. aureus ATCC 25923, S. aureus ATCC 29213, methicillin resistant S. aureus 43866 and S. epidermidis RP62A) along with Gram-negative pathogens (P. aeruginosa ATCC9027 and E. coli ATCC25922). Ph2SnCl2(EtOH)2(dptp)2 showed good antibacterial activity with a MIC value of 5 \u3bcg mL-1 against S. aureus ATCC29213 and also resulted active against methicillin resistant S. epidermidis RP62A