Relevant bibliographies by topics / Inner derivation

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

Academic literature on the topic 'Inner derivation'

Author: Grafiati
Published: 2 June 2025
Last updated: 31 July 2025

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Journal articles on the topic "Inner derivation"

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Mohammed, N. F., Hanan F. Qasim, and Z. H. Maibed. "Outer derivations of low-dimensional diassociative algebras." Journal of Discrete Mathematical Sciences and Cryptography 28, no. 4-B (2025): 1369–73. https://doi.org/10.47974/jdmsc-2276.

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The work of outer derivations is extremely extensive research that occurs in many sorts of mathematics fields. This work is concerned with the description of outer derivation for diassociative algebras which is isomorphic to the first cohomology group. The inner derivation (resp. the derivation) interpreted as 1-coboundaries (resp. 1-cocycles). In this regard, we propose an algebraic method to get an important specific case of derivations, known as inner derivation. The procedure is used to determine the inner derivatives of diassociative algebras.
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Karimov, U. Sh. "Local derivations on real AW*-algebras." UZBEK MATHEMATICAL JOURNAL 68, no. 4 (2025): 96–99. https://doi.org/10.29229/uzmj.2024-4-10.

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In this article, we consider local derivations of real AW*-algebras. It has been proven that any derivation of real AW*-algebra whose complexification is (complex) AW*-algebra, is inner. Using a result by M. Bresar, we prove that any local derivation of a real AW*-algebra, whose enveloping algebra is a (complex) AW*-algebra, is a derivation. Consequently, any local derivation of a real W*- algebra is an inner derivation.
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Vladeva, Dimitrinka. "Derivations of polynomial semirings." International Journal of Algebra and Computation 30, no. 01 (2019): 1–12. http://dx.doi.org/10.1142/s0218196719500620.

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The aim of this paper is the investigation of derivations in semiring of polynomials over idempotent semiring. For semiring [Formula: see text], where [Formula: see text] is a commutative idempotent semiring we construct derivations corresponding to the polynomials from the principal ideal [Formula: see text] and prove that the set of these derivations is a non-commutative idempotent semiring closed under the Jordan product of derivations — Theorem 3.3. We introduce generalized inner derivations defined as derivations acting only over the coefficients of the polynomial and consider [Formula: s
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Mansuroğlu, Nil, and Mücahit Özkaya. "Almost inner derivations of Leibniz algebras." Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics 73, no. 4 (2024): 969–81. https://doi.org/10.31801/cfsuasmas.1485446.

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This work is presented the study on almost inner derivations of Leibniz algebras. In this note, we demonstrate the natural extensions of some general properties on derivations given for Lie algebras to Leibniz algebras with finite dimension, and also we investigate which statements a mapping have to hold to be an almost inner derivation.
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Luh, Jiang, and Youpei Ye. "Derivations of higher order in semiprime rings." International Journal of Mathematics and Mathematical Sciences 21, no. 1 (1998): 89–92. http://dx.doi.org/10.1155/s0161171298000106.

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LetRbe a2-torsion free semiprime ring with derivationd. Supposedd2nis a derivation ofR, wherenis a positive integer. It is shown that ifRis(4n−2)-torsion free or ifRis an inner derivation ofR, thend2n−1=0.
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Kaushik, Dr K. L. "Generalised Inner Derivations in Semi Prime Rings." International Journal of Innovative Science and Research Technology 5, no. 7 (2020): 372–74. http://dx.doi.org/10.38124/ijisrt20aug277.

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Let A be any ring and f(xy) = f(x)y+xha(y), where f be any generalised inner derivation(G.I.D ) a be the fixed element of A. In this paper, it is shown that (i) ha must necessarily be a derivation for semi prime ring A. (ii) ∃ no generalized inner derivations f : A → A such that f(x ◦ y) = x ◦ y or f(x ◦ y) + x ◦ y = 0 ∀ x,y ∈ A, We have proved Havala [2] def. p.1147, Herstein [3] Lemma 3.1 p. 1106 as corollaries, along with other results.
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Fošner, Ajda, and Tsiu-Kwen Lee. "Jordan *-Derivations of Finite-Dimensional Semiprime Algebras." Canadian Mathematical Bulletin 57, no. 1 (2014): 51–60. http://dx.doi.org/10.4153/cmb-2012-024-2.

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AbstractIn this paper, we characterize Jordan *-derivations of a 2-torsion free, finite-dimensional semiprime algebra R with involution *. To be precise, we prove the following. Let δ : R → R be a Jordan *-derivation. Then there exists a *-algebra decomposition R = U ⊕ V such that both U and V are invariant under δ. Moreover, * is the identity map of U and δ|U is a derivation, and the Jordan *-derivation δ|V is inner. We also prove the following. Let R be a noncommutative, centrally closed prime algebra with involution *, char R ≠ 2, and let δ be a nonzero Jordan *-derivation of R. If δ is an
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Erfanian, Attar, S. Barootkoob, and Vishki Ebrahimi. "On extension of bi-derivations to the bidual of Banach algebras." Filomat 30, no. 8 (2016): 2261–67. http://dx.doi.org/10.2298/fil1608261e.

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We present some necessary and sufficient conditions such that the (Arens) extensions of a bi-derivation on Banach algebras are again bi-derivations. We then examine our results for some Banach algebras. In particular, we show that the (Arens) extensions of a bi-derivation on C*-algebras are biderivations. Some results on extensions of an inner bi-derivation are also included.
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Patlertsin, Sutida, Suchada Pongprasert, and Thitarie Rungratgasame. "On Inner Derivations of Leibniz Algebras." Mathematics 12, no. 8 (2024): 1152. http://dx.doi.org/10.3390/math12081152.

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Leibniz algebras are generalizations of Lie algebras. Similar to Lie algebras, inner derivations play a crucial role in characterizing complete Leibniz algebras. In this work, we demonstrate that the algebra of inner derivations of a Leibniz algebra can be decomposed into the sum of the algebra of left multiplications and a certain ideal. Furthermore, we show that the quotient of the algebra of derivations of the Leibniz algebra by this ideal yields a complete Lie algebra. Our results independently establish that any derivation of a semisimple Leibniz algebra can be expressed as a combination
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Dadakhodjaev, R. A., and A. A. Rakhimov. "2-Local derivations of real AW*-algebras are derivation." Positivity 25, no. 4 (2021): 1351–56. http://dx.doi.org/10.1007/s11117-021-00815-8.

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Abstract2-Local derivations on real matrix algebras over unital semi-prime Banach algebras are considered. Using the real analogue of the result that any 2-local derivation on the algebra $$M_{2^n}(A)$$ M 2 n ( A ) ($$n\ge 2$$ n ≥ 2 ) is a derivation, it is shown that any 2-local derivation on real AW$$^*$$ ∗ -algebra for which the enveloping algebra is (complex) AW*-algebra, is a derivation, where A is a unital semi-prime Banach algebra with the inner derivation property.
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Dissertations / Theses on the topic "Inner derivation"

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Somerset, Douglas W. B. "Quasi-standard c*-algebras and norms of inner derivations." Thesis, University of Oxford, 1989. http://ora.ox.ac.uk/objects/uuid:ab71e110-d152-473e-ad13-8f09fcd7d7c4.

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In the first half of the thesis a necessary and sufficient condition is given for a separable C*-algebra to be *-isomorphic to a maximal full algebra of cross-sections over a base-space such that the fibre algebras are primitive throughout a dense subset. The condition is that the relation of inseparability for pairs of points in the primitive ideal space should be an open equivalence relation. In the second half of the thesis a characterisation is given of those C*- algebras A for which each self-adjoint inner derivation D(α, A) satisfies ∥D(α, A)∥ = 2 inf {∥α-z∥ : z ∈Z(A), the centre of A}.
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Rasmusson, Lars. "Network capacity sharing with QoS as a financial derivative pricing problem : algorithms and network." Doctoral thesis, SICS, 2002. http://urn.kb.se/resolve?urn=urn:nbn:se:ri:diva-22556.

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A design of an automatic network capacity markets, often referred to as a bandwidth market, is presented. Three topics are investigated. First, a network model is proposed. The proposed model is based upon a trisection of the participant roles into network users, network owners, and market middlemen. The network capacity is defined in a way that allows it to be traded, and to have a well defined price. The network devices are modeled as core nodes, access nodes, and border nodes. Requirements on these are given. It is shown how their functionalities can be implemented in a network. Second, a s
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Andersen, Érika. "As ideias centrais do teorema fundamental do cálculo mobilizadas por alunos de licenciatura em matemática." Pontifícia Universidade Católica de São Paulo, 2011. https://tede2.pucsp.br/handle/handle/10860.

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Made available in DSpace on 2016-04-27T16:57:05Z (GMT). No. of bitstreams: 1 Erika Andersen.pdf: 2001632 bytes, checksum: 36ee4401b855c9699524d7159f8bb771 (MD5) Previous issue date: 2011-05-27<br>The present study relates the results of a qualitative research that aimed to investigate which mental processes may intervene and be combined by students in the development of activities involving the expression = F( x ) = f ( t )dt . The research was based on the study titled Advanced Mathematical Thinking Processes of Tommy Dreyfus. The survey instrument was developed, implemented and analyzed us
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Darney, Keyvin. "Towards next generation risk assessment of chemicals : bayesian meta-analysis of human variability in metabolism and transporters and application for the derivation of pathway-related uncertainty factors Aggregate exposure of the adult French population to pyrethroids, in Toxicology and Applied Pharmacology 351, July 2018 Inter-ethnic differences in CYP3A4 metabolism: A Bayesian meta-analysis for the refinement of uncertainty factors in chemical risk assessment, in Computational Toxicology 12, November 2019 Bayesian meta-analysis of inter-phenotypic differences in human serum paraoxonase-1 activity for chemical risk assessment, in Environment International 138, May 2020 Human variability in influx and efflux transporters in relation to uncertainty factors for chemical risk assessment, in Food and Chemical Toxicology 140, June 2020." Thesis, Brest, 2020. http://www.theses.fr/2020BRES0013.

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Dans le monde moderne, les humains sont exposés à une vaste gamme de produits chimiques tout au long de leur vie. L'évaluation des risques des substances chimiques pour l'homme revêt une importance capitale pour la santé publique et permet de calculer des niveaux sûrs d'exposition aiguë et chronique pour des sous-groupes de la population humaine, notamment les nouveau-nés, les enfants, les personnes âgées et les populations d’origine géographique et de polymorphismes génétiques différents. L'application des données cinétiques liées aux voies métaboliques pour tenir compte de la variabilité hum
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Nakajima, Katsushi, and 克志 中島. "Derivative Pricing Models with Inter-commodity Price Relations." Thesis, 2013. http://hdl.handle.net/10086/25784.

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Chieh, Chang Shih, and 張世傑. "Living.Ego.Liberation-A Study on Inner Reality Derivative from Paper Works to Animation." Thesis, 2007. http://ndltd.ncl.edu.tw/handle/73403742442777014288.

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碩士<br>實踐大學<br>時尚與媒體設計研究所<br>96<br>Abstract This study is preceded by examining the process of producing the art works to explore the creator’s inner world and self recognition. According to the analysis from the researches and creator’s perspectives, the life experiences and personal points of view are reflected on the pieces of the work. In the study, handmade creations turning into an animation are the ways chosen to discover the producer’s personality and self-concept. With compare and contrast to the producing process and the final works, ‘the true oneself’ will be revealed and explored. T
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Matos, Bruno Guilherme Gonçalves de. "Controlador e accionador para motor DC em malha fechada." Master's thesis, 2008. http://hdl.handle.net/1822/65413.

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Dissertação submetida à Universidade do Minho para obtenção do grau de Mestre em Electrónica Industrial e Computadores<br>Os motores CC (corrente contínua) têm actualmente como principal aplicação a motorização de plataformas móveis, onde a fonte de energia é feita com o recurso a baterias. O controlo da velocidade de rotação destes motores vinha a ser feito com recurso a reóstatos variando a corrente do estator, sendo um processo eficaz no caso em que a carga e velocidade de rotação variem pouco. Com o desenvolvimento da electrónica surgiram controladores de velocidade, com gamas de variação
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Books on the topic "Inner derivation"

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Kissin, E. Representations on Krein spaces and derivations of C*-algebras. Longman Scientific & Technical, 1997.

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Kissin, E. Representations on Krein spaces and derivations of C-algebras. Longman, 1997.

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Foster, John E. Inter-cusp ion and electron transport in a NSTAR-derivative ion thruster. National Aeronautics and Space Administration, Glenn Research Center, 2001.

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Arquiola, Elena Felíu. Morfología derivativa y semántica léxica: La prefijación de auto-, co- e inter-. Universidad Autónoma de Madrid, 2003.

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Foster, John E. Inter-cusp ion and electron transport in a NSTAR-derivative ion thruster. National Aeronautics and Space Administration, Glenn Research Center, 2001.

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Foster, John E. Inter-cusp ion and electron transport in a NSTAR-derivative ion thruster. National Aeronautics and Space Administration, Glenn Research Center, 2001.

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Foster, John E. Inter-cusp ion and electron transport in a NSTAR-derivative ion thruster. National Aeronautics and Space Administration, Glenn Research Center, 2001.

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D'Alessandro, Roberta, Irene Franco, and Ángel J. Gallego, eds. The Verbal Domain. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198767886.001.0001.

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The structure of the VP, its complexity, its semantics, its function, and the universality of the heads that it contains are a fascinating puzzle. A lot of progress has been made: this volume features cutting-edge research on the verbal domain, while tackling the problem of the nature and structure of the vP-VP domain. It includes some chapters based on papers presented at the “Little v” workshop which was held at Leiden University on October 25–26, 2013. The volume is divided into three main sections, representing the areas in which contemporary debate on the verbal domain is most active. The
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Simon, Grieser, and Mecklenburg Christian. 6 The Revision of the Credit Derivative Definitions in the Context of the Bank Recovery and Resolution Directive. Oxford University Press, 2016. http://dx.doi.org/10.1093/law/9780198754411.003.0006.

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This chapter examines the adaptation of the International Swaps and Derivatives Association (ISDA) framework by means of the 2014 ISDA Credit Derivatives Definitions (2014 Definitions) in addressing how the EU Bank Recovery and Resolution Directive (BRRD) affects credit derivatives. Particularly, it analyses how the institution or obligation to which the credit derivative is referenced becomes subject to the measures. Financial derivatives are financial instruments which provide for an immediate or future exchange of a reference value. Its price inter alia derives from the underlying reference
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Stephen, Connelly. 5 The Implementation of the EU Bank Recovery and Resolution Directive’s Derivative Bail-in Powers in the UK. Oxford University Press, 2016. http://dx.doi.org/10.1093/law/9780198754411.003.0005.

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This chapter explores the implementation of the EU Bank Recovery and Resolution Directive’s (BRRD) bail-in powers in the UK. Special bail-in powers have been in the legislative sights of HM Treasury (HMT) since the Independent Commission on Banking’s report in 2011. The chapter considers inter alia the following issues raised by the power to bail-in derivatives in the Banking Act 2009: what is a ‘derivative’ and does HMT agree with the BRRD’s definition; the critical role of close-out netting, and whether this is appropriate for all derivatives in the widest sense; the appropriateness of safeg
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Book chapters on the topic "Inner derivation"

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Mashreghi, Javad. "Angular Derivative." In Derivatives of Inner Functions. Springer New York, 2012. http://dx.doi.org/10.1007/978-1-4614-5611-7_4.

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García, Esther, Miguel Gómez Lozano, and Guillermo Vera de Salas. "Nilpotent Inner Derivations in Prime Superalgebras." In Springer Proceedings in Mathematics & Statistics. Springer Nature Singapore, 2022. http://dx.doi.org/10.1007/978-981-19-4751-3_53.

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Mashreghi, Javad. "The Derivative of Finite Blaschke Products." In Derivatives of Inner Functions. Springer New York, 2012. http://dx.doi.org/10.1007/978-1-4614-5611-7_3.

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Mashreghi, Javad. "The Derivative of a Blaschke Product." In Derivatives of Inner Functions. Springer New York, 2012. http://dx.doi.org/10.1007/978-1-4614-5611-7_7.

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Jorgensen, Palle. "Approximately Inner Derivations, Decompositions and Vector Fields of Simple C*-Algebras." In Mappings of Operator Algebras. Birkhäuser Boston, 1991. http://dx.doi.org/10.1007/978-1-4612-0453-4_2.

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Knox, S. A. R. "By Reaction with a Group-IVb Transition- or Inner Transition-Metal Derivative." In Inorganic Reactions and Methods. John Wiley & Sons, Inc., 2007. http://dx.doi.org/10.1002/9780470145265.ch36.

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Lupu, Radu, Eugen Borcoci, and Tinku Rasheed. "Identity-Based Key Derivation Method for Low Delay Inter-domain Handover Re-authentication Service." In Information Security Technology for Applications. Springer Berlin Heidelberg, 2012. http://dx.doi.org/10.1007/978-3-642-29615-4_12.

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Knox, S. A. R. "By Reaction with a Transition- or Inner Transition-Metal Halide Derivative in the Presence of Triethylamine." In Inorganic Reactions and Methods. John Wiley & Sons, Inc., 2007. http://dx.doi.org/10.1002/9780470145265.ch31.

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Cahn, Jean, Marie-Gilberte Borzeix, and Gino Toffano. "Effect of GMI Ganglioside and of Its Inner Ester Derivative in a Model of Transient Cerebral Ischemia in the Rat." In Gangliosides and Neuronal Plasticity. Springer New York, 1986. http://dx.doi.org/10.1007/978-1-4757-5309-7_35.

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Lundström, Håkan, and Jan-Olof Svantesson. "Expressives in Kammu singing." In Capturing Expressivity. Oxford University PressOxford, 2025. https://doi.org/10.1093/oso/9780192858931.003.0003.

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Abstract The Kammu people are mainly rice farmers and live on mountain slopes in northern Laos. The chapter studies the expressives of the Yùan Kammu dialect from the combined perspectives of linguistics and ethnomusicology. Kammu expressives are derived from monosyllabic roots, which each carry the general meaning of those expressives that are derived from that root. The morphological processes involved in the derivation of expressives are reduplication, prefixation, and infixation. The chapter discusses the use of expressives in songs. The expressives, and also other techniques—verb reduplic
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Conference papers on the topic "Inner derivation"

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Selfridge, Justin M. "Centrifugally Stiffened Rotor: A Complete Derivation and Simulation of the Inner Loop Controller." In AIAA Guidance, Navigation, and Control Conference. American Institute of Aeronautics and Astronautics, 2015. http://dx.doi.org/10.2514/6.2015-0073.

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Dai, Yuanjun, Yigong Lu, Yuxin Sun, and Yan He. "Wind energy resource assessment and derivation of characteristic wind speed in wind field of Inner Mongolia." In 2011 International Conference on Consumer Electronics, Communications and Networks (CECNet). IEEE, 2011. http://dx.doi.org/10.1109/cecnet.2011.5769431.

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Kase, Wataru, and Yasuhiko Mutoh. "A simple derivation of right interactor for tall plant and its application to inner-outer factorization." In Automation (MED 2011). IEEE, 2011. http://dx.doi.org/10.1109/med.2011.5983099.

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Churnside, James H. "Approximate refractive turbulence spectrum." In OSA Annual Meeting. Optica Publishing Group, 1989. http://dx.doi.org/10.1364/oam.1989.wf2.

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In calculations of optical propagation through the turbulent atmosphere, one often requires knowledge of the spectrum of the refractivity fluctuations. Several forms of the spectrum are typically used. The Kolmogorov spectrum is a pure −11/3 power law in wavenumber and does not include inner scale effects. The Tatarski spectrum includes a Gaussian taper at high wavenumbers to account for inner scale effects. The Hill spectrum is the result of a rigorous derivation of the spectrum, and includes a bump near the inner scale. Unfortunately, it does not have an analytic form. However, it can be rea
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Haahr, Marie, Jonas Gudme, Jacob Sonne, Sten Overby, Torben Nielsen, and Adam Rubin. "H2S Consumption and the Derivation of a New Annulus Prediction Model for Offshore Flexible Pipes." In ASME 2016 35th International Conference on Ocean, Offshore and Arctic Engineering. American Society of Mechanical Engineers, 2016. http://dx.doi.org/10.1115/omae2016-54472.

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This paper presents the outcome of investigations on the effects of H2S consumption in the annulus of a flexible pipe. Low-molecular gases, such as CH4, H2S, H2O and CO2, permeate slowly from the bore through the inner liner into the annular space between the inner liner and outer sheath of a flexible pipe. This space is densely packed with carbon steel armour wires leaving a very limited free volume. In the presence of water, a corrosive environment for the armour wires is generated and a risk of sour service cracking is introduced. H2S concentration in the annulus is traditionally calculated
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Leishear, Robert A. "Derivations for Hoop Stresses Due to Shock Waves in a Tube." In ASME 2007 Pressure Vessels and Piping Conference. ASMEDC, 2007. http://dx.doi.org/10.1115/pvp2007-26722.

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Equations describing the hoop stresses in a pipe due to water hammer have been presented in the literature in a series of papers, and this paper discusses the complete derivation of the pertinent equations. The derivation considers the pipe wall response to a water hammer induced shock wave moving along the inner wall of the pipe. Factors such as fluid properties, pipe wall materials, pipe dimensions, and damping are considered. These factors are combined to present a single, albeit rather complicated, equation to describe the pipe wall vibrations and hoop stresses as a function of time. This
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Popov, I. A., A. N. Skrypnik, and A. V. Schelchkov. "Derivation of the generalized correlations for heat transfer and friction factor for the flow in pipes with inner helical finning." In THE 1ST INTERNATIONAL CONFERENCE ON PHYSICS AND APPLIED PHYSICS (THE 1ST ICP&AP) 2019: Fundamental and Innovative Research for Improving Competitive Dignified Nation and Industrial Revolution 4.0. AIP Publishing, 2020. http://dx.doi.org/10.1063/5.0000552.

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Loukis, E., P. Wetta, K. Mathioudakis, A. Papathanasiou, and K. Papailiou. "Combination of Different Unsteady Quantity Measurements for Gas Turbine Blade Fault Diagnosis." In ASME 1991 International Gas Turbine and Aeroengine Congress and Exposition. American Society of Mechanical Engineers, 1991. http://dx.doi.org/10.1115/91-gt-201.

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The exploitation of different unsteady quantity measurements for identifying various blade faults is examined in this paper. Measurements of sound emission, casing vibration, shaft displacement and unsteady inner wall pressure are considered. It is demonstrated that particular measurements are sensitive to specific faults. The suitability of measuring each of the above physical quantities for tracing the existence of each kind of fault is discussed. The advantage of combining different measurements originates from the possibility of extending the fault repertory covered when only one particula
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Aretakis, N., and K. Mathioudakis. "Radial Compressor Fault Identification Using Dynamic Measurement Data." In ASME 1996 International Gas Turbine and Aeroengine Congress and Exhibition. American Society of Mechanical Engineers, 1996. http://dx.doi.org/10.1115/96-gt-102.

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The exploitation of different unsteady quantity measurements for identifying mechanical alterations on a radial compressor with a vaned diffuser is examined in this paper. Measurements of sound emission, casing vibration and unsteady inner wall pressure are performed. The mechanical alterations considered have been chosen in order to reproduce or simulate faults in the compressor. They include the insertion of an inlet obstruction, an obstruction in a diffuser passage, variation of impeller tip clearance, and impeller fouling. Processing these measurement data leads to the derivation of fault
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Hojo, Kiminobu, Mayumi Ochi, Seiji Ioka, and Shiro Kubo. "Development and Validation of Inverse Analysis of Heat Conduction and Thermal Stress for Elbow: Part II." In ASME 2013 Pressure Vessels and Piping Conference. American Society of Mechanical Engineers, 2013. http://dx.doi.org/10.1115/pvp2013-97620.

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A heat conduction inverse method for piping elbow was developed to estimate the temperature and stress distribution on the inner surface by measuring the outer surface temperature. In the Part I paper, the derivation and verification of the heat conduction inverse method were described. In the Part II paper, the accuracy for the thermal stress calculation was confirmed by assuming several thermal stratification patters and comparing with the reference results from normal FE heat conduction and thermal stress analyses. As a result, in the case of the measured-basis fluid temperature input from
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Reports on the topic "Inner derivation"

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Wright, Jesse. A Primer on Risk Management: Applications to Latin America and the Caribbean. Inter-American Development Bank, 1997. http://dx.doi.org/10.18235/0008776.

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Bank involvement with derivative activities--both for risk management and as a line of business--is considerable, and supervisory authorities are paying heightened attention to securities markets. This paper discusses the components of risk management in terms of legal, operational, credit and market risks, and identifies best practices in risk management. Finally, it suggests areas in which the Inter-American Development Bank may assist official regulators and the private sector to create the legal and institutional infrastructure for these markets and to instill best practices.
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