Repeat Space Theory Applied to Carbon Nanotubes and Related Molecular Networks. I

The present article is the first part of a series devoted to extending the Repeat Space Theory (RST) to apply to carbon nanotubes and related molecular networks. Four key problems are formulated whose affirmative solutions imply the formation of the initial investigative bridge between the research field of nanotubes and that of the additivity and other network problems studied and solved by using the RST. All of these four problems are solved affirmatively by using tools from the RST. The Piecewise Monotone Lemmas (PMLs) are cornerstones of the proof of the Fukui conjecture concerning the additivity problems of hydrocarbons. The solution of the fourth problem gives a generalized analytical formula of the pi-electron energy band curves of nanotube (a, b), with two new complex parameters c and d. These two parameters bring forth a broad class of analytic curves to which the PMLs and associated theoretical devices apply. Based on the above affirmative solutions of the problems, a central theorem in the RST, called the asymptotic linearity theorem (ALT) has been applied to nanotubes and monocyclic polyenes. Analytical formulae derived in this application of the ALT illuminate in a new global context (i) the conductivity of nanotubes and (ii) the aromaticity of monocyclic polyenes; moreover an analytical formula obtained by using the ALT provides a fresh insight into Hückel’s (4n+2) rule. The present article forms a foundation of the forthcoming articles in this series. The present series of articles is closely associated with the series of articles entitled ‘Proof of the Fukui conjecture via resolution of singularities and related methods’ published in the JOMC.     

Proof of the Fukui conjecture via resolution of singularities and related methods. II $$^{\star}$$

The present article is a direct continuation of the first part of this series. We reduce a proof of the Fukui conjecture (concerning the additivity problem of the zero-point vibrational energies of hydrocarbons) to that of a proposition related to the theory of algebraic curves, so that we can focus on the key mechanism of the additivity phenomena. Namely, by establishing what is called the Basic Piecewise Monotone Theorem (BPMT), we reduce a proof of the Fukui conjecture to that of a proposition, called the Local Analyticity Proposition, Version 1 (LAP1), which admits a proof via resolution of singularities. By LAP1, the essential part of the mechanism of the asymptotic linearity phenomena is extracted and is elucidated by using tools from the mathematical theory of algebraic curves, whose language is of vital importance in analyzing the crux of the additivity mechanism. Dedicated to the memory of Prof. Kenichi Fukui (1918–1998).     

Haplotype studies support slippage as the mechanism of germline mutations in short tandem repeats

Germline mutations of human short tandem repeat (STR) loci are expansions or contractions of repeat arrays which are not well understood in terms of the mechanism(s) underlying such mutations. Although polymerase slippage is generally accepted as a mechanism capable to explain most features of such mutations, it is still possible that unequal crossing over plays some role in those events, as most studies in humans could not exclude unequal crossing over (UCO). Crossing over can be studied by analyzing haplotypes using flanking markers. To check for UCO in mutations, we have analyzed 150 paternity cases for which more than the usual trio (mother, child, and father) were available for testing by analyzing 16 STR loci. In a total of 4900 parent-child allele transfers four mutations were observed at different loci (D8S1179, D18S51, D21S11, and SE33/ACTBP2). To identify the mutated allele and to check for UCO, we typed at least four informative loci flanking the mutated locus and used the pedigree data to establish haplotypes. By doing so we were able to exclude UCO in each case. Moreover, we were able to identify the mutations as one-repeat contractions/expansions. Our data thus support slippage as the mechanism of germline mutations in STRs.     

Match statistics for sequence-based alleles in profiles from forensic PCR-mps kits

The first autosomal sequence-based allele (aka SNP-STR haplotype) frequency database for forensic massively parallel sequencing (MPS) has been published, thereby removing one of the remaining barriers to implementing MPS in casework. The database was developed using a specific set of flank trim sites. If different trim sites or different kits with different primers are used for casework, then SNP-STR haplotypes may be detected that do not have frequencies in the database. We describe a procedure to address calculation of match probabilities when casework samples are generated using an MPS kit with different trim sites than those present in the relevant population frequency database. The procedure provides a framework for comparison of any MPS kit or database combination while also accommodating comparison of MPS and CE profiles.     

Characterization of sequence variation at 30 autosomal STRs in Chinese Han and Tibetan populations

Massively parallel sequencing (MPS) technologies have the ability to reveal sequence variations within STR alleles as well as their nominal allele lengths, which have traditionally been detected by CE instruments. Recently, Thermo Fisher Scientific has updated the MPS-STR panel, named the Precision ID GlobalFiler next-generation sequencing (NGS) STR Panel version 2, with primers redesigned to add two pentanucleotide tandem repeat loci and profile interpretation supported by the Converge software. Using the Ion Chef System, the Ion S5XL System, and the Converge software, genetic variations were characterized within STR repeat and flanking regions of 30 autosomal STR markers in 115 unrelated individuals from two Chinese population groups (58 Tibetans and 57 Hans). Nineteen STRs demonstrated a relative increase in diversity with the variant sequence alleles compared with those of traditional nominal length alleles. In total, 390 alleles were identified by their sequences compared with 258 alleles that were identified by length. Of these 92 sequence variants found within the STR repeat regions, 40 variants were located in STR flanking regions. Additionally, the agreement of the results with CE data was evaluated, as was the ability of this new MPS panel to analyze case-type (11 samples) and artificially degraded samples (seven samples in triplicate). The results generated from this study illustrate that extensive sequence variation exists in commonly used STR markers in the selected population samples and indicate that this NGS STR panel has the potential to be used as an effective tool for human forensics.     

Molecularly engineered graphene surfaces for sensing applications: A review

Graphene is scientifically and commercially important because of its unique molecular structure which is monoatomic in thickness, rigorously two-dimensional and highly conjugated. Consequently, graphene exhibits exceptional electrical, optical, thermal and mechanical properties. Herein, we critically discuss the surface modification of graphene, the specific advantages that graphene-based materials can provide over other materials in sensor research and their related chemical and electrochemical properties. Furthermore, we describe the latest developments in the use of these materials for sensing technology, including chemical sensors and biosensors and their applications in security, environmental safety and diseases detection and diagnosis.     

Structural analysis of certain linear operators representing chemical network systems via the existence and uniqueness theorems of spectral resolution. VI*

The present Part VI of this series of articles provides a mathematical and methodical link between (i) fundamental methodology in the repeat space theory (RST), which is referred to as the approach via the aspect of form and general topology and which has universal unifying power to handle additivity problems of molecules that have many identical moieties, and (ii) frontier electron theory of reactivity indices. Using theoretical tools required to link (i) and (ii), we establish a theorem from which the Generalized Alpha Existence Theorem (a theorem essential in the RST and proved in the previous Part V) directly follows. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem 84: 389–400, 2001     

The discrete-time preemptive repeat identical priority queue

Priority queueing systems come natural when customers with diversified delay requirements have to wait to get service. The customers that cannot tolerate but small delays get service priority over customers which are less delay-sensitive. In this contribution, we analyze a discrete-time two-class preemptive repeat identical priority queue with infinite buffer space and generally distributed service times. Newly arriving high-priority customers interrupt the on-going service of a low-priority customer. After all high-priority customers have left the system, the interrupted service of the low-priority customer has to be repeated completely. By means of a probability generating functions approach, we analyze the system content and the delay of both types of customers. Performance measures (such as means and variances) are calculated and the impact of the priority scheduling is discussed by means of some numerical examples.     

Discrete-Time Geo X /G/1 Queue with Preemptive Repeat Different Priority

This paper considers a discrete-time Geo ^X /G/1 queue accepting two classes of messages with preemptive repeat different priority. Service times of messages of each priority class are i.i.d. according to a general discrete distribution function that may differ between two classes. The completion time and the stability condition for our system are investigated. By using the supplementary variable method and the generating function technique, we derive the joint system contents distributions at various observation instants and also compute the probability distribution for the unfinished work.