(完整版)83-Define-and-Use-Zero-and-Negative-Exponents83定义与使用零与负指数

EGM2008 - WGS 84 VersionIntroductionThe official Earth Gravitational Model EGM2008 has been publicly released by the U.S. National Geospatial-Intelligence Agency (NGA) EGM Development Team. This gravitational model is complete to spherical harmonic degree and order 2159, and contains additional coefficients extending to degree 2190 and order 2159. Full access to the model's coefficients and other descriptive files with additional details about EGM2008 are provided herein.Those wishing to use EGM2008 to compute geoid undulation values with respect to WGS 84,may do so using the self-contained suite of coefficient files, FORTRAN software, and pre-computed geoid grids provided on this web page. For other applications, the previous release of the full 'Geoscience' package for EGM2008 can be accessed through the link at the bottom of this web page.The WGS 84 constants used to define the reference ellipsoid, and the associated normal gravity field, to which the geoid undulations are referenced are:•a=6378137.00 m (semi-major axis of WGS 84 ellipsoid)•f=1/298.257223563 (flattening of WGS 84 ellipsoid)•GM=3.986004418 x 1014 m3s-2 (Product of the Earth's mass and the Gravitational Constant)•Ï‰=7292115 x 10-11 radians/sec (Earth's angular velocity)All synthesis software, coefficients, and pre-computed geoid grids listed below assume a Tide Free system, as far as permanent tide is concerned.Note that the harmonic synthesis software provided below applies a constant,zero-degree term of -41 cm to all geoid undulations computed using EGM2008 with the height_anomaly-to-geoid_undulation correction model (also provided). Similarly, all pre-computed geoid undulations incorporate this constant zero-degree term. This term converts geoid undulations that are intrinsically referenced to an ideal mean-earth ellipsoid into undulations that are referenced to WGS 84. The value of -41 cm derives from a mean-earth ellipsoid for which the estimated parameters in the Tide Free system are: a=6378136.58 m and 1/f=298.257686.Description of Software and DataTo compute point geoid undulations from spherical harmonic synthesis of the EGM2008 Tide Free Spherical Harmonic Coefficients and its associated Correction Model, at any WGS 84 latitude/longitude coordinate pair listed in a coordinate input file (such as INPUT.DAT), use the FORTRAN harmonic synthesis program, hsynth_WGS84.f.At present, we are also providing two grids of pre-computed geoid undulations: one at 1 x 1-minute resolution and one at 2.5 x 2.5-minute resolution. To interpolate geoid undulations from the 1 x 1-Minute Geoid Undulation Grid file, for any WGS 84 latitude/longitude coordinate pair listed in a coordinate input file (such as INPUT.DAT), use the FORTRAN interpolation program interp_1min.f. Similarly, the FORTRAN interpolation program interp_2p5min.f, will interpolate geoid undulations from the 2.5 x 2.5-Minute Geoid Undulation Grid file.Filenames containing a ".gz" in the suffix have been compressed using the Unix "gzip" command.All files that are SMALL ENDIAN format are highlighted in green. All files that are BIG ENDIAN format are highlighted in purple.Please carefully review the README_WGS84 file for a complete description of coefficient and data files for computing EGM2008 geoid undulations with respect to WGS 84. Also, before using any of the data files on this web page, please read the disclaimer.For GIS data formats, please visit the EGM2008 GIS Data Page.Software and Coefficients for WGS 84 Geoid Undulation Computations by Harmonic Synthesis•EGM2008 Harmonic Synthesis Program(hsynth_WGS84.f -184 KB) - Use this FORTRAN program to generate WGS 84 geoid undulations by spherical harmonic synthesis of EGM2008 and its associated height_anomaly-to-geoid_undulation correction model. This program requires that the coefficients for both EGM2008 and the correction model, and an INPUT.DAT file, all be located in the same directory as the hsynth_WGS84.f program.•Harmonic Synthesis Executable for Windows XP(hsynth_WGS84.exe - 696 KB) - Windows executable version of EGM2008 Harmonic Synthesis Program. Use as described above.•Spherical Harmonic Coefficients for Earth's Gravitational Potential (Tide Free System)-(EGM2008_to2190_TideFree.gz - 72 MB) - EGM2008 coefficients required by the harmonic synthesis program.•Correction Coefficients(Zeta-to-N_to2160_egm2008.gz - 50 MB) -Height_anomaly-to-geoid_undulation coefficients required by the harmonic synthesis program.•INPUT.DAT(4 KB) - sample input file of WGS 84 latitude/longitude coordinate pairs for testing the harmonic synthesis program.•OUTPUT.DAT(4 KB)- sample output file of geoid undulation values, generated by reading INPUT.DAT into the harmonic synthesis program (for test verification).*Users - Please verify results by comparing to OUTPUT.DAT file immediately above. Software and Grids for WGS 84 Geoid Undulation Computation by Interpolation*Note: The 1 x 1 minute interpolation program below requires a large PC RAM capacity.•Interpolation Program for 1 x 1-Minute Geoid Grid(interp_1min.f - 28 KB) - Use this FORTRAN program to interpolate geoid undulations from the 1 x 1-Minute Geoid Undulation Grid file, for any WGS 84 latitude/longitude coordinate pair listed in a coordinate input file (such as INPUT.DAT). When applied to the 1 x 1-Minute Geoid Undulation Grid file, this interp_1min.f program will generate geoid undulation values that match the corresponding values generated by harmonic synthesis (hsynth_WGS84.f or hsynth_WGS84.exe above) to within 1 mm. The interp_1min.f program requires that the 1 x 1-Minute Geoid Undulation Grid file and the INPUT.DAT are located in the same directory as interp_1min.f.• 1 x 1-Minute Grid Interpolation Executable for Windows XP(interp_1min.exe - 434 KB) - Windows executable version of the Interpolation Program for 1 x 1-Minute Geoid Grid. Use as described above. To be used with the SMALL ENDIAN Geoid Undulation Grid below.• 1 x 1-Minute Geoid Undulation Grid in WGS 84 - SMALL ENDIAN (Und_min1x1_egm2008_isw=82_WGS84_TideFree_SE.gz - 825 MB) - 1 x 1-minute global grid of pre-computed geoid undulations. This file has a SMALL ENDIAN internal binary representation.• 1 x 1-Minute Geoid Undulation Grid in WGS 84 - BIG ENDIAN(Und_min1x1_egm2008_isw=82_WGS84_TideFree.gz - 828 MB) - 1 x 1-minute global grid of pre-computed geoid undulations. This file has a BIG ENDIAN internal binary representation.•INPUT.DAT(4 KB) - sample input file of WGS 84 latitude/longitude coordinate pairs for testing the Interpolation Program for 1 x 1-Minute Geoid Grid.•OUTPUT.DAT(4 KB)- sample output file of geoid undulation values, generated by reading INPUT.DAT into the Interpolation Program for 1 x 1-Minute Geoid Grid above (for test verification).*Users - Please verify results by comparing to OUTPUT.DAT file immediately above.•Interpolation Program for 2.5 x 2.5 Minute Geoid Grid(interp_2p5min.f - 28 KB) - Use this FORTRAN program to interpolate geoid undulations from the 2.5 x2.5-Minute Geoid Undulation Grid file, for any WGS 84 latitude/longitude coordinatepair listed in a coordinate input file (such as INPUT.DAT). When applied to the 2.5 x2.5-Minute Geoid Undulation Grid file, this interp_2p5min.f program will generategeoid undulation values that match the corresponding values generated by harmonic synthesis (hsynth_WGS84.f or hsynth_WGS84.exe above) to within 1 cm. The interp_2p5min.f program requires that the 2.5 x 2.5-Minute Geoid Undulation Grid file and the INPUT.DAT are located in the same directory as interp_2p5min.f.• 2.5 x 2.5-Minute Interpolation Executable for Windows XP(interp_2p5min.exe - 434 KB) - Windows executable version of the Interpolation Program for 2.5 x 2.5 Minute Geoid Grid. Use as described above. To be used with the SMALL ENDIAN Geoid Undulation Grid below.• 2.5 x 2.5-Minute Geoid Undulation Grid in WGS 84 - SMALL ENDIAN (Und_min2.5x2.5_egm2008_isw=82_WGS84_TideFree_SE.gz - 135 MB) - 2.5 x2.5-minute global grid of pre-computed geoid undulations. This file has a SMALLENDIAN internal binary representation.• 2.5 x 2.5-Minute Geoid Undulation Grid in WGS 84 - BIG ENDIAN(Und_min2.5x2.5_egm2008_isw=82_WGS84_TideFree.gz - 135 MB) - 2.5 x2.5-minute global grid of pre-computed geoid undulations. This file has a BIG ENDIANinternal binary representation.•INPUT.DAT(4 KB) - sample input file of WGS 84 latitude/longitude coordinate pairs for testing the Interpolation Program for 2.5 x 2.5-Minute Geoid Grid.•OUTPUT.DAT(4 KB)- sample output file of geoid undulation values, generated by reading INPUT.DAT into the Interpolation Program for 2.5 x 2.5-Minute Geoid Grid above (for test verification).*Users - Please verify results by comparing to OUTPUT.DAT file immediately above. Software to Extract WGS 84 Geoid Undulations from Grid Files (No Interpolation)•Extract 1 x 1-Minute Grid Program(gridget_1min.f -20 KB) - Use this FORTRAN program to extract a user-defined sub-rectangle of geoid undulations from the 1 x 1-Minute Geoid Undulation Grid file. This program prompts the user for a rectangular area of interest and requires the 1 x 1-Minute Geoid Undulation Grid file to be located in the same directory as the gridget_1min.f program.•Extract 1 x 1-Minute Grid Executable for Windows XP(gridget_1min.exe - 401 KB) - Windows executable version of the Extract 1 x 1-Minute Grid Program. Use as described above. To be used with the SMALL ENDIAN Geoid Undulation Grid below.• 1 x 1-Minute Geoid Undulation Grid in WGS 84 - SMALL ENDIAN (Und_min1x1_egm2008_isw=82_WGS84_TideFree_SE.gz - 825 MB) - 1 x 1-minute global grid of pre-computed geoid undulations. This file has a SMALL ENDIAN internal binary representation.• 1 x 1-Minute Geoid Undulation Grid in WGS 84 - BIG ENDIAN(Und_min1x1_egm2008_isw=82_WGS84_TideFree.gz - 828 MB) - 1 x 1-minute global grid of pre-computed geoid undulations. This file has a BIG ENDIAN internal binary representation.•Extract 2.5 x 2.5-Minute Grid Program(gridget_2p5min.f - 20 KB) - Use thisFORTRAN program to extract a user-defined sub-rectangle of geoid undulations from the 2.5 x 2.5-Minute Geoid Undulation Grid file. This program prompts the user for a rectangular area of interest and requires the 2.5 x 2.5-Minute Geoid Undulation Grid file to be located in the same directory as the gridget_2p5min.f program.•Extract 2.5 x 2.5-Minute Grid Executable for Windows XP(gridget_2p5min.exe - 397 KB) - Windows executable version of the Extract 2.5 x 2.5-Minute Grid Program.Use as described above. To be used with the SMALL ENDIAN Geoid Undulation Grid below.• 2.5 x 2.5-Minute Geoid Undulation Grid in WGS 84 - SMALL ENDIAN (Und_min2.5x2.5_egm2008_isw=82_WGS84_TideFree_SE.gz - 135 MB) - 2.5 x2.5-minute global grid of pre-computed geoid undulations. This file has a SMALLENDIAN internal binary representation.• 2.5 x 2.5-Minute Geoid Undulation Grid in WGS 84 - BIG ENDIAN(Und_min2.5x2.5_egm2008_isw=82_WGS84_TideFree.gz - 135 MB) - 2.5 x2.5-minute global grid of pre-computed geoid undulations. This file has a BIG ENDIANinternal binary representation.Additional Information•Original Release of the EGM2008 Model Coefficients from EGU General Assembly, Vienna, Austria 2008: Geoscience Package•An Earth Gravitational Model to Degree 2160: EGM2008(NPavlis&al_EGU2008.ppt - 18.5 MB) - Presentation given at the 2008 European Geosciences Union General Assembly held in Vienna, Austria, April13-18, 2008.•Background Papers from Earlier Symposiums on the New EGM。

Biogeochemical processes in intensive zero-effluent marine fish culture with recirculating aerobic and anaerobic biofiltersAmir Neori a,⁎,Michael D.Krom b ,Jaap van Rijn caIsrael Oceanographic and Limnological Research,The National Centre for Mariculture,P .O.Box 1212,Eilat 88112,Israel bEarth and Biosphere Institute,School of Earth and Environment,Leeds University,Leeds LS29JT,United KingdomcDepartment of Animal Sciences,Faculty of Agricultural,Food and Environmental Quality Sciences,The Hebrew University of Jerusalem,Rehovot 76100,IsraelReceived 14December 2006;received in revised form 11April 2007;accepted 20May 2007AbstractThe biogeochemical processes that drive nutrient transformations and recycling in organic marine sediment –water environments were studied for 17months in a zero-effluent intensive recirculating culture system.The system consisted of a 10m 3gilthead seabream (Sparus aurata )tank coupled to aerobic and anaerobic water treatment elements.Nutrients and alkalinity were measured in the system to quantify the main biogeochemical processes.Fractions of the carbon fed in feed were found in fish (18.3%)and in sludge (11%);the missing carbon was respired by fish (45%)and by aerobic (8.4%)and anaerobic (7.7%)microorganisms.Fractions of the nitrogen fed in feed were found in fish (15.4%)and in sludge (14.3%);the missing nitrogen was eliminated by nitrification –denitrification.Most of the phosphorus and ash fed in feed and not found in fish accumulated within the sludge in the system.The rates of nitrification,denitrification and sulphate reduction increased with time,reaching 0.3g N m −2d −1,53g N m −2d −1and 145g S m −2d −1,respectively.Nitrification developed more rapidly than denitrification,leading at first to nitrate accumulation (to 20mmol NO 3l −1by day 200)and a decrease in alkalinity.Once denitrification surpassed nitrification,nitrate concentrations decreased,eventually being reduced to b 0.3mmol NO 3l −1by day 510,and alkalinity stabilized.Toxic hydrogen sulphide,generated within the anaerobic sludge,was oxidized by oxygen and nitrate as it diffused through the anaerobic –aerobic sediment –water interface.When nitrate levels in the water above the sludge dropped below 2mmol l −1,sulphide was also oxidized in the fluidized bed reactor.Denitrification reduced nitrate in the water,respired (jointly with sulphate reduction)carbon in the sludge,oxidized the hydrogen sulphide,and contributed to stabilization of alkalinity and accumulation of polyphosphate in bacteria as a major sink of labile P.©2007Elsevier B.V .All rights reserved.Keywords:Alkalinity;Fish waste treatment;Nitrification –denitrification;Nutrients;Polyphosphate accumulation;Sludge;Sparus aurata ;Sulphate reduction1.IntroductionThe same microbial processes that occur naturally in organic-rich aerobic and anaerobic environments also occur in intensive aquaculture systems (van Rijn,1996).The observations from controlled fish culture systems provide insights into microbial processes and interactions driving the environmental situation in heavily-loaded natural ecosystems.The present study in a novel seawater fish culture system that does not require water discharge,quantified biogeochemical processes by long-term nutri-ent and alkalinity profiles and budgets.This approach has been useful to understand water quality processes andJournal of Experimental Marine Biology and Ecology 349(2007)235–247/locate/jembe⁎Corresponding author.Tel.:+97286361445;fax:+97286375761.E-mail address:aneori@ (A.Neori).0022-0981/$-see front matter ©2007Elsevier B.V .All rights reserved.doi:10.1016/j.jembe.2007.05.023problems in intensive aquaculture systems(e.g.,Krom and Neori,1989;Thoman et al.,2001).Specifically,such studies enable pinpointing of the principle biogeochem-ical processes within such systems(e.g.,Krom,1991). Several of the main processes occurring in sediments involve stoichiometric changes in alkalinity,a feature that provides additional information in the quantitative elucidation of the biogeochemical processes taking place(Lazar et al.,1989).Modern fish mariculture is increasingly criticized for its non-sustainability(Aldhous,2004).It is practiced almost exclusively in flow-through systems such as cages and ponds often in a narrow and heavily populated belt along the coast.Mariculture is subject to public controversy,since it generally discharges effluents without pollutant removal(Naylor et al.,1998).Recircu-lating aquaculture technology can overcome many of aquaculture's economic and environmental limitations, because it combines good regulation of the water quality characteristics with high fish yields,low water use and minimal nutrient export(Saylor et al.,1991;van Rijn, 1996;Gutierrez-Wing and Malone,2006).The advanced biofilter system of recirculating aqua-culture used in the present study is attractive for both freshwater fish(Shnel et al.,2002)and marine fish (Gelfand et al.,2003).The design consists of a fish basin stocked at high fish density.Water from this basin circulates through an aerobic nitrifying filter and through an anaerobic loop,with a sedimentation/digestion basin and a fluidized bed reactor.A particular novelty of the design is the use of the organic fish waste as the carbon and energy source for nitrate reduction(van Rijn,1996). This feature results in minimal environmental pollution and near zero water discharge,without addition of foreign chemicals(e.g.,methanol)to enhance microbial respira-tion.It is possible to operate this system for prolonged periods with water quality parameters remaining within the range of values acceptable for intensive fish culture (Shnel et al.,2002;Gelfand et al.,2003).An important limitation in the large-scale application of the recirculating fishculture approach is the scarcity of information on the microbial populations and processes that drive them(Blancheton,2000).Previous studies on this particular system have looked in detail at several individual microbial processes that occur in various modules of the system(e.g.,Barak et al.,2003;Gelfand et al.,2003;Cytryn et al.,2003,2005,2006).In the present study,biogeochemical processes were quantified through long-term nutrient and alkalinity profiles and budgets.It was found that the closed environment of the fish culture system allowed a fairly accurate estimation of the contribution of the main biochemical processes involved in carbon,nitrogen,sulphur and phosphorus transformations.Processes and rates were compared to those occurring in natural hypertrophic marine environments.2.Materials and methods2.1.Fish culture system2.1.1.ConfigurationThe physical setup has been described in detail (Gelfand et al.,2003;Cytryn et al.,2005).It comprised the following components(Fig.1):(1)a round10,000l polypropylene fish production basin(FT),3m diameter×1.5m depth;(2)a sedimentation basin(SB), 9.5m length×0.5m width×0.3–0.4m depth;working volume:1.5–2m3;(3)a cube-shaped trickling filter(TF), containing3.8m3of a PVC cross-flow medium with a specific surface area of240m2m−3(Jerushalmi Ltd., Israel);(4)a fluidized bed reactor(FBR),made of a Perspex column,200l volume,2m height,36cm diameter.The reactor was filled as bacterial carrier material with several kg sand(N97%SiO2)of1.1mm grain diameter(15%of grains N1.4mm and10%of grains b0.85mm).The system was designed to handle up to6kg d−1of fish feed input.Gilthead seabream(Sparus aurata)were cultured from May2000to August2001. The fish were stocked several times to replace mortalities and to increase fish density as the system matured and water qualitystabilized.Fig.1.A basic schematic of the main components(not to scale), dimensions and water flow rates of the studied facility.Solid arrows mark water flows.236 A.Neori et al./Journal of Experimental Marine Biology and Ecology349(2007)235–2472.1.2.OperationSeawater pumped from the nearby ultra-oligotrophic Gulf of Aqaba(Eilat)filled the system and compensated for water spills and dilutions of the culture water.Tap water diluted the seawater to36parts per thousands(ppt) from its N40ppt natural salinity.Tap water compensated for evaporation loss,which was intense in the summer. Two water loops constituted the water recirculation (Fig.1).In the aerobic loop,water from the top of the FT was pumped and sprinkled over the TF at a rate of15–20m3h−1and drained back to the top of the FT.In the anaerobic loop,the sediment-laden water from the bottom-center of the FT was continuously withdrawn through a standpipe into the SB.From there it was pumped at a rate of1.8–2.4m3h−1upward through the FBR and into the TF intake.A flow of about2.4m3h−1of the FTwater was diverted from the main pump's flow into a foam fractionator(model:TF8AZ,Top Fathom Ltd, USA)for clarification.The force of the water returning from the trickling filter drove a circular water velocity of between30and40cm s−1at the circumference of the FT. Ozone(Pacific Ozone Model G11ozone generator) injected into the foam fractionator disinfected all makeup water.The TF outflow water was enriched,when necessary,with pure oxygen from a liquid oxygen tank. A1hp blower drove air upward through the trickling filter for cooling.An automatic monitoring/alarm system(Point Four,Port Moody,BC,Canada)monitored water level, DO,pH and temperature.Additional measurements of DO were made with a portable Handy Gamma Meter (OxyGuard International A/S,Blokken,Denmark).The fish were fed a commercial feed(Matmor Ltd.,Evtach Israel)with8%water(105°C),44%carbon,45%protein (7.2%N),19.5%lipid,1.4%phosphorus(P),and5kcal energy g−1.Feed was offered according to a standard feeding table(Lupatsch and Kissil,1998,2001),based on the average fish density,size and average water temperature.Seabream of the size range used in this study and fed this diet contain on average317g OM, 212g C,27.2g N,7.2g P,43g ash and2100mega-cal energy per kilogram live weight(Lupatsch and Kissil, 1998,2001).These values were used in mass budgets.A fraction of feed(a total of100kg)was initially put daily into the SB to prime the anaerobic processes until a sufficient quantity of fish waste accumulated.2.2.Analytical procedures2.2.1.Solids analysisAt the end of the study,the sludge accumulated in the SB and the sand-sludge flocks in the FBR were collected and air-dried.The TF plastic medium was disassembled,air-dried and then each piece was strongly beaten over a container to release the dried sludge.For protein analysis, 50g sub samples of the residual solids from each of the biofilters was rinsed with tap water and oven-dried at 60°C for24h followed by drying at105°C to constant weights(%dw content)before TKN(total Kjeldahl N) analysis(Scheiner,1976).For total P content,the oven-dried sludge was pulverized;then500mg of the dry powder was mixed with9ml65%nitric acid and2ml HCl in120ml Teflon PFA digestion vessels.Samples and blanks were prepared for analysis by microwave-assisted digestion(10min in500W and another10min in580W of microwave radiation).Liquid residues were supple-mented with deionized water to a final volume of25ml. Analyses were conducted on portions of the solutions versus certified standards by inductively coupled plasma atomic emission spectrometry(ICP-AES,Spectro Ana-lytical Instruments GmbH&Co.,KG Boschstr.10,47533 Kleve,Germany),equipped with cross-flow nebulizers (precision:b2%;accuracy:b5%).2.2.2.Dissolved nutrients and other compoundsSeveral times a week,at08:00and14:00h,water samples for nutrient analyses were collected from the different components of the system.Nutrients were analyzed by an Auto Analyzer II(Technicon Instruments Co.,Tarrytown,New York).Total ammonia N(TAN)was analyzed by a modification of the Berthelot phenol reaction(Krom et al.,1985).Nitrate and nitrite were analyzed after Solorzano and Sharp(1980).Orthophos-phate was analyzed after Glibert and Loder(1977).Water samples for total hydrogen sulphide were sampled with minimum air exposure.They were preserved immediately with a stabilizing solution of sodium carboxymethyl cellulose and cadmium sulphate,sealed and then analyzed colorimetrically by a sulphide Auto Analyzer II cartridge (Bran+Luebbe GmbH,Werkstraβe4,22844Norder-stedt,Schleswig-Holstein,Germany),using a reaction with dimethyl-p-phenylenediamine dihydrochloride and ferric chloride to yield methylene blue(Method No.G-193-97of Bran+Luebbe,based on Grasshoff et al., 1983).The pH was measured with a HI8424pH meter (Hanna Instruments Ltd.,Bedfordshire,England)and total alkalinity(precision b2%)was determined by titration with hydrochloric acid(Parsons et al.,1984). 3.Results and discussion3.1.Feed,water use,fish growthThe fish were fed in total1158kg feed(8%moisture). An additional100kg feed was added directly to the SB in237A.Neori et al./Journal of Experimental Marine Biology and Ecology349(2007)235–247order to speed up the development of anaerobic conditions(i.e.,b1mg l−1DO),so that the total dw feed input was1157kg(Table1).Filling the tanks and making up for various water losses,mostly by evaporation(up to 0.5m3d−1in summer),consumed a total of only25m3of seawater and105m3of tap water.Specific total use of water was only0.265m3kg−1of gross fish production. Overall,699kg fish biomass was stocked and490kg biomass was produced.Over the entire512day study,the daily production rate averaged1.2kg(0.41%fish weight d−1)for an average seabream standing stock of271kg (partially reported in Gelfand et al.,2003).3.2.Carbon,nitrogen and phosphorus recovery in fish and sludgeThe fractions of the nutrients introduced into the system as feed that were recovered in fish growth were 18.3%C and15.4%N(Table1),at the low end of the range of15–30%reported from nutrient budgets of this fish(Krom and Neori,1989;Lupatsch and Kissil,1998). Higher values of C and N recovery in seabream growth have more recently been obtained in technically im-proved systems of this design(van Rijn,unpublished; Neori,unpublished).Only11%of the carbon budget and 14%of the nitrogen budget were recovered in sludge, mainly in the oxic sludge.The rest of the nutrients–over 70%of carbon and nitrogen inputs–were missing from this total budget.In contrast,nearly all added P was recovered,with21%in fish,5%in dissolved phosphate and the rest in sludge(Table1).The budgets for dry weight and for energy content paralleled those for C and N,while the ash budget paralleled the P budget.A comparison of the overall quantities of sludge production,composition and processes found in this study(Table1)with budgets reviewed by Chen et al. (1997)for several different recirculating culture systems with different fish,highlights similarities and differ-ences.The fraction of feed dw recovered in sludge in the present data(22.5%)was in the low range of the recoveries defined as“typical”in Chen et al.(1997)of 20%–50%;it is an expression of the high rate of sludge decomposition in the SB.The fraction of feed TKN recovered as sludge TKN in the present data was12.1%, compared with a recovery of9.8%that can be calculated from the data in Chen et al.(1997)for a channel catfish (Ictalurus punctatus)recirculating N content of4.2%in the accumulated sludge in the present study is similar to the mean of4.0%in Chen et al.(1997). However,while in the present system the remaining 70%N not assimilated by fish or in sludge was denitrified to N2(see below),in the single-loop culture systems reviewed in Chen et al.(1997)this N was discharged with effluents as nitrate and created a pollution hazard.An additional striking difference between the present data and other recirculating fish culture systems is in the fraction of feed P that was recovered in the sludge.In the present study nearly all P not found in fish was recovered as sludge,whose P content in dw totaled 5.9%(Table1).In other aquaculture systems,on the contrary,most of waste P was discharged as dissolved P while sludge P content averaged only0.7%(Chen et al.,1997).3.3.Dissolved nutrients dynamics and transformationsThe nutrients entered in the seawater and tap water constituted b1%of feed input for each of the nutrients, both because of the low water use and the low concentration of nutrients in both sources of water.Table1Budgets of the main ingredients of the fish feed introduced to the fish culture system in17monthsIngredient:DW C N P Ash Energy('000kcal) Budget component1.Feed inputa.(kg)11575579117.61456374b.%100100100100100100 Outputs2.In produced fish(%of line1)15.818.315.42115.216.83.In waste(%of line1)a.Anoxic sludge(SB+FBR)10 3.95 4.450.648.3 3.2b.Oxic sludge(TF)12.5 6.87.736.944.17.3c.Water0b0.1 2.2 5.1004.Total waste(%of line1)22.510.7514.392.692.410.55.Missing nutrient fractions(lines2and4subtracted from line1)a%of line161.770.9570.3−13.6−7.672.7a A negative value indicates that the system contained a quantity of the substance larger than the input.238 A.Neori et al./Journal of Experimental Marine Biology and Ecology349(2007)235–247Ammonia (analyzed as TAN),a major excretion product of the fish,accumulated in the water immedi-ately after initiation of feeding.TAN concentration peaked at nearly 200μmol l −1at day 10,and gradually decreased below 50μmol l −1(Fig.2).It remained low until day 370,except for a peak of ∼2700μmol l −1,which developed on day 78following a partial drying of the TF due to a burnout of the main water pump.After day 370,the ammonia concentration circulating through the system increased to between 50and 250μmol l −1.Nitrate concentration reached 20mmol l −1by day 200(Fig.2).After day 270,it gradually dropped,to b 5mmol l −1by day 370and b 0.3mmol l −1at the end of the study.Based on our data,nitrification and denitrification were the dominant microbial N transformations within the system.Nitrification developed shortly after start up of the system.A dynamic balance between ammonia production and nitrification kept ammonia levels at this time by and large below 50μmol l − nitrification,calculated by the difference between expected quantities of ammonia (5%of total feed input —Lupatsch and Kissil,1998,2001)and measured ammonia in the watermultiplied by water volume,matched the increase in feeding.It increased from 5.9mol N d −1(1–270days)to 10.0mol N d −1(270–370days)and to 16.5mol N d −1for the last 130days (Table 2).The overall average nitrification rate for the entire period was 9mol N d −1.It is likely that nitrification occurred mostly in the TF and on other wet surfaces,including the top of the SB sludge and in the FBR,where ammonia produced by anaerobic respiration processes could become exposedtoFig. 2.Total ammonia N (TAN),nitrate and orthophosphate concentrations in the fish basin during the study.Table 2Total changes in titration alkalinity and associated parameters in the system over the following three periods:Period 1)from day 1to day 270(270days)when the system operated in net nitrification mode;Period 2)from day 270to day 370(100days)when the system operated in net denitrification mode;Period 3)from day 370to day 500(130days)when the system operated in net denitrification mode with leakage of ammonia and hydrogen sulphide in the anaerobic loop Measured parameters Period 1Period 2Period 31.Alkalinity in seawater inflow (equivalents)6715.5572.Alkalinity in tap water inflow (equivalents)6210.6553.Alkalinity in water outflow (equivalents)−59−8.7504.Alkalinity added as bicarbonate (equivalents)321005.Measured net change in alkalinity (equivalents)5.28.5−9.86.Dissolved N excreted by fish asammonia and reactive DON (moles)142883219787.Dissolved N released from labile organic matter in SB (moles)175172165.58.Nitrate accumulation in FT water,Fig.2(moles)263.5−99−61.53Calculated parameters (see text)9.Total nitrification (TN)(moles)(Eq.(6))16031004214310.Total denitrification (TDN)(moles)(Eq.(8))13401103220511.Presumed net alkalinity production,sum of lines 1–5(equivalents)3871012212Calculated surplus of alkalinityproduced,line 11,but not matched by the accumulation of nitrate,line 8(equivalents)(Eq.(11))123109183Calculated daily rates of microbial processes 13.Total nitrification rate (mol d −1) 5.910.016.514.Total denitrification rate (mol d −1) 5.011.017.015.Alkalinity surplus (sulphate reduction)rate (equivalents d −1)0.461.1 1.4All units are equivalents (for alkalinity)or moles (for dissolved nitrogen).239A.Neori et al./Journal of Experimental Marine Biology and Ecology 349(2007)235–247dissolved oxygen in the overlying water(Blackburn, 1986).The average and maximum nitrification rates divided by the total area of the TF and other wet surfaces were0.126and0.3g N m−2d−1,respectively.Values of up to0.28g N m−2d−1have been reported for other marine nitrifying trickling filters(Nijhof and Boven-deur,1990;Eding et al.,2006).Denitrification devel-oped more slowly than nitrification(Tables2,3),so that a shift from net accumulation to net consumption of nitrate occurred only after day270.This was twice the time it took in freshwater systems of similar design (Shnell et al.,2002;Gelfand et al.,2003).Dissolved orthophosphate is produced by fish and microbial respiration.Orthophosphate concentrations in the water(Fig.2)gradually increased to a value of2.5mmol l−1by day300.Then,during the period of intense denitrification,dissolved phosphate levels decreased to1.2mmol l−1by day370.After nitrate dropped below5mmol l−1and begun to rise again, phosphate concentration also rose,to about3mmol l−1 by day410(Fig.2).There was no detectable hydrogen sulphide circulat-ing in the water until about day370(data not shown). Then as nitrate levels dropped below2mmol l−1, hydrogen sulphide gradually appeared in the water overlying the sludge in the SB.Concentrations of up to 500μmol l−1were measured there at the end of the study(Fig.3).Microbial processes in the FBR con-sumed most of the hydrogen sulphide that escaped the SB before the water returned to the main water loop. Hydrogen sulphide that escaped the FBR disappeared in the trickling filter,by either evaporation or oxidation, leading to levels considered safe for fish in Bagarinao and Lantin-Olaguer(1998)of b5μmol l−1in the fish basin water(Fig.3).The rate of hydrogen sulphide release to the SB water was inversely related to nitrate concentration(Fig.4). Sulphide release started at∼350mmol S d−1at nitrate concentrations below2mmol l−1increasing to18mol S d−1when nitrate concentration dropped below0.2mmol l−1(Fig.4).Nitrate concentrations above2mmol l−1have been shown to totally inhibit sulphate reduction in organically-rich aquatic sediments(Lucassen et al., 2004).The removal of sulphide and nitrate together in the FBR suggests that a principle process for sulphide oxidation was autotrophic denitrification(Cytryn et al., 2005,2006).Oxidation of hydrogen sulphide in the FBR was proportional to its concentration in the water leaving the SB(Fig.5),reaching a rate of over100mmol S l−1 d−1.Considering the extreme toxicity to fish of hydrogen sulphide(96h LC50=60μmol S l−1in Bagarinao and Lantin-Olaguer,1998),it can be calculated that without itsTable3Net rates of denitrification in the system,assuming it took place in the SB,based on maximal rates of nitrate disappearance from the water during the indicated periods(negative linear slopes in Fig.2nitrate curve)Denitrification rates Period:June–July2000July–August2000January–March2001Totalg N d−180187249mol N d−1 5.713.417.8Areal in SB ag N m−2d−1173953mol N m−2d−1 1.2 2.8 3.8 Volumetric in SB amg N l−1d−146106143m mol N l−1d−1 3.37.610.2a Per surface area and volume(4.75m2and1750l,respectively)of theSB.Fig.3.Hydrogen sulphide concentration in the water exiting the fish basin,sedimentation basin(SB)and FBR during the final months of thestudy.Fig.4.Concentration of hydrogen sulphide(open triangles,right Y axis)and rate of production(closed circles,left Y axis)in the SB as a function of nitrate concentration in the system water.240 A.Neori et al./Journal of Experimental Marine Biology and Ecology349(2007)235–247oxidation hydrogen sulphide concentration in the FT could have reached toxic levels in 1–2h.Significant sulphate reduction to sulphide apparently occurred in the lower layers of the SB all the time.Sulphate reduction estimates based on alkalinity surplus (explained below)increased with time in parallel with the increase in denitrification (Table 2).Hydrogen sulphide levels of up to 6mmol l −1were determined in the pore waters of the bottom layers of the SB anoxic sludge in a very similar system (Cytryn et al.,2003).However,during the first 370days of the present study,no sulphide was detected in the overlying water of the SB.This was probably due to oxidation in the boundary layer between sludge and water,where white patches of elemental S were often visible.Reduced solutes –sulphides and ammonia –escaped into the overlying water only when limits were reached with regard to the oxidative capacity of the boundary layer.Microbial redox recycling processes are characteristic of the aerobic/anaerobic boundary in organic marine sedi-ments and are particularly important in the oxidation of sulphide (Blackburn,1986).Typically,more than 90%of the sulphide produced in marine sediments is reoxidized in this boundary layer (Jorgensen,1980;Canfield et al.,1993).Oxidation by dissolved oxygen is generally considered the most important oxidative process occurring in marine sediments.However,here the artificial conditions created very high-nitrate con-centrations in the SB sludge's overlying water,a situation that is rare in nature.In natural marine sediments,autotrophic denitrification has been reported at nitrate concentration of only 0.025mmol l −1.Even at those low concentrations,the process was considered of quantitative importance for the C,N and S cycles in thatsediment (Fossing et al.,1995).With nitrate concentra-tions orders of magnitude higher,it is likely that oxidation by nitrate was the most significant process for the oxidation of sulphide in the present system.3.4.Alkalinity and the biogeochemical processes 3.4.1.Alkalinity dynamicsAlkalinity of the initial seawater was 2.3mEq l −1while the alkalinity of the tap water was 1.2mEq l −1.Alkalinity values for most of the first 3months of the study are missing.However,during this time only 72kg feed was introduced to the system out of a total of 1258kg during the entire 17months,and,therefore,the error introduced to the quantitative analyses of the processes should be minor.Initially,when nitrate accumulated in the system,pH dropped below 6.0while also alkalinity steeply declined,reaching a low point after 3months (Fig.6).Brief but sharp rises in alkalinity and pH (with a simultaneous sharp dropinFig.5.Relationship between concentration of hydrogen sulphide in the water leaving the SB into the FBR and the rate of change in hydrogen sulphide (negative units)in the FBR;data,solid circles;regression,line (Y =−29X −376;r 2=0.912;p b0.0001).Fig.6.Alkalinity in water exiting the trickling filter and the SB,and pH in the fish basin during the study.241A.Neori et al./Journal of Experimental Marine Biology and Ecology 349(2007)235–247nitrate concentration)followed the temporary removalof the fish between days 99and 128.Sodium bicarbonate buffer (totaling 27kg,321equivalents)was added occasionally between days 74to 99and days 194to 293to restore alkalinity.During the initial 3months period,once measured,alkalinity levels in the various modules of the system were low and did not differ significantly between the modules (Figs.6,7).Between day 200and 270,when denitrification matched and then surpassed nitrification (Table 2),alkalinity rose to 3mEq l −1and pH stabilized (Fig.6).After day 270,as a result of the intense denitrification,alkalinity gradually increased to 4–5mEq l −1.Typically,during this latter period there was an increase in alkalinity of 0.2–0.3mEq l −1in the water as it passed through the anaerobic loop (Fig.7),indicating large net alkalinity generation by anaerobic processes.After day 370,alkalinity generation in the SB was balanced by consumption elsewhere in the system and the overall alkalinity stabilized at close to 3mEq l −1without any further addition of bicarbonate buffer.From day 200until the end of the experiment,the pH of the entire fish culture system stabilized at just above 7with minor fluctuations (Fig.6).3.4.2.Alkalinity budgets and contingent biogeochem-ical processesAlkalinity budgets (Table 2)were calculated over the following three biogeochemically distinct periods:(1)period 1(first 270days)net nitrification (accumu-lation of nitrate);(2)period 2(from day 270to 370)net denitrification (nitrate levels dropped sharply);and (3)period 3(day 370–500)net denitrification accom-panied by leakage of ammonia and hydrogen sulphidefrom the SB sludge into the overlying SB water.The budgets were calculated using the following assumptions and equations (derived from Lazar et al.,1989;van Rijn et al.,2006).During microbial nitrification of ammonia 1equivalent (Eq)of alkalinity is consumed per mol of ammonia oxidized (Eq.(1)below).During denitrifica-tion 1Eq of alkalinity is produced per mol of nitrate reduced to nitrogen gas,whether the process involved is heterotrophic denitrification (Eq.(2))or autotrophic denitrification with H 2S (Eq.(3))coupled to sulphate reduction to produce H 2S (Eq.(4)).The rearrangement of Eqs.(3)and (4)in Eq.(5)assumes that dissimilatory reduction of nitrate to ammonia is not significant.Nitrification:NH 3þ2O 2¼NO −3þH þþH 2Oð1ÞHeterotrophic denitrification2NO −3þ½5H 2 þ2H þ¼N 2þ6H 2Oð2ÞAutotrophic denitrification with H 2S5H 2S þ8NO −3→5SO 2−4þ4N 2þ4H 2O þ2Hþð3ÞSulphate reduction to produce H 2S5SO 2−4þ½20H 2 þ10H þ→5H 2S þ20H 2Oð4ÞA rearrangement of Eqs.(3)and (4)8NO −3þ½20H 2 þ8H þ→4N 2þ24H 2Oð5ÞFor each of the three defined time periods,with insignificant amounts of ammonia accumulating in the system,total nitrification (TN)is the sum of the ammonia produced from fish excretion (N fish )and from decompo-sition of organic matter in the SB (N SB ;Eq.(6))TN ¼N fish þN SBð6ÞTotal net nitrification (TNN)is total nitrification (TN)minus total denitrification (TDN)and equals the amount of nitrate accumulated during the period considered.TNN ¼TN −TDN ¼▵NO 3ð7ÞRearranging Eq.(7)TDN ¼TN −▵NO 3ð8ÞAssuming that the only microbial processes that result in a change of alkalinity are nitrification and denitrification,then the change in alkalinity (▵Alk)Fig.7.The differences in water alkalinity between the trickling filter and fish basin and between the sedimentation basin and the fish basin during the study.242 A.Neori et al./Journal of Experimental Marine Biology and Ecology 349(2007)235–247。

软件编码规范中国人民银行清算总中心支付系统开发中心注：变化状态：A—增加，M—修改，D—删除目录第一篇C/C++编码规范 (6)第一章代码组织 (6)第二章命名 (8)2.1文件命名 (8)2.2变量命名 (8)2.3常量与宏命名 (9)2.4类命名 (9)2.5函数命名 (9)2.6参数命名 (10)第三章注释 (11)3.1文档化注释 (11)3.2语句块注释 (16)3.3代码维护注释 (19)第四章编码风格 (21)4.1排版风格 (21)4.2头文件 (25)4.3宏定义 (26)4.4变量与常量 (29)4.5条件判断 (31)4.6空间申请与释放 (32)4.7函数编写 (32)4.8类的编写 (35)4.9异常处理 (38)4.10特殊限制 (38)第五章编译 (40)第六章ESQL/C编码 (45)第二篇JAVA编码规范 (46)第一章代码组织 (47)第二章命名 (50)2.1包命名 (50)2.2类命名 (50)2.3接口命名 (50)2.4方法命名 (50)2.5变量命名 (50)2.6类变量命名 (50)2.7常量命名 (51)2.8参数命名 (51)第三章注释 (52)3.1文档化注释 (52)3.2语句块注释 (56)3.3代码维护注释 (57)第四章编码风格 (59)4.1排版风格 (59)4.2包与类引用 (64)4.3变量与常量 (64)4.4类编写 (65)4.5方法编写 (66)4.6异常处理 (69)4.7特殊限制 (69)第五章编译 (71)第六章JSP编码 (72)6.1文件命名及存放位置 (72)6.2内容组织 (72)6.3编码风格 (73)6.4注释 (76)6.5缩进与对齐 (76)6.6表达式 (77)6.7JavaScript (77)第三篇POWERBUILDER编码规范 (78)第一章代码组织 (79)第二章命名 (80)2.1文件命名 (80)2.2对象命名 (80)2.3变量命名 (82)2.4常量命名 (83)2.5函数与事件命名 (83)2.6参数命名 (83)第三章注释 (83)3.1文档化注释 (83)3.2语句块注释 (85)3.3代码维护注释 (86)第四章编码风格 (87)4.1界面风格 (87)4.2排版风格 (90)4.3变量与常量 (93)4.4条件判断 (93)4.5空间申请与释放 (94)4.6函数编写 (94)4.7特殊限制 (94)第五章SQL编码 (95)前言程序编码是一种艺术，既灵活又严谨，充满了创造性与奇思妙想。

PUBLICATIONS DE L’INSTITUT MATH´EMATIQUENouvelle s´e rie,tome80(94)(2006),171–192DOI:10.2298/PIM0694171HREGULARLY V ARYING FUNCTIONSAnders Hedegaard Jessen and Thomas MikoschIn memoriam Tatjana Ostrogorski.Abstract.We consider some elementary functions of the components of aregularly varying random vector such as linear combinations,products,min-ima,maxima,order statistics,powers.We give conditions under which thesefunctions are again regularly varying,possibly with a different index.1.IntroductionRegular variation is one of the basic concepts which appears in a natural way in different contexts of applied probability theory.Feller’s[21]monograph has certainly contributed to the propagation of regular variation in the context of limit theory for sums of iid random variables.Resnick[50,51,52]popularized the notion of multivariate regular variation for multivariate extreme value theory.Bingham et al.[3]is an encyclopedia where onefinds many analytical results related to one-dimensional regular variation.Kesten[28]and Goldie[22]studied regular variation of the stationary solution to a stochastic recurrence equation.These resultsfind natural applications infinancial time series analysis,see Basrak et al.[2]and Mikosch[39].Recently,regular variation has become one of the key notions for modelling the behavior of large telecommunications networks,see e.g.Leland et al.[35],Heath et al.[23],Mikosch et al.[40].It is the aim of this paper to review some known results on basic functions acting on regularly varying random variables and random vectors such as sums, products,linear combinations,maxima and minima,and powers.These results are often useful in applications related to time series analysis,risk management, insurance and telecommunications.Most of the results belong to the folklore but they are often wide spread over the literature and not always easily accessible.We will give references whenever we are aware of a proved result and give proofs if this is not the case.2000Mathematics Subject Classification:Primary60G70;Secondary62P05.Mikosch’s research is also partially supported by the Danish Research Council(SNF)Grant 21-04-0400.Jessen’s research is partly supported by a grant from CODAN Insurance.171172JESSEN AND MIKOSCHWe focus on functions offinitely many regularly varying random variables. With a few exceptions(the tail of the marginal distribution of a linear process, functionals with a random index)we will not consider results where an increasing or an infinite number of such random variables or vectors is involved.We exclude distributional limit results e.g.for partial sums and maxima of iid and strictly stationary sequences,tail probabilities of subadditive functionals acting on a regu-larly varying random walk(e.g.ruin probabilities)and heavy-tailed large deviation results,tails of solutions to stochastic recurrence equations.We start by introducing the notion of a multivariate regularly varying vector in Section2.Then we will consider sum-type functionals of regularly varying vectors in Section3.Functionals of product-type are considered in Section4.In Section5 wefinally study order statistics and powers.2.Regularly varying random vectorsIn what follows,we will often need the notion of a regularly varying random vector and its properties;we refer to Resnick[50]and[51,Section5.4.2].This notion was further developed by Tatjana Ostrogorski in a series of papers,see [42,43,44,45,46,47].Definition2.1.An R d-valued random vector X and its distribution are said to be regularly varying with limiting non-null Radon measureµon the Borelσ-field B(R d0)of R d0=R d {0}if(2.1)P(x−1X∈·)P(|X|>x)v→µ,x→∞.Here|·|is any norm in R d and v→refers to vague convergence on B(R d0).Sinceµnecessarily has the propertyµ(t A)=t−αµ(A),t>0,for someα>0 and all Borel sets A in R d0,we say that X is regularly varying with indexαand limiting measureµ,for short X∈RV(α,µ).If the limit measureµis irrelevant we also write X∈RV(α).Relation(2.1)is often used in different equivalent disguises. It is equivalent to the sequential definition of regular variation:there exist c n→∞such that n P(c−1n X∈·)v→µ.One can always choose(c n)increasing and suchthat n P(|X|>c n)∼1.Another aspect of regular variation can be seen if one switches in(2.1)to a polar coordinate representation.Writing x=x/|x|for any x=0and S d−1={x∈R d:|x|=1}for the unit sphere in R d,relation(2.1)is equivalent to(2.2)P(|X|>x t, X∈·)P(|X|>x)w→t−αP(Θ∈·)for all t>0,whereΘis a random vector assuming values in S d−1and w→refers to weak conver-gence on the Borelσ-field of S d−1.Plugging the set S d−1into(2.2),it is straightforward that the norm|X|is regularly varying with indexα.REGULARLY V ARYING FUNCTIONS173 The special case d=1refers to a regularly varying random variable X with indexα 0:(2.3)P(X>x)∼p x−αL(x)and P(X −x)∼q x−αL(x),p+q=1, where L is a slowly varying function,i.e.,L(cx)/L(x)→1as x→∞for every c>0.Condition(2.3)is also referred to as a tail balance condition.The cases p=0or q=0are not excluded.Here and in what follows we write f(x)∼g(x)as x→∞if f(x)/g(x)→1or,if g(x)=0,we interpret this asymptotic relation as f(x)=o(1).3.Sum-type functions3.1.Partial sums of random variables.Consider regularly varying ran-dom variables X1,X2,...,possibly with different indices.We writeS n=X1+···+X n,n 1,for the partial sums.In what follows,we write G=1−G for the right tail of a distribution function G on R.Lemma3.1.Assume|X1|is regularly varying with indexα 0and distribution function F.Assume X1,...,X n are random variables satisfying(3.1)limx→∞P(X i>x)F(x)=c+iand limx→∞P(X i −x)F(x)=c−i,i=1,...,n,for some non-negative numbers c±iandlim x→∞P(X i>x,X j>x)F(x)=limx→∞P(X i −x,X j>x)F(x)=limx→∞P(X i −x,X j −x)F(x)=0,i=j.(3.2) Thenlim x→∞P(S n>x)F(x)=c+1+···+c+n and limx→∞P(S n −x)F(x)=c−1+···+c−n.In particular,if the X i’s are independent non-negative regularly varying random variables then(3.3)P(S n>x)∼P(X1>x)+···+P(X n>x).The proof of(3.3)can be found in Feller[21,p.278],cf.Embrechts et al. [18,Lemma1.3.1].The general case of possibly dependent non-negative X i’s was proved in Davis and Resnick[14,Lemma2.1];the extension to general X i’s follows along the lines of the proof in[14].Generalizations to the multivariate case are given in Section3.6below.The conditions in Lemma3.1are sharp in the sense that they cannot be sub-stantially improved.A condition like(3.1)with not all c±i ’s vanishing is neededin order to ensure that at least one summand X i is regularly varying.Condition (3.2)is a so-called asymptotic independence condition.It cannot be avoided as the174JESSEN AND MIKOSCHtrivial example X2=−X1for a regularly varying X1shows.Then(3.1)holds but(3.2)does not and S2=0a.s.A partial converse follows from Embrechts et al.[17].Lemma3.2.Assume S n=X1+···+X n is regularly varying with indexα 0 and X i are iid non-negative.Then the X i’s are regularly varying with indexαand (3.4)P(S n>x)∼n P(X1>x),n 1.Relation(3.4)can be taken as the definition of a subexponential distribution. The class of those distributions is larger than the class of regularly varying distri-butions,see Embrechts et al.[18,Sections1.3,1.4and Appendix A3].Lemma3.2 remains valid for subexponential distributions in the sense that subexponentiality of S n implies subexponentiality of X1.This property is referred to as convolution root closure of subexponential distributions.Proof.Since S n is regularly varying it is subexponential.Then the regular variation of X i follows from the convolution root closure of subexponential dis-tributions,see Proposition A3.18in Embrechts et al.[18].Relation(3.4)is a consequence of(3.3).An alternative proof is presented in the proof of Proposition4.8in Fa¨y et al.[20].It strongly depends on the regular variation of the X i’s:Karamata’s Tauberian theorem(see Feller[21,XIII,Section5])is used.In general,one cannot conclude from regular variation of X+Y for independent X and Y that X and Y are regularly varying.For example,if X+Y has a Cauchy distribution,in particular X+Y∈RV(1),then X can be chosen Poisson,see Theorem6.3.1on p.71in Lukacs[37].It follows from Lemma3.12below that Y∈RV(1).3.2.Weighted sums of iid regularly varying random variables.We assume that(Z i)is an iid sequence of regularly varying random variables with indexα 0and tail balance condition(2.3)(applied to X=Z i).Then it follows from Lemma3.1that for any real constantsψiP(ψ1Z1+···+ψm Z m>x)∼P(ψ1Z1>x)+···+P(ψm Z1>x).Then evaluating P(ψi Z1>x)=P(ψ+i Z+i>x)+P(ψ−iZ−i>x),where x±=0∨(±x)we conclude the following result which can be found in various books,e.g. Embrechts et al.[18,Lemma A3.26].Lemma3.3.Let(Z i)be an iid sequence of regularly varying random variables satisfying the tail balance condition(2.3).Then for any real constantsψi and m 1,(3.5)P(ψ1Z1+···+ψm Z m>x)∼P(|Z1|>x)mi=1p(ψ+i)α+q(ψ−i)α.The converse of Lemma3.3is in general incorrect,i.e.,regular variation of ψ1Z1+···+ψm Z m with indexα>0for an iid sequence(Z i)does in generalREGULARLY V ARYING FUNCTIONS 175not imply regular variation of Z 1,an exception being the case m =2with ψi >0,Z i 0a.s.,i =1,2,cf.Jacobsen et al.[27].3.3.Infinite series of weighted iid regularly varying random vari-ables.The question about the tail behavior of an infinite series(3.6)X =∞i =0ψj Z jfor an iid sequence (Z i )of regularly varying random variables with index α>0and real weights occurs for example in the context of extreme value theory for linear processes,including ARMA and FARIMA processes,see Davis and Resnick[11,12,13],Kl¨u ppelberg and Mikosch [29,30,31],cf.Brockwell and Davis[5,Section 13.3],Resnick [51,Section 4.5],Embrechts et al.[18,Section 5.5and Chapter 7].The problem about the regular variation of X is only reasonably posed if the infinite series (3.6)converges a.s.Necessary and sufficient conditions are given by Kolmogorov’s 3-series theorem,cf.Petrov [48,49].For example,if α>2(then var(Z i )<∞),the conditions E (Z 1)=0and i ψ2i <∞are necessary and sufficient for the a.s.convergence of X .The following conditions from Mikosch and Samorodnitsky [41]are best pos-sible in the sense that the conditions on (ψi )coincide with or are close to the con-ditions in the 3-series theorem.Similar results,partly under stronger conditions,can be found in Lemma 4.24of Resnick [51]for α 1(attributed to Cline [7,8]),Theorem 2.2in Kokoszka and Taqqu [32]for α∈(1,2).Lemma 3.4.Let (Z i )be an iid sequence of regularly varying random variables with index α>0which satisfy the tail balance condition (2.3).Let (ψi )be a sequence of real weights.Assume that one of the following conditions holds:(1)α>2,E (Z 1)=0and ∞i =0ψ2i <∞.(2)α∈(1,2],E (Z 1)=0and ∞i =0|ψi |α−ε<∞for some ε>0.(3)α∈(0,1]and ∞i =0|ψi |α−ε<∞for some ε>0.Then(3.7)P (X >x )∼P (|Z 1|>x )∞ i =0p (ψ+i )α+q (ψ−i )α .The conditions on (ψj )in the case α∈(0,2]can be slightly relaxed if one knows more about the slowly varying L .In this case the following result from Mikosch and Samorodnitsky [41]holds.Lemma 3.5.Let (Z i )be an iid sequence of regularly varying random variables with indexα∈(0,2]which satisfy the tail balance condition (2.3).Assume that ∞i =1|ψi |α<∞,that the infinite series (3.6)converges a.s.and that one of thefollowing conditions holds:(1)There exist constants c,x 0>0such that L (x 2) c L (x 1)for all x 0<x 1<x 2.176JESSEN AND MIKOSCH(2)There exist constants c,x 0>0such that L (x 1x 2) c L (x 1)L (x 2)for allx 1,x 2 x 0>0Then (3.7)holds.Condition (2)holds for Pareto-like tails P (Z 1>x )∼c x −α,in particular for α-stable random variables Z i and for student distributed Z i ’s with αdegrees of freedom.It is also satisfied for L (x )=(log k x )β,any real β,where log k is the k th time iterated logarithm.Classical time series analysis deals with the strictly stationary linear processesX n =∞i =0ψi Z n −i ,n ∈Z ,where (Z i )is an iid white noise sequence,cf.Brockwell and Davis [5].In the case of regularly varying Z i ’s with α>2,var(Z 1)and var(X 1)are finite and there-fore it makes sense to define the autocovariance function γX (h )=cov(X 0,X h )=var(Z 1) i ψi ψi +|h |,h ∈Z .The condition i ψ2i <∞(which is necessary for the a.s.convergence of X n )does not only capture short range dependent sequences(such as ARMA processes for which γX (h )decays exponentially fast to zero)but also long range dependent sequences (X n )in the sense that h |γX (h )|=∞.Thus Lemma 3.4also covers long range dependent sequences.The latter class in-cludes fractional ARIMA processes;cf.Brockwell and Davis [5,Section 13.2],and Samorodnitsky and Taqqu [56].Notice that (3.7)is the direct analog of (3.5)for the truncated series.The proof of (3.7)is based on (3.5)and the fact that the remainder term ∞i =m +1ψi Z i is negligible compared to P (|Z 1|>x )when first letting x →∞and then m →∞.More generally,the following result holds:Lemma 3.6.Let A be a random variable and let Z be positive regularly varying random variable with index α 0.Assume that for every m 0there exist finite positive constants c m >0,random variables A m and B m such that the representa-tion A d =A m +B m holds and the following three conditions are satisfied:P (A m >x )∼c m P (Z >x ),as x →∞,c m →c 0,as m →∞,lim m →∞lim sup x →∞P (B m >x )P (Z >x )=0and A m ,B m are independent for every m 1or lim m →∞lim sup x →∞P (|B m |>x )P (Z >x )=0.Then P (A >x )∼c 0P (Z >x ).Proof.For every m 1and ε∈(0,1).P (A >x ) P (A m >x (1−ε))+P (B m >εx ).REGULARLY V ARYING FUNCTIONS 177Hencelim sup x →∞P (A >x )P (Z >x ) lim sup x →∞P (A m >x (1−ε))P (Z >x )+lim sup x →∞P (B m >εx )P (Z >x )=c m (1−ε)−α+ε−αlim sup x →∞P (B m >εx )P (Z >εx )→c 0(1−ε)−αas m →∞→c 0as ε↓0.Similarly,for independent A m and B m ,lim inf x →∞P (A >x )P (Z >x ) lim inf x →∞P (A m >x (1+ε),B m −εx )P (Z >x )=lim inf x →∞P (A m >x (1+ε))P (Z >x )P (B m −εx )=c m (1+ε)−α→c 0,as m →∞,ε↓0.If A m and B m are not necessarily independent a similar bound yieldslim inf x →∞P (A >x )P (Z >x ) lim inf x →∞P (A m >x (1+ε),|B m | εx )P (Z >x )lim inf x →∞P (A m >x (1+ε))P (Z >x )−lim sup x →∞P (|B m |>εx )P (Z >x )=c m (1+ε)−α→c 0,as m →∞,ε↓0.Combining the upper and lower bounds,we arrive at the desired result.We also mention that Resnick and Willekens [53]study the tails of the infinite series i A i Z i ,where (A i )is an iid sequence of random matrices,independent of the iid sequence (Z i )of regularly varying vectors Z i .3.4.Random sums.We consider an iid sequence (X i )of non-negative ran-dom variables,independent of the integer-valued non-negative random variable K .Depending on the distributional tails of K and X 1,one gets rather different tail behavior for the random sum S K = K i =1X i .The following results are taken from Fa¨y et al.[20].Lemma 3.7.(1)Assume X 1is regularly varying with index α>0,EK <∞and P (K >x )=o (P (X 1>x )).Then,as x →∞,(3.8)P (S K >x )∼EK P (X 1>x ).(2)Assume K is regularly varying with index β 0.If β=1,assume that EK <∞.Moreover,let (X i )be an iid sequence such that E (X 1)<∞and P (X 1>x )=o (P (K >x )).Then,as x →∞,(3.9)P (S K >x )∼P (K >(E (X 1))−1x )∼(E (X 1))βP (K >x ).(3)Assume S K is regularly varying with index α>0and E (K 1∨(α+δ))<∞for some positive δ.Then X 1is regularly varying with index αand P (S K >x )∼EK P (X 1>x ).178JESSEN AND MIKOSCH(4)Assume S K is regularly varying with indexα>0.Suppose that E(X1)<∞and P(X1>x)=o(P(S K>x))as x→∞.In the caseα=1and E(S K)=∞, assume that x P(X1>x)=o(P(S K>x))as x→∞.Then K is regularly varying with indexαandP(S K>x)∼(E(X1))αP(K>x).(5)Assume P(K>x)∼c P(X1>x)for some c>0,that X1is regularly varying with indexα 1and E(X1)<∞.ThenP(S K>x)∼(EK+c(E(X1))α)P(X1>x).Relations(3)and(4)are the partial converses of the corresponding relations (1)and(2).The law of large numbers stands behind the form of relation(3.9), whereas relation(3.8)is expected from the results in Section3.1.Relations of type(3.8)appear in a natural way in risk and queuing theory when the summands X i are subexponential and K has a moment generating function in some neighborhood of the origin,see for example the proof of the Cram´e r-Lundberg ruin bound in Section1.4of Embrechts et al.[18].Forα∈(0,2)some of the results in Lemma3.7can already be found in Resnick[50]and even in the earlier papers by Stam[57],Embrechts and Omey [19].The restriction toα<2is due to the fact that some of the proofs depend on the equivalence between regular variation and membership in the domain of attraction of infinite variance stable distributions.Resnick[50]also extends some of his results to the case when K is a stopping time.In the following example the assumptions of Lemma3.7are not necessarily satisfied.Assume(X i)is a sequence of iid positiveα-stable random variables for someα<1.Then S K d=K1/αX1and P(X1>x)∼c x−αfor some c>0;cf. Feller[21]or Samorodnitsky and Taqqu[56].If EK<∞then Breiman’s result (see Lemma4.2below)yields P(S K>x)∼EKP(X>x)in agreement with(3.8). If EK=∞we have to consider different possibilities.If K is regularly varying with index1,then K1/α∈RV(α).Then we are in the situation of Lemma4.2 below and S K is regularly varying with indexα.If we assume that K∈RV(β)for someβ<1,then K1/α∈RV(βα)and the results of Lemma4.2ensure that P(S K>x)∼E(Xαβ)P(K1/α>x).The latter result can be extended by using a Tauberian argument.Lemma3.8.Assume that K,X1>0are regularly varying with indicesβ∈[0,1) andα∈[0,1),respectively.ThenP(S K>x)∼P(K>[P(X>x)]−1)∼P(M K>x),where M n=max i=1,...,n X i.Proof.By Karamata’s Tauberian theorem(see Feller[21,XIII,Section5]) 1−E(e−sK)∼sβL K(1/s)as s↓0provided that P(K>x)=x−βL K(x)for some slowly varying function L.In the same way,1−E(e−tX1)∼tαL X(1/t)as t↓0.REGULARLY V ARYING FUNCTIONS179Then1−E(e−t S K)=1−EexpK logEe−t X1∼−logEe−t X1βL K(1/[−logEe−t X1])∼1−Ee−t X1βL K1/1−Ee−t X1∼[tαL X(1/t)]βL K([tαL X(1/t)]−1) =tαβL(1/t),where L(x)=LβX (x)L K(xα/L X(x))is a slowly varying function.Again by Kara-mata’s Tauberian theorem,P(S K>x)∼x−αβL(x).Notice that the right-hand side is equivalent to the tail P(K>[P(X1>x)]−1)∼P(M K>x).The latter equivalence follows from(5.1)below.3.5.Linear combinations of a regularly varying random vector.As-sume X∈RV(α,µ)and let c∈R d,c=0,be a constant.The set A c={x: c x>1}is bounded away from zero andµ(∂A c)=0.Indeed,it follows from the scaling property ofµthatµ({x:c x=y})=y−αµ({x:c x=1}),y>0.If µ({x:c x=1})>0this would contradict thefiniteness ofµ(A c).Therefore,from(2.1),P(x−1X∈A c) P(|X|>x)=P(c X>x)P(|X|>x)→µ(A c).We conclude the following,see also Resnick[52],Section7.3.Lemma3.9.For c∈R,c=0,c X is regularly varying with indexαifµ(A c)= 0.In general,P(c X>x)P(|X|>x)→µ({x:c x>1}),where the right-hand side possibly vanishes.In particular,ifµ({x:c i x>1})>0 for the basis vector c i=(0,...,0,1,0,...,0) with1in the i th component then (X i)+is regularly varying with indexα.A natural question arises:given that(3.10)P(c X>x)L(x)x−α=C(c)for all c=0and C(c)=0for at least one cholds for some function C,is then X regularly varying in the sense of Definition2.1? This would yield a Cram´e r–Wold device analog for regularly varying random vec-tors.The answer to this question is not obvious.Here are some partial answers.The first three statements can be found in Basrak et al.[1],the last statements are due to Hult and Lindskog[26].Statement(5)was already mentioned(without proof) in Kesten[28].Lemma3.10.(1)(3.10)implies that X is regularly varying with a unique spec-tral measure ifαis not an integer.180JESSEN AND MIKOSCH(2)(3.10)when restricted to c∈[0,∞)d {0}implies that X is regularly varying with a unique spectral measure if X has non-negative components andαis positive and not an integer,(3)(3.10)implies that X is regularly varying with a unique spectral measure if X has non-negative components andαis an odd integer.(4)(1)and(2)cannot be extended to integerαwithout additional assumptions on the distribution of X.There exist regularly varying X1and X2both satisfying (3.10)with the same function C but having different spectral measures.(5)For integerα>0,there exist non-regularly varying X satisfying(3.10).3.6.Multivariate extensions.In this section we assume that X1and X2are random vectors with values in R d.The following result due to Hult and Lindskog [24],see also Resnick[52,Section7.3],yields an extension of Lemma3.1to regularly varying vectors.Lemma3.11.Assume that X1and X2are independent regularly varying such that n P(c−1n X i∈)v→µi,i=1,2,for some sequence c n→∞and Radon measures µi,i=1,2.Then X1+X2is regularly varying and n P(c−1n(X1+X2)∈·)v→µ1+µ2.The following lemma is often useful.Lemma3.12.Assume X1∈RV(α,µ)and P(|X2|>x)=o(P(|X1|>x))as x→∞.Then X1+X2∈RV(α,µ).Proof.It suffices to show that(3.11)P(x−1(X1+X2)∈A)∼P(x−1X1∈A),where A is any rectangle in R d bounded away from zero.The latter class of sets generates vague convergence in B(R d0)and satisfiesµ(∂A)=0.Assume that A= [a,b]={x:a x b}for two vectors a<b,where<, are defined in the natural componentwise way.Write a±ε=(a1±ε,···,a d±ε)and define b±εcorrespondingly.Define the rectangles A−ε=[a−ε,bε]and Aε=[aε,b−ε]in the same way as A.For smallεthese sets are not empty,bounded away from zero and Aε⊂A⊂A−ε.For smallε>0,P(x−1(X1+X2)∈A)=P(x−1(X1+X2)∈A,x−1|X2|>ε)+P(x−1(X1+X2)∈A,x−1|X2| ε) P(|X2|>xε)+P(x−1X1∈A−ε).Thenlim sup x→∞P(x−1(X1+X2)∈A)P(|X1|>x)lim supx→∞P(|X2|>xε)P(|X1|>x)+lim supx→∞P(x−1X1∈A−ε)P(|X1|>x)=µ(A−ε)↓µ(A)asε↓0.REGULARLY V ARYING FUNCTIONS181 In the last step we used that A is aµ-continuity set.Similarly,P(x−1(X1+X2)∈A) P(x−1X1∈Aε,x−1|X2| ε)P(x−1X1∈Aε)−P(|X2|>εx).Thenlim inf x→∞P(x−1(X1+X2)∈A)P(|X1|>x)lim infx→∞P(x−1X1∈Aε)P(|X1|>x)=µ(Aε)↑µ(A)asε↓0.In the last step we again used that A is aµ-continuity set.Now collecting the upper and lower bounds,we arrive at the desired relation(3.11).4.Product-type functionsProducts are more complicated objects than sums.Their asymptotic tail be-havior crucially depends on which tail of the factors in the product dominates.If the factors have similar tail behavior the results become more complicated.Assume for the moment d=2.The set A={x:x1x2>1}is bounded away from zero and therefore regular variation of X implies that the limitP(X1X2>x2) P(|X|>x)=P(x−1(X1,X2)∈A)P(|X|>x)→µ(A)exists.However,the quantityµ(A)can be rather non-informative,for example,in the two extreme cases:X=(X,X)for a non-negative regularly varying X with indexαand X=(X1,X2),where X1and X2are independent copies of X.In the former case,with the max-norm|·|,µ(A)=1,and in the latter caseµ(A)=0 sinceµis concentrated on the axes.Thus,the knowledge about regular variation of X is useful whenµ(A)>0, i.e.,when the components of X are not(asymptotically)independent.However,if µ(A)=0the regular variation of X is too crude in order to determine the tails of the distribution of the products of the components.4.1.One-dimensional results.In the following result we collect some of the well known results about the tail behavior of the product of two independent non-negative random variables.Lemma4.1.Assume that X1and X2are independent non-negative random variables and that X1is regularly varying with indexα>0.(1)If either X2is regularly varying with indexα>0or P(X2>x)= o(P(X1>x))then X1X2is regularly varying with indexα.(2)If X1,X2are iid such that E(Xα1)=∞then P(X1X2>x)/P(X1>x)→∞.(3)If X1,X2are iid such that E(Xα1)<∞,then the only possible limit of P(X1X2>x)/P(X1>x)as x→∞is given by2E(Xα1)which is attained under the conditionlim M→∞lim supx→∞P(X1X2>x,M<X1X2 x/M)1=0.182JESSEN AND MIKOSCH(4)Assume P (X 1>x )∼c αx −αfor some c >0.Then for iid copiesX 1,...,X n of X 1,n 1,P (X 1···X n >x )∼αn −1c nαx −αlog n −1x.Proof.(1)was proved in Embrechts and Goldie [16,p.245].(2)The following decomposition holds for any M >0:(4.1)P (X 1X 2>x )P (X 1>x )= (0,M ]P (X 2>x/y )P (X 1>x )dP (X 1 y )+ (M,∞)P (X 2>x/y )P (X 1>x )dP (X 1 y )=I 1+I 2.By the uniform convergence theorem,P (X 1>x/y )/P (X 1>x )→y −αuniformly for y ∈(0,M ].HenceI 1→My αdP (X 1 y ),x →∞,→E (X α1),M →∞.Hence,if E (X α1)=∞,(2)applies.(3)It follows from Chover et al.[6]that the only possible limits of P (X 1X 2>x )/P (X 1>x )are 2E (X α1).The proof follows now from Davis and Resnick [12,Proposition 3.1].(4)We start with the case when P (Y i /c >x )=x −α,for x 1and an iid sequence (Y i ).Then n i =1log(Y i /c )is Γ(α,n )distributed:P n i =1log(Y i /c )>x =αn (n −1)! x 0y n −1e −αy dy ,x >0.Then,by Karamata’s theorem,P n i =1(Y i /c )>x/c n =αn (n −1)! log(x/c n )0y n −1e −αy dy=αn (n −1)! x/c n 1(log z )n −1z −α−1dz ∼αn −1(n −1)!(log(x/c n ))n −1(x/c n )−α∼αn −1c nα(n −1)!(log x )n −1x −α.(4.2)Next consider an iid sequence (X i )with P (X 1>x )∼c αx −α,independent of (Y i ),and assume without loss of generality that c =1.Denote the distribution function of n i =2Y i by G and let h (x )→∞be any increasing function such that x/h (x )→∞.ThenREGULARLY V ARYING FUNCTIONS 183PX 1ni =2Y i >x = ∞0P (X 1>x/y )dG (y )=h (x )0P (X 1>x/y )P (Y 1>x/y )P (Y 1>x/y )dG (y )+∞h (x )P (X 1>x/y )dG (y )=I 1(x )+I 2(x ).For any ε>0,sufficiently large x and y ∈(0,h (x )),1−εP (X 1>x/y )P (Y 1>x/y ) 1+ε.HenceI 1(x )∼h (x )0P (Y 1>x/y )dG (y ).Now choose,for example,h (x )=x/log log x .ThenI 2(x ) G (x/log log x )=O ((x/(log log x ))−αlog n −2x )=o (x −αlog n −1x ).A similar argument yields ∞h (x )P (Y 1>x/y )dG (y )=o (x −αlog n −1x ).In view of (4.2)we conclude that P X 1n i =2Y i >x ∼I 1(x )∼P n i =1Y i >xA similar argument shows that we may replace in the left probability any Y i ,i =2,...,n ,by X i .This proves (4).Under the assumption lim sup x →∞x αP (X i >x )<∞upper bounds similar to(4)were obtained by Rosi´n ski and Woyczy´n ski [54].The tail behavior of products of independent random variables is then also reflected in the tail behavior of poly-nomial forms of iid random variables with regularly varying tails and in multiple stochastic integrals driven by α-stable L´e vy motion;see Kwapie´n and Woyczy´n ski[34].In the following results for the product X 1X 2of non-negative independent random variables X 1and X 2we assume that the tail of one of the factors dominates the tail of the other one.Lemma 4.2.Assume X 1and X 2are non-negative independent random vari-ables and that X 1is regularly varying with index α>0.(1)If there exists an ε>0such that E (X α+ε2)<∞,then (4.3)P (X 1X 2>x )∼E (X α2)P (X 1>x ).。

Unicode Nearly Plain-Text Encoding of MathematicsVersion 3Murray Sargent IIIPublisher Text Services, Microsoft Corporation10-Mar-101.Introduction (2)2.Encoding Simple Math Expressions (3)2.1Fractions (4)2.2Subscripts and Superscripts (6)2.3Use of the Blank (Space) Character (7)3.Encoding Other Math Expressions (8)3.1Delimiters (8)3.2Literal Operators (10)3.3Prescripts and Above/Below Scripts (11)3.4n-ary Operators (12)3.5Mathematical Functions (13)3.6Square Roots and Radicals (13)3.7Enclosures (14)3.8Stretchy Characters (15)3.9Matrices (16)3.10Accent Operators (16)3.11Differential, Exponential, and Imaginary Symbols (17)3.12Unicode Subscripts and Superscripts (18)3.13Concatenation Operators (18)3.14Comma, Period, and Colon (18)3.15Ordinary Text Inside Math Zones (19)3.16Space Characters (19)3.17Phantoms and Smashes (21)3.18Arbitrary Groupings (22)3.19Equation Arrays (22)3.20Math Zones (22)3.21Equation Numbers (23)3.22Linear Format Characters and Operands (23)3.23Equation Breaking and Alignment (26)3.24Size Overrides (26)4.Input Methods (27)4.1Character Translations (27)4.2Math Keyboards (29)4.3Hexadecimal Input (29)4.4Pull-Down Menus, Toolbars, Context Menus (29)4.5Macros (30)4.6Linear Format Math Autocorrect List (30)4.7Handwritten Input (30)5.Recognizing Mathematical Expressions (31)ing the Linear Format in Programming Languages (32)6.1Advantages of Linear Format in Programs (33)6.2Comparison of Programming Notations (34)6.3Export to TeX (36)7.Conclusions (37)Acknowledgements (37)Appendix A. Linear Format Grammar (38)Appendix B. Character Keywords and Properties (39)Version Differences (48)References (48)1.IntroductionGetting computers to understand human languages is important in increasing the utility of computers. Natural-language translation, speech recognition and gen-eration, and programming are typical ways in which such machine comprehension plays a role. The better this comprehension, the more useful the computer, and hence there has been considerable current effort devoted to these areas since the early 1960s. Ironically one truly international human language that tends to be ne-glected in this connection is mathematics itself.With a few conventions, Unicode1 can encode many mathematical expressions in readable nearly plain text. Technically this format is a “lightly marked up format”; hence the use of “nearly”. The format is linear, but it can be displayed in built-up presentation form. To distinguish the two kinds of formats in this paper, we refer to the nearly plain-text format as the linear format and to the built-up presentation format as the built-up format. This linear format can be used with heuristics based on the Unicode math properties to recognize mathematical expressions without the aid of explicit math-on/off commands. The recognition is facilitated by Unicode’s strong support for mathematical symbols.2Alternatively, the linear format can be used in “math zones” explicitly controlled by the user either with on-off characters as used in TeX or with a character format attribute in a rich-text environment. Use of math zones is desirable, since the recognition heuristics are not infallible.The linear format is more compact and easy to read than [La]TeX,3,4or MathML.5 However unlike those formats, it doesn’t attempt to include all typograph-ical embellishments. Instead we feel i t’s useful to handle some embellishments in the higher-level layer that handles rich text properties like text and background col-ors, font size, footnotes, comments, hyperlinks, etc. In principle one can extend the notation to include the properties of the higher-level layer, but at the cost of re-duced readability. Hence embedded in a rich-text environment, the linear format can faithfully represent rich mathematical text, whereas embedded in a plain-text environment it lacks most rich-text properties and some mathematical typograph-ical properties. The linear format is primarily concerned with presentation, but it has some semantic features that might seem to be only content oriented, e.g., n-aryands and function-apply arguments (see Secs. 3.4 and 3.5). These have been in-cluded to aid in displaying built-up functions with proper typography, but they also help to interoperate with math-oriented programs.Most mathematical expressions can be represented unambiguously in the line-ar format, from which they can be exported to [La]TeX, MathML, C++, and symbolic manipulation programs. The linear format borrows notation from TeX for mathe-matical objects that don’t lend themselves well to a mathematical linear notation, e.g., for matrices.A variety of syntax choices can be used for a linear format. The choices made in this paper favor a number of criteria: efficient input of mathematical formulae, suffi-cient generality to support high-quality mathematical typography, the ability to round trip elegant mathematical text at least in a rich-text environment, and a for-mat that resembles a real mathematical notation. Obviously compromises between these goals had to be made.The linear format is useful for 1) inputting mathematical expressions,6 2) dis-playing mathematics by text engines that cannot display a built-up format, and 3) computer programs. For more general storage and interchange of math expressions between math-aware programs, MathML and other higher-level languages are pre-ferred.Section 2 motivates and illustrates the linear format for math using the fraction, subscripts, and superscripts along with a discussion of how the ASCII space U+0020 is used to build up one construct at a time. Section 3 summarizes the usage of the other constructs along with their relative precedences, which are used to simplify the notation. Section 4 discusses input methods. Section 5 gives ways to recognize mathematical expressions embedded in ordinary text. Section 6explains how Unicode plain text can be helpful in programming languages. Section 7 gives conclu-sions. The appendices present a simplified linear-format grammar and a partial list of operators.2.Encoding Simple Math ExpressionsGiven Unicode’s strong support for mathematics2 relative to ASCII, how much better can a plain-text encoding of mathematical expressions look using Unicode? The most well-known ASCII encoding of such expressions is that of TeX, so we use it for comparison. MathML is more verbose than TeX and some of the comparisons ap-ply to it as well. Notwithstanding TeX’s phenomenal success in the science and engi-neering communities, a casual glance at its representations of mathematical expres-sions reveals that they do not look very much like the expressions they represent. It’s not easy to make algebraic calculations by hand directly using TeX’s notation. With Unicode, one can represent mathematical expressions more readably, and the resulting nearly plain text can often be used with few or no modifications for such calculations. This capability is considerably enhanced by using the linear format in a system that can also display and edit the mathematics in built-up form.The present section introduces the linear format with fractions, subscripts, and superscripts. It concludes with a subsection on how the ASCII space character U+0020 is used to build up one construct at a time. This is a key idea that makes the linear format ideal for inputting mathematical formulae. In general where syntax and semantic choices were made, input convenience was given high priority.2.1FractionsOne way to specify a fraction linearly is LaTeX’s \frac{numerator}{denominator}. The { } are not printed when the fraction is built up. These simple rules immediately give a “plain text” that is unambiguous, but looks quite different from the corre-sponding mathematical notation, thereby making it harder to read.Instead we define a simple operand to consist of all consecutive letters and decimal digits, i.e., a span of alphanumeric characters, those belonging to the L x and Nd General Categories (see The Unicode Standard 5.0,1 Table 4-2. General Category). As such, a simple numerator or denominator is terminated by most nonalphanumer-ic characters, including, for example, arithmetic operators, the blank (U+0020), and Unicode characters in the ranges U+2200..U+23FF, U+2500..U+27FF, and U+2900 .. U+2AFF. The fraction operator is given by the usual solidus / (U+002F). So the sim-ple built-up fractionabc d .appears in linear format as abc/d. To force a display of a normal-size linear fraction, one can use \/ (backslash followed by slash).For more complicated operands (such as those that include operators), paren-theses (), brackets [], or braces {} can be used to enclose the desired character combinations. If parentheses are used and the outermost parentheses are preceded and followed by operators, those parentheses are not displayed in built-up form, since usually one does not want to see such parentheses. So the plain text (a + c)/d displays asa+cd.In practice, this approach leads to plain text that is easier to read than LaTeX’s, e.g., \frac{a + c}{d}, since in many cases, parentheses are not needed, while TeX requires {}’s. To force the display of the outermost parentheses, one encloses them, in turn, within parentheses, which then become the outermost parentheses. For example, ((a + c))/d displays as(a+c).A really neat feature of this notation is that the plain text is, in fact, often a legit-imate mathematical notation in its own right, so it is relatively easy to read. Contrast this with the MathML version, which (with no parentheses) reads as<mfrac><mrow><mi>a</mi><mo>+</mo><mi>c</mi></mrow><mi>d</mi></mfrac>Three built-up fraction variations are available: the “fraction slash” U+2044 (which one might input by typing \sdiv) builds up to a skewed fraction, the “division slash” U+2215 (\ldiv) builds up to a potentially large linear fraction, and the circled slash ⊘ (U+2298, \ndiv) builds up a small numeric fraction (although characters other than digits can be used as well). The three kinds of built-up fractions are illus-trated byab+c de+f ,ab+cde+f⁄,(ab+c)(de+f)⁄When building up the large linear fraction, the outermost parentheses should not be removed.The same notational syntax is used for a “stack” which is like a fraction with no fraction bar. The stack is used to create binomial coefficients and the stack operator is ‘¦’ (\atop). For example, the binomial theorem(a+b)n=∑(nk)a k b n−knk=0in linear format reads as (see Sec. 3.4 for a discussion of the n-aryand “glue” opera-tor ▒)(a + b)^n = ∑_(k=0)^n▒ (n ¦ k) a^k b^(n-k),where (n ¦ k) is the binomial coefficient for the combinations of n items grouped k at a time. The summation limits use the subscript/superscript notation discussed in the next subsection.Since binomial coefficients are quite common, TeX has the \choose control word for them. In the linear format Version 3, this uses the \choose operator ⒞in-stead of the \atop operator ¦. Accordingly the binomial coefficient in the binomial theorem above can be written as “n\choose k”, assuming that you type a space after the k. This shortcut is included primarily for compatibility with TeX, since (n¦k) is pretty easy to type.When / is followed by an operator, it’s highly unlikely that a fraction is intend-ed. This fact leads to a simple way to enter negated operators like ≠, namely, justtype /= to get ≠. A list of such negated operator combinations is given in Section 4.1. To enter ≠, you can also type TeX’s name, \ne, but /= is slightly simpler. And the TeX names for the other negated operators in Section 4.1 are harder to remember. One other trick with fractions is that a period or comma in between two digits or in be-tween the slash and a digit is considered to be part of a number, rather than being a terminator. For example 1/3.1416 builds up to 13.1416, rather than 13.1416.2.2 Subscripts and SuperscriptsSubscripts and superscripts are a bit trickier, but they’re still quite readable. Specifically, we introduce a subscript by a subscript operator, which we display as the ASCII underscore _ as in TeX. A simple subscript operand consists of the string of one or more characters with the General Categories L x (alphabetic) and Nd (decimal digits), as well as the invisible comma. For example, a pair of subscripts, such as δ is written as δ_μν. Similarly, superscripts are introduced by a superscript operator, which we display as the ASCII ^ as in TeX. So a ^b means a b . A nice enhancement for a text processing system with build -up capabilities is to display the _ as a small sub-script down arrow and the ^ as a small superscript up arrow, in order to convey the semantics of these build -up operators in a math context.Compound subscripts and superscripts include expressions within parenthe-ses, square brackets, and curly braces. So δ is written as δ_(μ+ν). In addition it is worthwhile to treat two more operators, the comma and the period, in special ways. Specifically, if a subscript operand is followed directly by a comma or a period that is, in turn, followed by whitespace, then the comma or period appears on line, i.e., is treated as the operator that terminates the subscript. However a comma or period followed by an alphanumeric is treated as part of the subscript. This refine-ment obviates the need for many overriding parentheses, thereby yielding a more readable linear -format text (see Sec. 3.14 for more discussion of comma and period).Another kind of compound subscript is a subscripted subscript, which works using right -to -left associativity, e.g., a_b_c stands for a b c . Similarly a ^b ^c stands for a b c .Parentheses are needed for constructs such as a subscripted superscript like a b c , which is given by a ^(b_c ), since a ^b_c displays as a c b (as does a _c ^b ). The build -up program is responsible for figuring out what the subscript or superscript base is. Typically the base is just a single math italic character like the a in these examples. But it could be a bracketed expression or the name of a mathematical function like sin as in sin^2 x , which renders as sin 2x (see Sec. 3.5 for more discussion of this case). It can also be an operator, as in the examples +1 and =2. In Indic and other clus-ter -oriented scripts the base is by default the cluster preceding the subscript or su-perscript operator.As an example of a slightly more complicated example, consider the expression W δ1ρ1σ23β, which can be written with the linear format W^3β_δ1ρ1ς2, where Unicode numeric subscripts are used. In TeX, one types$W^{3\beta}_{\delta_1\rho_1\sigma_2}$The TeX version looks simpler using Unicode for the symbols, namely $W^{3β}_{δ_1 ρ_ς_2}$ or $W^{3β}_{δ1ρ1ς2}$, since Unicode has a full set of decimal subscripts and superscripts. As a practical matter, numeric subscripts are typically entered using an underscore and the number followed by a space or an operator, so the major simpli-fication is that fewer brackets are needed.For the ratio α23β23+γ23 the linear -format text can read as α₂³/( β₂³ + γ₂³), while the standard TeX version reads as$$\alpha_2^3 \over \beta_2^3 + \gamma_2^3$$·The linear -format text is a legitimate mathematical expression, while the TeX ver-sion bears no resemblance to a mathematical expression.TeX becomes cumbersome for longer equations such asW δ1ρ1σ23β=U δ1ρ13β+12∫dα2′[U δ1ρ12β−α2′U ρ1σ21βU ρ1σ2]α2α1 A linear -format version of this reads asW_δ1ρ1ς2^3β=U_δ1ρ1^3β+1/8π^2 ∫_α1^α2▒dα’2 [(U_δ1ρ1^2β-α’2U_ρ1ς2^1β)/U_ρ1ς2^0β]while the standard TeX version reads as$$W_{\delta_1\rho_1\sigma_2}^{3\beta}= U_{\delta_1\rho_1}^{3\beta} + {1 \over 8\pi^2}\int_{\alpha_1}^{\alpha_2} d\alpha_2’ \left[{U_{\delta_1\rho_1}^{2\beta} - \alpha_2’U_{\rho_1\sigma_2}^{1\beta} \overU_{\rho_1\sigma_2}^{0\beta}} \right] $$ .2.3 Use of the Blank (Space) CharacterThe ASCII space character U+0020 is rarely needed for explicit spacing of built-up text since the spacing around operators should be provided automatically by the math display engine (Sec. 3.16 discusses this automatic spacing). However the space character is very useful for delimiting the operands of the linear-format notation. When the space plays this role, it is eliminated upon build up. So if you type \alphafollowed by a space to get α, the space is eliminated when the α replaces the \alpha. Similarly a_1 b_2 builds up as a1b2 with no intervening space.Another example is that a space following the denominator of a fraction is eliminated, since it causes the fraction to build up. If a space precedes the numerator of a fraction, the space is eliminated since it may be necessary to delimit the start of the numerator. Similarly if a space is used before a function-apply construct (see Sec.3.5) or before above/below scripts (see Sec. 3.3), it is eliminated since it delimits the start of those constructs.In a nested subscript/superscript expression, the space builds up one script at a time. For example, to build up a^b^c to a b c, two spaces are needed if spaces are used for build up. Some other operator like + builds up the whole expression, since the operands are unambiguously terminated by such operators.In TeX, the space character is also used to delimit control words like \alpha and does not appear in built-up form. A difference between TeX’s usage and the lin-ear format’s is that in TeX, blanks are invariably eliminated in built-up display, whereas in the linear format blanks that don’t delimit operands or keywords do re-sult in spacing. Additional spacing characters are discussed in Sec. 3.16.One displayed use for spaces is in overriding the algorithm that decides that an ambiguous unary/binary operator like + or − is unary. If followed by a space, the operator is considered to be binary and the space isn’t displayed. Spaces are also used to obtain the correct spacing around comma, period, and colon in various con-texts (see Sec. 3.14).3.Encoding Other Math ExpressionsThe previous section describes how we encode fractions, subscripts and super-scripts in the linear format and gives a feel for that format. The current section de-scribes how we encode other mathematical constructs using this approach and ends with a more formal discussion of the linear format.3.1DelimitersBrackets [], braces {}, and parentheses () represent themselves in the Unicode plain text, and a word processing system capable of displaying built-up formulas should be able to enlarge them to fit around what’s inside them. In general we refer to such characters as delimiters. A delimited pair need not consist of the same kinds of delimiters. For example, it’s fine to open with [ and close with } and one sees this usage in some mathematical documents. The closing delimiter can have a subscript and/or a superscript. Delimiters are called fences in MathML.These choices suffice for most cases of interest. But to allow for use of a delim-iter without a matching delimiter and to overrule the open/close character of delim-iters, the special keywords \open and \close can be used. These translate to the box-drawings characters├ and ┤, respectively. Box drawings characters are used for theopen/close delimiters because they aren’t likely to be used as mathematical charac-ters and they are readily available in fonts. If used before any character that isn’t a delimiter of the opposite sense, the open/close delimiter acts as an invisible delimit-er, defining the corresponding end of a delimited expression. A common use of this is the “cases” equation, such as,|x|={x if x≥ 0−x if x<0which has the linear format “|x| = {█ (&x" if "x ≥ 0@−&x" if "x < 0)┤" (see Sec. 3.19 for a discussion of the equation-array operator █ ).Because the cases construct is fairly common, TeX has the \cases control word for it. This can be implemented in the linear format Version 3 with the \cases opera-tor Ⓒ. With this the equation above can be written as “|x| = Ⓒ(&x" if "x ≥ 0@−&x" if "x < 0)", which is still a little strange, but you don’t have to type the opening curly brace and \close.The open/close delimiters can be used to overrule the normal open/close character of delimiters as in the admittedly strange, but nevertheless sometimes used, expression “]a + b[”, which has the linear format “├]a+b┤[”. Note that a blank following an open or close delimiter is “eaten”. This is to allow an open delimiter to be followed by a normal delimiter without interpreting the pair as a single delimiter. See also Sec. 3.18 on how to make arbitrary groupings. If a├ needs to be treated as an empty open delimiter when it appears before a delimiter like | or ], follow the├by a space to force the open-delimiter interpretation.To suppress automatic sizing and to choose specific sizes,├is followed by a digit ‘0’ –‘4’ with the meanings in the following tableIt’s rarely necessary to use explicit sizes if the display system can break equations within bracketed expressions.The usage of open and close delimiters in the linear format is admittedly a compromise between the explicit nature of TeX and the desire for a legitimate math notation, but the flexibility can be worth the compromise especially when interoper-ating with ordinarily built-up text such as in a WYSIWYG math system. TeX uses \left and \right for this purpose instead of \open and \close. We use the latter since they apply to right-to-left mathematics used in many Arabic locales as well as to the usual left-to-right mathematics.Absolute values are represented by the ASCII vertical bar | (U+007C). The evenness of its count at any given bracket nesting level typically determines whetherthe vertical bar is a close |. Specifically, the first appearance is considered to be anopen | (unless subscripted or superscripted), the next a close | (unless following anoperator), the next an open |, and so forth.Nested absolute values can be handled unambiguously by discarding theoutermost parentheses within an absolute value. For example, the built-up expres-sion ||x| - |y|| can have the linear format |(|x|−|y|)|. Some cases, such as this one, can be parsed without the clarifying parentheses by noting that a vertical bar | directlyfollowing an operator is an open |. But the example |a|b−c|d| needs the clarifying pa-rentheses since it can be interpreted as either (|a|b)−(c|d|) or |a(|b−c|)d|. The usualalgorithm gives the former, so if one wants the latter without the inner parentheses,one can type |(a|b−c|d)|.Another case where we treat | as a close delimiter is if it is followed by a space(U+0020). This handles the important case of the bra vector in Dirac notation. Forexample, the quantum mechanical density operator ρ has the definition,ρ=∑Pψ|ψ⟩⟨ψ|ψwhere the vertical bars can be input using the ASCII vertical bar.If a | is followed by a subscript and/or a superscript and has no correspondingopen |, it is treated as a script base character, i.e., not a delimiter. Its built-up sizeshould be the height of the integral sign in the current display/inline mode.The Unicode norm delimiter U+2016 (‖ or \norm) has the same open/closedefinitions as the absolute value character | except that it’s always considered to be adelimiter.Delimiters can also have separators within them. Version 2 of the linear formatdoesn’t formalize the comma separators of function arguments (MathML does), butit supports the vertical bar separator \vbar, which is represented by the box draw-ings light vertical character│(U+2502). We tried using the ASCII | (U+007C) for this purpose too, but the resulting ambiguities are insurmountable in general. One case using U+007C as a separator that can be deciphered is that of the form (a|b), where a and b are mathematical expressions. But (a|b|c) interprets the vertical bars as the absolute value. And one might want to interpret the | in (a|b) as an open delimiter with ) as the corresponding close delimiter, while the ( isn’t yet matched. If so, pre-cede the | by├, i.e., (├|b). The vertical bar separator grows in size to match the size of the surrounding brackets. In Version 3, other operators can be treated as separa-tors by preceding them with \middle (║— U+2551).Another common separator is the \mid character ∣ (U+2223), commonly usedin expressions like {x | f(x)=0}. This separator also grows in size to match the sur-rounding brackets and is spaced as a relational operator.3.2Literal OperatorsCertain operators like brackets, braces, parentheses, superscript, subscript, in-tegral, etc., have special meaning in the linear-format notation. In fact, even a charac-ter like ‘+’, which displays the same glyph in linear format as in built-up form (aside from a possible size reduction), plays a role in the linear format in that it terminates an operand. To remove the linear-format role of such an operator, we precede it by the “literal operator”, for which the backslash \ is handy. So \[ is displayed as an or-dinary left square bracket, with no attempt by the build-up software to match a cor-responding right square bracket. Such quoted operators are automatically included in the current operand.Linear format operators always consist of a single Unicode character, although a control word like \open may be used to input the character. Using a single charac-ter has the advantage of being globalized, since the control word typically looks like English. Users can define other control words that look like words in other lan-guages just so long as they map into the appropriate operator characters. A slight exception to the single-character operator rule occurs for accent operators that are applied to two or more characters (see Sec. 3.10). For these the accent combining mark may be preceded by a no-break space for the sake of readability. Another ad-vantage of using operator characters rather than control words is that the build-up processing is simplified and therefore faster. And one should delight in the fact that the operator characters look like the operators they represent, while the control words do not.3.3Prescripts and Above/Below ScriptsA special parenthesized syntax is used to form prescripts, that is, subscripts and superscripts that precede their base. For this (_c^b)a creates the prescripted variable a c b. Variables can have both prescripts and postscripts (ordinary subscripts and superscripts).In Version 3 of the linear format, you can use a prescript notation similar to TeX’s. Just type a subscript and/or a superscript not preceded by a base and then follow it with a character that can be used as a base. For the c b a example, you type _c^b a. Note that you need to terminate the superscript with a space. If a variable precedes the prescript, you also need to precede the prescript with a space. A com-mon use of prescripts is for the confluent hypergeometric functions, such as F11. In Version 3, this can be input as _1 F_1 or as (_1^)F_1.Below scripts and above scripts are represented in general by the line drawing operators \below (┬) and \above (┴), respectively. Hence the expression im n a n can be represented by lim┬ (n→∞) a_n. Since the operations det, gcd, inf, lim, lim inf, lim sup, max, min, Pr, and sup are common, their below scripts are also accessible by the usual subscript operator _. So in display mode, im n a n can also be represented by lim_(n→∞) a_n, which is a little easier to type than lim┬(n→∞) a_n.Although for illustration purposes, the belowscript examples are shown here in-line with the script below, ordinarily this choice is only for display-mode math. When inline, below- and abovescripts entered with _ and ^ are shown as subscripts。

A General Treatment of Dynamic ConstraintsE.O. de BrockNovember 1997SOM theme A: Intra-firm coordination and changeAbstractThis paper introduces a general, formal treatment of dynamic constraints, i.e., constraints on the state changes that are allowed in a given state space. Such dynamic constraints can be seen as representations of "real world" constraints in a managerial context. The notions of transition, reversible and irreversible transition, and transition relation will be introduced. The link with Kripke models (for modal logics) is also made explicit. Several (subtle) examples of dynamic constraints will be given. Some important classes of dynamic constraints in a database context will be identified, e.g., various forms of cumulativity, non-decreasing values, constraints on initial and final values, life cycles, changing life cycles, and transition and constant dependen-cies. Several properties of these dependencies will be treated. For instance, it turns out that functional dependencies can be considered as "degenerated" transition dependencies. Also, the distinction between primary keys and alternate keys is reexamined, from a dynamic point of view.Keywords: Dynamic constraints, transition (relation), (ir)reversibility, transition dependencies, constant dependencies, cumulativity, (changing) life cycles, Kripke modelsE.O. de BrockFaculty of Management and OrganizationState University of GroningenP.O. Box 800, 9700 AV GroningenThe NetherlandsTel. +31.50.3637315Fax +31.50.3632275Email: e.o.de.brock@bdk.rug.nlIntroductionIn data modelling, constraints play an important role. We can distinguish between static and dynamic constraints. The requirements on the states are called static constraints, and the requirements on the state transitions are called dynamic constraints. Traditionally, most emphasis in the constraints literature is on static constraints. This paper is dedicated to a general, formal treatment of dynamic constraints.Although there exist many publications on temporal databases in the literature (see for example the bibliography in [Kl 93]), the related topic of dynamic constraints received much less attention. The papers on dynamic constraints usually concentrate on certain standardized forms of dynamic constraints (see [Vi 87], [Na 89], [Wij 95], and [Je 94], for instance). In our paper, we address (and solve) the problem of the development of a general theory of dynamic constraints in which all kinds of dynamic constraints can be neatly treated. We will also illustrate the richness and relevance of our theory with various (classes of) practical examples. From a management point of view, it is sometimes inconvenient to treat dynamic constraints as "ironclad rules". After all, the situation might arise that in the end a certain transaction appeared to be erroneous or unintentional, while it cannot be undone because of the dynamic constraints. We say in this case that the transition is irreversible; if it can be undone, the trans-ition is called reversible. In many cases of non-reversible transition relations, a dynamic constraint is not used as a "law" but rather as a warning ("Do you really want this?") or as a condition pertaining to special authorization required to carry out the modification.All in all, this paper contributes to both the theory and application of dynamic constraints, among others by establishing a general theoretical foundation for solutions to practical problems, even in subtle and complex situations.The paper is organized as follows. The notions of transition, reversible and irreversible trans-ition, and transition relation will be introduced in Section 1. The link with Kripke models (for modal logics) is also made explicit here. Several examples of dynamic constraints will be given in Section 2. Here, the distinction between primary keys and alternate keys is also reexamined, from a dynamic point of view. The special classes of transition and constant dependencies will be identified in Section 3. Several properties of these dependencies will be treated here. In the Appendix our basic notions and notations are established.1. Transition relations and (ir)reversibility1By defining a database universe U (or a state space in general), we establish which states are formally allowed in an organization. In general, however, not all state transitions between those (in itself) allowed states are admissible in that organization. We can establish the set of admissible transitions by means of a subset R of U × U, having the following intuitive meaning:(v;v N) 0 R ] the direct transition from state v to state v N is allowed.If (v;v N) ó R then it might still be possible that v N is indirectly reachable from v, namely if v N is reachable from v via a number of allowed intermediate steps (or formally, if (v;v N) 0 Tcl(R), where Tcl(R) denotes the transitive closure of R).An element of U × U is called a transition within U. A set of admissible transitions for the state space U is called a transition relation on U. Formally:Definition 1:If U is a set, then:(a) p is a transition within U ] p 0 U × U;(b) R is a transition relation on U ] R f U × U.In summary, the requirements on the states in an organization are called static constraints and determine a state space U, while the requirements on the transitions are called dynamic constraints and determine a transition relation R on U.The readers who are familiar with Kripke models for modal logic, will recognize the pair (U;R) as a so-called world system, where U is a set of possible worlds and R is an accessibility relation on that set of possible worlds. This means that expressions like "it is necessary that .." and "it is possible that .." can get a meaning in a given state or "world" v (see [Kr 59] or [Le 77], for example).2Example 1:We define a simple database universe EXU (concerning employees and departments) for a fictitious company called Dyncons. First, the set-valued functions FE and FD introduce the (employee and department) attributes and their corresponding value sets. Then WE and WD determine the set of allowed employee tables and department tables, respectively. The function HF introduces the table names EMPL and DEP and their corresponding sets of allowed tables. Finally, EXU determines the set of allowed database states, expressing for instance that each department manager must be an employee with the proper maturity. (Our notations are defined in the Appendix.)FE = { (NO ; ù),.employee number(NAME ; Chs(40)),.employee name(SAL ; ù),.salary(SEX ; {'M', 'F'}),.sex(DEPNO ; [1 .. 99]),.department number(MAT ; {'jun','anal','sr-an',.specialisation/maturity'mgr','prog','sr-pr'})}; .FD = { (DNO ; ù),.department number(NAME ; Chs(45)),.department name(MANNO; ù)};.manager numberWE = {T * T f J(FE) and .the set of allowed employee tables {NO} is u.i. in T};.WD = {T * T f J(FD) and .the set of allowed department tables {DNO} is u.i. in T and .{NAME} is u.i. in T};.HF = { (EMPL ; WE),.employees(DEP ; WD)};.departmentsEXU = {v * v 0J(HF) and{(DEPNO; DNO)} connects v(EMPL) with v(DEP) and{(MANNO; NO)} connects v(DEP) with{t * t 0 v(EMPL) and t(MAT) = 'mgr'} }.3Suppose now that the following dynamic constraints should hold:(DC1)Existing department numbers must not expire(although, for example, the name of a department is allowed to change).(DC2)An employee must always remain within the same department.(DC3)Salaries of employees must not decrease.(DC4)Specialisation/maturity growth develops along the following lines:jun 6 anal 6 sr-an 6 mgr` prog 6 sr-pr _expressing that a jun(ior) can become an anal(ist) or a prog(rammer), an analist can become a s(enio)r-an(alist), a senior-analist can become a manager, etc.These constraints can be formally represented by the transition relation EXR on EXU, as defined below. In order to formalize the constraints (DC2), (DC3), and (DC4), we will assume that "in the course of time" an employee retains the same identity number.EXR = {(v;v N) * (v;v N) 0 EXU × EXU andv(DEP) Þ{DNO} f v N(DEP) Þ{DNO} and(DC1)œt 0 v(EMPL): œt N0 v N(EMPL):[if t(NO) = t N(NO)then (t(DEPNO) = t N(DEPNO) and (DC2)t(SAL) # t N(SAL) and(DC3)if t(MAT) … t N(MAT)(DC4)then (t(MAT); t N(MAT)) 0 { ('jun' ;'anal'),('anal' ;'sr-an'),('sr-an';'mgr'),('jun' ;'prog'),('prog';'sr-pr'),('sr-pr';'mgr')} )]}~ Example 1.4Usually, the transition from each state v 0 U to v itself should be allowed. Such a "trivial" transition might arise when we try to delete something, e.g. an employee tuple, that does not appear to occur in v. In that case, the state remains the same and the "transition" at hand happens to be (v;v). In other words, usually we want such a transition relation R to be reflexive on U, i.e., œv 0 U: (v;v) 0 R.Note that the transition relation EXR is reflexive on the database universe EXU.We note that it is sometimes inconvenient to treat dynamic constraints as "ironclad rules". After all, the situation might arise that in the end a certain transaction appeared to be erroneous or unintentional, yet cannot be undone because of the dynamic constraints. If for example department numbers (or order numbers) that are already present must not expire at any time, then the erroneous addition of a department (or an order) cannot be undone. We say in this case that the transition is irreversible; if it can be undone, the transition is called reversible:Definition 2:If (x;y) is any ordered pair and R is a set of ordered pairs, then:(a) (x;y) is reversible in R ] (y;x) 0 Tcl(R);(b) (x;y) is irreversible in R] (y;x) ó Tcl(R).If R contains only reversible elements, then R itself is also called reversible:Definition 3:If R is a set of ordered pairs, then:R is reversible]œp 0 R: p is reversible in R.We note that the transition relation EXR on EXU from Example 1 is not reversible, due to the dynamic constraint (DC1).In many cases of non-reversible transition relations, a dynamic constraint is not used as a "law" but rather as a warning ("Do you really want this?") or as a condition pertaining to special authorization required to carry out the modification. In these cases this means that these dynamic constraints cannot be used as genuine transition invariants.2. Some special classes of dynamic constraints5In this section we discuss some general forms of dynamic constraints, each introduced by some real world situation. Each form will also be illustrated by the database universe EXU or the transition relation EXR from Example 1.Suppose we have a database skeleton g, a database universe U over g, E 0 dom(g), S f g(E), B f g(E), C f g(E), a 0 g(E), and b 0 g(E), where the a-values for E are numbers. We will express each form of constraint in terms of an arbitrary transition (v;v N) within U.(F1)Cumulativity of tuplesSome data should not be changed or deleted anymore once they are in the database,e.g. historical data. This dynamic constraint belongs to the following general class:The E-table is cumulative, i.e., no E-data will be deleted or modified. In brief, no E-tuples must expire:v(E) f v N(E).It is interesting to note that this dynamic constraint can also be regarded as a connection requirement concerning two different "points in time":id(g(E)) connects v(E) with v N(E).This would be quite a severe demand for E = DEP in EXU, implying that a depart-ment (number) must not expire, that the corresponding department name must not be altered, and also that the manager of that department must not be replaced. In fact the insertion of completely new departments is the only allowed modification of the department-table.If the E-table is intended for recording historical data or logging activities for example, then this severe demand of cumulativity can be realistic, however.(F2)Cumulativity of all primary attribute value combinationsLet us now suppose that a department number should not expire and the correspon-ding department name may not change in our company Dyncons from Example 1, but that the manager can be replaced (and new departments can also be added). This dynamic constraint is a special case of the following:6No single combination of values of primary attributes - i.e. attributes belonging to a minimal key - may expire in the E-table. When we denote the set of all primary attributes of the table index E in the database universe U by Prim(E,U), we can write this dynamic constraint as follows:id( Prim(E,U) ) connects v(E) with v N(E).For E = DEP in EXU this means:id({DNO,NAME}) connects v(DEP) with v N(DEP).In other words, a department number should not expire and the corresponding department name may not change; however, the manager can be replaced (and new departments can also be added).Note that the requirement (F1) implies the requirement (F2). In other words, (F2) is weaker than (F1).(F3)Cumulativity of attribute value combinationsSometimes, entities such as orders or payments should not be deleted anymore once they are recorded in the database - which indicates cumulativity of object identifica-tions - and, moreover, some of their attribute values may not be changed either, e.g.creation date. This dynamic constraint belongs to the following class:The E-table is cumulative with respect to S, that is, no S-values are allowed to expire:v(E) Þ S f v N(E) Þ S.Again, this can be rephrased as a connection requirement:id(S) connects v(E) with v N(E).In our order example above, S will consist of order number and creation date.Another example of a dynamic constraint of this form is the requirement (DC1) from Example 1, where E = DEP and S = {DNO}.If S contains only primary attributes, e.g., if S is a minimal key of E in U, then S f Prim(E,U); in that case the requirement (F2) implies the requirement (F3).(F4)Transition dependency7Sometimes entities such as orders or payments may be deleted once they are in the database, but the initial values of some of their attributes may not be changed anymo-re, e.g. creation date. This dynamic constraint is of the following general form:For each B-value remaining in the E-table, the corresponding C-values must remain constant:œt 0 v(E): œt N0 v N(E): if t j B = t N j B then t j C = t N j C.In our order example above, B will consist of the order number and C of the creation date. The requirement (DC2) from Example 1 is also a special case of this form.(Take U = EXU, E = EMPL, B = {NO} and C = {DEPNO}.)If B is a key of E in U and S = B c C in (F3), then that requirement (F3) implies (F4),i.e., then (F4) is weaker than (F3).We will treat this class of dynamic constraints in more detail in Section 3.(F5)Non-decreasing attribute valuesThe values of "cumulatively counting" attributes (such as Total number of treated patients of a department) should typically be non-decreasing. In general:The a-value of each E-tuple in the new state must be greater than or equal to the a-value of each E-tuple with the same S-value in the old state:œt 0 v(E): œt N0 v N(E): if t j S = t N j S then t(a) # t N(a).Here, S is usually a key of E in U, e.g., as in requirement (DC3) in Example 1.If B = S and C = { a } in (F4), then that requirement (F4) implies the requirement (F5).(F6)Non-decreasing number of tuplesSuppose that the number of courses that our faculty offers is not allowed to decrease (although any course can be replaced by an other one), then we might have a dynamic constraint of the form below.The number of E-tuples is not allowed to decrease:*v(E)*#*v N(E)*.8For E = DEP in EXU this would mean that departments cannot expire all of a sudden, although they can apparently be replaced by other departments.If S in (F3) is a key of E in U, then the requirement (F3) implies the requirement (F6), i.e., then (F6) is weaker than (F3).(F7)Constraints on initial valuesA constraint such as the requirement that each new invoice must have the status"open" is a dynamic constraint, with the following general form:For each initial value for the key S of the E-table, the tuple concerned has to satisfy a certain requirement n (in our case the requirement that the status is "open"). In other words, the S-values in the E-table of the new state v N for which the tuples do not satisfy n must already have occurred in the E-table of the old state v. This dynamic constraint on the admissible initial values can therefore be formulated as a connection requirement:id(S) connects { t N0 v N(E) * ¬ n(t N) } with v(E).Our invoice-example above would result in something like:id({INV-NO}) connects { t N0 v N(INV) * t N(STATUS) … "open" } with v(INV).We will also illustrate this class of dynamic constraints (concerning initial values) with a concrete instance based on Example 1. Suppose that we require that no new employee can immediately start as a department manager. Here,E = EMPL, S = {NO}, and n(t N) = (t N(NO) ó {a(MANNO) * a 0 v N(DEP)}). Thisresults in the following dynamic constraint:id({NO}) connects { t N0 v N(EMPL) * t N(NO) 0 {a(MANNO) * a 0 v N(DEP)} } with v(EMPL).(F8)Constraints on final valuesThe requirement that an invoice can only be deleted when the status is "ready" or "exit" is also a dynamic constraint, with the following general form:A value for the key S of the E-table can only disappear when the tuple concernedsatisfies a certain requirement n. In other words, the S-values in the E-table of the old state v for which the tuples do not satisfy n must still occur in the E-table of the new9state v N. This dynamic constraint on the final values can therefore also be formulated as a connection requirement:id(S) connects { t 0 v(E) * ¬ n(t) } with v N(E).Our invoice-example above would result in something like:id({INV-NO}) connects { t 0 v(INV) * t(STATUS) ó {"ready", "exit"} }with v N(INV).We will also illustrate this class of dynamic constraints (concerning final values) witha concrete instance based on Example 1. Suppose that we require that a departmentcan only disappear when it has no employees anymore. Here,E = DEP, S = {DNO}, and n(t) = (t(DNO) ó {y(DEPNO) * y 0 v(EMPL)}). Thisresults in the following dynamic constraint:id({DNO}) connects { t 0 v(DEP) * t(DNO) 0 {y(DEPNO) * y 0 v(EMPL)} } with v N(DEP).Note that this dynamic constraint formally rules out the possibility of a proper casca-ding delete here (considered as one atomic transaction).(F9)Life cyclesSometimes, tuples contain some kind of "status" attribute for which only certain status transitions are allowed during their "life time", thus enforcing certain "life cycle"constraints. A classical example of such an attribute is marital status.Generally, for a "status" attribute b (with value set M), proper status transitions are only allowed within a given L f M × M (for old and new tuples with the same value for a key S of the E-table):{ (t(b);t N(b)) * (t;t N) 0 v(E) × v N(E) and t j S = t Nj S and t(b) … t N(b)} f L.An example of a dynamic constraint of this form is the requirement (DC4) from Example 1, where b = MAT, E = EMPL, S = {NO}, andM = {'jun','anal','sr-an','mgr','prog','sr-pr'} and10L = { ('jun';'anal'), ('anal' ;'sr-an'), ('sr-an';'mgr'),('jun';'prog'), ('prog';'sr-pr'), ('sr-pr';'mgr')}(F10)Changing life cyclesLife cycle constraints may be subject to change as well. E.g., our company in Example 1 may decide to add the possibility of the career step "prog 6 anal". If life cycles are occasionally subject to change then it is better to add a table containing the currently allowed (proper) status transitions in our database universe (i.e., we mean: better than to adapt our transition relation over and over again). In our example, that table would look like:FROM TOjun analanal sr-ansr-an mgrjun progprog sr-prsr-pr mgrGenerally, we would extend our database skeleton g with an ordered pair, say (ST;{FROM, TO}), while the value set for the attributes FROM and TO is M (or perhaps something of the form Chs(n) if the status set M as such might be subject to change as well). This results in the following dynamic constraint allowing changing life cycles:{ (t(b); t N(b)) * (t;t N) 0 v(E) × v N(E) and t j S = t Nj S and t(b) … t N(b)} f{ (y(FROM); y(TO)) * y 0 v(ST)}.Note that we used v(ST) in stead of v N(ST) here. This means that status transitions have to obey the status transition conditions in the "old" state v (which is relevant in the rare situation that status transitions and status transition conditions are updated in one and the same "atomic" transaction).Even if our life cycles are not subject to change, one might argue that this solution is11more elegant than the one used in Example 1 anyway.Note that for v = v N, the constraint forms (F1), (F2), (F3), (F6), (F7), and (F8) are always satis-fied. Hence, these constraint forms can not hinder the reflexivity of any transition relation in which they appear. Also if B in (F4) and S in (F5), in (F9), and in (F10) are keys of E in U then those constraint forms are satisfied for v = v N as well.We notice that sometimes one of the minimal keys of E in U plays a special role, such as S in the dynamic constraints in (F3), (F5), (F7), (F8), (F9), and (F10). The intuition behind this is that in the course of time the values for that special key continue to correspond one-to-one to the "real world" objects they intend to represent. Such a special minimal key is sometimes called a primary key. The other minimal keys are called alternate keys in this regard. This topic is discussed in Chapter 3 of [Da 86], for example. Although in the literature the distincti-on between primary keys and alternate keys is usually made on the basis of other (often vague) criteria, the explicit role in dynamic constraints as mentioned above is in our view the most significant and concrete argument for such a distinction.We also noticed in this section that the concept of a connection requirement is useful not only for the formulation of static constraints, but for the formulation of dynamic constraints as well. In those cases, the connecting attribute transformation is obviously an identity function.3. Transition dependency andconstant dependencyFor the special class of dynamic constraints encountered in (F4) of Section 2, we shall introduce special names and notations. In the case below we will call C transition dependent on B at (T;T N), by analogy to momentary dependency.Definition 4:If A, B, and C are sets and T and T N are tables over A, then:B ²C at (T;T N) ]œt 0T: œt N0T N: if t j B = t N j B then t j C = t N j C.The requirement (F4) in Section 2 can now be written as: B ² C at (v(E);v N(E)).12The next five lemmas describe some basic properties of transition dependency. We note that the properties in Lemma 1 constitute the analogon to the Armstrong axioms for momentary dependencies; see [Ar 74].Lemma 1:If A is a set and T and T N are tables over A, and B f A and C f A and D f A, then:(a) if C f B then B ² C at (T;T N);(b) if B ² C at (T;T N) and C ² D at (T;T N) then B ² D at (T;T N);(c) B ² C at (T;T N) ]œc 0C: B ² { c } at (T;T N).Lemma 1 can be proven by simply applying Definition 4. The following lemma can be proven directly from Lemma 1, i.e., without reverting to the actual definition of transition dependency.Lemma 2:If A is a set, T and T N are tables over A, B f A, C f A, D f A, and E f A, then:(a) B ² B at (T;T N);(b) if B ² C at (T;T N) and B f D then D ² C at (T;T N);(c) if B ² C at (T;T N) and D f C then B ² D at (T;T N);(d) if B ² C at (T;T N) and D ² E at (T;T N) then B c D ² C c E at (T;T N).The following lemma treats some special cases (concerning the empty set), and follows directly from Definition 4.Lemma 3:If A is a set and T and T N are tables over A and B f A and C f A, then:(a) i² C at (T;T N) ] T = i or T N = i or *T Þ C c T NÞ C*# 1;(b) B ²i at (T;T N);(c) B ² C at (i;T N);(d) B ² C at (T;i).Lemma 4 relates transition dependency to momentary dependency (and to itself in (b)) and follows directly from Definition 4 as well.Lemma 4:If A is a set and T and T N are tables over A, and B f A and C f A, then:(a) B 6 C in T c T N] B ² C at (T;T N) and B 6 C in T and B 6 C in T N;13(b) B ² C at (T;T N) ] B ² C at (T N;T);(c) B 6 C in T ] B ² C at (T;T).The proofs of the foregoing lemmas are left to the reader. Although these lemmas all have simple proofs, they are useful enough to be stated explicitly.To illustrate part (a) of Lemma 4, we note that because of the uniqueness condition in the definition of WE in Example 1, the dynamic constraint (DC2) is equivalent to the dynamic constraint that {NO} 6 {DEPNO} in v(EMPL) c v N(EMPL).The following lemma shows how transition dependency interferes with some well-known table operations.Lemma 5:If A is a set and T and T N are tables over A, and B f A and C f A, then:(a) if B ² C at (T;T N) and X f T and X N f T N, then B ² C at (X;X N);(b) if B ² C at (T;T N), then B 6 C in (T ® (T NÞB)) c (T N® (TÞB));(c) if B is u.i. in T N, then:(B ² C at (T;T N) and T Þ B f T NÞ B) ] T Þ(B c C) f T NÞ(B c C).Proof:(a)This follows directly from Definition 4.(b)Let X = (T ® T NÞB) and X N = (T N® TÞB); now X f T and X N f T N, soB ²C at (T;T N) implies B ² C at (X;X N) according to part (a).We want to prove that B 6 C in X and B 6 C in X N,after which we can apply Lemma 4(a) to conclude that B 6 C in X c X N.Now, let x 0 X and y 0 X and x j B = y j B;then x 0 T and (›x N0 T N: x j B = x N j B) and y 0 T;hence, B ² C at (T;T N) implies that x j C = x N j C and y j C = x N j C(since y j B = x j B = x N j B);so, x j C = y j C. Thus, B 6 C in X.Similarly, we can prove that B 6 C in X N.(c) Y:T Þ B f T NÞ B, so œt 0 T: ›t N0 T N: t j B = t N j B;therefore, t j C = t N j C (since B ² C at (T;T N));hence, t j (B c C) = t N j (B c C).Thus, œt 0 T: ›t N0 T N: t j (B c C) = t N j (B c C),i.e., T Þ(B c C) f T NÞ(B c C).14Z:T Þ B f T NÞ B follows directly from T Þ(B c C) f T NÞ(B c C).Now, let t 0 T and t N0 T N and t j B = t N j B;we have to prove that t j C = t N j C.T Þ(B c C) f T NÞ(B c C), thus ›t NN0 T N: t j (B c C) = t NNj (B c C) since t 0 T.Now, t N j B = t j B = t NN j B, and B is u.i. in T N, so t N = t NN;hence, t j C = t NN j C = t N j C, which we had to prove.~We would like to define the concept of transition dependency also (and particularly) at the level of transition relations. We will call C constantly dependent on B in E at R iff C is transition dependent on B at (v(E);v N(E)) for each pair (v;v N) in R:Definition 5:If g is a set function, U is a database universe over g, E 0 dom(g), B and C are sets, and R f U × U, then:B ²C in E at R ]œ(v;v N) 0 R: B ² C at (v(E);v N(E)).The following five lemmas describe some basic properties of constant dependency; they are easily derived from the first five lemmas. We note that the properties in Lemma 6 again constitute the analogon to the Armstrong axioms; see [Ar 74].Lemma 6:If g is a set function, U is a database universe over g, E 0 dom(g), B f g(E), C f g(E), D f g(E), and R f U × U, then:(a) if C f B then B ² C in E at R;(b) if B ² C in E at R and C ² D in E at R then B ² D in E at R;(c) B ² C in E at R ]œc 0C: B ² { c } in E at R.The following lemma can be proven directly from Lemma 6, i.e., without reverting to the actual definition of constant dependency.Lemma 7:If g is a set function, U is a database universe over g, E 0 dom(g), A f g(E), B f g(E), C f g(E), D f g(E), and R f U × U, then:(a) B ² B in E at R;15。

zero-sum game英文解释A zero-sum game is a concept in game theory where the total gains and losses of all players involved in the game equal zero. This means that in a zero-sum game, what one player wins, another player must lose. In other words, the gains and losses of all players in a zero-sum game must add up to zero, making it a competitive and purely antagonistic scenario.In a zero-sum game, there is a fixed amount of resources or utility to be divided among the players, and the gain of one player directly corresponds to the loss of another player. This type of game is commonly found in competitive sports, gambling, and negotiations, where one party's success is directly tied to another party's failure.One of the classic examples of a zero-sum game is poker, where players compete for a fixed pot of money and the winnings of one player are directly taken from the losses of others. In this scenario, the sum of the gains and losses of all players involved in the game must add up to zero.Zero-sum games can be found in various real-world situations, such as business competition, international relations, and political negotiations. In these cases, parties involved areusually in a competitive and adversarial relationship, each striving to maximize their gains at the expense of others.It is important to note that not all situations in life arezero-sum games. In fact, many scenarios can be characterized as non-zero-sum games, where cooperation and collaboration can lead to mutual gains for all parties involved.In conclusion, a zero-sum game is a competitive scenario where the gains and losses of all players involved add up to zero. While these types of games are prevalent in various aspects of life, it is essential to recognize that cooperation and collaboration can lead to win-win outcomes in many situations. By understanding the dynamics of zero-sum games and seeking opportunities for mutual benefit, we can work towards creating a more harmonious and successful world for everyone.。