Relevant bibliographies by topics / Mahler measure

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  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers

Academic literature on the topic 'Mahler measure'

Author: Grafiati
Published: 4 June 2021
Last updated: 29 January 2023

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Journal articles on the topic "Mahler measure"

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Issa, Zahraa, and Matilde Lalín. "A Generalization of a Theorem of Boyd and Lawton." Canadian Mathematical Bulletin 56, no. 4 (December 1, 2013): 759–68. http://dx.doi.org/10.4153/cmb-2012-010-2.

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Abstract.The Mahler measure of a nonzero n-variable polynomial P is the integral of log |P| on the unit n-torus. A result of Boyd and Lawton says that the Mahler measure of a multivariate polynomial is the limit of Mahler measures of univariate polynomials. We prove the analogous result for different extensions of Mahler measure such as generalized Mahler measure (integrating the maximum of log |P| for possibly different P’s), multiple Mahler measure (involving products of log |P| for possibly different P’s), and higher Mahler measure (involving logk |P|).
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Sasaki, Yoshitaka. "Zeta Mahler measures, multiple zeta values and L-values." International Journal of Number Theory 11, no. 07 (October 21, 2015): 2239–46. http://dx.doi.org/10.1142/s1793042115501006.

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The zeta Mahler measure is the generating function of higher Mahler measures. In this article, explicit formulas of higher Mahler measures, and relations between higher Mahler measures and multiple zeta (star) values are showed by observing connections between zeta Mahler measures and the generating functions of multiple zeta (star) values. Additionally, connections between higher Mahler measures and Dirichlet L-values associated with primitive quadratic characters are discussed.
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Kurokawa, Nobushige. "A $q$-Mahler measure." Proceedings of the Japan Academy, Series A, Mathematical Sciences 80, no. 5 (May 2004): 70–73. http://dx.doi.org/10.3792/pjaa.80.70.

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Everest, G. R., and Bríd Ní Fhlathúin. "The elliptic Mahler measure." Mathematical Proceedings of the Cambridge Philosophical Society 120, no. 1 (July 1996): 13–25. http://dx.doi.org/10.1017/s0305004100074624.

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In this paper, we are going to introduce an elliptic analogue of the classical Mahler measure of an integral polynomial. The measure is shown to vanish if and only if the roots of the polynomial are attached to division points of the curve. This is the exact analogue of the statement that the Mahler measure of an integral polynomial vanishes if and only if all of its roots are division points of the circle (in other words, roots of unity). The proof of this result exploits an integral representation for the local canonical heights on the elliptic curve. The integral is that of a simple polynomial function over the complete curve in the appropriate valuation. Attention then shifts to the calculation of local integrals of arbitrary rational functions on elliptic curves. Results are proved which show these integrals may be computed as effective limits of Riemann sums. Finally, consideration is given to analogous behaviour for abelian varieties. We give a method, in principle, for computing the global canonical height of a rational point on an abelian variety denned over an algebraic number field, once again exploiting the integral representation of the local heights. The methods used include recent inequalities for linear forms in elliptic and abelian logarithms.
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Fei, Jiarui. "Mahler measure of 3D Landau–Ginzburg potentials." Forum Mathematicum 33, no. 5 (July 28, 2021): 1369–401. http://dx.doi.org/10.1515/forum-2020-0339.

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Abstract We express the Mahler measures of 23 families of Laurent polynomials in terms of Eisenstein–Kronecker series. These Laurent polynomials arise as Landau–Ginzburg potentials on Fano 3-folds, sixteen of which define K ⁢ 3 {K3} hypersurfaces of generic Picard rank 19, and the rest are of generic Picard rank less than 19. We relate the Mahler measure at each rational singular moduli to the value at 3 of the L-function of some weight-3 newform. Moreover, we find ten exotic relations among the Mahler measures of these families.
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Guilloux, Antonin, and Julien Marché. "Volume function and Mahler measure of exact polynomials." Compositio Mathematica 157, no. 4 (April 2021): 809–34. http://dx.doi.org/10.1112/s0010437x21007016.

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We study a class of two-variable polynomials called exact polynomials which contains $A$ -polynomials of knot complements. The Mahler measure of these polynomials can be computed in terms of a volume function defined on the vanishing set of the polynomial. We prove that the local extrema of the volume function are on the two-dimensional torus and give a closed formula for the Mahler measure in terms of these extremal values. This formula shows that the Mahler measure of an irreducible and exact polynomial divided by $\pi$ is greater than the amplitude of the volume function. We also prove a K-theoretic criterion for a polynomial to be a factor of an $A$ -polynomial and give a topological interpretation of its Mahler measure.
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SILVER, DANIEL S., and SUSAN G. WILLIAMS. "MAHLER MEASURE OF ALEXANDER POLYNOMIALS." Journal of the London Mathematical Society 69, no. 03 (May 24, 2004): 767–82. http://dx.doi.org/10.1112/s0024610704005289.

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PINNER, CHRISTOPHER. "Bounding the elliptic Mahler measure." Mathematical Proceedings of the Cambridge Philosophical Society 124, no. 3 (November 1998): 521–29. http://dx.doi.org/10.1017/s0305004198002795.

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Mossinghoff, Michael J. "Polynomials with small Mahler measure." Mathematics of Computation 67, no. 224 (October 1, 1998): 1697–706. http://dx.doi.org/10.1090/s0025-5718-98-01006-0.

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Amoroso, Francesco. "Mahler measure on Galois extensions." International Journal of Number Theory 14, no. 06 (July 2018): 1605–17. http://dx.doi.org/10.1142/s1793042118500963.

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We study the Mahler measure of generators of a Galois extension with Galois group the full symmetric group. We prove that two classical constructions of generators give always algebraic numbers of big height. These results answer a question of Smyth and provide some evidence to a conjecture which asserts that the height of such a generator grows to infinity with the degree of the extension.
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Dissertations / Theses on the topic "Mahler measure"

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Rogers, Mathew D. "Hypergeometric functions and Mahler measure." Thesis, University of British Columbia, 2008. http://hdl.handle.net/2429/1420.

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The logarithmic Mahler measure of an n-variable Laurent polynomial, P(x1,...,xn) is defined by [expression]. Using experimental methods, David Boyd conjectured a large number of explicit relations between Mahler measures of polynomials and special values of different types of L-series. This thesis contains four papers which either prove or attempt to prove conjectures due to Boyd. The introductory chapter contains an overview of the contents of each manuscript.
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Staines, Matthew. "On the inverse problem for Mahler Measure." Thesis, University of East Anglia, 2012. https://ueaeprints.uea.ac.uk/48118/.

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We investigate a number of aspects of the inverse problem for Mahler Measure. If β is an algebraic unit, we demonstrate how to determine if there are any reciprocal numbers with measure β. We also give a formula for the number of integer polynomials with measure β and given degree.
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Chern, Shey-jey. "Estimates for the number of polynomials with bounded degree and bounded Mahler measure /." Digital version accessible at:, 2000. http://wwwlib.umi.com/cr/utexas/main.

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De, Silva Dilum P. "Lind-Lehmer constant for groups of the form Z[superscript]n[subscript]p." Diss., Kansas State University, 2013. http://hdl.handle.net/2097/16244.

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Mohamed, Ismail Mohamed Ishak. "Lower bounds for heights in cyclotomic extensions and related problems." Diss., Manhattan, Kan. : Kansas State University, 2009. http://hdl.handle.net/2097/2274.

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Mehrabdollahei, Mahya. "La mesure de Mahler d’une famille de polynômes exacts." Thesis, Sorbonne université, 2022. https://accesdistant.sorbonne-universite.fr/login?url=https://theses-intra.sorbonne-universite.fr/2022SORUS170.pdf.

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Dans cette thèse, nous étudions la suite de mesures de Mahler d’une famille de polynômes à deux variables exacts et réguliers, que nous notons Pd := P0≤i+j≤d xiyj . Elle n’est bornée ni en volume, ni en genre de la courbe algébrique sous-jacente. Nous obtenons une expression pour la mesure de Mahler de Pd comme somme finie de valeurs spéciales du dilogarithme de Bloch-Wigner. Nous utilisons SageMath pour approximer m(Pd) pour 1 ≤ d ≤ 1000. En recourant à trois méthodes différentes, nous prouvons que la limite de la suite de mesures de Mahler de cette famille converge vers 92π2 ζ(3). De plus, nous calculons le développement asymptotique de la mesure de Mahler de Pd et prouvons que sa vitesse de convergence est de O(log dd2 ). Nous démontrons également une généralisation du théorème de Boyd-Lawton, affirmant que les mesures de Mahler multivariées peuvent être approximéess en utilisant les mesures de Mahler de dimension inférieure. Enfin, nous prouvons que la mesure de Mahler de Pd pour d arbitraire peut être écrite comme une combinaison linéaire de fonctions L associées à un caractère de Dirichlet primitif impair. Nous calculons finalement explicitement la représentation de la mesure de Mahler de Pd en termes de fonctions L, pour 1 ≤ d ≤ 6
In this thesis we investigate the sequence of Mahler measures of a family of bivariate regular exact polynomials, called Pd := P0≤i+j≤d xiyj , unbounded in both degree and the genus of the algebraic curve. We obtain a closed formula for the Mahler measure of Pd in termsof special values of the Bloch–Wigner dilogarithm. We approximate m(Pd), for 1 ≤ d ≤ 1000,with arbitrary precision using SageMath. Using 3 different methods we prove that the limitof the sequence of the Mahler measure of this family converges to 92π2 ζ(3). Moreover, we compute the asymptotic expansion of the Mahler measure of Pd which implies that the rate of the convergence is O(log dd2 ). We also prove a generalization of the theorem of the Boyd-Lawton which asserts that the multivariate Mahler measures can be approximated using the lower dimensional Mahler measures. Finally, we prove that the Mahler measure of Pd, for arbitrary d can be written as a linear combination of L-functions associated with an odd primitive Dirichlet character. In addition, we compute explicitly the representation of the Mahler measure of Pd in terms of L-functions, for 1 ≤ d ≤ 6
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Santos, Jefferson Marques. "Altura e equidistribuição de pontos algébricos." Universidade Federal de Goiás, 2017. http://repositorio.bc.ufg.br/tede/handle/tede/7564.

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The concept of roots of a polynomial is quite simple but has several applications. This concept extends more generally to the case of "small" algebraic points sequences in a curve. This dissertation aims to estimate the size of algebraic numbers by means of Weil height. In addition to showing that they are distributed evenly around the unit circle, through Bilu Equidistribution Theorem.
O conceito de raízes de um polinômio é bastante simples mas possui várias aplicações. Este conceito se estende de forma mais geral para o caso de sequências de pontos algébricos “pequenos” em uma curva. Esta dissertação tem por objetivo estimar o tamanho de números algébricos por meio da altura de Weil. Além de mostrar que os mesmos se distribuem uniformemente em torno do círculo unitário, por meio do Teorema de Equidistribuição de Bilu.
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Fhlathuin, Brid ni. "Mahler's measure on Abelian varieties." Thesis, University of East Anglia, 1995. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.296951.

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This thesis is a study of the integration of proximity functions over certain compact groups. Mean values are found of the ultrametric valuation of certain rational functions associated with a divisor on an abelian variety, and it is shown how these may be expressed in terms of an integral, thus finding the analogue, for an abelian variety, of Mahler's definition of the measure of a polynomial. These integrals are shown to arise in a manner which mimics classical Riemann sums, and their relation with the global canonical height is investigated. It is shown that the measure is a rational multiple of log p. Similar results are given for elliptic curves, taking the divisor to be the identity of the group law, and somewhat stronger mean value theorems proven in this more specific case by working directly with local canonical heights rather than approaching them through related functions. Effective asymptotic formulae for the local height are derived, first for the kernel of reduction of a curve and then, via a detailed analysis of the local reduction of the curve, for the group of rational points. The theory of uniform distribution is used to show that the mean value also takes an integral form in the case of an archimedean valuations, and recent inequalities for elliptic forms in logarithms are used to give error terms for the convergence towards the measure. This is undertaken first for the local height on an elliptic curve, and then, in terms of general theta-functions, on an abelian variety. We then seek to exploit these generalisations of the Mahler measure to yield an alternative method to that of Silverman and Tate for the determining of the global height. The integration over a cyclic group of the laws satisfied locally by the height allows us to reformulate our theorems in a manner conducive to practical application. It is demonstrated how our asymptotic formulae may be used together with an appropriate computer software package, PARI in our case, to calculate the mean value of heights, and, more generally, of rational functions, on an elliptic curve and on abehan varieties of higher genus. Some such calculations are displayed, with comments on their efficacy and their possible future development.
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Condon, John Donald. "Mahler measure evaluations in terms of polylogarithms." Thesis, 2004. http://hdl.handle.net/2152/1218.

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Condon, John Donald Rodríguez Villegas Fernando. "Mahler measure evaluations in terms of polylogarithms." 2004. http://wwwlib.umi.com/cr/utexas/fullcit?p3142710.

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Books on the topic "Mahler measure"

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Alta.) WIN (Conference) (2nd 2011 Banff. Women in Numbers 2: Research directions in number theory : BIRS Workshop, WIN2 - Women in Numbers 2, November 6-11, 2011, Banff International Research Station, Banff, Alberta, Canada. Edited by David Chantal 1964-, Lalín Matilde 1977-, and Manes Michelle 1970-. Providence, Rhode Island: American Mathematical Society, 2013.

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Around the Unit Circle: Mahler Measure, Integer Matrices and Roots of Unity. Springer International Publishing AG, 2021.

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Brunault, François. Many Variations of Mahler Measures. Cambridge University Press, 2020.

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Zudilin, Wadim, and François Brunault. Many Variations of Mahler Measures: A Lasting Symphony. University of Cambridge ESOL Examinations, 2020.

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Book chapters on the topic "Mahler measure"

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Everest, Graham, and Thomas Ward. "The Elliptic Mahler Measure." In Heights of Polynomials and Entropy in Algebraic Dynamics, 117–44. London: Springer London, 1999. http://dx.doi.org/10.1007/978-1-4471-3898-3_6.

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McKee, James, and Chris Smyth. "Restricted Mahler Measure Results." In Universitext, 205–18. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-80031-4_11.

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McKee, James, and Chris Smyth. "The Mahler Measure of Nonreciprocal Polynomials." In Universitext, 219–36. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-80031-4_12.

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Villegas, F. Rodriguez. "Modular Mahler Measures I." In Topics in Number Theory, 17–48. Boston, MA: Springer US, 1999. http://dx.doi.org/10.1007/978-1-4613-0305-3_2.

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Szpiro, L., and T. J. Tucker. "Equidistribution and generalized Mahler measures." In Number Theory, Analysis and Geometry, 609–38. Boston, MA: Springer US, 2011. http://dx.doi.org/10.1007/978-1-4614-1260-1_26.

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McKee, James, and Chris Smyth. "Mahler Measures of Polynomials in One Variable." In Universitext, 1–24. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-80031-4_1.

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McKee, James, and Chris Smyth. "Mahler Measures of Polynomials in Several Variables." In Universitext, 25–56. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-80031-4_2.

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Everest, Graham, and Thomas Ward. "Mahler’s Measure in Many Variables." In Heights of Polynomials and Entropy in Algebraic Dynamics, 51–80. London: Springer London, 1999. http://dx.doi.org/10.1007/978-1-4471-3898-3_3.

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Baake, Michael, Michael Coons, and Neil Mañibo. "Binary Constant-Length Substitutions and Mahler Measures of Borwein Polynomials." In Springer Proceedings in Mathematics & Statistics, 303–22. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-36568-4_20.

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"Mahler measure, Eisenstein series and dimers." In Mirror Symmetry V, 151–58. Providence, Rhode Island: American Mathematical Society, 2006. http://dx.doi.org/10.1090/amsip/038/08.

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Conference papers on the topic "Mahler measure"

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Chesi, Graziano. "On the Mahler measure of matrix pencils." In 2013 American Control Conference (ACC). IEEE, 2013. http://dx.doi.org/10.1109/acc.2013.6580630.

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Morisawa, Takayuki, and Takao Komatsu. "Mahler measure of the Horie unit and Weber’s Class Number Problem in the Cyclotomic Z[sub p]-extension of Q." In DIOPHANTINE ANALYSIS AND RELATED FIELDS—2010: DARF—2010. AIP, 2010. http://dx.doi.org/10.1063/1.3478179.

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Borwein, Jonathan. "Mahler measures, short walks and log-sine integrals." In the 2011 International Workshop. New York, New York, USA: ACM Press, 2011. http://dx.doi.org/10.1145/2331684.2331685.

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Warth, Marco, Boris Lerch, Adam Loch, and Alfred Elsaesser. "MAHLE Advanced EGR Systems for Commercial Diesel Engines to Meet Future Emission Demands." In ASME 2007 Internal Combustion Engine Division Fall Technical Conference. ASMEDC, 2007. http://dx.doi.org/10.1115/icef2007-1639.

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Given the ever more stringent emission regulations modern diesel engines undergo these days, the need for advanced EGR systems becomes crucial in all major applications, in particular on- & off-road commercial diesel engines. One of the key aspects of these so-called advanced EGR systems thereby is to reliably provide the engine with the appropriate, high amounts of EGR over the entire range of operating conditions. Whereas common systems are either optimized for low-torque/low-speed operating conditions, or a narrow range around one specific engine speed, the advanced systems aim to both cover the entire operating range and significantly increase the current level of EGR. The advanced EGR systems developed at MAHLE make use of two types of fast acting devices in a modular approach. Depending on the engine size/layout and the amount of EGR needed, the devices are either placed directly in the EGR line or the intake manifold. Using the latest technical advances in mechatronics, the oscillating valves can be opened or closed within less than 3ms, which makes it not only possible to accurately control the amount of EGR fed back into the engine, it also allows to boost the amount of EGR using the exhaust pressure oscillations. In addition to these oscillating valves, rotational flaps have been developed to significantly reduce the complexity of the systems, while still offering similar benefits in terms of EGR rates and variability. Shown hereafter are the results from thorough investigations conducted on both European and US heavy-duty diesel engines. Focusing on some of the most common engine characteristics, such as EGR rates, emissions of nitrogen oxide and fuel consumption, significant benefits can be seen using the newly developed technologies. Compared to conventional measures, such as increased exhaust backpressure and/or constant charge-air throttling, the advanced systems prove to be both more efficient and flexible in terms of EGR rates, as well as beneficial regarding some of the most important engine characteristics.
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Chen, Yu, and Carol Lynn Deck. "Accurate Specific Fuel Consumption Measurement and its Application on Piston Friction Reduction for a Heavy-Duty Diesel Engine." In ASME 2011 Internal Combustion Engine Division Fall Technical Conference. ASMEDC, 2011. http://dx.doi.org/10.1115/icef2011-60219.

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In recent years the attention of the internal combustion engine industry has been on improving fuel economy. These changes not only decrease the amount of fuel used and improve the efficiency of the engine, but also save the end-user on fuel costs, reduce engine emissions, and aid in the achievement of future government fuel economy regulations. An approach to decreasing fuel consumption is through improvements to engine mechanical and thermal efficiency. MAHLE has developed a testing method to accurately measure engine specific fuel consumption (SFC). SFC is an indicator of engine efficiency, hence it is directly effected by a reduction in friction. Since changes in SFC are small, considerable precision was required to measure it. To achieve this high level of accuracy key engine parameters were controlled along with boundary parameters. This study utilized a firing heavy-duty diesel engine running on a dynamometer. Results are presented to depict the repeatability of the technique over speed and load.
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Schreer, Kai, Ingo Roth, Simon Schneider, and Holger Ehnis. "Analysis of Aluminum and Steel Pistons: Comparison of Friction, Piston Temperature, and Combustion." In ASME 2013 Internal Combustion Engine Division Fall Technical Conference. American Society of Mechanical Engineers, 2013. http://dx.doi.org/10.1115/icef2013-19114.

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While steel pistons have been in use for a long time in commercial vehicle diesel engines, the first series production applications for passenger car diesel engines are currently imminent. The main reason for the use of steel pistons in high speed diesel engines is not, as maybe initially hypothesized, the increasing requirements on the component strength due to increasing mechanical loads, but rather challenges based on the actual CO2-legislation. The increasing requirements to reduce the fuel consumption necessitate new innovative technologies. The imminent penalties for exceeding the prescribed CO2 emissions seem to make the steel piston a viable alternative today, despite its higher manufacturing costs. So far, the CO2-benefits using steel pistons were mainly ascribed to the reduced friction between piston and cylinder liner due to no thermal interference. Fuel consumption measurements at vehicle manufacturer and research institutes hypothesize also an influence of the steel piston on the thermodynamic efficiency. MAHLE uses engine tests to investigate one piston variant made of aluminum (series production piston with cooled ring carrier) and one of steel (MAHLE TopWeld) in a detailed system comparison. Using a fully indicated engine, a combustion process analysis is performed and used as the basis for a loss analysis. The engine set-up parameters can be adjusted fully variable using a flexible ECU. The effect that the piston variant has on the combustion process is captured and balanced, e.g., by adjusting the parameters to obtain identical emissions. The analysis records the potential of the variants for each engine operating map area. The thermal conditions for the piston and the piston wall temperature on the combustion chamber side are varied over a wide range using a conditioning device for piston cooling. The influence of this intervention on the thermal load of the piston and the combustion and also the influence of different combustion mappings is measured directly by telemetric piston temperature measurement. MAHLE recently completed a system comparison [3] between aluminum and steel pistons with detailed measurements on a fully indicated engine, covering friction and temperature behavior as well as influences on combustion.
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Abdo, Jamil, and Elhanafi Shamseldin. "Modeling of Contact Area, Contact Force, and Contact Stiffness of Mechanical Systems With Friction." In ASME 2005 International Mechanical Engineering Congress and Exposition. ASMEDC, 2005. http://dx.doi.org/10.1115/imece2005-82980.

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It is well recognized that the contact stiffness, true contact area, and the contact force are among the key features in the study of friction system behavior. This paper presents the development of formulae for the mechanical component of dry-friction at the interface of two microscopic rough surfaces. Elastic deformation under the influence of the contact forces is considered. The elastic contact model formulation between interacting asperities is not assumed to occur only at asperity peaks, thus allowing the possibility of oblique contacts wherein the local contact surfaces are no longer parallel to the mean planes of the mating surfaces. It is shown that the approach enables the separation of the contact area into its normal and tangential projections and the contact force into its normal and tangential components. The mathematical model of contact is utilized to develop formulae for normal and tangential contact stiffness. The analytical method is used to estimate contact stiffness components. Contact parameter values for the sample are derived from the surface profile data taken from a 1.0-mm by 10-mm test area. The profile is measured using a Mahr profilometer. A computer program is written and used to analyze the profile data. The analysis yields the asperity density, average asperity radius, and the standard deviation for each test area.
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