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The Use of Chemical Composition and Additives to Classify Petrol and Diesel Using Gas Chromatography–Mass Spectrometry And

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The Use of Chemical Composition and Additives to Classify Petrol and Diesel Using Gas Chromatography–Mass Spectrometry And Spec. Matrices 2019; 7:1–19 Research Article Open Access View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by University of Lincoln Institutional Repository Kazumasa Nomura* and Paul Terwilliger OpenJo Chem.,urnal xyz 2019; 2017; 17:1 (2): 183– 122–135197 The FirstSelf-dual Decade (1964-1972) Leonard pairs Research Article Open Access Researchhttps://doi.org/10.1515/spma-2019-0001 Article Praew Suppajariyawat, Ana Flavia BelchiorReceived de Andrade, May 8, 2018; Mathieu accepted Elie, September Mark 22, Baron, 2018 Max Musterman, Paul Placeholder Jose Gonzalez–Rodriguez* WhatAbstract: Is SoLet DifferentF denote a eld About and let V denote a vector space over F with nite positive dimension. Consider a pair A, A∗ of diagonalizable F-linear maps on V, each of which acts on an eigenbasis for the other one in an The Use Of Chemical Neuroenhancement?Compositionirreducible tridiagonal fashion.And Such a pair is called a Leonard pair. We consider the self-dual case in which Was ist so anders am Neuroenhancement? Additives To Classify Petrolthere exists And an automorphism Diesel of theUsing endomorphism algebra of V that swaps A and A∗. Such an automorphism is unique, and called the duality A A∗. In the present paper we give a comprehensive description of this ↔ Pharmacologicalduality. In particular, and Mental we display Self-transformation an invertible F in-linear Ethic map T on V such that the map X TXT− is the duality Gas Chromatography–MassComparison Spectrometry And → A A∗. We express T as a polynomial in A and A∗. We describe how T acts on ags, decompositions, Pharmakologische↔ und mentale Selbstveränderung im and 24 bases for V. Chemometric Analysis:ethischen A Uk Vergleich Study Keywords: Leonard pair, tridiagonal matrix, self-dual https://doi.org/10.1515/chem-2019-0021 https://doi.org/10.1515/xyz-2017-0010 received February 9, 2013; accepted March 25, 2013; published online July 12, 2014 received August 27, 2018; accepted November 26, 2018. Classication:to be sufficient17B37,15A21 to achieve a high level of classification among the different fuel samples. Leave–one–out cross– Abstract: In the concept of the aesthetic formation of knowledge and its as soon Abstract: The identification of the fuel found inas apossible crime andvalidation success-oriented was applied application, in order insights to validate and profits the results. without the scene and establishing its source is important forreference forensic to the arguments developed around 1900. The main investigation also investigations. In this study petrol and dieselincludes samples theIntroduction periodKeywords: between petrol;the entry diesel; into force fuel and additives; the presentation principal in its current were analysed in order to identify chemical compositionversion. Their functioncomponent as part analysis; of the literary gas chromatography–mass portrayal and narrative technique. and additives with the aim of obtaining a set of markersKeywords:Let orF Function, denotespectrometry. transmission, a eld and investigation, let V denote principal, a vector period space over F with nite positive dimension. We consider a compounds that will allow forensic scientists to identifypair A , A∗ of diagonalizable F-linear maps on V, each of which acts on an eigenbasis for the other one in an fuel sources. The study was performed using petrolDedicatedirreducible and to Paul Placeholder tridiagonal fashion. Such a pair is called a Leonard pair (see [13, Denition 1.1]). The Leonard pair diesel sold in the city of Lincoln (UK) during four seasons A, A∗ is1 said Introduction to be self-dual whenever there exists an automorphism of the endomorphism algebra of V that (winter, spring, summer, autumn).Diesel samples, swaps A and A∗. In this case such an automorphism is unique, and called the duality A A∗. representing eight different brands were analyzed1 Studies by Petrol, and sometimesInvestigations referred to as gasoline, and diesel are ↔ gas chromatography−mass spectrometry (GC−MS) andThe the literature most important contains petroleum many examples derived of liquid self-dual products Leonard pairs. For instance (i) the Leonard pair associ- principal component analysis (PCA). From theThe GC–MS mainated investigation withused an in irreducible alsocombustion includes module theengines. period for Theybetween the Terwilligerare the a complexentry into algebra mixtureforce and of the hypercube (see [4, Corollaries 6.8, 8.5]); (ii) a analytical results, MTBE and ETBE were identifiedthe presentationLeonard as consistingpair in its of current Krawtchouk mostly version. of Their typeorganic function (see hydrocarbons [10, as Denitionpart of the literaryenhanced 6.1]); por (iii) - the Leonard pair associated with an irreducible trayal and narrative technique. the additive more preferably added in super unleadedmodule with for thea variety Terwilliger of special algebra additives of a distance-regular such as oxygenates, graph that has a spin model in the Bose-Mesner alge- petrol samples that can be used to aid in separationbra and (see antiknock, [1, Theorem], antioxidants, [3, Theorems anti–corrosion, 4.1, 5.5]); (iv) ananti–icing appropriately normalized totally bipartite Leonard pair identification. In diesel, the distribution of FAME*Max contents Musterman:(see [11,and Institute Lemma cetane of Marine 14.8]); improvers. Biology, (v) the National All Leonard of Taiwan these Ocean pair additives University, consisting are 2 Pei-Ning used of any to two of a modular Leonard triple A, B, C (see [2, showed the effect of seasonal variation as these wereRoad foundKeelungDenition 20224,improve Taiwan 1.4]); (R.O.C),engine (vi) the e-mail: capacity Leonard [email protected] [1] pair and consisting can vary significantly of a pair of opposite generators for the q-tetrahedron alge- in all spring, summer and autumn samples, but Paulnot Placeholder:found depending Institute of Marine on countryBiology, National and brand.Taiwan Ocean Petrol University, and diesel 2 Pei-Ning have Road Keelungbra, acting 20224, Taiwan on an (R.O.C), evaluation e-mail: [email protected] module (see [5, Proposition 9.2]). The example (i) is a special case of (ii), and the in all winter ones. The selection of a reduced number different chemical properties, boiling points and chemical examples (iii), (iv) are special cases of (v). of key fuel compounds and additives was also Open shown Access. © 2017components. Mustermann and Most Placeholder, fuel published products by De vary Gruyter. depending Thison work the is licensed underLet the CreativeA, A∗ Commonsdenote Attribution-NonCommercial-NoDerivatives a Leonard pair on V. We 4.0 can License. determine whether A, A∗ is self-dual in the following way. original source of the crude oils used, the refining process, d By [13, Lemmamarket 1.3]demand, each eigenspacelegislative ofrequirements,A, A∗ has dimension and the one. Let θi i= denote an ordering of the eigen- { } d valuesparticular of A. For product ≤ i ≤ requirementsd let vi denote with a respectθi-eigenvector to the fuels for A. The ordering θi i= is said to be standard *Corresponding author: Jose Gonzalez–Rodriguez, School of d { } d wheneveruse.A Give∗ acts this, on thethe basiscontentv iofi =variousin an types irreducible of additives tridiagonal fashion. If the ordering θi i= is standard Chemistry, University of Lincoln, Joseph Banks Laboratories, Green d { } { } Lane, Lincoln, Lincolnshire LN6 7DL, United Kingdom, E-mail: then theor orderingblending agentsθd−i i =helpis alsoto differentiate standard, and characterize no further ordering is standard. Similar comments apply to d { } d [email protected] A∗. Let differentθ denote fuel products. a standard ordering of the eigenvalues of A. Then A, A∗ is self-dual if and only if θ { i}i= { i}i= Praew Suppajariyawat, Ana Flavia Belchior de Andrade, Mathieuis a standardThe ordering identification of the eigenvaluesand classification of A∗ (see of [7,fuel Proposition is 8.7]). Elie, Mark Baron: School of Chemistry, University of Lincoln, Joseph extremely important in a forensic context as it may help Banks Laboratories, Green Lane, Lincoln, Lincolnshire LN6 7DL, police forces to identify fuel sources in various fuel related United Kingdom Praew Suppajariyawat: Central Institute of Forensic Science, offenses. These offenses can range from fuel fraud and Ministry of Justice, 499 Sukpraprute Building 16th Floor, *Correspondingspills through Author: to Kazumasaarson and Nomura:terrorism.Tokyo For example, Medical and petrol Dental University, Ichikawa, 272-0827,Japan, Prachachuen Rd, Bang–Sue, Bangkok 10800, Thailand E-mail: [email protected] diesel provide powerful heat energy and are the choice Ana Flavia Belchior de Andrade: Criminalistic Institute, Federal Paul Terwilliger:for arsonistsDepartment due to its of Mathematics,high ignition University power. Additionally, of Wisconsin, Madison, WI53706, USA, E-mail: District Civil Police, Secao de Pericia e Analises Laboratoriais–SPAL–[email protected] can be combined with ammonium nitrate to make SPO, Conjunto A, Lote 23, Bloco E, CEP 70.610–907 Brasilia, DF, Brazil Journal xyz 2017; 1 (2): 122–135 Open Access. © 2019 Praew Suppajariyawat et al., published byOpen De Gruyter. Access. ©This2019 work Kazumasa is licensed Nomura under and the Paul Creative Terwilliger, Commons published by De Gruyter. This work is licensed under the Creative Commons Attribution alone 4.0 License. Attribution alone 4.0 License. Brought to you by | University
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