Academic literature on the topic 'Higher Operators'

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Journal articles on the topic "Higher Operators"

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Okello, Boaz, Fredrick Nyamwala, and David Ambogo. "Symmetric operator extensions of composites of higher order difference operators." Annals of Mathematics and Computer Science 24 (July 9, 2024): 99–116. http://dx.doi.org/10.56947/amcs.v24.352.

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In this paper we have considered two higher order difference operators generated by two higher order difference functions on the Hilbert space of square summable functions. By allowing the leading coefficients to be unbounded and the other coefficients as constant functions, we have shown that the composites of two symmetric difference operators are symmetric if the leading coefficients are scalar multiple of each other and the common divisor of their orders is 1. Using examples, we have shown that these conditions of symmetry cannot be weakened. Furthermore, We have shown that the deficiency
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Baskakov, A. G., and V. D. Kharitonov. "Spectral analysis of operator polynomials and higher-order difference operators." Mathematical Notes 101, no. 3-4 (2017): 391–405. http://dx.doi.org/10.1134/s0001434617030026.

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Alshammari, Hadi Obaid. "Higher order hyperexpansivity and higher order hypercontractivity." AIMS Mathematics 8, no. 11 (2023): 27227–40. http://dx.doi.org/10.3934/math.20231393.

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<abstract><p>As a natural extension of the concept of $ (m, p) $-hyperexpansive and $ (m, p) $-hypercontractive of a single operator, we introduce and study the concepts of $ (m, p) $-hyperexpansivity and $ (m, p) $-hypercontractivity for $ d $-tuple of commuting operators acting on Banach spaces. These concepts extend the definitions of $ m $-isometries and $ (m, p) $-isometric tuples of bounded linear operators acting on Hilbert or Banach spaces, which have been introduced and studied by many authors.</p></abstract>
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Talamucci, F. "Lagrangian Operators with Higher Derivatives." Journal of Advances in Mathematics and Computer Science 29, no. 4 (2018): 1–12. http://dx.doi.org/10.9734/jamcs/2018/44539.

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Pomraning, G. C. "Higher Order Fokker-Planck Operators." Nuclear Science and Engineering 124, no. 3 (1996): 390–97. http://dx.doi.org/10.13182/nse96-a17918.

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Yurko, V. A. "On higher-order difference operators." Journal of Difference Equations and Applications 1, no. 4 (1995): 347–52. http://dx.doi.org/10.1080/10236199508808033.

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Razumov, A. V. "Monodromy operators for higher rank." Journal of Physics A: Mathematical and Theoretical 46, no. 38 (2013): 385201. http://dx.doi.org/10.1088/1751-8113/46/38/385201.

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Harman, C. J., and N. I. Robinson. "Higher order geometric difference operators." International Journal of Computer Mathematics 63, no. 3-4 (1997): 265–74. http://dx.doi.org/10.1080/00207169708804565.

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Salehi, Maryam, and Karim Hedayatian. "On higher order selfadjoint operators." Linear Algebra and its Applications 587 (February 2020): 358–86. http://dx.doi.org/10.1016/j.laa.2019.11.009.

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De Schepper, H., D. Eelbode, and T. Raeymaekers. "Twisted Higher Spin Dirac Operators." Complex Analysis and Operator Theory 8, no. 2 (2013): 429–47. http://dx.doi.org/10.1007/s11785-013-0295-5.

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Dissertations / Theses on the topic "Higher Operators"

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Paoloni, Lorenzo. "On higher-derivative operators in supersymmetric theories." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2022. http://amslaurea.unibo.it/25119/.

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The occurrence of higher-derivative terms both in classical and in quantum theories can be problematic. In case of impossibility of eliminating them by performing an integration by parts they give rise, in fact, to multiple problems, like extra unphysical degrees of freedom, negative unbounded energies and pathological negative norm states called ghosts. The possibility of curing them comes from a procedure developed by Jaén, Llosa and Molina in a classical Hamiltonian framework. It is applied when these terms are introduced as perturbative corrections to a healthy leading order theory multipl
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Muller, Jacob. "Higher order differential operators on graphs." Licentiate thesis, Stockholms universitet, Matematiska institutionen, 2020. http://urn.kb.se/resolve?urn=urn:nbn:se:su:diva-178070.

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This thesis consists of two papers, enumerated by Roman numerals. The main focus is on the spectral theory of <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?n" />-Laplacians. Here, an <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?n" />-Laplacian, for integer <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?n" />, refers to a metric graph equipped with a differential operator whose differential expression is the <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?2n" />-th derivative. In Paper I, a classification of all vertex conditions corresponding to self-a
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Fedosov, Boris, Bert-Wolfgang Schulze, and Nikolai N. Tarkhanov. "The index of higher order operators on singular surfaces." Universität Potsdam, 1998. http://opus.kobv.de/ubp/volltexte/2008/2512/.

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The index formula for elliptic pseudodifferential operators on a two-dimensional manifold with conical points contains the Atiyah-Singer integral as well as two additional terms. One of the two is the 'eta' invariant defined by the conormal symbol, and the other term is explicitly expressed via the principal and subprincipal symbols of the operator at conical points. In the preceding paper we clarified the meaning of the additional terms for first-order differential operators. The aim of this paper is an explicit description of the contribution of a conical point for higher-order differential
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Baier, Herbert [Verfasser], Klaus [Gutachter] Wagner, and Gerd [Gutachter] Wechsung. "Operators of Higher Order / Herbert Baier ; Gutachter: Klaus Wagner, Gerd Wechsung." Würzburg : Universität Würzburg, 2016. http://d-nb.info/112092507X/34.

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Claire, Narinder Singh. "Spectral theory and heat kernel estimates for higher order differential operators." Thesis, King's College London (University of London), 2000. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.395811.

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Gioev, Dimitri. "Generalizations of Szego Limit Theorem : Higher Order Terms and Discontinuous Symbols." Doctoral thesis, KTH, Mathematics, 2001. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-3123.

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Abed, Jamil. "An iterative approach to operators on manifolds with singularities." Phd thesis, Universität Potsdam, 2010. http://opus.kobv.de/ubp/volltexte/2010/4475/.

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We establish elements of a new approach to ellipticity and parametrices within operator algebras on manifolds with higher singularities, only based on some general axiomatic requirements on parameter-dependent operators in suitable scales of spaes. The idea is to model an iterative process with new generations of parameter-dependent operator theories, together with new scales of spaces that satisfy analogous requirements as the original ones, now on a corresponding higher level. The "full" calculus involves two separate theories, one near the tip of the corner and another one at the conical ex
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ZONGO, WEND BENEDO EMMANUEL. "BOUNDARY VALUE PROBLEMS FOR QUASI-LINEAR AND HIGHER-ORDER ELLIPTIC OPERATORS AND APPLICATION TO BIFURCATION AND STABILIZATION." Doctoral thesis, Università degli Studi di Milano, 2022. http://hdl.handle.net/2434/892091.

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In this thesis, we are interested in the study of nonlinear eigenvalue problem and the controllability of partial differential equations in a smooth bounded domain with boundary. The first part is devoted to the analysis of an eigenvalue problem for quasilinear elliptic operators involving homogeneous Dirichlet boundary conditions. We investigate the asymptotic behaviour of the spectrum of the related problem by showing on the one hand the bifurcation results from trivial solutions using the Krasnoselski bifurcation theorem and bifurcation from infinity using the Leray-Schauder degree on t
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Ugur, Omur. "Boundary Value Problems For Higher Order Linear Impulsive Differential Equations." Phd thesis, METU, 2003. http://etd.lib.metu.edu.tr/upload/686691/index.pdf.

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_I The theory of impulsive di&reg<br>erential equations has become an important area of research in recent years. Linear equations, meanwhile, are fundamental in most branches of applied mathematics, science, and technology. The theory of higher order linear impulsive equations, however, has not been studied as much as the cor- responding theory of ordinary di&reg<br>erential equations. In this work, higher order linear impulsive equations at &macr<br>xed moments of impulses together with certain boundary conditions are investigated by making use of a Green&#039<br>s formula, constructed for p
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Maraventano, Sara. "Superspace formulation of higher-derivative actions from D-branes." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2018. http://amslaurea.unibo.it/15828/.

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Higher-derivative operators play a fundamental role in the development of effective field theories. This thesis deals with the study of such operators in the context of the effective low-energy theory of type IIB superstring theory. In particular, we revisit a past attempt to find the supersymmetric form of the higher-derivatives for the D7-brane modulus in an orbifold compactification at the level of a globally supersymmetric theory. These higher-derivatives descend from the Dirac-Born-Infeld action of the D7-brane. Starting from the component Lagrangian the task is to identify the supersym
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Books on the topic "Higher Operators"

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Unterberger, André. Automorphic pseudodifferential analysis and higher level Weyl calculi. Birkhauser Verlag, 2002.

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Deninger, Christopher. Higher order operations in Deligne cohomology. Mathematischen Instituts der Universität Münster, 1991.

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McGovern, James Joseph. Flight operations for higher harmonic control research. Naval Postgraduate School, 1991.

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J, Enright Thomas, and Stanke Ronald J. 1954-, eds. Differential operators and highest weight representations. American Mathematical Society, 1991.

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Eremeeva, Lyudmila. Intermodal and multimodal transport. INFRA-M Academic Publishing LLC., 2021. http://dx.doi.org/10.12737/993498.

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The textbook examines the theoretical and methodological foundations of intermodal and multimodal transportation, their organizational aspects necessary for effective management of transport processes in the supply chain in the context of modern development of economic systems using several modes of transport, taking into account the branching of transport networks, transport infrastructure, accessibility to the locations of shippers and consignees.&#x0D; It contains control questions on chapters, a list of topics for abstracts and reports, test and situational tasks, a glossary and appendices
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Sinuany-Stern, Zilla, ed. Handbook of Operations Research and Management Science in Higher Education. Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-74051-1.

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Unterberger, André. Automorphic Pseudodifferential Analysis and Higher Level Weyl Calculi. Birkhäuser Basel, 2003.

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Forum internazionale sulla didattica in criminologia applicata e sulla formazione degli operatori della sicurezza e del controllo sociale (1999 Centro residenziale universitario di Bertinoro). Didattica in criminologia applicata: Formazione degli operatori della sicurezza e del controllo sociale. CLUEB, 2000.

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New York (State). Legislature. Senate. Task Force on Government Efficiency. New York State Senate public forum on SUNY operations. s.n., 2010.

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Whitesitt, Larry. Higher than eagles: Spokane's World War II pilots. Marquette Books, 2006.

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Book chapters on the topic "Higher Operators"

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Murty, M. Ram, Michael Dewar, and Hester Graves. "Hecke Operators of Higher Level." In HBA Lecture Notes in Mathematics. Springer Singapore, 2016. http://dx.doi.org/10.1007/978-981-10-2651-5_18.

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Murty, M. Ram, Michael Dewar, and Hester Graves. "Hecke Operators of Higher Level." In HBA Lecture Notes in Mathematics. Springer Singapore, 2016. http://dx.doi.org/10.1007/978-981-10-2651-5_8.

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Yurko, V. A. "On Higher-Order Difference Operators." In Algebraic and Geometric Methods in Mathematical Physics. Springer Netherlands, 1996. http://dx.doi.org/10.1007/978-94-017-0693-3_31.

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Eelbode, David. "Clifford Analysis for Higher Spin Operators." In Operator Theory. Springer Basel, 2015. http://dx.doi.org/10.1007/978-3-0348-0667-1_23.

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Eelbode, David. "Clifford Analysis for Higher Spin Operators." In Operator Theory. Springer Basel, 2014. http://dx.doi.org/10.1007/978-3-0348-0692-3_23-1.

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Ding, Chao, Raymond Walter, and John Ryan. "Higher Order Fermionic and Bosonic Operators." In Trends in Mathematics. Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-23854-4_17.

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Sanchez, Eduardo, Christopher Paolini, Peter Blomgren, and Jose Castillo. "Algorithms for Higher-Order Mimetic Operators." In Lecture Notes in Computational Science and Engineering. Springer International Publishing, 2015. http://dx.doi.org/10.1007/978-3-319-19800-2_39.

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Gohberg, Israel, Seymour Goldberg, and Nahum Krupnik. "Regularized Determinants of Higher Order." In Traces and Determinants of Linear Operators. Birkhäuser Basel, 2000. http://dx.doi.org/10.1007/978-3-0348-8401-3_12.

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Rabczuk, Timon, Huilong Ren, and Xiaoying Zhuang. "Higher Order Nonlocal Operator Method." In Computational Methods Based on Peridynamics and Nonlocal Operators. Springer International Publishing, 2023. http://dx.doi.org/10.1007/978-3-031-20906-2_5.

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Cialdea, Alberto, and Vladimir Maz’ya. "Higher-order Differential Operators in $$L^p$$." In Semi-bounded Partial Differential Operators. Springer International Publishing, 2014. http://dx.doi.org/10.1007/978-3-319-04558-0_6.

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Conference papers on the topic "Higher Operators"

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Fieltsch, Wolfgang, and J. Scott Farquharson. "Innovative Mitigation Strategies to Address AC Corrosion at Higher Frequencies." In CONFERENCE 2023. AMPP, 2023. https://doi.org/10.5006/c2023-19280.

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Abstract A 2018 field investigation on an NPS 6, approximately 4 km long, liquid pipeline identified a rectifier as the likely source of the elevated DC and AC current densities, which resulted in AC corrosion anomalies detected during in-line inspections (ILI). Initial findings indicated that the 120 Hz rectifier ripple may have contributed to the accelerated AC corrosion at this location. A literature review identified a gap in the industry with no existing standards that address AC corrosion at these higher frequencies, and very little research on the topic. This resulted in additional fiel
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Millet, Cécile, Florent Decultieux, Christelle Gomes, and Seungmin Song. "New Supermartensitic Alloy for High Pressure High Temperature Applications." In CORROSION 2019. NACE International, 2019. https://doi.org/10.5006/c2019-12782.

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Abstract In High Pressure High Temperature “HPHT” wells (pressure above 690bar, temperature above 150°C) which contain CO2 and H2S, operators need to select corrosive resistant materials. High strength grades are usually required to meet collapse and burst properties. The aim of the end users is to get the specific grade which can resist to corrosion while minimizing the cost. The API 5CRA standard defines corrosion resistant alloy (CRA) grades for casings and tubings, from group 1 named “Super 13Cr 13-5-2” (suitable up to 180°C), to group 2 “Duplex” grades 22-5-3 (232°C) or “Super Duplex” 25-
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Iyer, Parth, Yoko Nakazato, Han Wu, Millan Sen, and Len Krissa. "Estimating the Financial Risk Reductions Associated with Cathodic Protection Programs." In CORROSION 2020. NACE International, 2020. https://doi.org/10.5006/c2020-14546.

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Abstract On an increasingly frequent basis, pipeline operators are using risk-based decision making to prioritize cross-company expenditures. Due to the long-term mitigation benefits of Cathodic Protection (CP), when planning external corrosion mitigation activities, pipeline operators typically prioritize mitigation of deeper anomalies for integrity expenditures due to their higher Probability of Failure (PoF). However, anomalies that are not receiving adequate CP or those experiencing electrical interference may remain unaddressed using this rationale. This paper presents both a qualitative
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Emmanuel, J. I. "Application of 5-M Approach to Control Internal Corrosion for Crude Oil Transmission Pipelines." In CORROSION 2016. NACE International, 2016. https://doi.org/10.5006/c2016-07048.

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Abstract The 5-M methodology for controlling internal corrosion for wet crude oil transmission pipeline with a Gas to Oil Ratio (GOR) of zero and API gravity higher than 31.1°API (less than 870 kg/m3) is discussed with industry practice as the fulcrum of the application. The steps of the 5-M approach are modelling corrosion failure mechanism, mitigation, monitoring, maintenance and management. While this approach is not new, the discrepancies in application by operators have led to inconsistency and inferior results, hence the need to improve understanding and application. This paper places em
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Hamza, Ameer, and Samer S. Mulla. "Corrosion Challenges in Hydrocracker Unit: Comprehensive Review." In CONFERENCE 2025. AMPP, 2025. https://doi.org/10.5006/c2025-00546.

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Abstract Hydrocracking units are versatile units in the refinery application. Their common goals are producing low-sulfur refinery products and cracking larger gasoil feeds into higher-demand jet or diesel range hydrocarbons. This unit utilizes hydrogen and operates at high elevated temperatures and pressures to improve the quality of the different feedstock. Hydrogen sulfide (H2S) ammonium bisulfide (NH4HS) and ammonia (NH3) are produced from the reaction of hydrogen with sulfur and other nitrogen compounds present in the hydrocarbon feed. Several other reactions can also occur, including con
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Antoniadis, I., E. Dudas, D. Ghilencea, P. Tziveloglou, Pyungwon Ko, and Deog Ki Hong. "Higher Dimensional Operators in the MSSM." In SUPERSYMMETRY AND THE UNIFICATION OF FUNDAMENTAL INTERACTIONS. AIP, 2008. http://dx.doi.org/10.1063/1.3051901.

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Šmíd, Dalibor. "Conformally invariant higher order higher spin operators on the sphere." In 9TH INTERNATIONAL CONFERENCE ON MATHEMATICAL PROBLEMS IN ENGINEERING, AEROSPACE AND SCIENCES: ICNPAA 2012. AIP, 2012. http://dx.doi.org/10.1063/1.4765596.

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Skouroupathis, Apostolos. "Higher loop renormalization of fermion bilinear operators." In The XXV International Symposium on Lattice Field Theory. Sissa Medialab, 2008. http://dx.doi.org/10.22323/1.042.0254.

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Ota, Toshihiko, B. S. Acharya, Maury Goodman, and Naba K. Mondal. "Neutrino mass generation from higher dimensional operators." In 12TH INTERNATIONAL WORKSHOP ON NEUTRINO FACTORIES, SUPERBEAMS, AND BETABEAMS: NuFact10. AIP, 2011. http://dx.doi.org/10.1063/1.3644284.

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Reyes, Carlos M., and Luis F. Urrutia. "Lorentz-Invariance Violation with Higher-Order Operators." In Seventh Meeting on CPT and Lorentz Symmetry. WORLD SCIENTIFIC, 2017. http://dx.doi.org/10.1142/9789813148505_0063.

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Reports on the topic "Higher Operators"

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Kolda, Tamara Gibson. Multilinear operators for higher-order decompositions. Office of Scientific and Technical Information (OSTI), 2006. http://dx.doi.org/10.2172/923081.

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Briones, Roehlano, and Isabel Espineli. Towards Competitive Livestock, Poultry, and Dairy Industries: Consolidated Benchmarking Study. Philippine Institute for Development Studies, 2022. https://doi.org/10.62986/dp2022.20.

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This benchmarking study compares the performance of the Philippine livestock, poultry, and dairy (LPD) industries with other large LPD producers and consumers in Asia: China, Thailand, and Viet Nam, supplemented with figures from major global players. In the case of the Philippine swine industry, commercial farms’ unit cost of production is lower than that of backyard farms due to economies of scale. Moreover, the cost per unit of commercial farms in the Philippines is among the highest among the countries studied, primarily due to the higher feed and grower stock cost. As with swine, economie
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Winikoff, Justin B., and Karen Maguire. The role of commercial energy payments in agricultural producer income. Economic Research Service, U.S. Department of Agriculture, 2024. http://dx.doi.org/10.32747/2024.8374827.ers.

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A significant portion of energy production in the United States takes place on farmland, which can have substantial economic implications for the farmers who host such developments. This report analyzes energy payments made to farmers for the production of oil, natural gas, and wind energy on their land. The authors used 10 years of data (2011-20) from the USDA's Agricultural Resource Management Survey (ARMS) in their analysis. Results show that 3.5 percent of farm operations received energy payments and that the average annual payment to the operators was more than $30,000 (in 2020 dollars),
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Hayes. L51633 Investigation of WIC Test Variables. Pipeline Research Council International, Inc. (PRCI), 1991. http://dx.doi.org/10.55274/r0010107.

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The Welding Institute of Canada (WIC) has developed a test for pipelines which determine threshold preheat levels necessary to avoid hydrogen cracking in the root region of circumferential welds. This test is simple to perform, reproducible and reasonably inexpensive. Experience has shown, however, that preheat levels determined by the WIC test are typically higher than normally required to achieve, sound, crack-free welds in the field. Consequently, there is a need to characterize the WIC test in more detail even though significant work was performed by other researchers. Through further char
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Koppenhoefer. L51936 Factors Influencing Girth Weld Reliability in Older Pipelines. Pipeline Research Council International, Inc. (PRCI), 2003. http://dx.doi.org/10.55274/r0010210.

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Many older pipelines were constructed without either the benefit of detailed weld inspection or the understanding of the requirements for girth weld performance that currently exists. Consequently, they often contain discontinuities that exceed today's workmanship standards. Operators of pipeline facilities are now being required, by recent legislation and regulation, to provide a higher level of assurance of the integrity of their pipeline systems. Older pipelines present some unique challenges in meeting these requirements. The objective of this project was to develop a technical basis for d
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Robinett, Fred. PR-471-14207-R02 Field Pump Performance Testing Procedure Phase II Proof of Concept Testing. Pipeline Research Council International, Inc. (PRCI), 2018. http://dx.doi.org/10.55274/r0011491.

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Proof of Concept Testing was performed to verify the Field Pump Performance Testing Procedure developed in Phase I of the project before applying it in the field where costs are higher. Testing was performed in Sulzer's pump test laboratory, in Winterthur Switzerland, using a model pump with similar specific speed and design as used in many crude oil pipeline installations. Testing included a baseline test with water and five tests with mineral oil at viscosities varying from 95 to 500 cSt. To validate field pump performance testing techniques several measurements were taken using different me
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Kingston, A. W., A. Mort, C. Deblonde, and O H Ardakani. Hydrogen sulfide (H2S) distribution in the Triassic Montney Formation of the Western Canadian Sedimentary Basin. Natural Resources Canada/CMSS/Information Management, 2022. http://dx.doi.org/10.4095/329797.

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The Montney Formation is a highly productive hydrocarbon reservoir with significant reserves of hydrocarbon gases and liquids making it of great economic importance to Canada. However, high concentrations of hydrogen sulfide (H2S) have been encountered during exploration and development that have detrimental effects on environmental, health, and economics of production. H2S is a highly toxic and corrosive gas and therefore it is essential to understand the distribution of H2S within the basin in order to enhance identification of areas with a high risk of encountering elevated H2S concentratio
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Mosalam, Khalid, Issac Pang, and Selim Gunay. Towards Deep Learning-Based Structural Response Prediction and Ground Motion Reconstruction. Pacific Earthquake Engineering Research Center, 2025. https://doi.org/10.55461/ipos1888.

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This research presents a novel methodology that uses Temporal Convolutional Networks (TCNs), a state-of-the-art deep learning architecture, for predicting the time history of structural responses to seismic events. By leveraging accelerometer data from instrumented buildings, the proposed approach complements traditional structural analysis models, offering a computationally efficient alternative to nonlinear time history analysis. The methodology is validated across a broad spectrum of structural scenarios, including buildings with pronounced higher-mode effects and those exhibiting both line
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Koopmann, Patrick. Ontology-Mediated Query Answering for Probabilistic Temporal Data with EL Ontologies (Extended Version). Technische Universität Dresden, 2018. http://dx.doi.org/10.25368/2022.242.

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Especially in the field of stream reasoning, there is an increased interest in reasoning about temporal data in order to detect situations of interest or complex events. Ontologies have been proved a useful way to infer missing information from incomplete data, or simply to allow for a higher order vocabulary to be used in the event descriptions. Motivated by this, ontology-based temporal query answering has been proposed as a means for the recognition of situations and complex events. But often, the data to be processed do not only contain temporal information, but also probabilistic informat
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Lacerda Silva, P., G. R. Chalmers, A. M. M. Bustin, and R. M. Bustin. Gas geochemistry and the origins of H2S in the Montney Formation. Natural Resources Canada/CMSS/Information Management, 2022. http://dx.doi.org/10.4095/329794.

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The geology of the Montney Formation and the geochemistry of its produced fluids, including nonhydrocarbon gases such as hydrogen sulfide were investigated for both Alberta and BC play areas. Key parameters for understanding a complex petroleum system like the Montney play include changes in thickness, depth of burial, mass balance calculations, timing and magnitudes of paleotemperature exposure, as well as kerogen concentration and types to determine the distribution of hydrocarbon composition, H2S concentrations and CO2 concentrations. Results show that there is first-, second- and third- or
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