Algebraic Models of Hadron Structure II. Strange Baryons

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Sediment Transport Modeling Review—Current and

Sediment Transport Modeling Review—Current andFuture DevelopmentsAthanasios N.͑Thanos ͒PapanicolaouAssociate Professor,Univ.of Iowa,IIHR-Hydroscience and Engineering,Dept.of Civil and Environmental Engineering,Iowa City,IA 52241.E-mail:apapanic@Mohamed ElhakeemResearch Associate Engineer,Univ.of Iowa,IIHR-Hydroscience and En-gineering,Dept.of Civil and Environmental Engineering,Iowa City,IA 52241.E-mail:melhakee@George KrallisERM Inc.,Exton,PA.E-mail:george.krallis@Shwet PrakashERM Inc.,Exton,PAJohn EdingerFaculty Research Associate,Bryn Mawr College,Dept.of Geology.E-mail:jeedgr@IntroductionThe use of computational models for solving sediment transport and fate problems is relatively recent compared with the use of physical models.Several considerations govern the choice be-tween physical and computational models;namely,the nature of the problem that needs to be solved,the available resources,and the overall cost associated with the problem solution.In some speciﬁc problems,a combination of physical and computational models can be used to obtain a better understanding of the pro-cesses under investigation ͑de Vries 1973͒.Using computational hydrodynamic/sediment transport models,in general,involves the numerical solution of one or more of the governing differential equations of continuity,momentum,and energy of ﬂuid,along with the differential equation for sediment continuity.An advan-tage of computational models is that they can be adapted to dif-ferent physical domains more easily than physical models,which are typically constructed to represent site-speciﬁc conditions.An-other advantage of computational models is that they are not sub-ject to distortion effects of physical models when a solution can be obtained for the same ﬂow conditions ͑identical Reynolds and Froude numbers,same length scale in the three directions,etc.͒as those present in the ﬁeld.With the rapid developments in numerical methods for ﬂuid mechanics,computational modeling has become an attractive tool for studying ﬂow/sediment transport and associated pollutant fate processes in such different environments as rivers,lakes,and coastal areas.Representative processes in these environments in-clude bed aggradation and degradation,bank failure,local scour around structures,formation of river bends,ﬁning,coarsening and armoring of streambeds,transport of point source and nonpointsource pollutant attached to sediments,such sediment exchange processes as settling,deposition,and self-weight consolidation;coastal sedimentation;and beach processes under tidal currents and wave action.Over the past three decades,a large number of computational hydrodynamic/sediment transport models have been developed ͑Fan 1988;Rodi 2006͒.Extensive reviews of different hydrodynamic/sediment transport models can be found in Nicollet ͑1988͒,Nakato ͑1989͒,Onishi ͑1994͒,Przedwojski et al.͑1995͒,Spasojevic and Holly ͑2000͒,and the ASCE Sedimentation Engi-neering Manual no.110͑2007͒.Broadly speaking,these models can be classiﬁed on the basis of the range of their applications ͑e.g.,suspended load versus bed-load;physical versus chemical transport ͒;and their formulation in the spatial and temporal con-tinua ͑e.g.,one-dimensional model ͑1D ͒;two-dimensional model ͑2D ͒;or three-dimensional model ͑3D ͒;and steady versus un-steady ͒.The choice of a certain model for solving a speciﬁc prob-lem depends on the nature and complexity of the problem itself,the chosen model capabilities to simulate the problem adequately,data availability for model calibration,data availability for model veriﬁcation,and overall available time and budget for solving the problem.The objectives of this article are twofold.First,the article aims to trace the developmental stages of current representative ͑1D,2D,and 3D ͒models and describe their main applications,strengths,and limitations.The article is intended as a ﬁrst guide to readers interested in immersing themselves in modeling and at the same time sets the stage for discussing current limitations and future needs.Second,the article provides insight about future trends and needs with respect to hydrodynamic/sediment transport models.In preparing this article,the authors may have uninten-tionally omitted some models,since including all the available models found in the literature is impossible.Finally,this article is mainly focused on multidimensional computational models ͑2D and 3D models ͒;however,a brief overview of the 1D models is also included for providing a rational comparison of the 1D model features with the main features of the 2D and 3D models.Description of ModelsThis section provides information about the model formulation,the spatial and temporal characteristics,the coupling/linkage of the hydrodynamic and sediment components,and the model’s predictive capabilities.Tables 1–3complement this description by providing useful information about the model capabilities to handle unsteady ﬂows,bed load and suspended load,sediment exchange processes,type of sediment ͑cohesive versus cohesion-less ͒,and multifractional sediment rmation about model acronyms,language,availability,and distribution is also provided in Tables 1–3.Tables 4–6summarize examples of the different model applications.The reader can use these case stud-JOURNAL OF HYDRAULIC ENGINEERING ©ASCE /JANUARY 2008/1D o w n l o a d e d f r o m a s c e l i b r a r y .o r g b y Z h e j i a n g U n i v e r s i t y o n 01/26/13. C o p y r i g h t A S CE .F o r p e r s o n a l u s e o n l y ; a l l r i g h t s r e s e r v e d .ies as a reference guide for model setup,calibration,and veriﬁ-cation.One-Dimensional ModelsSince the early 1980s,1D models have been used with some success in research and engineering practice.Most of the 1D models are formulated in a rectilinear coordinate system and solve the differential conservation equations of mass and momen-tum of ﬂow ͑the St.Venant ﬂow equations ͒along with the sedi-ment mass continuity equation ͑the Exner equation ͒by using ﬁnite-difference schemes.Some representative models that are developed on the basis of the previously mentioned equations include MOBED by Krishnappan ͑1981͒,IALLUVIAL by Karim and Kennedy ͑1982͒,CHARIMA by Holly et al.͑1990͒,SEDI-COUP by Holly and Rahuel ͑1990͒,3ST1D by Papanicolaou et al.͑2004͒.The HEC-6formulation by Thomas and Prashum ͑1977͒is also presented in a rectilinear coordinate and is dis-cretized by using ﬁnite-difference schemes;but it solves the differential conservation equation of energy instead of the mo-mentum equation.Among other 1D models that use different coordinate systems,equations or schemes of solution are FLUVIAL 11by Chang ͑1984͒,GSTARS by Molinas and Yang ͑1986͒,and OTIS by Runkel and Broshears ͑1991͒.Chang ͑1984͒used a curvilinear coordinate system to solve the governing equations of his model.Molinas and Yang ͑1986͒implemented the theory of minimum stream power to determine the optimum channel width and ge-ometry for a given set of hydraulic and sediment conditions.Runkel and Broshears ͑1991͒modiﬁed the 1D advection-diffusion equation with additional terms to account for lateral inﬂow,ﬁrst-order decay,sorption of nonconservative solutes,and transient storage of these solutes.Most of the 1D models that are presented here can predict the basic parameters of a particular channel,including the bulk-velocity,water surface elevation,bed-elevation variation,and sediment transport load.All of them,except OTIS,can also pre-dict the total sediment load and grain size distribution of nonuni-form sediment.3ST1D by Papanicolaou et al.͑2004͒cannot differentiate the total sediment load into bed load and suspended load.HEC-6by Thomas and Prashum ͑1977͒and IALLUVIAL by Karim and Kennedy ͑1982͒are not applicable to unsteady ﬂow conditions.Table 1contains the complete model reference and acronym explanation and summarizes the main features for each model.Some of these 1D models have additional speciﬁc features.HEC-6by Thomas and Prashum ͑1977͒,for example,decom-poses energy losses into form loss and skin friction loss.MOBED by Krishnappan ͑1981͒can predict the sediment characteristics of a streambed as a function of time and distance for different ﬂow hydrographs.FLUVIAL 11by Chang ͑1984͒accounts for the presence of secondary currents in a curved channel by adjusting the magnitude of the streamwise velocity.The same model can predict changes in the channel bed proﬁle,width,and lateral mi-gration in channel bends.CHARIMA by Holly et al.͑1990͒andHEC-6:Hydraulic Engineering Center;Thomas and Prashum ͑1977͒V .4.2͑2004͒Steady Yes Yes Yes No Entrainment and deposition PDPDF77MOBED:MObile BED;Krishnappan ͑1981͒—Unsteady Yes Yes Yes No Entrainment and deposition C C F90IALLUVIAL:Iowa ALLUVIAL;Karim and Kennedy ͑1982͒—Quasi-steady Yes Yes Yes No Entrainment and deposition C C FIV FLUVIAL 11;Chang ͑1984͒—Unsteady Yes Yes Yes No Entrainment and deposition C P FIV GSTARS:Generalized sediment transport models for alluvial River simulation͑Molinas and Yang,1986͒V .3͑2002͒Unsteady Yes Yes Yes No Entrainment and deposition PDPDF90/95CHARIMA:Acronym of the word CHARiage which means bedload in FrenchHolly et al.͑1990͒—Unsteady Yes Yes Yes Yes Entrainment and deposition C C F 77SEDICOUP:SEDIment COUPled;Holly and Rahuel ͑1990͒—Unsteady Yes Yes Yes No Entrainment and deposition C C F77OTIS:One-dimensional transport with inﬂow and storage;Runkel and Broshears ͑1991͒V .OTIS-P ͑1998͒Unsteady No Yes No No Advection-diffusion PDPDF 77EFDC1D:Environmental ﬂuid dynamics code;Hamrick ͑2001͒—Unsteady Yes Yes Yes Yes Entrainment anddeposition PD PD F773STD1,steep stream sediment Transport 1D model;Papanicolaou et al.͑2004͒—Unsteady a Yes aYes Yes No Entrainmentand deposition C P F90Note:V ϭversion;C ϭcopyrighted;LD ϭlimited distribution;P ϭproprietary;PD ϭpublic domain;and F ϭFORTRAN.aTreated as a total load without separation.2/JOURNAL OF HYDRAULIC ENGINEERING ©ASCE /JANUARY 2008D o w n l o a d e d f r o m a s c e l i b r a r y .o r g b y Z h e j i a n g U n i v e r s i t y o n 01/26/13. C o p y r i g h t A S CE .F o r p e r s o n a l u s e o n l y ; a l l r i g h t s r e s e r v e d .OTIS by Runkel and Broshears ͑1991͒can treat transport and fate of conservative contaminants and heat.EFDC1D by Hamrick ͑2001͒can be applied to stream networks.3ST1D by Papanico-laou et al.͑2004͒is capable of capturing hydraulic jumps and simulating supercritical ﬂows;therefore,it is applicable to un-steady ﬂow conditions that occur over transcritical ﬂow stream reaches,such as ﬂows over step-pool sequences in mountain streams.Because of their low data,central processor unit ͑CPU ͒re-quirements,and simplicity of use,1D models remain useful pre-dictive tools even today,especially in consulting,for rivers and stream ecological applications where 2D or 3D models may not be needed and are computationally expensive.Table 4presents examples of different 1D model applications.Two-Dimensional ModelsSince the early 1990s,there has been a shift in computational research toward 2D models.Most of the 2D models are currently available to the hydraulic engineering community as interface-based software to allow easy data input and visualization of re-sults.This added capability has made these models user-friendly and popular.2D models are depth-averaged models that can pro-vide spatially varied information about water depth and bed elevation within rivers,lakes,and estuaries,as well as the mag-nitude of depth-averaged streamwise and transverse velocity com-ponents.Most 2-D models solve the depth-averaged continuity and Navier-Stokes equations along with the sediment mass bal-ance equation with the methods of ﬁnite difference,ﬁnite element,or ﬁnite volume.Table 2shows the complete model reference and explanation of the model acronym and summarizes the characteristics of selected 2D hydrodynamic/sediment trans-port models.The main speciﬁc features of each model are de-scribed below:SERATRA:A ﬁnite-element sediment-contaminant transport model developed by Onishi and Wise ͑1982͒.The model includes general advection-diffusion equations and incorporates sink/source terms.The model can predict overland ͑terrestrial ͒and in-stream pesticide migration and fate to assess the potential short-and long-term impacts on aquatic biota in receiving streams.SUTRENCH-2D:A ﬁnite-volume hydrodynamic and sediment transport model developed by van Rijn and Tan ͑1985͒for simu-SERATRA:SEdiment and RAdionuclide TRAnsport;Onishi and Wise ͑1982͒—Unsteadya Yes a Yes No Yes Advection-diffusion CC/LDFIVSUTRENCH-2D:SUspended sediment transport in TRENCHes;van Rijn and Tan ͑1985͒—Quasisteadya Yes a Yes No No Advection-diffusion C LD F90TABS-2;Thomas and McAnally ͑1985͒—Unsteadya Yes a Yes No Yes Entrainment and deposition C C F77MOBED2:MObile BED;Spasojevic and Holly ͑1990a ͒—Unsteady Yes Yes Yes No Entrainment and depositionC C F77ADCIRC:ADvanced CIRCulation;Luettich et al.͑1992͒—Unsteady a Yes aYes No Yes Advection-diffusionC/LD C/LD F90MIKE 21:Danish acronym of the word microcomputer;Danish Hydraulic Institute ͑1993͒—Unsteady a Yes aYes No Yes Entrainment and deposition CPF90UNIBEST-TC:UNIform BEach Sediment Transport—Transport Cross-shore;Bosboom et al.͑1997͒—Quasi-steadya Yes a Yes No No Entrainment and advection C LD F90USTARS:Unsteady Sediment Transport models for Alluvial Rivers Simulations;Lee et al.͑1997͒—Unsteady Yes Yes Yes No Entrainment and deposition P P F90FAST2D:Flow Analysis Simulation Tool;Minh Duc et al.͑1998͒—Unsteady Yes Yes No No Entrainment and deposition LD P F90FLUVIAL 12;Chang ͑1998͒—Unsteady Yes Yes Yes No Entrainment and deposition C P F77Delft 2D;Walstra et al.͑1998͒—Unsteady Yes Yes No Yes Advection-diffusion C LD F90CCHE2D:The National Center for Computational Hydroscience and Engineering;Jia and Wang ͑1999͒V .2.1͑2001͒Unsteady Yes Yes Yes No Advection-diffusion PD/CLDF77/F90Note:V ϭversion;C ϭcopyrighted;LD ϭlimited distribution;P ϭproprietary;PD ϭpublic domain;F ϭFORTRAN.aTreated as a total load without separation.JOURNAL OF HYDRAULIC ENGINEERING ©ASCE /JANUARY 2008/3D o w n l o a d e d f r o m a s c e l i b r a r y .o r g b y Z h e j i a n g U n i v e r s i t y o n 01/26/13. C o p y r i g h t A S CE .F o r p e r s o n a l u s e o n l y ; a l l r i g h t s r e s e r v e d .lating sediment transport and associated bed level change under conditions of combined quasi-steady currents and wind-induced waves over a sediment bed.The model solves the general advection-diffusion equations by incorporating a lag coefﬁcient to account for the settling of sediments.TABS-2:A group of ﬁnite-element based hydrodynamic and sediment transport computer codes developed by the USACE Wa-terways Experimental Station ͑Thomas and McAnally 1985͒that currently operates by using the SMS v.9.0windows interface.These codes are applicable to rivers,reservoirs,and estuaries.The main components of TABS-2are the hydrodynamic component,RMA2;the sediment transport component,SED2D ͑formally STUDH ͒;and the water quality component,RMA4.MOBED2:A ﬁnite-difference hydrodynamic and sediment transport model used in a curvilinear coordinate system,devel-oped by Spasojevic and Holly ͑1990a ͒.The model can simulate water ﬂow,sediment transport,and bed evolution in natural wa-terways such as reservoirs,estuaries,and coastal environments where depth averaging is appropriate.ADCIRC-2D:A ﬁnite-element hydrodynamic and sediment transport model developed by Luettich et al.͑1992͒in a rectilin-ear coordinate system for simulating large-scale domains ͑e.g.,the entire East Coast of the United States ͒by using 2D equationsfor the “external mode”but using the “internal mode”for obtain-ing detailed velocity and stress at localized areas.The internal mode is achieved by specifying the momentum dispersion and the bottom shear stress in terms of the vertical velocity proﬁle.The wave-continuity formulation of the shallow-water equations is used to solve the time-dependent,free-surface circulation and transport processes.MIKE2:A ﬁnite-difference model in a rectilinear coordinate system developed by the Danish Hydraulic Institute ͑1993͒for simulating transport and fate of dissolved and suspended loads discharged or accidentally spilled in lakes,estuaries,coastal areas,or in the open sea.The system consists of four main model groups ͑modules ͒,namely,the hydrodynamic and wave models,the sediment process model,and the environmental hydrody-namic model groups.The hydrodynamic and wave models are relevant to the types of physical processes considered in ﬂood-plain mapping.The sediment process models are used to simulate shoreline change and sand transport,whereas the environmental hydrodynamic models are used to examine water quality issues.UNIBEST-TC2:A ﬁnite-difference hydrodynamic and sedi-ment transport model in a rectilinear coordinate system to de-scribe the hydrodynamic processes of waves and currents in the cross-shore direction by assuming the presence of uniform meanECOMSED:Estuarine,Coastal,and Ocean Model—SEDiment transport;Blumberg and Mellor ͑1987͒V .1.3͑2002͒Unsteady aYesaYes No YesEntrainment and deposition PDPDF77RMA-10:Resource Management Associates;King ͑1988͒—Unsteady aYesaYes No Yes Entrainment and deposition C P F77GBTOXe:Green Bay TOXic enhancement;Bierman et al.͑1992͒—Unsteady No Yes No Yes Entrainment and deposition NA NA F77EFDC3D:Environmental Fluid Dynamics code;Hamrick ͑1992͒—Unsteady Yes Yes Yes Yes Entrainment and deposition PD P F77ROMS:Regional Ocean Modeling System;Song and Haidvogel ͑1994͒V .1.7.2͑2002͒Unsteady Yes Yes Yes No Entrainment and deposition LD LD F77CH3D-SED:Computational Hydraulics 3D-SEDiment;Spasojevic and Holly ͑1994͒—Unsteady Yes Yes Yes Yes Entrainment and deposition C C F90SSIIM:Sediment Simulation In Intakes with Multiblock options;Olsen ͑1994͒V .2.0͑2006͒Steady Yes Yes Yes No Advection-diffusion PD P C-Langua.MIKE 3:Danish acronym of the word Microcomputer;Jacobsen and Rasmussen ͑1997͒—Unsteady aYesaYes No Yes Entrainment and deposition C P F90FAST3D:Flow Analysis Simulation Tool;Landsberg et al.͑1998͒V .Beta-1.1͑1998͒Unsteady Yes Yes No No Entrainment and depositionLD P F90Delft 3D;Delft Hydraulics ͑1999͒V .3.25.00͑2005͒Unsteady Yes Yes No Yes Entrainment anddepositionC LD F77TELEMAC;Hervouet and Bates ͑2000͒—Unsteady a Yes a Yes No Yes Entrainment anddepositionC P F90Zeng et al.͑2005͒—UnsteadyYes Yes No No Entrainment anddepositionPPF90Note:V ϭversion;C ϭcopyrighted;LD ϭlimited distribution;P ϭproprietary;PD ϭpublic domain;F ϭFORTRAN.aTreated as a total load without separation.4/JOURNAL OF HYDRAULIC ENGINEERING ©ASCE /JANUARY 2008D o w n l o a d e d f r o m a s c e l i b r a r y .o r g b y Z h e j i a n g U n i v e r s i t y o n 01/26/13. C o p y r i g h t A S CE .F o r p e r s o n a l u s e o n l y ; a l l r i g h t s r e s e r v e d .longshore currents along the beach ͑Bosboom et al.1997͒.The bed load and suspended load transport processes are modeled by assuming local equilibrium conditions ͑no lag effects are consid-ered between ﬂow and sediment ͒.USTARS:A modiﬁed form of ͑GSTARS ͒that is also based on the stream tube concept ͑Lee et al.1997͒.The hydrodynamic and sediment equations are solved with a ﬁnite-difference scheme in a rectilinear coordinate system.As in GSTARS,the theory of mini-mum stream power is used here to determine the optimum chan-nel width and geometry for a given set of hydraulic,geomorpho-logic,sediment,and man-made constraints.FAST2D:A ﬁnite-volume hydrodynamic and sediment model with boundary-ﬁtted grids in a curvilinear coordinate system to simulate sediment transport and morphodynamic problems in al-luvial channels ͑Minh Duc et al.1998͒.The model accounts in-directly for secondary effects attributed to the complexity of the domain.FLUVIAL 12:A ﬁnite-difference hydrodynamic and sediment model in a curvilinear coordinate system developed by Chang ͑1998͒.The combined effects of ﬂow hydraulics,sediment trans-port,and river channel changes can be simulated for a given ﬂow period.The model is a mobile-bed model and simulates changes in the channel-bed proﬁle,width,and sediment bed composition induced by the channel curvature.DELFT-2D:A ﬁnite-difference hydrodynamic and sediment transport model simulating waves and currents ͑Walstra et al.1998͒.The model couples the hydrodynamics with computed bot-tom morphological changes in a time-dependent way.The model can simulate bed-load and suspended load transport by using ei-ther a local equilibrium or a nonequilibrium ͑i.e.,the lag effects between ﬂow and sediment ͒approach.The model can also show the effects of wave motion on transport magnitude and HE2D:A ﬁnite-element hydrodynamic and sediment model developed by Jia and Wang ͑1999͒.The model simulates the sus-pended sediment by solving the advection-diffusion equation and the bed-load transport by empirical functions ͑e.g.,Yalin 1972;van Rijn 1993͒.The model accounts for the secondary ﬂow effect in curved channels.All the aforementioned models are applicable to unsteady ﬂow conditions,except SUTRENCH-2D by van Rijn and Tan ͑1985͒and UNIBEST-TC2by Bosboom et al.͑1997͒.All models can predict the total sediment transport load;but only MOBED2,USTARS,FLUVIAL 12,and CCHE2D can handle multifrac-tional sediment transport and can decompose the total sediment load into bedload and suspended load.DELFT-2D and FAST2D can also separate the total sediment load into bedload and sus-HEC-6;Thomas and Prashum ͑1977͒Prediction of the ﬂow and sediment transport along with the bed level change of the Saskatchewan River below Gardiner Dam,Canada ͑Krishnappan 1985͒Prediction of the bed proﬁle for the eroded and redeposited delta sediment upstream from Glines Canyon Dam,Washington ͑U.S.Department of Interior,Bureau of Reclamation 1996͒MOBED;Krishnappan ͑1981͒Comparison of MOBED results with HEC-6results for the ﬂow and sediment transport along with the bed-level change for the Saskatchewan River below Gardiner Dam,Canada ͑Krishnappan 1985͒Prediction of ﬁne sediment transport under ice cover in the Hay River in Northwest Territories,Canada IALLUVIAL;Karim and Kennedy ͑1982͒Simulation of ﬂow and sediment processes in the Missouri River,Nebraska ͑Karim and Kennedy 1982͒Simulation of ﬂow and sediment processes downstream of the Gavins Point Dam on the Missouri River,Nebraska ͑Karim 1985͒FLUVIAL 11;Chang ͑1984͒Simulation of ﬂow and sediment processes of the San Dieguito River,Southern California ͑Chang 1984͒Simulation of ﬂow and sediment processes of the San Lorenzo River,Northern California ͑Chang 1985͒GSTARS;Molinas and Yang ͑1986͒Prediction of the scour depth and pattern at the Lock and Dam No.26replacement site on the Mississippi River,Illinois ͑Yang et al.1989͒Prediction of the variation in channel geometry for the unlined spillway downstream Lake Mescalero Reservoir,New Mexico ͑Yang and Simões 2000͒CHARIMA;Holly et al.͑1990͒Mobile-bed dynamics in the Missouri River from Ft.Randall to Gavins Point Dam,South Dakota ͑Corps of Engineers ͒Mobile-bed dynamics in the Missouri River from Gavins Point Dam to Rulo,Nebraska ͑National Science Foundation ͒SEDICOUP;Holly and Rahuel ͑1990͒Modeling of long-term effects of rehabilitation measures on bed-load transport at the Lower Salzach River,Germany ͑Otto 1999͒Long-term modeling of the morphology of the Danube River,Germany ͑Belleudy 1992͒OTIS;Runkel and Broshears ͑1991͒Simulation of ﬁeld experiments conducted by Bencala and Walters ͑1983͒for the change in chloride concentration of the Uvas Creek,California.Estimation of the travel times and mixing characteristics of the Clackamas River,Oregon,using the slug of rhodamine data of Laenen and Risley ͑1997͒EFDC1D;Hamrick ͑2001͒Simulation of the ﬂow and sediment transport processes in the Duwamish River and Elliott Bay,Washington ͑Schock et al.1998͒Development of a water quality model for the Christina River,Delaware ͑USEPA 2000͒3STD1;Papanicolaou et al.͑2004͒Prediction of the grain size distribution and bed morphology of the Cocorotico River,Venezuela ͑Papanicolaou et al.2004͒.Prediction of the grain size distribution and bed-load rate of the Alec River,Alaska ͑Papanicolaou et al.2006͒JOURNAL OF HYDRAULIC ENGINEERING ©ASCE /JANUARY 2008/5D o w n l o a d e d f r o m a s c e l i b r a r y .o r g b y Z h e j i a n g U n i v e r s i t y o n 01/26/13. C o p y r i g h t A S CE .F o r p e r s o n a l u s e o n l y ; a l l r i g h t s r e s e r v e d .pended load,but they are limited to uniform sediment sizes.Ex-amples of the different 2D model applications are summarized in Table 5.Three-Dimensional ModelsIn many hydraulic engineering applications,one has to resort to 3D models when 2D models are not suitable for describing cer-tain hydrodynamic/sediment transport processes.Flows in the vi-cinity of piers and near hydraulic structures are examples in which 3D ﬂow structures are ubiquitous and in which 2D models do not adequately represent the physics.With the latest develop-ments in computing technology—such as computational speed,parallel computing,and data storage classiﬁcation—3D hydrodynamic/sediment transport models have become much more attractive to use.Most 3D models solve the continuity and the Navier-Stokes equations,along with the sediment mass bal-ance equation through the methods of ﬁnite difference,ﬁnite ele-ment,or ﬁnite-volume.The Reynolds average Navier-Stokes ͑RANS ͒approach has been employed to solve the governing equations.The RANS models can be separated into hydrostatic and non-hydrostatic models.The hydrostatic models ͑e.g.,Gessler et al.1999͒are not able to accurately predict ﬂow and transport phe-nomena in regions where the ﬂow is strongly 3D and where large adverse pressure gradients or massive separation are present ͑e.g.,river bends containing hydraulic structures ͒.On the contrary,non-hydrostatic RANS models have been shown to adequately de-scribe intricate features of secondary ﬂows in complex domains ͑e.g.,Wu et al.2000;Ruther and Olsen 2005͒.Table 3shows the complete model reference and explanation of the acronym and summarizes the characteristics of 12selected 3D hydrodynamic/sediment transport models.The main speciﬁc features of each model are described below:ECOMSED:A fully integrated 3D ﬁnite-difference hydrody-namic,wave,and sediment transport model in an orthogonal cur-SERATRA;Onishi and Wise ͑1982͒Investigation of the effects of sediment on the transport of radionuclides in Cattaraugus and Buttermilk Creeks,New York ͑Walters et al.1982͒Simulation of the hydrogeochemical behavior of radionuclides released to the Pripyat and Dnieper rivers from the Chernobyl Nuclear Power Plant in Ukraine ͑V oitsekhovitch et al.1994͒SUTRENCH-2D;van Rijn and Tan ͑1985͒Simulation of sand transport processes and associated bed-level changes along dredged pits and trenches at the lower Dutch coast,The Netherlands ͑Walstra et al.1998͒Modeling sediment transport and coastline development along the Iranian coast,Caspian Sea ͑Niyyati and Maraghei 2002͒TABS-2;Thomas and McAnally ͑1985͒Simulation of the ﬂow and sediment transport processes in the Black Lake,Alaska ͑Papanicolaou et al.2006͒Evaluation of the hydraulic performance of different structures found in the Missouri River for creating new shallow water habitat ͑Papanicolaou and Elhakeem 2006͒MOBED2;Spasojevic and Holly ͑1990a ͒Simulation of mobile-bed dynamics in the Coralville Reservoir on the Iowa River,Iowa ͑Spasojevic and Holly 1990b ͒ADCIRC;Luettich et al.͑1992͒Simulation of the ﬂow and sediment transport processes of the natural cap in the Matagorda Bay,Texas ͑Edge 2004͒Simulation of sand transport processes at Scheveningen Trial Trench,The Netherlands ͑Edge 2004͒MIKE 21;Danish Hydraulic Institute ͑1993͒Prediction of the spreading of dredged spoils in the Øresund Link,Denmark-Sweden Prediction of sediment transport rate at ebb ﬂow in a tidal inlet,Grådyb,Denmark UNIBEST-TC;Bosboom et al.͑1997͒Coastal study for the impacts of constructing Kelantan Harbor,MalaysiaCoastal study for shoreline protection of Texel region,The NetherlandsUSTARS;Lee et al.͑1997͒Simulation of sand transport processes and associated bed-level changes of a reach in the Keelung River,Taiwan ͑Lee et al.1997͒Routing of ﬂow and sediment of the Shiemen Reservoir,upstream Tan-Hsui River,Taiwan ͑Lee et al.1997͒FAST2D;Minh Duc et al.͑1998͒Simulation of sediment transport processes and associated bed level changes of a reach in the Bavarian Danube River,Germany ͑Minh Duc et al.1998͒Flood analysis and mitigation on the Orlice River,Poland ͑Beck et al.2003͒FLUVIAL 12;Chang ͑1998͒Simulation of ﬂow and sediment processes of the San Dieguito River,Southern California ͑Chang 1994͒Simulation of ﬂow and sediment processes of the Feather River,Northern California ͑Chang et al.1996͒Delft 2D;Walstra et al.͑1998͒Simulation of sand transport processes and associated bed-level changes along dredged pits and trenches at the lower Dutch coast,The Netherlands ͑Walstra et al.1998͒Simulation of the ﬂow ﬁeld and sediment transport processes of the Pannerdense Kop and IJssel Kop bifurcations in the Rhine River,The Netherlands ͑Sloff 2004͒CCHE2D;Jia and Wang ͑1999͒Investigation of the effects of the rock pile and the submerged dikes downstream of the Lock and Dam No.2of the Red River Waterway,LouisianaInvestigation of the effects of large woody debris structures on the ﬂuvial processes in the Little Topashaw Creek,Mississippi ͑Wu et al.2005͒6/JOURNAL OF HYDRAULIC ENGINEERING ©ASCE /JANUARY 2008D o w n l o a d e d f r o m a s c e l i b r a r y .o r g b y Z h e j i a n g U n i v e r s i t y o n 01/26/13. C o p y r i g h t A S CE .F o r p e r s o n a l u s e o n l y ; a l l r i g h t s r e s e r v e d .。

Algebraic Structure of a Master Equation with Generalized Lindblad Form

(4)
where κ is a complex constant satisfying the condition µν ≥ |κ|2 which ensures the positivity. See for example [7]1 . Then we examine an algebraic structure related to the Lie algebras su(1, 1) and su(2), and construct interesting approximate solutions by use of it. In order to solve the equation we use the method in [1] once more. For that we review a matrix representation of a and a† on the usual Fock space F = VectC {|0 , |1 , |2 , |3 , · · · }; like 0 0 1 √ 1 0 0 2 √ √ † −iθ iθ , a = e a=e 3 2 0 0 √ .. . 0 3 0 .. .. .. . . . 0 1 † N =aa= 2 3 .. . (5) (a† )n |n = √ |0 n!

X = (xij ) −→ X = (x11 , x12 , x13 , · · · ; x21 , x22 , x23 , · · · ; x31 , x32 , x33 , · · · ; · · · · · · )T where T means the transpose. The following formula AXB = (A ⊗ B T )X holds for A, B, X ∈ M (F ). Then (4) is transformed into ∂ ρ(t) = H ρ(t) ∂t where H = −iω (N ⊗ 1 − 1 ⊗ N ) =⇒ ρ(t) = etH ρ(0)

Twisted Quantum Affine Superalgebra $U_q[sl(22)^{(2)}]$, $U_q[osp(22)]$ Invariant R-matrice

$Twisted Quantum Affine Superalgebra $U_q[sl(22)^{(2)}]$, $U_q[osp(22)]$ Invariant R-matrice$

becomes sl(2|2) invariant. Using this R-matrix, we will derive a new Uq [osp(2|2)] invariant
aﬃne superalgebra sl(2|2)(2) and its quantized version Uq [sl(2|2)(2) ], respectively. The
Abstract We describe the twisted aﬃne superalgebra sl(2|2)(2) and its quantized version Uq [sl(2|2)(2) ].
We investigate the tensor product representation of the 4-dimensional grade star represen-
model of strongly correlated electrons which is integrable on a one dimension lattice. This model has diﬀerent interaction terms from the ones in the models [3, 4, 5]. This paper is organized as follows. In section 2 and section 3, we study the twisted tensor product representation of the 4-dimensional grade star representation for the ﬁxed
subsuperalgebra Uq [osp(2|2)] is also investigated in details, and basis and its dual for this

Particles and Quantum Symmetries

a r X i v :m a t h /9805122v 1 [m a t h .Q A ] 27 M a y 1998Particles and Quantum SymmetriesW l adys l aw Marcinek 11Institute of Theoretical Physics,University of Wroc l aw Poland Abstract A system of interacting particles equipped with quantum symmetry is described in an abstract algebraic way.The concept of quantum commutativity is used for description of the algebra of quantum states of the system.The graded commu-tativity and particles in singular magnetic ﬁeld is considered as an example.The generalized Pauli exclusion principle is also mentioned.1.INTRODUCTION In this report particle systems intereacting with certain external ﬁeld and equipped with a quantum symmetry are considered.Our study is based on the assumption that the system is characterized by a given Hopf algebra H which act or coact on an unital and associative algebra A .The Hopf algebra H describes quanta of external ﬁeld and the algebra A describes physical states of the system of particles.The action of H on A represents the process of emission of quanta of the external ﬁeld.The coaction correspond for the process of absorption.If the algebra A is quantum commutative with respect to the given action or coaction of Hopf algebra H ,then we say that the system posses a quantum symmetry .Note that quanum commutative algebras have been studied previously by several authors,see [1,2,3,4]for example.In this paperwe assume that the Hopf algebra H is a group algebra kG ,where G is an abelian group and k ≡C is the ﬁeld of complex numbers.In this particular case the coaction of H on A is equivalent to the G –gradation of A and the quantum commutativity is equivalent to the well–known graded commutativity [3,4].A purely algebraic model of particles in singular magnetic ﬁeld corresponding to the ﬁlling factor v =1N,where n is the number of chargedparticles per N ﬂuxes.Note that there is an approach in which the algebra A play the role of noncommutative Fock space.In this attempt the creation and anihilation operators act on the algebra A such that the creation operators act as the multiplication in A and the anihilation ones act as a noncommutative contraction (noncommutativepartial derivatives)[6,7,8].One can use such formalism in order to give the algebraic Fock space representation[9,10,11].2.MATHEMATICAL PRELIMINARIESLet us start with a brief review on the concept of quantum commutativity[1,4].It is well known that for a quasitriangular Hopf algebra H there is the so–called R–matrix R=ΣR1⊗R2∈H⊗H stisﬁng the quantum Yang–Baxter equation(R⊗id)(id⊗R)(R⊗id)=(id⊗R)(R⊗id)(id⊗R).(2.1) An algebra A equipped with an action of⊲:H⊗A−→A is said to be quantum commutative if the following conditionab=Σ(R2⊲b)(R1⊲a)(2.2) is satisﬁed for every a,b∈A,see[1,2]for example.It is interesting that there is a dual notion of quantum commutativity.It is based on the coquasitriangular Hopf algebras(CQHA)[3,4].Let H be a Hopf algebra over oﬁeld k.We use the following notation for the coproduct in H:if h∈H,then△(h):=Σh1⊗h2∈H⊗H.A coquasitriangular Hopf algebra(a CQTHA)is a Hopf algebra H equipped with a bilinear form −,− :H⊗H−→k such thatΣ h1,k1 k2h2=Σh1k1 h2,k2 ,(2.3)h,kl =Σ h1,k h2,l ,hk,l =Σ h,l2 k,l1for every h,k,l∈H.If such bilinear form b exists for a given Hopf algebra H,then we say that there is a coquasitriangular structure on H.Let H be a CQTHA with coquasitriangular structure given by a biform −,− : H⊗H−→k and let A be a(right)H-comodule algebra with coactionρ.Then the algebra A is said to be quantum commutative with respect to the coaction of(H,b)if an only if we have the relationa b=Σ a1,b1 b0a0,(2.4) whereρ(a)=Σa0⊗a1∈A⊗H,andρ(b)=Σb0⊗b1∈A⊗H for every a,b∈A,see[4]. The Hopf algebra H is said to be a quantum symmetry for A.Let G be an arbitrary group,then the group algebra H:=kG is a Hopf algebra for which the comultiplication, the counit,and the antypode are given by the formulae△(g):=g⊗g,η(g):=1,S(g):=g−1for g∈G. respectively.If H≡kG,where G is an abelian group,k≡C is theﬁeld of complex numbers,then the coquasitriangular structure on H is given as a bicharacter on G[3]. Note that for abelian groups we use the additive notation.A mappingǫ:G×G−→C\{0}is said to be a bicharacter on G if and only if we have the following relations ǫ(α,β+γ)=ǫ(α,β)ǫ(α,γ),ǫ(α+β,γ)=ǫ(α,γ)ǫ(β,γ)(2.5) forα,β,γ∈G.If in additionǫ(α,β)ǫ(β,α)=1,(2.6)for α,β∈G ,then ǫis said to be a normalized bicharacter .Note that this mapping are also said to be a commutation factor on G and it has been studied previously by Scheunert [12])in the context of color Lie algebras.We restrict here our attention to normalized bicharacters only.It is interesting that the coaction of H on certain space E ,where H ≡kG is equivalent to the G -gradation of E ,see [3].In this case the quantum commutative algebra A becomes graded commutativeab =ǫ(α,β)ba (2.7)for homogeneous elements a and b of grade αand β,respectively.Let H be a CQTHA with coquasitriangular structure −,− .The family of all H -comodules forms a category C =M H .The category C is braided monoidal.The braid symmetry Ψ≡{ΨU,V :U ⊗V −→V ⊗U ;U,V ∈Ob C}in C is deﬁned by the equationΨU,V (u ⊗v )=Σ v 1,u 1 v 0⊗u 0,(2.8)where ρ(u )=Σu 0⊗u 1∈U ⊗H ,and ρ(v )=Σv 0⊗v 1∈V ⊗H for every u ∈U,v ∈V .It is known that in an arbtrary braided monoidal category C there is an algebra A with braid commutative multiplication m ◦Ψ=m .Obviously this algebra is quantum commutative.3.FUNDAMENTAL ASSUMTIONSLet E be a ﬁnite dimensional Hilbert space equipped with a basis Θi ,i =1,...,N =dimH .We assume that E is a space of single particle quantum states for certain system of particles.We also assume that there is a coaction ρE of the Hopf algebra H on the space E ,i.e.a linear mapping ρE :E −→E ⊗H ,which deﬁne a (right-)H -comodule structure on E .Then there is a braided monoidal category C ≡C (E,ρH )which contains the ﬁeld C ,the space E ,all tensors product and direct sums of these spaces,and some quotients.An algebra deﬁned as the quotient A ≡A (E ):=T E/I b ,where I b is an ideal in T E generated by elements of the formu ⊗v −Σ v 1,u 1 v 0⊗u 0(3.1)and is said to be −,− -symmetric algebra over E ,in the category C [3].The algebra A describe partitions of particles dressed with quanta of certain physical external ﬁeld.Now let us assume that H =C G .In this particular case the category C =M H becomes symmetric.The category C =M H ,where H :=C G for certain abelian group G and −,− is a bicharacter like above is denoted by C ≡C (G, −,− ).Observe that the symmetry Ψin the formula (2.8)can be understood as a cointertwiner for corepresentations ρU and ρV .Note that if E is a H -comodule,where H =C G ,then E is also a G -graded vector space,i.e E =α∈GE α.This means that the above coaction is equivalent to G -gradation.The normalized bicharacter on the grading group G has the following formǫ(α,β)=(−1)(α|β)q <α|β>,(3.2)where (−|−)is an integer-valued symmetric bi-form on G ,and <−|−>is a skew-symmetric integer-valued bi-form on G ,q is some complex parameter [13].We use here the so-called standard gradation [14].This means that the grading group is G ≡Z N :=Z ⊕...⊕Z (N -sumands)for arbitrary q .If q =exp (2πi 3,then the grading groupG≡Z N can be reduced to G=Z n⊕...⊕Z n.If q=±1,then the grading group G can be reduced to the group Z2⊕...⊕Z2.In the standard gradation(see[14])the algebra A is generated by the relationΘiΘj=ǫijΘjΘi,(3.3)whereǫij:=ǫ(σi,σj)for i,j=1,...,N,σi is the i-th generator of the group G.In this gradation the i–th generatorΘa i is a homogeneous element of gradeσi and describes a particle dressed with a single quantum.Here the quantum commutativity with respect to the algebra H≡C G is the abovew graded commutativity[12,13,14].Every element Θαof the algebra A can be given in the formΘα=Θα11,...,ΘαN N(3.4)forα=ΣN i=1a iσi,x0≡1,where1is the unit in A.4.PARTICLES IN SINGULAR MAGNETIC FIELDNow let us consider an algebraic model for a system of charged particles moving in two-dimensional space M with perpendicular magneticﬁeld.Our fundamental assumption here is that the magneticﬁeld is completely concentrated in vertical lines(ﬂuxes).Every charged particle such as electron moving under inﬂuence of such singular magneticﬁeld is transformed into a composite system which consists a charge and certain number of magneticﬂuxes.We assume that there is in average n electrons per Nﬂuxes.This means that theﬁlling factor is v=nwhereΩij=−Ωji=1,for i=j,Ωii=0,i,j=1,2,...,N.This means that the grading group isG≡Z N2:=Z2⊕...⊕Z2(Nsumands)(4.3) Now we describe the the algebra A corresponding to our model.In this case the algebra A is generated byΘa i(i=1,...,N;a=1,...n)and relationsΘa iΘa j=ǫijΘa jΘa i for a=bΘa iΘb j=−ǫijΘb jΘa i for a=b(Θa i)2=0,(4.4) whereǫis given by the formula(4.2).The G–gradation of A is given by the relationgradeΘa i=σi,(4.5)whereσi=(0...1...0)(1on the i–th place),is the i–th generator of the grading group G=Z N2.In our physical interpretation the generatorΘa i desribes the partition containing one particle dressing with the i–thﬂux(a quasiparticle)and N−1quasiholes. The statistics of quasiparticles is desrcibed by normalized bicharacter.We also have here the generalized Pauli exclusion principle.According to this principle partitions containing two or more anticommuting quasiparticles must be excluded.If N is even, then we obtain the following relationsΘa iΘa j=−(−1)ΩijΘa jΘa i for a=bΘa iΘb j=(−1)ΩijΘb jΘa i for a=b(Θa i)2=0.(4.6) For odd N we obtainΘa iΘa j=(−1)ΩijΘa jΘa i for a=bΘa iΘb j=−(−1)ΩijΘb jΘa i for a=b(Θa i)2=0(4.7)Let us consider a few examples.Example1.Let us assume that N=2and n=1.In this case the algebra A is generating by two generatorsΘ1andΘ2satisﬁng the following relationsΘ1Θ2=Θ2Θ1,Θ21=Θ22=0(4.8)and describes the systems containing in average one electron per two magneticﬂuxes,i.e. theﬁlling factor is v=13and the algebra A isgenmerated by three generatorsΘ1,Θ2,Θ3and relationsΘiΘj=−ΘjΘi,for i=j,Θ21=Θ22=Θ23=0.(4.9)We have here three partitionsΘ1,Θ2,andΘ3which contain one quasiparticle and two quasiholes.Observe that the following partitionsΘ1Θ2,Θ1Θ3,Θ2Θ3which containtwo quasiparticles and one quasihole and the partitionΘ1Θ2Θ3which contains three quasiparticles are not physically admissible by the generalized Pauli exclusion princi-ple.Hence in this case the single quasiparticle partitions with two quasiholes can be physically realized!Example3.For N=3and n=2we have theﬁlling factor v=2。

克隆巴赫系数的英文

克隆巴赫系数的英文English:The Bernstein–Sato polynomial of a germ of a complex analytic function at a point is an object of central importance in singularity theory and complex analysis. It encodes crucial information about the singular behavior of the function near the given point. The Bernstein–Sato polynomial is intimately related to the Bernstein–Sato ideal, which is a fundamental concept in the study of D-modules and p-adic differential equations. The study of Bernstein–Sato polynomials and ideals has deep connections to various areas of mathematics, including algebraic geometry, representation theory, and harmonic analysis. These polynomials have been extensively studied in the context of local zeta functions, where they play a crucial role in understanding the structure of singularities and in the computation of zeta functions associated with singularities. In recent years, there has been significant progress in understanding the properties and applications of Bernstein–Sato polynomials, leading to new insights into the nature of singularities and their interactions with other mathematical objects.中文翻译:克隆巴赫多项式是复解析函数在某一点处的一个基本概念，它在奇点理论和复分析中具有重要的地位。

The algebraic structure of the Onsager algebra

a rX iv:c ond-ma t/2418v1[c ond-m at.stat-m ec h]27Feb2THE ALGEBRAIC STRUCTURE OF THE ONSAGER ALGEBRA 1Etsuro Date 23Department of Mathematics,Osaka University ,Osaka 560,Japan Shi-shyr Roan 45Institute of Mathematics ,Academia Sinica ,Taipei ,Taiwan Abstract We study the Lie algebra structure of the Onsager algebra from the ideal theoretic point of view.A structure theorem of ideals in the Onsager algebra is obtained with the connection to the ﬁnite-dimensional representations.We also discuss the solvable algebra aspect of the Onsager algebra through the formal Lie algebra theory.1Introduction In this note,we are going to discuss the algebraic structure of a certain inﬁnite-dimensional Lie algebra which appeared in the seminal work of Onsager of 1944on the solution of 2D Ising model [1].Due to some other simpler and powerful methods introduced later in the study of Ising model,the algebra in the Onsager’s original work,now called the Onsager algebra,had not received enough attention in the substantial years until the 1980-s when its new-found role appeared in the superintegrable chiral Potts model,[2-12](for a review of the subject,see [13]and references therein ).Since the work of Onsager it was known that there exists an intimate relationship between the Onsager algebra and sl 2,a connection now clariﬁed in [8,14].Namely,the Onsager algebra is isomorphic to the ﬁxed subalgebra of sl 2-loop algebra (or alternatively,its central extension A (1)1,)by a certain involution.A generalization of the Onsager algebra to other Kac-Moody algebras was later introduced in [13,15],with some interesting relations with integrable motions.Due to the close relationship of the Onsager algebra with 2D integrable models in statistical mechanics,especially in the chiral Potts model,a thorough mathematical understanding of this inﬁnite-dimensional algebra is desirable to warrant the further investigation.In this note,we summarize our recent results on the algebraic study of the structure of the Onsager algebra.Detailed derivations,as well as extended references to the literature,may be found in [16].We have established the structure theorem of a certain class of (Lie)-ideals of the Onsager algebra,originated from the study of (reducible orirreducible)ﬁnite dimensional representations of the Onsager algebra,through the theory of thereciprocal polynomials.In the process,we have found a profound structure of the Onsager algebra with some interesting applications to solvable and nilpotent Lie algebras.It suggests that the further study of the ideals in the Onsager algebra and its possible generalization should provide certain useful information on the theory of solvable Lie algebras.The mathematical results obtained would be expected to have some feedback to the original physical theory.Such a program is now under progress and partial results are promising.2Hamiltonian of Superintegrable Chiral Potts ModelThe superintegrable chiral Potts N-state spin chain Hamiltonian has the following form of a pa-rameter k′,H(k′)=H0+k′H1,with H0,H1the Hermitian operators acting on the vector space of L-tensor of I C N,deﬁned byH0=−2Ll=1N−1n=1(1−ω−n)−1X n l,H1=−2Ll=1N−1n=1(1−ω−n)−1Z n l Z N−nl+1,whereω=e2πi2,−√2H(k′)=Ll=1(e−πi6X2l)+k′Ll=1(e−πi6Z2l Z l+1),which was studied by Howes,Kadanoﬀand M.den Nijs[11].Note that the Potts model is given bythe chiral angles,ϕ=φ=0.For a general N,H(k′)was constructed in a paper of G.von Gehlen and R.Rittenberg[10],in which the models were shown to be”superintegrable”in the sense that the Dolan-Grady(DG)condition[9]is satisﬁed for A0=−2N−1H0,A1=−2N−1H0,[A1,[A1,[A1,A0]]]=16[A1,A0],[A0,[A0,[A0,A1]]]=16[A0,A1].For a pair of operators,A0,A1,we denote,4G1=[A1,A0],and deﬁne an inﬁnite sequence of operators,A m,G m,(m∈Z Z),by relations,A m−1−A m+1=14[A m,A0].The DG-condition on A1,A0,are equivalent[8,14]to the statement that the collection of A m,G m forms the Onsager algebra,which means they satisfy the Onsager relations,[A m,A l]=4G m−l,[A m,G l]=2(A m−l−A m+l),[G m,G l]=0.(1)The above relations ensure that the k′-dependence eigenvalues of H(k′)have the following special form as in the Ising model,a+bk′+2Nnj=1 m jIt is easy to see that zeros of a reciprocal polynomial P(t)not equal to±1must occur in the reciprocal pairs,and I P(t)is invariant under the involution of OA,A m→A−m,G m→G−m. Through the Chinese remainder theorem,one can establish the a canonical(Lie-)isomorphism of the quotient algebras,OA/I P(t)∼−→Jj=1OA/I Pj(t),P(t):=Jj=1P j(t)where P j(t)are pairwise relatively prime reciprocal polynomials.The role of reciprocal polynomials in the study of the Onsager algebra is given by the following structure theorem of closed ideals of OA.Theorem1.An ideal I of OA is closed if and only if I=I P(t)for a reciprocal polynomial P(t) whose zero at t=±1are of the even multiplicity.The ideal I P(t)is characterized as the minimal closed ideal of OA containing P(t)e+P(t−1)f.2For an irreducible special representation of OA on aﬁnite dimensional vector space V,ρ:OA−→sl(V),the generating polynomial of the kernel Ker(ρ)is given by P(t)= n j=1U a j(t),where a j∈I C∗−{±1},a j=a±1i for j=i,and U a(t):=(t−a)(t−a−1).In this situation,the evaluation morphism of OA,induced from L(sl2),into the sum of n copies of sl2,e a1,...,a n :OA−→nsl2,X→(e a1(X),...,e a n(X)),gives rise the isomorphism between OA/Ker(ρ)andn sl2.Hence irreducible representations of the sl2-factors determine an irreducible representation of OA.4Completion of the Onsager Algebra and Solvable Lie Algebras By the previous discussion,the study of closed ideals in OA can be reduced to the case,I=I P(t) with P(t)=(t±1)L or U a(t)L for L∈Z Z≥0,a∈I C∗−{±1}.In this report,we shall only discuss the case,P(t)=(t±1)L.As the map,t→−t,gives rise an involution of OA,one has the isomorphism,OA/I(t−1)L≃OA/I(t+1)L.We may assume,P(t)=(t−1)L,(L∈Z Z≥0).DenoteπL,πKL the canonical projections,πL:OA−→OA/I(t−1)L,πKL:OA/I(t−1)L−→OA/I(t−1)K,L≥K≥0,and OA the projective limit of the projective system,(OA/I(t−1)L,πKL).Then OA is a Lie algebra and denote the canonical morphism,ψL: OA−→OA/I(t−1)L,L∈Z Z≥0,with the kernel, OA L:= Ker(ψL).We have, OA/ OA L≃OA/I(t−1)L.We have aﬁltration of ideals in OA,OA= OA0⊃ OA1⊃···⊃ OA L⊃···.There exists a morphism,π:OA−→ OA,withψLπ=πL.The Lie algebra OA is regarded as a completion of OA.For the convenience,we shall write the elementπ(X)of OA again by X for X∈OA if no confusion could arise.There exists the unique sequence of elements,X k ,Y k ,(k ∈Z Z ≥0),in OAsuch that the following identities hold in OA ,A m (=π(A m ))=k ≥0m (k )k !Y k ,where x (n )is the shifted factorial deﬁned by x (0):=1,x (n ):=x (x −1)···(x −n +1),n ∈Z Z >0.In fact,with the inﬁnite formal sum OA has X k ,Y k as the generators of the formal Lie algebra,while OA L is the ideal generated by X k ,Y k ,(k ≥L ).In OA /I (t −1)L ,one has,ψL (A m )=L −1 k =0m (k )k !ψL (Y k ),m ∈Z Z .The relations (1)for OA now become the following relations in OA, n,k ≥0a !b !k !s k a +b Y k , n,k (−1)n a !b !k !s k a +b X k ,[Y n ,Y k ]=0,where s n k ,S n k ,(n,k ∈Z Z ≥0),are the Stirling numbers,i.e.,the integers with the relations,x (n )= k ≥0x k s n k ,x n = k ≥0x (k )S n k .Note that s n n =S n n =1,and s n k =S n k =0for k >n .By the identities among Stirling’s numbers [18],k s a k S k b =δa b ,b !j !S j a = lk !(a +l )!k −n +1(2k 2n −1)Y 2k +1,where B j (j ≥1)are the Bernoulli numbers,B j =(−1)j −1b 2j ,with x j !x j .For thestudy of the Lie algebra structure of OA,it is more natural to use a local coordinate system near t =1for expressing elements in L (sl 2).A convenient variable is,λ:=t −t−1There is a formal Lie-algebra isomorphism,OA≃sl2 λ ,under which OA L is corresponding to sl2 λ L.As a consequence,≃sl2 λ /sl2 λ L.OA/I(t−1)L2By the above Theorem,OA/I(t−1)L is a solvable Lie algebra of dimension L+[L1+λ2,t−1−1=−λ−1+2)λ2j.jBy taking the expression of of A0,A1modularλL sl2[[λ]],one can construct reducible representationsof OA with the kernel ideal generated by(t−1)L.5Further RemarksFor the understanding of representations of the Onsager algebra,both the reducible and irreducible ones,the structure of OA/I P(t)with a reciprocal polynomial P(t)warrants the mathematical inves-tigation.But,just to keep things simple,we restrict our attention in this present report only to the case,P(t)=(t±1)L,L≥1,or U a(t),a=±1.With a similar argument,one can also obtain the structure of quotients for closed ideals generated by P(t)=U a(t)L,L∈Z Z>0,(for the details, see[16]).The results,together with the mixed types,should have some interesting applications and implications in solvable or nilpotent algebras,which we shall leave to future work.Here is the one example which appeared in[14]:Example.OA/I(t2+1)2≃sl2[ε]/ε2sl2[ε].Denote e k,f k,h k,(k=0,1)the basis elements of sl2[ε]/ε2sl2[ε] corresponding to the standard basis of sl2with the index k indicating the grade ofε.Deﬁne the linear map,ϕ:OA−→sl2[ε]/ε2sl2[ε],by the expression:ϕ(A m)=2(i m e0+i−m f0+m i m−1e1+m i−m+1f1),ϕ(G m)=(i m−i−m)h0+m(i m−1−i m−1)h1.Thenϕis a surjective morphism.Since sl2[ε]/ε2sl2[ε]has only trivial center,Ker(ϕ)is a closed ideal with(t2+1)2as the generating polynomial.2Generalizations of the Onsager algebra to other loop algebras or Kac-Moody algebras as in[13,15] should provide ample examples of solvable algebras of the above type.We hope that the further development of the study of the subject will eventually lead to some interesting results in Lie-theory with possible applications in quantum integrable models.References[1]L.Onsager:Phys.Rev.65(1944)117.[2]G.Albertini,B.M.McCoy,J.H.H.Perk,and S.Tang:Nucl.Phys.B314(1989)741.[3]G.Albertini,B.M.McCoy,and J.H.H.Perk:Phys.Lett.A135(1989)159;Phys.Lett.A139(1989)204;Adv.Stud.Pure Math.,vol.19,Kinokuniya Academic1989.[4]H.Au-Yang,B.M.McCoy,J.H.H.Perk,S.Tang,and M.L.Yan:Phys.Lett.123A(1987)219.[5]R.J.Baxter:Phys.Letts.A133(1988)185.[6]R.J.Baxter,J.H.H.Perk and H.Au-Yang:Phys.Letts.A128(1988)138.[7]R.J.Baxter,V.V.Bazhanov and J.H.H.Perk:Int.J.Mod.Phys.B4(1990)803.[8]B.Davies:J.Phys.A:Math.Gen.23(1990)2245;J.Math.Phys.32(1991)2945.[9]L.Dolan and M.Grady:Phys.Rev.D25(1982)1587.[10]G.von Gehlen and R.Rittenberg,Nucl.Phys.B257(1985)351.[11]S.Howes,L.P.Kadanoﬀand M.den Nijs:Nucl.Phys.B215(1983)169.[12]J.H.H.Perk:in P roc.Symp.inP ureMathematics,American Mathematical Society,1989,Vol.49,part1p.341.[13]D.B.Uglov and I.T.Ivanov:J.Stat.Phys.82(1996)87.[14]S.S.Roan:Onsager’s algebra,loop algebra and chiral Potts model,MPI91-70,Max-Planck-Institut f¨u r Mathematik,Bonn,1991.[15]C.Ahn and K.Shigemoto:Modern Phys.Lett.A6(1991)3509.[16]E.Date and S.S.Roan:The structure of quotients of the Onsager algebra by closed ideals,Preprint math.QA/9911018.[17]J.B.Kogus:Rev.Mod.Phys.51(1979)659and references therein.[18]J.Riordan:An introduction to combinatorial analysis,Wiley publication in statistics1958;Combinatorial identities,Wiley series in probabilty and mathematical statistics1968.。

Transformer modeling for low- and mid-frequency transients - a review

Transformer Modeling for Low-and Mid-Frequency Transients—A Review Juan A.Martinez,Member,IEEE,and Bruce A.Mork,Member,IEEEAbstract—One of the weakest components of modern transient simulation software is the transformer model.Many opportunities exist to improve the simulation of its complicated behaviors,which include magnetic saturation of the core,frequency-dependency,ca-pacitive coupling,and topologically correctness of core and coil structure.This paper presents a review of transformer models for simulation of low-and mid-frequency transients.Salient points of key references are presented and discussed in order to give an ac-cessible overview of development,implementation and limitations of the most useful models proposed to date.Techniques for repre-sentation of the most important behaviors are examined,with the intent of providing the needed foundation for continued develop-ment and improvement of transformer models.Index Terms—Eddy currents,hysteresis,saturation,simulation, transformer modeling.I.I NTRODUCTIONT RANSFORMER representation can be very complicated due to the large number of core designs and the fact that some of the transformer parameters are both nonlinear and frequency dependent.Physical attributes whose behavior,de-pending on frequency,may need to be correctly represented by a transformer model are core and coil conﬁgurations,self-and mutual inductances between coils,leakageﬂuxes,skin effect and proximity effect in coils,magnetic core saturation,hys-teresis and eddy current losses in core,and capacitive effects. Models of varying complexity have been implemented in tran-sients tools to duplicate the transient behavior of transformers. This paper aims at presenting a review of transformer models for simulation of transient phenomena well below theﬁrst winding resonance(several kilohertz).They include ferroreso-nance,most switching transients,and harmonic interactions. The document is organized as follows.A summary of the main models proposed for representation of transformers in low-and mid-frequency power system transients is presented in Section II.The representation of nonlinear and frequency-de-pendent parameters,which can have a signiﬁcant inﬂuence on transformer behavior,is analyzed in Section III.The estimation of some parameters,which can be a bottleneck due to the lack of reliable procedure for their determination,is discussed inManuscript received September3,2003;revised December11,2003.This work was supported in part by the Bonneville Power Administration,in part by the U.S.Department of Energy,and in part by by the Keith Companies.Paper no.TPWRD-00455-2003.J. A.Martinez is with the Departament d’Enginyeria Elèctrica,Uni-versitat Politècnica de Catalunya,Barcelona08028,Spain(e-mail:mar-tinez@ee.upc.es).B.A.Mork is with the Department of Electrical and Computer Engineering, Michigan Technological University,Houghton,MI49931USA.Digital Object Identiﬁer10.1109/TPWRD.2004.833884Section IV.Finally,a discussion of all of these issues is included in Section V.II.T RANSFORMER M ODELSA transformer model can be separated into two parts:repre-sentation of windings and representation of the iron core.The ﬁrst part is linear,the second one is nonlinear,and both of them are frequency dependent.Each part plays a different role,de-pending on the study for which the transformer model is re-quired.For instance,core representation is critical in ferrores-onance simulations,but is usually neglected in load-ﬂow and short-circuit calculations.Several criteria can be used to classify transformer models: number of phases,behavior(linear/nonlinear,constant/fre-quency-dependent parameters),and mathematical models.This section presents a summary of the main principles and the assembly equations of the most popular low-frequency trans-former models.They have been classiﬁed into three groups. Theﬁrst group uses either a branch impedance or an admittance matrix.The second group is an extension of the saturable trans-former component model to multiphase transformers.Both types of models have been implemented in the Electromagnetic Transient Program(EMTP),and both of them have important limitations for simulating some core designs.Topology-based models form a larger group for which many approaches have been proposed.Their derivation is performed from the core topology and can represent very accurately any type of core design in low-frequency transients if parameters are properly determined.A.Matrix Representation(BCTRAN Model)The steady-state equations of a multiphase multiwinding transformer can be expressed using the branch impedancematrix(1) In transient calculations,(1)must be rewrittenas(2)whereand are,respectively,the real and the imagi-nary partof,whose elements can be derived from excitation tests.This approach includes phase-to-phase coupling,models terminal characteristics,but does not consider differences in core or winding topology since all core designs receive the same mathematical treatment.0885-8977/$20.00©2005IEEEFig.1.Basic BCTRAN-based model for two-winding transformer,with an externally attached core.There could be some accuracy problems with the above cal-culations since the branch impedancematrixcan become ill-conditioned for very small exciting currents or when these currents are totally ignored [1].In addition,the short-circuit im-pedances,which describe the more important transfer character-istics of a transformer,get lost in such excitation measurements.To solve these problems,an admittance matrix representation should beused(3)where does always exist and its elements can be obtained directly from standard short-circuit tests.For transientstudies,must be split up into resistive and inductive components.The transformer can be also described by the followingequation:(4)All of these models are linear;however,for many transient studies,it is necessary to include saturation and hysteresis ef-fects.Exciting current effects can be linearized and left in the matrix description,which can lead to simulation errors when the core saturates.Alternately,excitation may be omitted from the matrix description and attached externally at the models ter-minals in the form of nonlinear elements (see Fig.1).Such an externally attached core is not always topologically correct,but good enough in many cases.Although these models are theoretically valid only for the frequency at which the nameplate data were obtained,they are reasonably accurate for frequencies below 1kHz.B.Saturable Transformer Component (STC Model)This model is based on the star-circuit representation shown in Fig.2.The primary branch is treated as an uncoupled R-L branch,each of the other windings being handled as a two-winding transformer.The equation of a single-phase N-winding transformer,without including the core,has the same form of(4).However,the matrixproductis symmetric,which is not true in the general case [2].Saturation and hysteresis ef-fects are modeled by adding an extra nonlinear inductor at the star point.The STC model can be extended to three-phase units through the addition of a zero-sequence reluctance parameter,but its use-fulness is limited.The input data consist ofthevalues of each star branch,the turn ratios,and the informationforFig.2.Star-circuit representation of single-phase N-winding transformers.the magnetizing branch.This model has some important limi-tations:it cannot be used for more than three windings,sincethe star circuit is not validfor ,the magnetizing in-ductance withresistance ,in parallel,is connected to the star point,which is not always the correct topological con-necting point,and numerical instability has been reported for the three-winding case,although this problem has been identi-ﬁed as the use of a negative value for one short-circuit reactance [3],[4].C.Topology-Based ModelsThis group has been split up into two subgroups:models de-rived using duality (i.e.,models constructed with a circuit-based approach without any previous mathematical description,and geometrical models,for which a core topology is considered,but their solution passes through a mathematical description).1)Duality-Based Models:Topologically correct equivalent circuit models can be derived from a magnetic circuit model using the principle of duality [5],[6].This approach results in models that include the effects of saturation in each individual leg of the core,interphase magnetic coupling,and leakage ef-fects.In the equivalent magnetic circuit,windings appear as magnetomotive-force (MMF)sources,leakage paths appear as linear reluctances,and magnetic cores appear as saturable re-luctances.The mesh and node equations of the magnetic circuit are duals of the electrical equivalent node and mesh equations,respectively.To make models practically useful,the current sources re-sulting from the transformation are replaced with ideal trans-formers to provide primary-to-secondary isolation and coupling to the core,while preserving the overall primary to secondary turns ratios.Turns ratios are chosen so that core parameters are referenced to the low-voltage winding.The portion of a model inside the coupling transformers represents the core and leak-ages.Winding resistances and interconnection of the windings appears external to the coupling transformers.The advantage of this is that the derived core equivalent functions are independent of the winding con ﬁguration.Winding resistance,core losses,and capacitive coupling ef-fects are not obtained directly from the transformation,but can be added to this equivalent electrical circuit.MARTINEZ AND MORK:TRANSFORMER MODELING FOR LOW-AND MID-FREQUENCY TRANSIENTS—A REVIEW1627 Fig.3shows the equivalent circuit of a single-phase shellform transformer,with concentric windings derived using thismethodology.Key work performed during the last years is listed below.•In1981,Dick and Watson presented the derivation of themodel of a three-legged stacked core transformer[7].Themain contributions of their work were the proposal ofa new hysteresis model and the determination of trans-former parameters from measurements.•In1991,Arturi applied this technique for representing aﬁve-legged stepup transformer working in highly satu-rated conditions[8].•In1994,De León and Semlyen proposed a very com-plete transformer model that was derived from a hybridapproach,a combination of duality,which was used to ob-tain the iron core model,and their own technique for cal-culation of leakage inductances[9].•In1994,Narang and Brierley used duality to derive theequivalent circuit of the magnetic core,which is interfacedby means of a three-phaseﬁctitious winding to an admit-tance matrix that represents the correct magnetic couplingamong windings[10].•In1999,Mork presented the derivation of aﬁve-leggedwound core transformer model,which was validated byduplicating ferroresonant phenomena[11].2)Geometric Models:Topologically correct models can bebased on the followingformulation:(5)in which the coupling between magnetic and electrical equa-tions is made taking into account the core topology.A short summary of some of the models is presented below.1)The coupled magnetic model was developed by Yacaminiand Bronzeado for simulating inrush transients[12].Because the permeability of the ferromagnetic elementsvaries withﬂux density,the magnetic material is dividedinto sections,each of which has a substantially uni-formﬂux density.The link between magneticequations,,and(5)is the Ampere’s circuitallaw.2)The uniﬁed magnetic equivalent circuit model wasproposed by Arrillaga et al.[13].This model uses thenormalized core concept for derivation of the inductancematrix.Leakage permeances can be obtained from open-and short-circuit tests;the effective lengths and cross-sectional areas of their leakage paths are not required.3)GMTRAN was developed by Hatziargyriou et al.[14].The magnetic equations were included in(5)by meansof the inductancematrix.The most importantcontribution of this model was the derivation ofthematrix from the core topology.4)SEATTLE XFORMER was developed and implementedin the ATP by Chen[15].Flux linkages were chosen in thismodel as state variables.That is,the magnetic equationsin(5)are included by means of therelationship.The main contribution of this model is,therefore,the derivation ofthematrix.Fig.3.Duality-derived model for a single-phase shell-form transformer.(a)Core design.(b)Equivalent circuit.Many other models for representation of transformers duringlow-and mid-frequency transients have been proposed,see[16]–[20];since all of them are based on a mathematical de-scription of the core topology,they could be included in thesecond group of topology-based models.Table I presents a summary of equations and characteristicsof the models detailed in this section.III.N ONLINEAR AND F REQUENCY-D EPENDENT P ARAMETERSSome transformer parameters are nonlinear and/or frequencydependent due to three major effects:saturation,hysteresis,andeddy currents.Saturation and hysteresis introduce distortion inwaveforms;hysteresis and eddy currents originate losses.Satu-ration is the predominant effect in power transformers,but eddycurrent and hysteresis effects can play an important role in sometransients.A.Modeling of Iron Cores1)Introduction:Iron core behavior is usually representedby a relationship between the magneticﬂuxdensity and themagneticﬁeldintensity.A general difﬁculty in modeling themagnetization curves is the fact that each of the magneticﬁeldvalues is related to an inﬁnity of possible magnetizations de-pending on the history of the sample.To characterize the mate-rial behavior fully,a model has to be able to plot numerous asso-ciated curves,see Fig.4.A major hysteresis loop is the largestpossible loop whose ends enter into technical saturation.Anyother closed loop is called a minor loop with a distinction alsobeing made between symmetric and asymmetric minor loops.1628IEEE TRANSACTIONS ON POWER DELIVERY ,VOL.20,NO.2,APRIL 2005TABLE IS UMMARY OF T RANSFORMER M ODELS FOR L OW -AND M ID -F REQUENCY TRANSIENTSFig.4.Magnetization curves and hysteresis loops.Hysteresis loops of modern transformers have a negligible in ﬂuence on the magnitude of the magnetizing current,although hysteresis losses and residual ﬂuxes can have some in ﬂuence on some transients (e.g.,inrush transients).Magnetic saturation of an iron core can be represented by the anhysteretic curve,the-relationship that would be obtained if there was no hysteresis effect in the material.The saturation characteristic can be modeled by a two-term poly-nomial relationship between the magnetizing current and the ﬂuxlinkage(6)In transients programs,it is usually represented by a piece-wise linear inductance with two slopes.Increasing the number of slopes does not signi ﬁcantly improve the accuracy,except forsimulation of some nonlinear transients (e.g.,ferroresonance),for which a more accurate representation of the saturation char-acteristic is usually required.The slope in the saturated region above the knee is the air-core inductance,which is almost linear and very low compared with the slope in the unsaturated region.Although the speci ﬁcation of this inductance requires a curve relating to the ﬂuxlinkage to the current ,the information usually available is the root mean square (rms)voltage as a func-tion of the rms current.2)Hysteresis Modeling:Hysteresis can be caused by sev-eral types of phenomena,being the dominant cause depending on the material.Detailed models of hysteresis loops based on a rigorous physical basis are too time-consuming,so most prac-tical approaches are curve ﬁts that ignore magnetic material be-havior [22]–[27].A very complete review of these models was presented in [26].Although the previous approach has been the most common one for the power community,during the last years some macro-scopic models based on the predominant physical phenomena have been also used (e.g.,the Jiles –Atherton model [28]and the Preisach model [29]).The main dif ﬁculties related to macro-scopic models are their complexity and the determination of pa-rameters involved in the description of magnetization mecha-nisms.A method to represent the Preisach model based on the major hysteresis loop was proposed in [30].An introduction to the basic concepts of some classical models was presented in [31].3)Implementation in a Transients Program:In transients simulations,an iron core,with or without hysteresis,can be rep-resented by the equivalent circuit shown in Fig.5[23].It is sim-ilar to the equivalent circuit of a linear inductor;however,the re-sistance depends on the segment slope and needs to be updatedMARTINEZ AND MORK:TRANSFORMER MODELING FOR LOW-AND MID-FREQUENCY TRANSIENTS —A REVIEW1629Fig.5.Equivalent circuit for a nonlinearinductor.Fig.6.Iron core model.with changes of the operating segment.This requires partial re-triangularization of the nodal conductance matrix.The current source consists of the past recorded history and must be updated every time step,as would be done for a linear inductor.A more sophisticated model is presently assumed by some authors.Fig.6shows an iron core model,in which saturationis represented by means of the anhystereticinductor,while losses are modeled by a nonlinearresistor,which represents no-load losses.Excitation losses are mostly iron-core losses,and consist of hysteresis,eddy current (see next subsection),and other anomalous losses.In modern transformers,hysteresis losses are much smaller than eddy current losses.Note the addition ofcapacitor,which represents turn-to-turn winding capacitances.This parameter will have no effect in most low-frequency transients;however,its effect should not be neglected when calculating other core parameters,see,for instance,[32].The concept of instantaneous magnetizing resis-tance,whose value is a function of the magnetization ﬂux,has been proposed in [32].According to this concept,the represen-tation of the no-load losses should be based on a nonlinear re-sistance whose value is very sensitive to the applied voltage.An EMTP model was developed to account for this variation,using an approach different than that shown in Fig.5.For recent experiences on modeling hysteresis loops based on a macroscopic approach in transient simulations,see [33]–[36].B.Modeling of Eddy Current EffectsSeveral physical phenomena occur simultaneously in a loaded transformer that results in both a nonuniform distribu-tion of current in the conductors and a nonuniform distribution of ﬂux in the iron core [37].Eddy current effects and their representation in the windings and the iron core are summarized in the subsequent paragraphs.1)Eddy Current Models for Transformer Windings:They manifest as an increase in the effective resistance and winding losses with respect to those for direct current.There exist a number of analytical expressions for the calculation of losses in transformer windings.Those derived by De Le ón and Sem-lyen [38]were based on the following assumptions:themag-Fig.7.Series Foster equivalent circuit for windings.netic ﬁeld has only an axial component parallel to the axis of the windings,the conductors have a rectangular cross section,all conductors carry the same total current,there is no gap be-tween conductors,and the surface ﬁeld intensity is assumed to be undisturbed by the eddy currents.These assumptions imply that the magnetic ﬁeld at the lateral surfaces of the conductors is known and it can be used to specify the boundary conditions.Foster equivalent circuits have been selected to represent the fre-quency dependence of the windings (Fig.7).These circuits must be of in ﬁnite order to exactly reproduce the impedance at all frequencies;however,a computationally ef ﬁcient circuit can be obtained by ﬁtting at only certain pre-established frequencies.Several ﬁtting procedures to determine circuit parameters have been developed [39].For practical studies,a series model of order 3or less is ad-equate.Such a model neglects displacement currents,so it is valid at frequencies below the ﬁrst winding resonance (i.e.,up to tens of kilohertz,and it should be used only when currents are uniformly distributed).2)Eddy Current Models for Iron-Laminated Cores:The magnetization curves presented above are only valid for slowly varying phenomena as it has been assumed that the magnetic ﬁeld can penetrate the core completely.In general,this is not always true.A change in the magnetic ﬁeld induces eddy cur-rents in the iron.As a consequence of this,the ﬂux density will be lower than that given by the normal magnetization curve.As frequency changes,ﬂux distribution in the iron core lamination changes.For high frequencies,the ﬂux will be con ﬁned to a thin layer close to the lamination surface,whose thickness decreases as the frequency increases.This indicates that in-ductances representing iron path magnetization and resistances representing eddy current losses are frequency dependent [37].The circulation of these eddy currents introduces additional losses.To limit their in ﬂuence,a transformer core is built up from a large number of parallel laminations.Eddy current models intended for simulation of the frequency dependence of the magnetizing inductance as well as losses can be classi ﬁed into two categories obtained,respectively,by the realization of the analytical expression for the magnetizing impedance as a function of frequency,and by subdivision of the lamination into sublaminations and the generation of their elec-trical equivalents [40].Computationally ef ﬁcient models have been derived by syn-thesizing a Foster or a dual Cauer equivalent circuit to match the equivalent impedance of either a single lamination or a coil wound around a laminated iron core limb.By using a continued fraction expansion,a standard Cauer equivalent circuit is de-1630IEEE TRANSACTIONS ON POWER DELIVERY,VOL.20,NO.2,APRIL2005Fig.8.Cauer equivalent circuits for iron cores.(a)Standard Cauer equivalent.(b)Dual Cauer equivalent.rived from the Foster equivalent circuit,so theﬁnal result is a Cauer type in both cases(Fig.8).The accuracy of the standard Cauer representation over a deﬁned frequency range depends on the number of terms retained in a partial fraction expansion and,therefore,on the number of sections.To represent the fre-quency range up to200kHz with an error of less than5%,only four terms are required[39].Theﬁrst section governs its char-acteristics at frequencies up to a few kilohertz,each subsequent section comes into play as the frequency increases.Another form of the Cauer model would have shunt resis-tances and series inductances[Fig.8(b)],[38].The inductances represent(using duality)theﬂux paths,and the resistances are in the path of eddy currents.The high-frequency response is de-ﬁned by the blocks near the terminals,while in the standard Cauer circuit,the high-frequency behavior is governed by the inner blocks.The blocks of this model can be thought as being a discretization of the lamination.The parameters of this circuit can be calculated by means of an iterative method that could be seen as an optimization of the discretizing distances for the used ﬁtting frequencies.Below200kHz,small errors of less than1%, can be achieved with a model order of4[38].A model order of 3or less will sufﬁce for transients below10kHz.Inductive components of the models representing the magne-tizing reactances have to be made nonlinear to account for the hysteresis and saturation effects.Since the inductances and re-sistances in this model do not represent any physical part of the iron lamination,the incorporation of these effects does not seem to be an easy task.However,since the high-frequency compo-nents do not contribute appreciably to theﬂux in the transformer core,it can be assumed that only low-frequency components are responsible for driving the core into saturation.It may,therefore, be justiﬁable to represent as nonlinear only theﬁrst section of the model.IV.P ARAMETER D ETERMINATIONAlthough no agreement has been reached to date on the best representation for three-phase core transformers,it is generally recognized that it should be based on the core topology,and in-clude eddy current effects and saturation/hysteresis representa-tion.In addition,a very careful representation and calculation of leakage inductances is required.Finally,coil capacitances have to be included for an accurate simulation of some transients. Several works have been presented in literature to calculate some parameters from transformer geometry[41],[42].How-ever,irrespective of the model chosen for a given study,param-eter estimation is usually based on nameplate data and standard tests[43],[44].Data usually available for any power transformer are:rat-ings,excitation,and short-circuit test data for direct and ho-mopolar sequences,saturation curve,and capacitances between terminals and between windings.Although procedures for de-termining transformer parameters from standard tests have been proposed,depending on the model selected to represent some effects,additional information will be usually required.That is, no standard procedures have been proposed to obtain all param-eters to be speciﬁed in some models.Some important works on parameter determination are listed below.•A good reference for the calculation of leakage induc-tances from standard test values is the work by Brandwajn et al.[1].•Important contributions on the calculation of parameters to be speciﬁed in duality-based models are the works by Dick and Watson[7],Stuehm[45],and Narang and Brieley[10].See also[8]and[11].•The inﬂuence of eddy current losses and the determination of resistances as a function of frequency have been studied by Fuchs et al.[46].•For the determination of the saturation characteristic,see[47]–[50],as well as the discussion of[48].•The determination of hysteresis parameters is very depen-dent on the selected hysteresis model.For determination of parameter to be speciﬁed in macroscopic models(e.g., the Jiles–Atherton),see[51]and[52].Parameter determi-nation from the manufacturer’s data has been presented in[27].V.D ISCUSSIONA model for representing transformers in low-and mid-fre-quency transients has to incorporate accurate representation of the transformer core,leakage inductances,and eddy current ef-fects in windings and core,saturation,and/or hysteresis effects. Terminal capacitances can also have an important effect in some transients[53].The development of a model that included all of these effects and could be used in any transient simulation for frequencies below10kHz is not immediately obvious.The computational burden for the most complicated core designs would be consid-erable,and the accurate determination of all parameters would be a difﬁcult task.To date,the most complete model was devel-oped by De León and Semlyen[9].An important issue for any transformer model is the nodes to which the core equivalent must be connected.For instance,it may be less important to which node an unsaturated inductance is connected to in the single-phase model shown in Fig.2,but it may make a difference when the inductance is saturated,be-cause of its low value.Ideally,the nonlinear inductance should。

the rising sea 代数几何发表

the rising sea 代数几何发表The Rising Sea: Algebraic Geometry in Modern ApplicationsIn the field of mathematics, algebraic geometry is a vibrant and rapidly growing discipline. It merges geometry and algebra, creating a powerful toolset for understanding and manipulating complex mathematical structures. The Rising Sea: Algebraic Geometry in Modern Applications is a groundbreaking textbook that delves into the depth and breadth of this fascinating field.The book opens with a discussion of the fundamental principles and building blocks of algebraic geometry. It introduces the reader to the basic concepts of polynomial rings, ideals, and varieties. The explanations are clear and accessible, laying a solid foundation for the more advanced topics that follow.The Rising Sea then delves into more advanced topics, such as abstract algebra, commutative algebra, and scheme theory. The text carefully explains these concepts, providing numerous examples and exercises to aid comprehension. It also introduces the reader to the powerful tools of homological algebra and algebraic geometry, further enriching their understanding of the subject.The book’s scope extends beyond the theoretical aspects of algebraic geometry. It also includes chapters on the applications of algebraic geometry in various fields, such as computer science, physics, and economics. These applications demonstrate the practical relevance and impact of algebraic geometry in modern society.The Rising Sea is an essential resource for students and researchers in mathematics, computer science, and related fields. Its clear and comprehensive approach makes it an excellent textbook for undergraduate and graduate courses on algebraic geometry. It will also serve as a valuable reference for professionals in industry and academia who seek a deeper understanding of this rapidly developing field.。

假如尼尔斯来到我们的中间英语作文500字

全文分为作者个人简介和正文两个部分：作者个人简介：Hello everyone, I am an author dedicated to creating and sharing high-quality document templates. In this era of information overload, accurate and efficient communication has become especially important. I firmly believe that good communication can build bridges between people, playing an indispensable role in academia, career, and daily life. Therefore, I decided to invest my knowledge and skills into creating valuable documents to help people find inspiration and direction when needed.正文：假如尼尔斯来到我们的中间英语作文500字全文共3篇示例，供读者参考篇1If Niels Bohr Came Among UsOh my gosh, can you imagine if the great Niels Bohr just randomly showed up at our school one day? That would be absolutely crazy! Niels Bohr was this brilliant physicist fromDenmark who revolutionized our understanding of atomic structure and quantum theory back in the early 20th century. He's definitely one of the most important scientists who ever lived.I've been learning all about Bohr's atomic model in physics class. It was a huge breakthrough at the time. Before Bohr, scientists kind of just accepted the plum pudding model that J.J. Thomson proposed. Basically, Thomson thought the atom was a big blob of positive charge with negative electrons scattered throughout it, kind of like raisins in a plum pudding dessert. But that didn't really fit with experimental evidence.Then Niels Bohr came along and said, "Hold up, I have a better idea!" He proposed that the atom has a positive nucleus in the center, with negative electrons orbiting around it in specific shells or energy levels. It was kind of like a tiny solar system. This atomic structure model made so much more sense and could actually explain the spectrum of light emitted by hydrogen atoms.Another huge contribution from Bohr was founding the basis for understanding quantum theory. He figured out that electrons can only exist in those discrete energy levels or shells, not just anywhere. And when they jump between those levels,that's what causes atoms to absorb or emit light of specific wavelengths. Wild, right?Bohr was awarded the Nobel Prize in Physics in 1922 for his revolutionary atomic model and work on quantum mechanics. He deserved it 100%. His insights transformed physics forever. Can you imagine if he just randomly popped into our classroom during physics period? I would be starstruck for sure!I bet Niels Bohr would blow all of our minds with his genius intellect and passionate lectures. He was totally obsessed with physics and finding the fundamental laws of the universe. I could picture him at the front of our class, messily scribbling equations on the chalkboard with chalk dust flying everywhere as he tried to explain these mind-bending quantum phenomena.Of course, he would probably get frustrated that we're still beginners struggling to understand basic concepts like atomic orbitals and blackbody radiation. "You must be joking!" he might exclaim in his thick Danish accent as we stared blankly at yet another equation he derived from first principles. I'm sure the pace of a Bohr lecture would be absolutely dizzying.At the same time, I'll bet Bohr would be an incredibly patient and caring teacher. From what I've read, he cultivated this famous "Bohr spirit" of free expression and open debate in hisresearch team. He embraced different perspectives and encouraged creative thinking. So in our classroom, Bohr would probably be very nurturing and want each of us to feel comfortable asking questions or proposing ideas, even if they seemed a bit half-baked.I could totally see him putting on a fun little demonstration to illustrate quantum principles too. Maybe he would set up a light source and use prisms or diffraction slits to show us the wave-particle duality and quantization of light. Or he might do an experiment involving atomic spectroscopy to drive home the point about discrete energy levels. Knowing Bohr, it would likely involve pipes, cables, and vacuum tubes sprawled across the lab bench in a chaotic mess that only a genius could understand. But it sure would make atomic physics feel real and alive!Just interacting with someone of Bohr's intellectual caliber and pioneering spirit would be so valuable and inspiring, even if he talked way over our heads at times. This was a man who shaped humanity's understanding of the fundamental nature of matter and energy. He wasn't afraid to challenge the status quo and think in completely new ways. That courage and creativity in the face of the unknown is what drives scientific revolutions.Part of me wonders if we mere high school students could even begin to comprehend the insights and discoveries of a mind like Niels Bohr's. His contributions to quantum theory and models of atomic structure were so profound and consequential that they're still mind-boggling a century later. Having the chance to learn from and engage with him directly would be a tremendous opportunity that could change our perspective forever.At the same time, maybe Bohr's genius wouldn't seem so alienating up close and in person. From the photos and recordings I've seen, he came across as down-to-earth, approachable, and full of playful humor despite his brilliance. Sure, he would be operating on another level intellectually. But at his core, Bohr was simply a man fascinated by the deepest mysteries of the universe, just like we kids are fascinated by even the basics of how atoms and matter work.So if Niels Bohr suddenly appeared in our classroom, I think the overall vibe would be charged with awe and excitement. This giant of 20th century science walking among us? His very presence would lend immense weight and importance to our studies of quantum phenomena. At the same time, I'm sure Bohr's warmth, passion and unwavering scientific ideals wouldinspire us to approach physics with renewed vigor and confidence in our ability to one day understand the deepest truths of nature, just like he did. It would be an unforgettable experience that could spark a lifelong love of science and discovery in all of us. I really hope it happens someday!篇2If Niles Came to Our MidstWhoa, you guys will never believe what just happened! You know that super old book we had to read for English class, A Connecticut Yankee in King Arthur's Court by Mark Twain? Well, something crazy like that totally went down at school today!It all started during Mr. Henderson's history class. We were learning about the medieval period and going over all the crazy stuff people believed back then. You know, like how they thought the world was flat and that diseases were caused by bad smells? Dumb, right?Anyway, right in the middle of Mr. H's lecture, there was this huge boom of thunder that shook the whole classroom even though it was sunny outside. Then this blinding flash of light exploded right in the center of the room. When the spots clearedfrom my eyes, there was this dude standing there dressed in the weirdest getup I've ever seen.He had on these tight leggings with a puffy shirt and this long jacket thing that went down to his knees. And get this - he was wearing tights! On his head was this funny hat with a feather sticking out the side. I thought it was bad when my little brother went through that Shakespeare phase and wouldn't stop talking in ye olde English. This guy looked like he had gone all out at one of those medieval fairs.Of course, everyone started cracking up at his ridiculous outfit. A few of the football players started chanting "Shakespeare in the park! Shakespeare in the park!" I guess they thought he was promoting some school play or something.The guy just looked around at all of us like we were the crazy ones, which made everyone laugh even harder. Finally, Mr. Henderson got the class settled down and asked the guy who he was and what he was doing here.In this super deep voice that definitely didn't match his goofy costume, he announced, "I am Niles Kalcheim, a most learned engineer from the 24th century. An unforeseen dysfunction has materialized in my chrono-displacement moduleduring its trial run, projecting me backward through thespace-time continuum."I'm not gonna lie, half of what he said went completely over my head. All I caught was "24th century" and I thought maybe this was some sort of stupid prank where one of the AV club nerds was trying to play dress up as a time traveler or something.But then the dude - Niles, I guess - went and proved he really wasn't from around here. He pulled out this shiny rectangular thing from his pocket - which actually looked kinda like one of the smartphones we're finally allowed to have at school next year. Only this one didn't have a screen or buttons or anything. It was completely smooth on both sides.Niles must have done something to activate it though because all of a sudden it projected this hologram image that hung in the air in front of him! It looked just like one of those 3D projectors they use for video games and stuff, except whatever tech he was using was a million times better. The colors were brighter and more realistic than anything I've ever seen before.The hologram was of the most bizarre contraption I've ever laid eyes on. It looked like a jungle gym designed by an insane person, with all these twisting metal tubes and giant spheres interconnected in some nutso pattern. As the hologram slowlyrotated, more and more crazy details became visible and my mind was completely blown."This is a prototype for a molecular disassembler," Niles proclaimed, like that was supposed to mean something to those of us living in the modern age rather than the 24th century.He started rambling on about how this "disassembler" could break down any object on an atomic level and convert it into elemental components or just pure energy. He claimed with enough of these crazy machines, his century had unlimited recycling and could rearrange matter itself however they wanted!Even Mr. Henderson looked dumbfounded by all this super advanced science Niles was spewing out. I figured either this guy was legitimately insane or he really was some kind of visitor from the future.That's when Niles said the words that convinced me this wasn't just an elaborate prank: "Perhaps a demonstration would render my displacement more fathomable."Before anyone could stop him, he aimed that little shiny rectangle at Mr. Henderson's desk and some kind of energy beam shot out of it. The heavy wooden desk just...disappeared!Vanished into thin air like it had never existed! All that was left behind were little sparkling particles slowly wafting through the space the desk used to occupy before they faded away completely.You can imagine the chaos that erupted after that. The girls started screaming, a few guys nearly fainted, and pretty much everyone dove for cover like Niles was about to disintegrate us all next. Even Mr. Henderson looked terrified out of his mind, sprawled there on the floor clutching his teacher's edition like it could protect him from whatever power this madman possessed.For his part, Niles just watched everyone's freaked-out reactions with an expression that seemed more confused than threatening. He tried to tell us not to be afraid, that he meant no harm, but no one was listening at that point. A few seconds later, the room was swarmed by campus security rushing in to subdue the supposed lunatic.I'm not sure what happened to Niles after that. They might have hauled him off to jail - or maybe an insane asylum is more likely considering his crazy claims of being a time traveler. Either way, I'm just glad no one else got disintegrated or anything!Can you even imagine how mind blowing it would be if Niles was telling the truth? Like, think about all the insane things wecould have in our time if his future inventions were real! Unlimited energy, the ability to just rearrange atoms however we wanted...I don't think the world today is ready for that level of technological advancement. We'd probably just use it for stupid stuff like binge watching shows without worrying about electricity bills or creating endless amounts of junk food!Still, I can't stop thinking about what might be possible in the 24th century. Just the fact that Niles could travel hundreds of years through time is crazy enough. But being able to disassemble matter into its basic components with the push of a button? If that's for real, it makes you wonder what other miraculous technologies might exist in the future. Maybe they'll have figured out how to teleport between planets or have mastered human cloning or something. Heck, maybe they'll even have figured out how to go into suspended animation so you can just sleep for 300 years and wake up in the future!Whether Niles was an actual visitor from the 24th century or just a highly convincing loon, the whole experience has me looking at the world through a different lens. For so long, our history classes have been stuck looking backward - studying the primitive civilizations, the wars and power struggles of the past. But the truth is that the most important history hasn't happenedyet. The future is where the real game-changers are going to take place that'll make everything we know today look as outdated as those eurth-cultures we learned about carving wheels out of stone.Who knows what unbelievable wonders the 24th century might hold? Flying personal vehicles, artificial intelligence assistants, maybe even some kind of master computer network safeguarding the limitless knowledge of the future! What I wouldn't give for a peek at a history book from that era. I'll bet Niles' crazy desk-dematerializer barely even registers as a significant invention compared to whatever world-altering technologies are commonplace in his time.I just hope that whoever is in charge in the 24th century uses their insane science knowledge for good and not evil. Can you imagine someone like that Thanos guy from the Avengers movies getting his hands on Niles' molecular rearranging tech? He'd be able to disassemble entire planets with the push of a button! Not that we should be worrying about hypothetical supervillains from the future, I guess. We've got enough issues to deal with in the present without borrowing troubles from another millennium.Whew, okay, that's enough of me rambling about the metaphysics of technologies yet to be invented. Whether it was real or an illusion, having a so-called time traveler materialize out of nowhere in the middle of my history class was an experience I'll never forget. It's got me thinking bigger about what might be possible and has honestly made me a lot more excited to see what the future holds - even if it's just our current century rather than the reality-bending architectures of the 24th. We're living in a pivotal time where our wildest science fictions are slowly morphing into patentable realities.Who knows? Maybe a hundred years from now, people will look back and say the real game-changing invention was whatever allowed this written record to be preserved for their eyes to read - the first relic of a primitive篇3If Niels Came Among UsBy A StudentCan you imagine what it would be like if the great Danish physicist Niels Bohr just showed up at our school one day? I've thought about this a lot, and I think it would be totallymind-blowing!First of all, I'm sure nobody would even recognize him at first. He'd probably just look like some old dude with wild Einstein hair and a funny accent. But then once the science teachers figured out who he was, it would be pure pandemonium! They'd be freaking out trying to roll out the red carpet for one of the most important scientists of the 20th century.I can just picture Niels strolling down the hallway, looking totally confused at all the commotion surrounding him. He'd probably be like "What is this peculiar place? Why are all these young people carrying those strange flat objects?" And someone would have to explain to him that we're all students at a school in the 21st century, and those "flat objects" are laptop computers that we use to access vast repositories of human knowledge and dank memes.Once he got past the initial culture shock, I bet Niels would be lowkey blown away by how much science and technology has advanced since his day. A big part of his work was on quantum theory and atomic structure, which laid the foundations for all the crazy quantum computing, nanotechnology, and other cutting-edge fields we're just starting to explore now. He'd probably get a huge kick out of seeing kids coding quantumalgorithms or running atomic force microscope simulations on their laptops.At the same time, I think he'd also be lowkey disturbed by how we sometimes take science for granted or misuse it in problematic ways. Fromatingout of more fossil fuels to industrialized warfare to social media misinformation, I can imagine Niels shaking his head and lamenting how human folly always finds new ways to run amok despite our growing scientific knowledge. He seemed like a pretty philosophical and ethical guy from what I've read, so I bet he'd want to sit us all down for some real talk about using our smarts responsibly.But more than anything, I think having Niels here would totally reinvigorate how we think about and approach science. Too often, we treat it as this dead collection of facts and formulas that we just have to regurgitate onto tests and assignments. Having one of the OG scientific revolutionaries in our midst could breathe new life into it as this radical, living endeavor to constantly question, explore, and reshape our understanding of the universe. Niels literally helped overturn centuries of classical physics doctrine, so he could show us firsthand that science isn't about memorizing - it's about creative thinking, challenging orthodoxies, and pushing the boundaries of human knowledge.I'll never forget reading about Niels's famous quote that "An expert is a person who has found out by her own painful experience all the latest mistakes." To me, that just encapsulates the mindset of a true scientist. It's all about humbly admitting the limitations of our current knowledge, while boldly venturing into new intellectual frontiers and inevitably making new mistakes that eventually lead to new discoveries. That fearless, unorthodox spirit of curiosity is what Niels embodied, and what he could hopefully instill in all of us if he walked among us.Just having a real, flesh-and-blood scientific titan like that in the classroom, wowing us with his brilliance while also showing he was still just a humble, curious human being in search of truth - it would be incredibly inspiring. We're all so used to science being this abstract collection of dusty old books and online resources. But having Niels physically present would make it more visceral and real in a whole new way. We could ask him anything we wanted about his work, his life, his mindset, his experiences - and get answers straight from the source instead of through some detached, sterilized secondhand account.Maybe Niels could even take over and lead some classes for a while, either lecturing on the latest developments in physics while he was alive or even learning about and weighing in onbrand new 21st century concepts. Just being taught by one of the most innovative scientific minds in history instead of a normal teacher would be utterly fascinating. We could get his unique perspective on not just physics, but anything from global politics to the nature of human consciousness. With Niels at the helm, our science classes would become this free-flowing Socratic dialogue where the greatest questions of the cosmos are pondered and no knowledge is too sacred to scrutinize or update as we make new empirical discoveries.Ultimately, having Niels Bohr visit our school wouldn't just be a cool celebrity cameo - it could fundamentally reshape how we experience and think about science itself. No longer would it be this dead, academic pursuit where we just absorb information. It would become an ethos - a living, evolving way of seeing and questioning the world around us with wonder, humility, and fearless curiosity. We'd go from being passive receptacles for established theories to active participants in the never-ending process of exploring, revising, and adding to human knowledge through scientific inquiry.Just picturing Niels Bohr hanging out and sharing his perspectives and life experiences with us has me brimming with excitement. Listening to that pioneering voice - the voice thathelped spark a revolutionary leap in our understanding of the universe - could imbue us with a passion for constantly questioning, challenging, and advancing our scientific narratives. We'd be connected to that grand tradition of intellectual fearlessness that shows no law or dogma is too sacrosanct once the empirical evidence points a new way. The abstract would become flesh. The dead words on a page would breathe with the vitality of the living mind that gave birth to them.Having Niels Bohr walk among us would inject a jolt of life, meaning, and inspiration into how we experience science. We'd glimpse the soul behind the formulas. We'd make a personal connection with one of the great intellectual pioneers who showed us that the universe is an ever-evolving, ever-mysterious place that constantly demands we shed our blinders and seek new truths. Just sharing the same hallways with such a luminary presence could elevate all of our scientific pursuits from rote and tedious textbook repetition to a vibrant, radical mission to boldly meet the unknown and use our minds to unravel its deepest secrets. That's what science is really about - and having Niels Bohr here could finally make us feel that in our bones.。

顶点算子代数模范畴的grothendick群(英文)

顶点算子代数模范畴的grothendick群(英文)In mathematics, the Grothendieck group of a vertex operator algebra is a group whose elements are formal linear combinations of irreducible modules of the algebra, modulo a suitable notion of equivalence. It is named in honour of Alexander Grothendieck, the famous mathematician whopioneered algebraic geometry and founded the Grothendieck school.The Grothendieck group of a vertex operator algebra is related to the representation theory of the algebra; in particular, it captures the structure of its irreducible modules, as well as their tensor product decompositions. Itis also closely related to the K-theory of vector bundles, which is one of Grothendieck's most important contributionsto mathematics.The Grothendieck group of a vertex operator algebra can be defined in several ways. One definition is based on the notion of a vertex operator algebra as a special kind of commutative algebra, and on the notion of a generic module as a module which is both projective and injective. Thisdefinition is then used to define isomorphism classes of irreducible vertex operator algebra modules, and to definethe Grothendieck group of the algebra.Another definition of the Grothendieck group uses the notion of a vertex operator algebra as a module over itself. This approach defines isomorphism classes of irreducible modules of the algebra, and then uses those classes toconstruct the Grothendieck group of the algebra.Finally, the Grothendieck group of a vertex operator algebra can be defined using the representation theory of the algebra. In this approach, the Grothendieck group is constructed from the structure constants of the irreducible modules of the algebra.The Grothendieck group of a vertex operator algebra has several properties which make it a useful tool in algebraic geometry and representation theory. For example, it is a finitely generated abelian group, which means that it is amenable to computation. It also provides a useful way to classify and study the irreducible modules of the algebra. Finally, it is related to several other groups, such as theK-theory of vector bundles and the Burnside ring of a group.Overall, the Grothendieck group of a vertex operator algebra provides a fundamental tool in the representation theory and algebraic geometry of such algebras. It is an important concept which continues to be studied, and which has many applications in mathematics.。

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a rXiv:n ucl-t h /434v113Apr2Algebraic Models of Hadron Structure II.Strange Baryons R.Bijker Instituto de Ciencias Nucleares,U.N.A.M.,A.P.70-543,04510M´e xico D.F.,M´e xico F.Iachello Center for Theoretical Physics,Sloane Laboratory,Yale University,New Haven,CT 06520-8120,U.S.A.A.Leviatan Racah Institute of Physics,The Hebrew University,Jerusalem 91904,Israel April 3,2000Abstract The algebraic treatment of baryons is extended to strange resonances.Within this framework we study a collective string-like model in which the radial excitations are interpreted as rotations and vibrations of the strings.We derive a mass formula and closed expressions for strong and electromagnetic decay widths and use these to analyze the available experimental data.PACS numbers:14.20.Jn,13.30.Eg,13.40.Hq,03.65.FdAnnals of Physics (N.Y.),in press11IntroductionIn the last few years there has been renewed interest in hadron spectroscopy.Especially the development of dedicated experimental facilities to probe the structure of hadrons in the nonperturbative region of QCD with far greater precision than before has generated a considerable amount of experimental and theoretical activity[1].This has stimulated us to reexamine hadron spectroscopy in a novel approach in which both internal(spin-ﬂavor-color)and space degrees of freedom of hadrons are treated algebraically. The new ingredient is the introduction of a space symmetry or spectrum generating algebra for the radial excitations which for mesons was taken as U(4)[2]and for baryons as U(7)[3].The algebraic approach uniﬁes the harmonic oscillator quark model,U(4)⊃U(3)for mesons and U(7)⊃U(6)for baryons,with collective string-like models of hadrons.In theﬁrst paper of this series[3]we have introduced U(7)to study the properties of nonstrange baryons,such as the mass spectrum,electromagnetic couplings[4]and strong decays[5].In this article we extend these studies to hyperons and present a systematic study of both nonstrange and strange baryons in the framework of a collective string-like qqq model in which the orbital excitations are treated as rotations and vibrations of the strings.The algebraic structure of the model enables us to obtain transparent results (mass formula,selection rules and decay widths for strong and electromagnetic couplings)that can be used to analyze and interpret the experimental data,and look for evidence of the existence of unconventional (i.e.non qqq)conﬁgurations of quarks and gluons,such as hybrid quark-gluon states qqq-g or multiquark meson-baryon bound states qqq-qrelative motion of the three constituent parts of this conﬁguration are provided by the relative Jacobi coordinates which we choose as[6]ρ=12( r1− r2),λ=1m21+m22+(m1+m2)2[m1 r1+m2 r2−(m1+m2) r3].(2.2) Here m i and r i(i=1,2,3)denote the mass and coordinate of the i-th constituent.When two of the constituents have equal mass(m1=m2),the above choice reduces toρ=12( r1− r2),λ=16( r1+ r2−2 r3).(2.3) Since the quark masses satisfy to a good approximation m u=m d=m s,the Jacobi coordinates of Eq.(2.3)are relevant for all baryons be it with strangeness S=0,−1,−2or−3.Instead of a formulation in terms of coordinates and momenta,we use the method of bosonic quantization in which we introduce a dipole boson with L P=1−for each independent relative coordinate,and an auxiliary scalar boson with L P=0+[3]b†ρ,m,b†λ,m,s†(m=−1,0,1).(2.4) The scalar boson does not represent an independent degree of freedom,but is added under the restriction that the total number of bosons N is conserved.This procedure leads to a compact spectrum generating algebra for the radial(or orbital)excitationsG r=U(7).(2.5) The U(7)algebra enlarges the U(6)algebra of the harmonic oscillator quark model[7],but still describes the dynamics of two vectors.For a system of interacting bosons the model space is spanned by the symmetric irreducible representation[N]of U(7).This representation contains all oscillator shells with n=nρ+nλ=0,1,2,...,N.The value of N determines the size of the model space and,in view of conﬁnement,is expected to be large.2.2Basis statesThe full algebraic structure is obtained by combining the spatial part G r of Eq.(2.5)with the internal spin-ﬂavor-color part G i of Eq.(2.1)G=G r⊗SU sf(6)⊗SU c(3).(2.6) The spatial part of the baryon wave function has to be combined with the spin-ﬂavor and color part,in such a way that the total wave function is antisymmetric.Since the color part of the wave function is antisymmetric(color singlet),the remaining part(space-spin-ﬂavor)has to be symmetric.A convenient set of basis states is provided by the case of three identical constituents,for which the spatial and spin-ﬂavor parts of the baryon wave function are in addition labeled by their transformation properties under3the permutation group S3:t=S for the symmetric,t=A for the antisymmetric and t=M for the two-dimensional(Mρ,Mλ)mixed symmetry representation.A set of basis states for the spin-ﬂavor part is provided by the decomposition of SU sf(6)into itsﬂavor and spin parts.(2.7)↓↓↓↓↓SU sf(6)⊃SU f(3)⊗SU s(2)⊃SU I(2)⊗U Y(1)⊗SU s(2)[f1f2f3][g1g2]S I YHere[f1f2f3]and[g1g2]represent the Young tableaux,S denotes the spin,I the isospin and Y the hypercharge.The representations of the spin-ﬂavor groups are often labeled by their dimensions(rather than by their Young tableaux)(f1−f2+1)(f1−f3+2)(f2−f3+1)(f1+5)!(f2+4)!(f3+3)!dim[f1f2f3]=(g1−g2+1)(g1+2)(g2+1),2dim[S]=2S+1.(2.8) For three constituent parts the allowed values of[f1f2f3]are[300](t=S),[210](t=M)and[111] (t=A)with dimensions56,70and20,respectively.Theﬂavor part is characterized by[g1g2]=[30], [21]or[00]with dimensions10(decuplet),8(octet)or1(singlet),respectively.In the notation of[8]the ﬂavor wave functions are labeled by(p,q)=(g1−g2,g2).Finally,the total spin of three spin-1/2objects is S=3/2or S=1/2.The decomposition of representations of SU sf(6)into those of SU f(3)⊗SU s(2)is the standard oneS↔[56]⊃28⊕410,M↔[70]⊃28⊕48⊕210⊕21,A↔[20]⊃28⊕41,(2.9) where we have denoted the irreducible representations by their dimensions.Eachﬂavor multiplet consists of families of baryons which are characterized by their isospin I and hypercharge Y(see Table I).The electric charge is given by the Gell-Mann and Nishijima relationYQ=I3+In this decomposition,the behavior in three-dimensional coordinate space SU(3)⊃SO(3)is separated from that in index space SU(2)⊃SO(2).The allowed values of the quantum numbers can be obtained from the branching rules.For the decomposition of U(6)we use the complementarity relationship between the groups SU(3)and SU(2)within the symmetric irreducible representation U(6).As a consequence, the labels of SU(3)are determined by those of SU(2).The branching rules aren=0,1,...,N,F=n,n−2,...,1or0,(n1,n2)=(n+F2),m F=−F,−F+2,...,F.(2.12)The reduction from the coupled harmonic oscillator group to the rotation group SU(3)⊃SO(3)is given by[10]K=min{λ,µ},min{λ,µ}−2,...,1or0,K=0:L=max{λ,µ},max{λ,µ}−2,...,1or0,K>0:L=K,K+1,...,K+max{λ,µ}.(2.13) Here(λ,µ)=(n1−n2,n2)=(F,(n−F)/2).The label K is an extra label that has to be introduced to label the states uniquely[10].The SO(2)group in Eq.(2.11)is related to the permutation symmetry [3,9,7].The states with good S3symmetry are given by the linear combinations|ψ1 =−i2(1+δmF ,0)[|φmF −|φ−m F ],|ψ2 =(−1)ν2(1+δmF ,0)[|φmF +|φ−m F ].(2.14)Here we have introduced the labelνby m F=ν(mod3).The wave functions|ψ1 (|ψ2 )transform for ν=0as t=A(S),and forν=1,2as t=Mρ(Mλ).Summarizing,the basis states are characterized uniquely by|N,n,F,m F,K,L P t ,(2.15) where P is the parity of the basis states P=(−)n.Finally,the quark orbital angular momentum L is coupled with the spin S to the total angular momentum J of the baryon.In Appendix B we present the space-spin-ﬂavor baryon wave functions with S3symmetry.3Mass operatorThe mass operator depends both on the spatial and the internal degrees of freedom.For the spatial part we adopt a collective model of the nucleon in which the baryons are interpreted as rotational and vibrational excitations of the string conﬁguration of Fig.1.For two identical constituent parts(as is the case for strange baryons)the vibrations are described by[6]ˆM2vib=A P†1P1+B P†2P2+C P†3P3+D(P†1P2+P†2P1),(3.1)5withP†1=R2s†s†−b†ρ·b†ρ−b†λ·b†λ,P†2=(cosβ)2b†ρ·b†ρ−(sinβ)2b†λ·b†λ,P†3=b†ρ·b†λ.(3.2) Here R is related to the hyperspherical radius1+R2√2DN sin(2β)R2/4+d ˆC1(U Y(1))−1 +e ˆC2(U Y(1))−1 +f ˆC2(SU I(2))−3The eigenvalues of the Casimir operators in the basis states of Eq.(2.7)areˆC2(SU sf(6)) =2 f1(f1+5)+f2(f2+3)+f3(f3+1)−12 g1(g1+2)+g22−1(f1+f2+f3)2−453+b 33(g1+g2)2 −9 +c S(S+1)−34 .(4.1) The coeﬃcient M20is determined by the nucleon mass M20=0.882GeV2.The remaining nine coeﬃcients are obtained in a simultaneousﬁt to the three and four star resonances of Tables III and IV which have been assigned as octet and decuplet states.Weﬁnd a good overallﬁt for48resonances with an r.m.s. deviation ofδ=33MeV.The values of the parameters are given in Table II.In the last column we show for comparison the parameters that were obtained in aﬁt to25nucleon and delta resonances with a r.m.s.deviation toδ=39MeV[3].In comparison with Table II of[3]the∆(1900)S31was left out, since it has been downgraded from a three to a two star resonance[13].Since for nonstrange resonances Y=1,the d and e terms in Eq.(4.1)do not contribute.Theﬂavor and isospin dependent terms that determine the mass splitting between the nucleon and∆resonances can be combined into a single b term 1with strength b+Tables III and IV and Figs.3–6show that the mass formula of Eq.(4.1)provides a good overall description of both positive and negative baryon resonances belonging to the N,∆,Σ,Λ,ΞandΩfamilies.There is no need for an additional energy shift for the positive parity states and another one for the negative parity states,as in the relativized quark model[14].4.1Octet and decuplet resonancesThe results are presented in Fig.2for the ground state baryon octet with J P=1/2+and the baryon decuplet with J P=3/2+.In Tables III and IV we show a comparison with all three and four star resonances.In our calculation we have assigned the N(1440),∆(1600),Σ(1660)andΛ(1600)resonances to the vibration characterized by(v1,v2)=(1,0),and the N(1710),Σ(1940)andΛ(1810)resonances to the(v1,v2)=(0,1)vibration.The remaining resonances are assigned as rotational members of the ground band with(v1,v2)=(0,0).We have followed the quark model assignments of Table13.4of[13],with the exception of the Σ(1750)S11resonance which we have assigned as281/2[70,1−],the lowest S11state with a mass of1711 MeV.In our calculation the lowest four S11Σstates occur at1711,1755,1822and1974MeV(for the assignments we refer to Tables V and VI;the second state belongs to the decuplet).In the nucleon andΛfamilies the Roper resonance lies below theﬁrst excited negative parity resonance.We expect the same to be true for theΣhyperons.With our assignment,Σ(1750)is the octet partner of N(1535)andΛ(1670), which is also supported by theirηdecay properties[15,16].In[13]the two starΣ(1620)resonance has been assigned as the281/2[70,1−]state,and theΣ(1750)resonance instead as the481/2[70,1−]state. Our assignment ofΣ(1750)coincides with that of[17].In the relatived quark model there are three low-lying S11Σresonances at1630,1675and1695MeV[14].Theﬁrst one was associated with the two starΣ(1620)resonance,and the next one with theΣ(1750)resonance.TheΣ(1940)D13resonance was not assigned in Table13.4of[13],whereas in[17]it was tentatively assigned as483/2[70,1−],the octet partner of N(1700).In our calculation the lowest four D13Σstates are the spin-orbit partners of the S11states at1711,1755,1822and1974MeV,theﬁrst one of which has been associated with theΣ(1670)resonance.We have assigned theΣ(1940)resonance as a member of the(v1,v2)=(0,1)vibrational band with283/2[56,1−]which occurs at1974MeV.This assignment is supported by its strong decay properties(see Sect.6).In the relativized quark model there are three low-lying D13Σstates at1655,1750and1755MeV[14],of which theﬁrst two were associated with the Σ(1670)andΣ(1940)resonances.4.2Singlet resonancesThere are three states which show a deviation of about100MeV or more from the data:theΛ∗(1405),Λ∗(1520)andΛ∗(2100)resonances are overpredicted by236,121and97MeV,respectively.These three resonances are assigned as singlet states in Table IV(and were not included in theﬁtting procedure). An additional energy shift for the singlet states(without eﬀecting the masses of the octet and decuplet states)can be obtained by adding to the mass formula of Eq.(4.1)a term that only acts on the singlet statesM2→M2+∆M2δg1,0δg2,0.(4.2)8This corresponds to a shift in the singlet masses of MK threshold. The inclusion of the coupling to the NK threshold[20].Such an interpretation is supported by the strong and electromagnetic couplings(see Sects.6and8).In a chiral meson-baryon Lagrangian approach with an eﬀective coupled-channel potential theΛ∗(1405)resonance emerges as a quasi-bound state of Nprovide a more sensitive tool to determine the most likely assignment(see Sect.6).In the relativized quark model a D13state has been predicted at1960MeV[14].5Strong couplingsStrong couplings provide an important test of baryon wave functions,and can be used to distinguish between diﬀerent models of baryon structure.Here we consider strong decays of baryons by the emission of a pseudoscalar mesonB→B′+M.(5.1)Several forms have been suggested for the form of the operator inducing the strong transition[26].We use here the simple form[27]H s=1(2π)3/2(2k0)1/26X M3 (gk−12h(s3,+ˆT−+s3,−ˆT+) ,(5.3) withˆU=e ik √3λz,ˆT m =122222/3λm→βˆDλ,m/X D[3,4,5].The result isˆU=e ikβˆDλ,z/X D,ˆT m =−im3k0βdistribution function g(β)of charge and magnetization over the entire volumeg(β)=β2e−β/a/2a3.(5.6) All spatial matrix elements can be expressed in terms of the collective form factorsF(k)= dβg(β) ψ′|ˆU|ψ ,G m(k)= dβg(β) ψ′|ˆT m|ψ .(5.7) Here|ψ and|ψ′ denote the spatial wave functions of the initial andﬁnal baryons.For strong decays in which the initial baryon B has angular momentum J= L+ S and in which the ﬁnal baryon B′has L′=0and thus J′=S′,the helicity amplitudes in the collective model are then given byAν(k)= dβg(β) Ψ′(0,S′,S′,ν)|H s|Ψ(L,S,J,ν) ,(5.8) Here|Ψ(L,S,J,M J) and|Ψ′(0,S′,S′,ν) denote the(space-spin-ﬂavor)angular momentum coupled wave functions of the initial andﬁnal baryons,respectively(see Appendix B).Theﬁnal baryon is a ground state baryon belonging either to the octet281/2[56,0+](0,0);0or the decuplet4103/2[56,0+](0,0);0. The helicity amplitudes can be expressed in terms of a spatial matrix element and a spin-ﬂavor matrix elementAν(k)=12 L,1,S,ν−1|J,ν ζ+Z−(k)+16hk)F(k)−6h G z(k),Z±(k)=−6h G±(k).(5.10)In Table VIII we present the collective form factor F(k)(for details of the derivation we refer to[3]). The form factors G m(k)are given byG z(k)=−δM,0m3k0a d F(k)L(L+1)F(k)(3,0)or (0,0)for the baryon ﬂavor octet,decuplet and singlet,respectively,and (p,q )=(1,1)or (0,0)for the meson ﬂavor octet and singlet,respectively.The spin-ﬂavor matrix elements ζm for a given isospin channel can be expressed asζm = γ(p f ,q f )(p,q )I f ,Y f I,Y (p i ,q i )γI i ,Y i αm,γ.(5.12)The sum over γis over diﬀerent multiplicities.The SU (3)isoscalar factor which appears in Eq.(5.12)depends on the ﬂavor multiplets (p,q),the isospin I and the hypercharge Y .A compilation of the SU (3)isoscalar factors relevant for strong decays of baryons can be found in [13].Results for a speciﬁc charge channel can be obtained by multiplying ζm with the appropriate isospin Clebsch-Gordan coeﬃcient I f ,M I f ,I,M I |I i ,M I i .In Tables IX–XII we present the coeﬃcients αm,γfor strong decays into octet or decuplet baryons emitting a pseudoscalar meson (either octet or singlet).For strong decays of nonstrange baryons into the Nπ,Nη,∆πand ∆ηchannels the coeﬃcients ζm are given explicitly in Tables III and IV of [5].Inspection of Tables IX–XII and the isoscalar factors on page 184of [13]yields some interesting selection rules:(i)the B 10→B 10+M 8decayΣ∗→Σ∗+η8,(5.13)is forbidden since the SU (3)isoscalar factor vanishes,and (ii)there is a spin-ﬂavor selection rule for the 48[70,L P ]→28[56,0+]+M 8decaysN→Λ+K ,Λ→N +2J +1 ν>0|A ν(k )|2.(6.1)Here we adopt the procedure of[26],in which the decay widths are calculated in the rest frame of the decaying resonance,and in which the relativistic expression for the phase space factorρas well as for the momentum k of the emitted meson are retained.The expressions for k andρare(m2B−m2B′+m2M)2k2=−m2M+(6.2)m Bwith E B′= m2M+k2.We consider here the strong decays of baryons in which a pseudoscalar meson(either octet or singlet)is emitted.The present calculation is an extension of[5] in which we only discussed nonstrange decays of nonstrange baryons.The calculated widths depend on the two parameters g and h in the transition operator of Eq.(5.3), and on the scale parameter a of Eq.(5.6).In accordance with[5]we keep g,h and aﬁxed for all resonances and all decay channels with the values g=1.164fm,h=−0.094fm and a=0.232fm(we note here that in[5]the values of g and h were given in GeV−1instead of in fm).The decay widths of resonances that have been interpreted as a vibrational excitation of the string conﬁguration of Fig.1depend on the coeﬃcientχ1for(v1,v2)=(1,0)(the N(1440),∆(1600),Σ(1660)andΛ(1600)resonances),orχ2for (v1,v2)=(0,1)(the N(1710),Σ(1940)andΛ(1810)resonances)(see Table VIII).These coeﬃcients are proportional to the intrinsic matrix element for each type of vibration.Here they are taken as constants with the valuesχ1=1.0[30]andχ2=0.7.For the pseudoscalarηmesons we introduce a mixing angle θP=−23◦between the octet and singlet mesons[13].This value is consistent with that determined in a study of meson spectroscopy[2].In comparison with other studies,we note that in the calculation in the nonrelativistic quark model of [27]an elementary emission model is used,just as in the present calculation,but with the diﬀerence that the decay widths are parametrized by four reduced partial wave amplitudes instead of the two elementary amplitudes g and h.Furthermore,the momentum dependence of these reduced amplitudes is represented by a constant.The calculations in the relativized quark model are done in a pair-creation model for the decay and involve a diﬀerent assumption on the phase space factor[31].Both the nonrelativistic and relativized quark model calculations include the eﬀects of mixing induced by the hyperﬁne interaction, which in the present calculations are not taken into account.It is important to note that we present a comparison of decay widths,rather than of decay amplitudes as was done in[27]and[31]for the nonrelativistic and relativized quark models.In Tables XIII–XVII we compare the experimental strong decay widths of three and four star baryon resonances from the most recent compilation by the Particle Data Group[13]with the results of our calculation for the nucleon,∆,Σ,ΛandΞfamilies.We have used the experimental value of the mass of the decaying baryon.Strong couplings of missing resonances belonging to theﬂavor octet,decuplet and singlet are presented in Tables XVIII–XXI,XXII–XXV and XXVI,respectively.The calculated decay widths are to a large extent a consequence of spin-ﬂavor symmetry and phase space.The use of the collective form factors of Table VIII introduces a power-law dependence on the meson momentum k,compared to,for example,an exponential dependence for harmonic oscillator form factors.Our results for the strong decay widths are in fair overall agreement with the available data, and show that the combination of a collective string-like qqq model of baryons and a simple elementary emission model for the decays can account for the main features of the data.There are a few exceptionswhich could indicate evidence for the importance of degrees of freedom which are outside the present qqq model of baryons.6.1Nucleon resonancesTheπandηdecays of nucleon resonances have already been discussed in[5].Whereas theπdecays are in fair agreement with the data,theηdecays of octet baryons show an unusual pattern:the S-wave states N(1535),Σ(1750)andΛ(1670)all are found experimentally to have a large branching ratio to theηchannel,whereas the corresponding phase space factor is very small[15].The small calculatedηwidths(<0.5MeV)for these resonances are due to a combination of spin-ﬂavor symmetry and the size of the phase space factor.The results of our analysis suggest that the observedηwidths are not due to a conventional qqq state,but may rather indicate evidence for the presence of a state in the same mass region of a more exotic nature,such as a pentaquark conﬁguration qqqqq just below or above threshold,bound by Van der Waals type forces(for example Nη,ΣK orΛK[21]).In order to answer this question one has to carry out an analysis,similar to the present one,of the other conﬁgurations.The K decays are suppressed with respect to theπdecays because of phase space.In addition,the decay of the N(1650),N(1675)and N(1700)resonances intoΛK is forbidden by the spin-ﬂavor selection rule for the decay of48[70,L P]nucleon states into this channel(see Sect.5).For N(1675)and N(1700) only an upper limit is known,whereas the N(1650)resonance has an observed width of12±7MeV. However,this resonance is just above theΛK threshold which may lead to a coupling to a quasi-bound meson-baryon S wave resonance.A study of the eﬀect of spin-ﬂavor symmetry breaking on these decays is in progress.6.2Delta resonancesThe strong decay widths of the∆resonances are in very good agreement with the available experimental data.The same holds for the other resonances that have been assigned as decuplet baryons:Σ∗(1385),Σ∗(2030)andΞ∗(1530).For the decuplet baryons there is no S state around the threshold of the various decay channels,so therefore there cannot be any coupling to quasi-molecular conﬁgurations.Just as for the nucleon resonances,theηand K decays are suppressed by phase space factors.6.3Sigma resonancesStrange resonances decay predominantly into theπandK andΣ∗πchannels,which is not supported by the data.6.4Lambda resonancesAlso forΛresonances theηand K decays are suppressed with respect to theπandK channel.Therefore,the calculated NK∗(892)[13].Since the mass of the resonance is just around the threshold of this channel,this could indicate a coupling with a quasi-molecular S wave.The NK threshold.It has been shown[20]that the inclusion of the coupling to the NK threshold.In a chiral meson-baryon Lagrangian approach with an eﬀective coupled-channel potential theΛ∗(1405)resonance emerges as aquasi-bound state of NK channels.Theηand K widths aresmall in comparison,due to the available phase space.Inspection of Tables XIX–XXI shows that the dominant decay channels of the missing octet baryons are N K forΣresonances,NK,ΞπforΞresonances.Similarly,we see from Tables XXIII–XXV that the missing decuplet baryons are most likely to couple toΛπ,Σ∗πforΣ∗resonances,toΣK,Ξπ,Σ∗K forΩresonances.Finally,Table XXVI shows that missing singlet baryons are most likely to decay intoΛ∗→Nπ(p j,−e−i k· r j+e−i k· r j p j,−) ,(7.2)2g jwhere r j, p j and s j are the coordinate,momentum and spin of the j-th constituent,respectively;k0is the photon energy,and k=kˆz denotes the momentum carried by the outgoing photon.The photon is emitted by the j-th constituent:q j→q′j+γ(see Figure8).The transition operator can be simpliﬁed by using the symmetry of the baryon wave functions,transforming to Jacobi coordinates and integrating over the baryon center-of-mass coordinate,to obtainH em=6 k0µ3e3 ks3,−ˆU−1πIn Tables XXVII and XXVIII we show the orbit-and spin-ﬂip amplitudes for some radiative hyperon decays.These results were obtained under the assumption of SU f(3)ﬂavor symmetry,i.e.µ3=µp and g3=g.The following selection rules apply:(i)the410[56]→28[56]+γdecaysΣ∗,−→Σ−+γ,Ξ∗,−→Ξ−+γ,(7.6) are forbidden by U-spin conservation[34].These decays can only occur ifﬂavor symmetry is broken (m d=m s).Also the21[70]→410[56]+γdecayΛ∗→Σ∗,0+γ,(7.7) is forbidden by U-spin selection rules but,contrary to the previous cases,remains forbidden in the case ofﬂavor symmetry breaking.8Comparison with experimental electromagnetic couplingsRadiative hyperon decay widths can be calculated from the helicity amplitudes as[26]Γ(B→B′+γ)=2πρ12J+1 ν>0|Aν(k)|2.(8.1)Just as for the strong couplings,the electromagnetic decay widths are calculated assuming SU sf(6)spin-ﬂavor symmetry and using the rest frame of the decaying resonancek=m2B−m2B′m B(8.2)with E B′=the decuplet to octet transitions[36,34].For the conventions used in Appendices A and B we obtain the relationsµΣ0Λ=13µp,µ∆+p=µ∆0n=−µΣ∗,+Σ+=−2µΣ∗,0Σ0=23µΣ∗,0Λ=−µΞ∗,0Ξ0=2√3µp,µΣ∗,−Σ−=µΞ∗,−Ξ−=0.(8.4)The numerical values are given in the third column of Table XXIX.A comparison with the last column shows the reduction of the transition magnetic moments due to the form factor F(k)=1/(1+k2a2)2.The experimental information on radiative decays of hyperons is very limited.In Table XXX we present the radiative decay widths of low-lying hyperon resonances,and compare wherever possible with the data.The∆+→p+γdecay width is underpredicted by35%,a common feature of all qqq constituent quark models.This discrepancy has been shown to be due to nonresonant meson-exchange mechanisms [40].Just as for the energies and the strong decays,theΛ∗(1405)resonance shows large deviations for the radiative decay widths,which once agains conﬁrms its uncertain nature as a qqq state.The forbidden decaysΛ∗(1405)→Σ∗,0+γandΛ∗(1520)→Σ∗,0+γhave not been observed.For comparison we also present the radiative decay widths of decuplet hyperons as obtained from lattice calculations[41]and from a chiral constituent quark model with electromagnetic exchange currents between quarks[42].The negative parity hyperon decay widths forΣ∗,−→Σ−+γandΞ∗,−→Ξ−+γhave a small nonvanishing value in[41,42],whereas in the present calculation they are forbidden byﬂavor symmetry selection rules. In a subsequent publication we plan to investigate the eﬀects of SU f(3)ﬂavor symmetry breaking on the electromagnetic couplings.9Summary and conclusionsWe have presented in this article a systematic analysis of spectra and transition rates of strange baryons in the framework of a collective string-like qqq model,in which the orbitally excited baryons are interpreted as collective rotations and vibrations of the strings.The algebraic structure of the model,both for the internal degrees of freedom of spin-ﬂavor-color and for the spatial degrees of freedom,has been used to derive transparent results,such as a mass formula,selection rules and closed expressions for strong and electromagnetic couplings.The situation is similar to that encountered for nonstrange baryons.While spectra are reasonably well described,transition rates,especially strong decay widths are only qualitatively described.The combination of a collective string-like qqq model of baryons and a simple elementary emission model for the decays can account for the main features of the data.The main discrepancies are found for the low-lying S-wave states,speciﬁcally N(1535),Σ(1750),Λ∗(1405),Λ(1670)andΛ(1800).All of these resonances have masses which are close to the threshold of a meson-baryon decay channel,and hence they could mix with a quasi-molecular S wave resonance of the form qqq−q。

Algebraic Models of Hadron Structure II. Strange Baryons

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Algebraic Models of Hadron Structure II. Strange Baryons

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