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This kit gives you everything you need for success. The kit includes a 231G Impact tool, a 170G 3/8 Ratchet, a socket set, and a blow-molded carrying case. Utilize the 231G when you need more power at 550 ft.
And use the 170G when you to fit into a tighter space. Exclusive twin hammer impact mechanism. Easy to use forward/reverse power regulator on 231G impact. Heat treated ratchet head for longer life. Air consumption at recommended operating pressure (CFM). No additional components or accessories are included. The item "1/2 Air Impact Wrench Gun 3/8 Pneumatic Ratchet Kit Bolting Socket Set with Case" is in sale since Thursday, March 23, 2017. This item is in the category "Home & Garden\Tools & Workshop Equipment\Air Tools\Air Tool Sets". The seller is "greatsavings4u2016" and is located in Plainfield, Indiana. This item can be shipped to United States. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,417 |
Q: How to run Solr Jetty in background I am using the Jetty/Solr build that comes with Solr and would like to run it in the background instead of in the terminal.
Right now I start it by java -jar start.jar but I would like it to log to a file and run in the background on the server so that I can close the terminal window.
I'm sure there is some java config that I can't find.
I have tried java -jar start.jar > log.txt & but no luck still outputs to the terminal window.
Thanks.
A: nohup is used to execute commands that runs after logout from a shell. What you need here is '2>&1'. This redirects standart error to the standart output. So everything will be logged to log.txt.
Try this
java -jar start.jar > log.txt 2>&1
Also you can add an '&' start it as a background process.
A: You can run it with screen if you are on unix.
A: You can properly install it as a linux service too.
cd to your jetty folder, for example mine is:
cd /home/spydon/jetty/
They have actually made most of the work with the jetty.sh file, so copy that one to /etc/init.d/
sudo cp ./bin/jetty.sh /etc/init.d/jetty
Then open the file with your favorite text editor, like vim or nano
sudo vim /etc/init.d/jetty
In the beginning simply uncomment (remove the hash(#)) three lines that says something like:
#chkconfig: 3 99 99
#description: Jetty 9 webserver
#processname: jetty
Meanwhile you have the text editor open, also add the jetty home directory to the beginning of the file, mine now looks like this:
#!/usr/bin/env bash
#
# Startup script for jetty under *nix systems (it works under NT/cygwin too).
JETTY_HOME=/home/spydon/jetty
# To get the service to restart correctly on reboot, uncomment below (3 lines):
# ========================
chkconfig: 3 99 99
description: Jetty 9 webserver
processname: jetty
# ========================
Now you should be able to start it with
sudo /etc/init.d/jetty start
And if you want it to run every time you reboot, simply add
sudo ln -s /etc/init.d/jetty /etc/rc1.d/K99jetty
sudo ln -s /etc/init.d/jetty /etc/rc2.d/S99jetty
This should work for most modern distros, but I've only tried it on debian based ones.
You could also consider doing a symlink to the jetty.sh so it will be easier to upgrade.
A: Try something like:
nohup yourcommand > output.log 2>&1 &
nohup will prevent yourcommand from being terminated in the event you log out.
& will run it in the background.
> output.log will send stdout to output.log
2>&1 will redirect stderr to stdout
A: You may want to try nohup, as explained in this previous answer.
| {
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{"url":"https:\/\/forum.bebac.at\/forum_entry.php?id=22099","text":"## Houston, we have a problem! [RSABE\u00a0\/ ABEL]\n\nDear Osama,\n\n\u00bb Thank you for the brilliant explanation.\n\nYou are welcome! In the meantime I added more stuff to my post.\nIf my interpretation of the GL is correct (is it?) and applied as such by members of the GCC, we have a problem if the CVwR observed in the study is \u226430%.\n\nWhat could be done?\n\u2022 Regulatory side\n\u2022 If high variability is suspected by the applicant, allow pre-specified wider limits for Cmax irrespective of the observed CV\u00a0\u2013 like currently in South Africa and acceptable by the EMEA prior to 2006.1 Does not even require a replicate design.\n\u2022 Implement ABEL instead. In line not only with the EMA but many other jurisdictions (the WHO, ASEAN States, Australia, Brazil, Canada, Chile, the East African Community, Egypt, the Eurasian Economic Union, New Zealand, the Russian Federation). A step towards glo\u00adbal harmonization. Lower inflation of the type\u00a0I error if CVwR \u226430% than with the current approach. However, inflation of the TIE also if CVwR >30% (up to ~45%), whereas there is none in the current approach. Many publications dealing with the issue; iteratively adjusting \u03b1 is provided by PowerTOST\u2019s function scABEL.ad(). Sample size estimation to compensate for the potential loss of power is provided by the function sampleN.scABEL.ad().\n\u2022 Implementing RSABE (USA, China) would be no good idea. Nasty inflation of the TIE if CVwR\u00a0<30% as well\u2026\n\u2022 Applicant\u2019s side\n\u2022 Ask the authority whether ABEL is an acceptable alternative to the current approach. Is it already?\n\u2022 If not, adjust \u03b1 with my -script. Be aware of the potential loss in power!\nMaybe (\u203c) I will implement it in scABEL.ad() and sampleN.scABEL.ad(). No promises.\nEdit: See this post.\n\u2022 Utopia\n\u2022 Within the last ten years many replicate studies were performed. Hence, we simply know a good number of drugs\u00a0\/ drug products which are highly variable and pose no safety concerns. Sometimes entire classes of drugs are highly variable (e.g., proton-pump inhi\u00adbi\u00adtors). Agencies could simply recommend widened limits in product-specific guidelines. No clinical justification2 needed by applicants, no replicate design needed, no issues with inflation of the TIE. Sigh.\n\n1. In the EU lots of accepted studies with 75.00\u2013133.33%. Prior to 2001 limits of 70\u2013143% were not uncommon for Cmax. Sometimes even for AUC\u2026\n2. An often overlooked detail. Regularly difficult to provide for generic companies with no access to the originator\u2019s data. Generally just a lot of in the protocol.\n\nDif-tor heh smusma\u00a0\ud83d\udd96\nHelmut Sch\u00fctz\n\nThe quality of responses received is directly proportional to the quality of the question asked.\u00a0\ud83d\udeae\nScience Quotes","date":"2022-05-27 13:01:50","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.4600069522857666, \"perplexity\": 4786.046413386959}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-21\/segments\/1652662647086.91\/warc\/CC-MAIN-20220527112418-20220527142418-00129.warc.gz\"}"} | null | null |
Q: PHP Errors on Making Local Copy of Server I have a site that I am trying to create a local copy of for test purposes (dusting off a past site). I was able to correctly get the db out and most of the site works great. However, I have places where things aren't loading correctly and are just giving me errors or messages in curly braces.
For example, a td class' name is being replaces with {BOX_HEADER} or on another page instead of a flash loading it just displays {PAGE} in the spot where the flash should be. Has anyone seen errors like this before?
I was thinking it could be PATH/echo errors but I was seeing if anyone had experienced problems like this before.
Thanks in advance.
A: There's a very good chance that you have some hard coded paths (and perhaps URL's also). When you copied the site, your docroot / home directory changed. So, if you had /home/elmer/public_html/site , you'll likely need to change that.
Look for an error_log file in all sub folders (especially where your template parser lives) and you should be able to root it out rather quickly. If all but DB configuration is stored in the DB, that's the first place to check.
A: It sounds like your templating solution is not working correctly. On the surface this is hard to diagnose because these are not typical php errors. You probably are going to have to do a little digging or provide some more information like which templating engine you're using.
| {
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{"url":"https:\/\/physics.stackexchange.com\/questions\/493169\/electric-potential-difference-between-coaxial-cables-clarification","text":"# Electric potential difference between coaxial cables clarification\n\nIn this video, he derives an expression for the potential difference between the two radii R2 and R1 where R2>R1\n\nHe arrives at the derivation by integrating the electric field with respect to radius. But his integration limits are from R1 to R2 (2:18 in video)\n\nWhats the reason behind this?\n\nI'm not understanding the basis of putting a lower value at the top of the integration and a higher value at the bottom, as I've always done the opposite.\n\n\u2022 You're aware that flipping the limits of integration simply multiplies the integral by $-1$, correct? \u2013\u00a0probably_someone Jul 23 '19 at 13:32\n\u2022 @probably_someone yes but is that all there is to it? In the video it seemed like there was a more of a reason behind it? But I guess the only reason he did it was so that the equation was positive at the end. \u2013\u00a0elliris Jul 23 '19 at 14:12","date":"2020-01-21 02:09:42","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7608078122138977, \"perplexity\": 242.7937921756135}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-05\/segments\/1579250601241.42\/warc\/CC-MAIN-20200121014531-20200121043531-00094.warc.gz\"}"} | null | null |
{"url":"https:\/\/www.kbwiki.ercoftac.org\/w\/index.php\/Test_Data_AC6-05","text":"# Annular compressor cascade with tip clearance\n\n## Overview of Tests\n\nExperimental data from the annular compressor cascade is available for four experiments. These correspond to tip-clearance sizes of 2% and 4% chord and with a rotating (N=6540rpm) or still (N=0) hub. Since all test cases correspond to the same mass flow rate, the main problem definition parameters (PDPs) are the tip-clearance size and the rotating hub speed. The test cases are summarized in Table EXP-A.\n\nCASE GNDP PDPs MPs\nRe ${\\displaystyle {(10^{6})}}$ m (kg\/s) t\/C (%) N (rpm) Raw Data DOAPs\nEXP-1 1.1 13.2 2 6540 ${\\displaystyle V_{axial},V_{per},V_{rad},{a},Pt,Ps,M}$ Circumferential mass average ${\\displaystyle V_{axial},V_{per},V_{rad},{a},Pt,Ps,M}$, PLC\nEXP-2 1.1 13.2 2 0 ${\\displaystyle V_{axial},V_{per},V_{rad},{a},Pt,Ps,M}$ Circumferential mass average ${\\displaystyle V_{axial},V_{per},V_{rad},{a},Pt,Ps,M}$, PLC\nEXP-3 1.1 13.2 4 6540 ${\\displaystyle V_{axial},V_{per},V_{rad},{a},Pt,Ps,M}$ Circumferential mass average ${\\displaystyle V_{axial},V_{per},V_{rad},{a},Pt,Ps,M}$, PLC\nEXP-4 1.1 13.2 4 0 ${\\displaystyle V_{axial},V_{per},V_{rad},{a},Pt,Ps,M}$ Circumferential mass average ${\\displaystyle V_{axial},V_{per},V_{rad},{a},Pt,Ps,M}$, PLC\n\nTable EXP-A: Summary of all test cases. Since we are dealing with a three-dimensional test case, and in order to reduce the amount of the experimental data, the competency of the calculations is judged on the basis of the circumferential mass average span-wise distributions of the flow quantities at measuring stations. Since the peripheral uniformity of the flow has been experimentally confirmed, the same distributions of the flow quantities at inlet and outlet stations provide the boundary conditions for the CFD calculations. The averaging of the flow quantities has been carried out using a simple averaging technique, with the mass flow rate being the weighting factor. The data corresponding to the mass average distributions is summarized in Table EXP-B.\n\nCASE DOAP 1 DOAP 2 DOAP 3\nSpan (%) Ps (Pa), Pt P(a), ${\\displaystyle V_{axial}\/V_{ref},V_{per}\/V_{ref},V_{rad}\/V_{ref},{a}{(^{0})},M}$, PLC at station 2 Span (%) Ps (Pa), Pt P(a), ${\\displaystyle V_{axial}\/V_{ref},V_{per}\/V_{ref},V_{rad}\/V_{ref},{a}{(^{0})},M}$, PLC at station 8 Span (%) Ps (Pa), Pt P(a), ${\\displaystyle V_{axial}\/V_{ref},V_{per}\/V_{ref},V_{rad}\/V_{ref},{a}{(^{0})},M}$, PLC at station 9\nEXP-1 exp11.dat exp12.dat exp13.dat\nEXP-2 exp21.dat exp22.dat exp23.dat\nEXP-3 exp31.dat exp33.dat\nEXP-4 exp41.dat exp43.dat\n\nTable EXP-B: Summary of all assessment parameters and available data files.\n\nDescription of experiment\n\nMany studies have been performed to investigate the effects of the tip-clearance flows on the performance of axial compressors and to study the structure of the flow-field, however there has been a lack of detailed measurements on high-speed machines. The experimental work conducted at NTUA\/LTT addresses this need by producing original data for an annular compressor cascade for two different tip-clearance sizes. In addition, and in order to study the effects that the relative end-wall\/blade motion introduces to the flow, measurements have been carried out for both a rotating (N=6540rpm) and a still hub for both tip-clearance configurations.\n\nTo provide as a complete a set of flow field data as possible, two types of measured data have been acquired:\n\n+ Measurements aiming at defining the operating point of the cascade and monitoring the operation of the facility. They have been carried out using long-nose 5-hole probes at measuring stations 2 and 9 (see Figure 5 for the definition of the measuring stations), which are regarded as the inlet and the outlet to the cascade, respectively. Measurements at the cascade inlet have been performed in order to obtain the actual flow conditions at the inlet to the test cascade, after the flow has developed within the annulus following the bent duct. Additionally, by examining the distributions of the measured quantities, the circumferential uniformity of the flow is assessed. In order to cover the whole blade-to-blade region, measurements have been performed by rotating the casing of the blade with respect to the probe that remained fixed at both stations.\n\nFigure 5: Definition of the measuring stations for the NTUA annular compressor cascade.\n\n+ Detailed measurements of the flow field at different axial locations on the blade passage, which provide the complete description of the three-dimensional flow field in the annular cascade. These measurements have been performed at five stations (4, 6, 8, 8a and 8b) using two long-nose 5-hole probes and at another five stations (3, 4, 5, 6 and 7) using a three-dimensional Laser Doppler Velocimetry (LDV) system. Two transparent glass windows enabled the optical access to the blade passage.\n\nWith respect to the pneumatic measurements, the probes were manufactured especially for this application according to the requirements posed by the geometry of the annulus and aiming at a minimal flow disturbances production. Their stem was inserted at the corresponding axial location, so that the nose head was located at the measuring plane. The probe nose was fixed on the stem at an angle greater than 90o in order to have the possibility of approaching the hub. On the other hand, this allowed coverage of a small part of the span-wise traverse, and to compensate for it a second probe was used. The second probe featured a smaller nose angle that was used to measure the flow field up to 50% of the span, since all phenomena of interest occur in this area. In order to verify the periodicity of the flow a circumferential extent of about 1.5 passages was covered at the measuring stations that were located downstream of the blade. At each axial plane 13-15 peripheral transverses of 17-24 peripheral locations were conducted, giving rise to a total of about 250 measuring locations at each station.\n\nThe data reduction was adapted to the particular requirements of the present geometry. The flow field quantities were derived from the pressure readings and the probe\u2019s calibration data. In this way the total and static pressure values were obtained, while the velocity vector referred to the coordinate system attached to the head of the probe. Thus a transformation was employed to derive the orientation of the vectors in a cylindrical coordinate system.\n\nThe three-dimensional LDV system used in the measurements comprises two independent systems, a one component and a two-component system, which employ independent laser sources, fibre optics and small diameter optical probes for the transmission and the collection of the scattered light from the three measuring volumes. Its layout is presented in Figure 6. The two-dimensional probe transmits two orthogonal beam-pairs and measures the axial and peripheral velocity components, while the one-dimensional component that transmits a third pair of laser beams is placed at an angle of 25o with respect to the two-dimensional component and therefore measures a component inclined with respect to the plane of the components measured by the two-dimensional component, allowing thus the measurement of the third component of the velocity vector. The flow was seeded with a TSI six-jet atomizer, using a solution of paraffin oil and benzene as a seeding medium. It is introduced at the inlet of the bellmouth, at a location of the stream-wise position on this plane, which corresponds to the test passage. The span-wise traversing is accomplished by means of a remotely controlled carriage, while the circumferential traversing is performed by means of rotating the cascade with respect to the fixed laser carriage.\n\nFigure 6: Layout of the 3-D Laser Doppler Velocimetry system.\n\nAcquisition of the data (frequencies) obtained by the laser is conducted via an IEEE data acquisition board. The processing of the data is achieved by a PC using custom-made LDV software. The acquisition of the analogue signals (DC voltage), obtained from the pressure transducers connected to the 5-hole probes, is conducted via a 16-channel AD data acquisition board (ADDVTECH PCL-818). The acquisition board, having a maximum total sampling frequency of 1MHz, variable input voltage range (maximum range \u00b110 V) and a 12-bit Analogue-to-Digital converter with a sensitivity of 4.88mV, feeds the data to a PC utilising in-house software for on-line processing.\n\nApart from the pneumatic measurement readings, several other quantities of interest are monitored at different locations of the facility during operation, namely:\n\n+ The rotational speed of both the compressor driving the cascade and the hub.\n\n+ The vibration levels of the compressor bearings, in order to ensure safe operation.\n\n+ The total and static pressure at the scroll inlet.\n\nBoundary Data\n\nAt the inlet boundary (Station 2) the flow is considered circumferentially uniform, as confirmed by the experiments. As such, all flow quantities included in the data files for this station can be used as boundary conditions. Although flow is turbulent, detailed measurements of turbulence quantities have not been performed and only estimates of these quantities can be reported (for a example a realistic value for turbulent intensity at inlet is 2%). With such estimates, the impact of the inlet turbulent quantities on the DOAPs is expected to be low. The outflow boundary (Station 9) is located far downstream and an adequate mixing of the flow has occurred. As such, the corresponding data files provide enough information on the boundary conditions. The hub and the casing are walls that are considered to be smooth. This is the case for the blade surfaces too, the pressure and suction sides, as well as the blade tip. Finally the remaining boundaries are periodic and must be treated as such.\n\nMeasurement Errors\n\nThe inaccuracy of angle measurements for the alignment of the LDV\/probe assembly yields a systematic error of the order of \u00b10.3o to the measured flow angles. For the LDV measurements, the statistical errors for the measured quantities were \u00b12% for the normal and shear stresses, \u00b11% and \u00b12.6% for the directly and indirectly, respectively, measured velocity components. For the 5-hole probes, the expected errors of the measured quantities were \u00b10.5o and \u00b11.3o for the yaw and the pitch angles, respectively, and \u00b11% for the total velocity and the static and total pressure. These figures are valid for all experimental cases.\n\nIt has already been mentioned that the flow in the cascade has been measured twice. In Figure 7 the circumferential mass average span-wise distributions of the flow quantities are plotted at Station 2 (inlet to the cascade). Qualitatively, these distributions are similar, despite the small differences, of the order of 2%, that appear. The differences appearing in these plots provide a realistic estimation of the overall accuracy of the experimental set of data. Nevertheless, the first set of measurements subjects to a larger degree of uncertainty, since an inspection of the test facility before the second measurements revealed large deposits of dust accumulated on the walls (particularly in the inlet section) as a result of construction work conducted at the laboratory at that time.\n\nFigure 7: Span-wise distributions of the circumferential mass average flow quantities at Station 2 for EXP-1. Red lines: First experimental set, green lines: second experimental set.\n\n## References\n\n+ Doukelis, A., Mathioudakis, K., Founti, M. and Papailiou, K. (1997), \u201c3-D LDA Measurements in a Annular Cascade for Studying Tip Clearance Effects,\u201d 90th on the Propulsion and Energetics Panel on Advanced non Intrusive Instrumentation for Propulsion Engines, Brussels Belgium, published by AGARD.\n\n+ Doukelis, A., Mathioudakis, K., and Papailiou, K., (1998a), \u201cThe Effect of Tip Clearance Gap Size and Wall Rotation on the Performance of a High-Speed Annular Compressor Cascade,\u201d ASME Paper 98-GT-38, Presented at the International Gas Turbine & Aeroengine Congress & Exhibition, Stockholm, Sweden, June 2-5, 1998.\n\n+ Doukelis, A., Mathioudakis, K., and Papailiou, K., (1998b), \u201cInvestigation of the 3-D Flow Structure in a High-Speed Annular Compressor Cascade for Tip Clearance Effects,\u201d ASME Paper 98-GT-39, Presented at the International Gas Turbine & Aeroengine Congress & Exhibition, Stockholm, Sweden, June 2-5, 1998.\n\n+ Doukelis, A., Mathioudakis, K., and Papailiou, K., (1999), \u201cEffect of Wall Rotation on the Performance of a High-Speed Compressor Cascade with Tip Clearence,\u201d ISABE Paper 99-7267, Presented at XIV ISABE, September 5-12, 1999, Florence, Italy.\n\n+ Mathioudakis, K., Papailiou, K., Neris, N., Bonhommet, C., Albrand, G., Wegner, U., (1997), \u201cAn Annular Cascade Facility for Studying Tip Clearance Effects in High Speed Flows,\u201d Proceedings of XIII ISABE, September 7-12, 1997, Chattanooga, Tennessee, U.S.A., Edited by F. S. Billig, Published by AIAA, Vol. 1, pp. 831-839.\n\n+ Pouagare, M. and Delaney, R. A. (1986), \u201cStudy of Three-Dimensional Viscous Flows in an Axial Compressor Cascade Included Tip Leakage Effects Using a SIMPLE-Based Algorithm,\u201d ASME Journal of Turbomachinery, Vol. 108, pp. 51-58.\n\n## Test Case EXP-1\n\nMeasured Data\n\nThis test case corresponds to a tip-clearance size t\/C=2%, a hub rotational speed N=6540rpm and a mass flow rate m=13.2kg\/s. Circumferential mass average span-wise distributions of flow quantities (axial component of velocity, Vaxial, peripheral component of velocity, Vper, radial component of velocity, Vrad, flow angle, a, Mach number, M, total pressure, Pt, and static pressure, Ps) derived from the corresponding measurements at three stations are provided. PLC distributions are derived using Equation (1). Velocity components are non-dimensional with respect to the peripheral velocity at hub.\n\nexp11.dat (ASCII file; headers: Station 2, t\/C=2%, N=6540rpm, m=13.2Kg\/s, columns: span, Ps, Pt, Vaxial\/Vref, Vper\/Vref, Vrad\/Vref, a, M, PLC)\n\nexp12.dat (ASCII file; headers: Station 8, t\/C=2%, N=6540rpm, m=13.2Kg\/s, columns: span, Ps, Pt, Vaxial\/Vref, Vper\/Vref, Vrad\/Vref, a, M, PLC)\n\nexp13.dat (ASCII file; headers: Station 9, t\/C=2%, N=6540rpm, m=13.2Kg\/s, columns: span, Ps, Pt, Vaxial\/Vref, Vper\/Vref, Vrad\/Vref, a, M, PLC)\n\n## Test Case EXP-2\n\nMeasured Data\n\nThis test case corresponds to a tip-clearance size t\/C=2%, a still hub (N=0rpm) and a mass flow rate m=13.2kg\/s. Circumferential mass average span-wise distributions of flow quantities (axial component of velocity, Vaxial, peripheral component of velocity, Vper, radial component of velocity, Vrad, flow angle, a, Mach number, M, total pressure, Pt, and static pressure, Ps) derived from the corresponding measurements at three stations are provided. PLC distributions are derived using Equation (1). Velocity components are non-dimensional with respect to the peripheral velocity at hub.\n\nexp21.dat (ASCII file; headers: Station 2, t\/C=2%, N=0rpm, m=13.2Kg\/s, columns: span, Ps, Pt, Vaxial\/Vref, Vper\/Vref, Vrad\/Vref, a, M, PLC)\n\nexp22.dat (ASCII file; headers: Station 8, t\/C=2%, N=0rpm, m=13.2Kg\/s, columns: span, Ps, Pt, Vaxial\/Vref, Vper\/Vref, Vrad\/Vref, a, M, PLC)\n\nexp23.dat (ASCII file; headers: Station 9, t\/C=2%, N=0rpm, m=13.2Kg\/s, columns: span, Ps, Pt, Vaxial\/Vref, Vper\/Vref, Vrad\/Vref, a, M, PLC)\n\n## Test Case EXP-3\n\nMeasured Data\n\nThis test case corresponds to a tip-clearance size t\/C=4%, a hub rotational speed N=6540rpm and a mass flow rate m=13.2kg\/s. Circumferential mass average span-wise distributions of flow quantities (axial component of velocity, Vaxial, peripheral component of velocity, Vper, radial component of velocity, Vrad, flow angle, a, Mach number, M, total pressure, Pt, and static pressure, Ps) derived from the corresponding measurements at three stations are provided. PLC distributions are derived using Equation (1). Velocity components are non-dimensional with respect to the peripheral velocity at hub.\n\nexp31.dat (ASCII file; headers: Station 2, t\/C=4%, N=6540rpm, m=13.2Kg\/s, columns: span, Ps, Pt, Vaxial\/Vref, Vper\/Vref, Vrad\/Vref, a, M, PLC)\n\nexp33.dat (ASCII file; headers: Station 9, t\/C=4%, N=6540rpm, m=13.2Kg\/s, columns: span, Ps, Pt, Vaxial\/Vref, Vper\/Vref, Vrad\/Vref, a, M, PLC)\n\n## Test Case EXP-4\n\nMeasured Data\n\nThis test case corresponds to a tip-clearance size t\/C=4%, a still hub (N=0rpm) and a mass flow rate m=13.2kg\/s. Circumferential mass average span-wise distributions of flow quantities (axial component of velocity, Vaxial, peripheral component of velocity, Vper, radial component of velocity, Vrad, flow angle, a, Mach number, M, total pressure, Pt, and static pressure, Ps) derived from the corresponding measurements at three stations are provided. PLC distributions are derived using Equation (1). Velocity components are non-dimensional with respect to the peripheral velocity at hub.\n\nexp41.dat (ASCII file; headers: Station 2, t\/C=4%, N=0rpm, m=13.2Kg\/s, columns: span, Ps, Pt, Vaxial\/Vref, Vper\/Vref, Vrad\/Vref, a, M, PLC)\n\nexp43.dat (ASCII file; headers: Station 9, t\/C=4%, N=0rpm, m=13.2Kg\/s, columns: span, Ps, Pt, Vaxial\/Vref, Vper\/Vref, Vrad\/Vref, a, M, PLC)","date":"2022-05-28 13:12:36","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 12, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6864496469497681, \"perplexity\": 4223.88515630141}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-21\/segments\/1652663016853.88\/warc\/CC-MAIN-20220528123744-20220528153744-00673.warc.gz\"}"} | null | null |
Q: Array copy in ubuntu Linux doesn't actually work I mean, this thing makes me feel like a naive programmer :')
I'll explain it.
For my buttons controller input I use two arrays, one for the current button pressed and one for the last button pressed. The thing I'm doing is, for every frame, copying the content of the "current" array to the "previous" array and then use glfw to get the buttons state into the "current" array.
Here's the code i'm talking about:
for(int j=0; j<Input::joyButtonsCount[i]; j++)
Input::joyPrevButtons[i][j] = Input::joyCurrButtons[i][j];
Input::joyCurrButtons[i] = (char*)glfwGetJoystickButtons(i, &Input::joyButtonsCount[i]);
Now, under Windows all works correctly, but in Linux it doesn't.
It results in the prev and curr arrays to be the same.
Can someone explain me why it occurs?
If you need this information I am using g++ under Linux and MinGW under Windows.
A: From what I see glfwGetJoystickButtons returns a pointer to some internal array. You are assigning it to Input::joyCurrButtons[i] and in next iteration you assume it will have a previous values. This might not be true (maybe its implementation defined), as system might have updated this array internally with new values. This way your first loop will copy to joyPrevButtons new values and not the previous ones.
glfwGetJoystickButtons returns a const pointer to make sure API user will not try to modify its values. So you should not cast it to non-const. Also its often a sign that you should not keep this pointer, but use it only to copy values instead.
My suggestion for you is not to store in pointer from glfwGetJoystickButtons in Input::joyCurrButtons[i], but instead copy values from glfwGetJoystickButtons to Input::joyCurrButtons[i] array.
A: Considering the code snippet you provided, there is no reason why the array copy does not perform the same way in Linux as in Windows. Your problem is likely elsewhere.
I suggest that you check the following points:
*
*is Input::joyButtonsCount[i] properly defined on both platform ?
*double check on both platforms that Input::joyCurrButtons[i] properly changes when you press button #i.
*check on both platform if the copy happens at time you expect; it's likely that, on Linux, the copy happens before you expect, so that when you check values, they are the same (because already copied earlier); then you will have to understand why copy happens before expected time.
A: For everyone else in my situation (such as noobs like me :P) I'll post the problem.
It seems that I had to copy the pointer to an array and not get the pointer and use that as my input data.
Here's how this code is now:
Input::joyPrevButtons[i] = (char*)memcpy(Input::joyPrevButtons[i], Input::joyCurrButtons[i], Input::joyButtonsCount[i]);
Input::joyCurrButtons[i] = (char*)memcpy(Input::joyCurrButtons[i], glfwGetJoystickButtons(i, &Input::joyButtonsCount[i]), Input::joyButtonsCount[i]);
Obviously I had to include the string header
#include <string.h>
But I still don't know why it doesn't work on Linux as it works on Windows.
| {
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Eric William Thomas Tindill (18 December 1910 – 1 August 2010) was a New Zealand sportsman. Tindill held a number of unique records: he was the oldest ever Test cricketer at the time of his death, the only person to play Tests for New Zealand in both cricket and rugby union (a so-called "double All Black"), and the only person ever to play Tests in both sports, referee a rugby union Test, and umpire a cricket Test: a unique "double-double".
Early life
Tindill was born in Nelson and was raised in Motueka. His family moved to Wellington in 1922, and he was educated at Wellington Technical College until 1925. He trained as an accountant, and worked as a civil servant for 40 years in the government audit office. He was nicknamed "Snowy" due to his fair hair. He married his wife Mary in 1937, shortly before he left on a cricket tour to England. An all-round sportsman, in addition to cricket and rugby, Tindill also played football for Wellington in 1927, and was a founder of the Wellington Table Tennis Association in 1932.
Cricket
In cricket, Tindill played club cricket for the Midland club (now Eastern Suburbs Cricket Club). He played domestic first-class cricket for Wellington from 1932–33 to 1949–50 as a wicket-keeper/batsman and left-handed opening batsman. He made a century on his first-class debut in January 1933, scoring 106 as an opening batsman in a Plunket Shield match against Auckland at Eden Park.
He also played five Tests for the New Zealand cricket team. He toured England under Curly Page in 1937, playing in 25 tour matches, including the three Test matches at Lord's, Old Trafford and the Oval. The report of the tour in the 1938 edition of Wisden Cricketers' Almanack stated that Tindill "did nothing out of the common with the bat, but as a wicket-keeper he was always worth his place". Later that year, in a match played against South Australia in Adelaide on the return journey to help cover the costs of the England tour, he caught Don Bradman for 11 – Bradman's only appearance against a New Zealand team – off the bowling of Jack Cowie in the opening over of the Saturday's play. Unfortunately, this caused large numbers of spectators who were queuing to enter the ground to leave, costing the New Zealand team the gate money and defeating the purpose of the game.
He served in the NZEF in the Second World War, in North Africa. He then played in the first two Test matches in New Zealand after the War. In the Test against Australia at Wellington in 1945–46, New Zealand were bowled out for 42 and then 54, Tindill making 1 and 13. He also played in the only Test against the touring English side at Lancaster Park, Christchurch, in 1946–47. His Test batting average of 9.12 did not reflect his talent.
He played his last first-class game for Wellington against the touring Australians in 1950. He scored six first-class centuries and finished with an average of 30.35 in 116 innings in 69 first-class matches. He reached his top score – 149 – playing for Wellington against Auckland in 1948. As a wicket-keeper, he took 96 catches and 33 stumpings.
Rugby
In rugby, Tindill alternated between half-back and first five-eighth for Wellington (Athletic) between 1932 and 1945, and was noted for his kicking of drop goals, then worth 4 points against 3 for a try. He made his debut for Wellington playing against the All Blacks in 1932, before they left on tour; he scored a try, and the provincial side beat the national team, 36–23.
The vast supply of midfield talent in Wellington during the 1930s made it difficult for him to get noticed, but the selectors for the All Blacks were remarkably thorough at seeking out talent—the trials for the team that would tour Britain in 1935–36 would see 188 players take part to fill 30 places. He was selected for that tour, and played in New Zealand's first loss to a club side against Swansea, 3–11, but scored two drop goals against London Counties on 26 December 1935, and was selected to play one Test against England at Twickenham on 4 January 1936. The match is most famous for the two tries scored by England's Alexander Obolensky – the son of a Russian émigré prince, playing in his first Test – and England beat New Zealand for the first time, 13–0.
Tindill was unable to play against South Africa in 1937, having already departed to tour England with the New Zealand cricket team, but he joined the All Blacks tour to Australia in 1938, playing in three games against state sides. The match against England in 1936 remained his only rugby Test. In all, he played 17 matches with the All Blacks, including one Test, and scored 24 points for 6 drop goals. Famed rugby commentator Winston McCarthy was certain that Tindill would have won selection for the 1940 tour of South Africa had it not been scrapped due to the outbreak of World War II.
Umpire and referee
After retiring from active sport, Tindill also refereed rugby at domestic and Test level. He was noted for his keen instinct of the game's progress; McCarthy would recall, "In one match in which six tries were scored I saw the six of them scored at Eric's feet as he waited for the player to ground the ball." The pinnacle of his career as a rugby referee came in 1950, when he oversaw the first two Tests of the series between the Lions and All Blacks, in Dunedin and Christchurch. He would later umpire a cricket Test at Lancaster Park in 1959, having refereed the rugby Test there in 1950. He also refereed the rugby match against Australia at Dunedin in 1955.
Tindill was later an umpire, standing in one Test at Lancaster Park in 1959 with Jack Cowie, which England won by an innings and 99 runs inside three days, thanks to Ted Dexter's 141 and Tony Lock's 5–31 and 6–53.
Later life
Tindill was secretary of the Wellington Cricket Association, and a selector for both Wellington and New Zealand cricket teams. He was a member of the panel that selected the New Zealand team that achieved the country's first Test victory, against the West Indies in 1956. He was also Treasurer of the New Zealand Boxing Council from 1973 to 1981. In the 1981 Queen's Birthday Honours, Tindill was appointed an Officer of the Order of the British Empire, for services to rugby and cricket. He was inducted as a member of New Zealand's Sports Hall of Fame in 1995. He received a Halberg award for services to sport in 2000.
He also wrote a best-selling book, The Tour of the Third All Blacks, about the 1935–36 tour, with fellow All Black Charlie Oliver, and wrote a history of Wellington Athletic Rugby Football Club in 1976. Of his five children, his son Paul played for Wellington in both rugby and first-class cricket, and another son Dennis played rugby for Wellington B in 1964.
Tindill died in Wellington in 2010 and was buried at Karori Cemetery.
Records
While Tindill is the only person to have played Tests in both cricket and rugby union for New Zealand, six other players have represented New Zealand in both rugby union and cricket – the others being George Dickinson and Curly Page (Tests in cricket only); Brian McKechnie, Charlie Oliver and Jeff Wilson (Tests in rugby only); and Bill Carson (no Test at either sport).
Tindill became the oldest living Test cricketer on 16 February 2004, on the death of Don Cleverley. On 8 November 2009, he surpassed Francis MacKinnon, who played one Test for England in 1879 and lived to 98 years and 324 days, as the oldest Test cricketer in history. His Test longevity record was surpassed on 23 March 2011 by Norman Gordon.
He became the oldest living All Black on 8 October 2001, on the death of Ray Williams, and was the last surviving All Black who played a Test before World War II. The oldest ever Test rugby player remains Scotland's Mac Henderson, who died on 5 March 2009 aged 101 years and 309 days.
After Tindill's death, the oldest living Test cricketer became former South African fast bowler Norman Gordon, and the oldest living All Black was Morrie McHugh.
Notes and references
External links
Eric Tindill at Cricket Archive
1910 births
2010 deaths
New Zealand Test cricketers
Wellington cricketers
New Zealand international rugby union players
New Zealand Test cricket umpires
New Zealand rugby union referees
Rugby union fly-halves
Rugby union scrum-halves
People educated at Wellington High School, New Zealand
New Zealand Officers of the Order of the British Empire
Burials at Karori Cemetery
North Island cricketers
New Zealand military personnel of World War II
Wicket-keepers | {
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Recent events are raising expectations among members of the "9/11 Truth movement" that an earthshaking 9/11 disclosure is imminent. The Russian-language Pravda.ru, an offshoot of Pravda, the former official mouthpiece of the Communist Party Of the Russian Federation, recently echoed the buildup of hopes and expectations in the "9/11 truther" community.
In a Feb. 7 article, the website claimed that the Russian government was preparing to release evidence that 9/11 was a "false flag" operation orchestrated by the U.S. government and its intelligence services.
According to Pravda, the evidence against the U.S. government consists of satellite imagery in the possession of the Russian authorities.
The alternative news website Veterans Today published a gabbled Google translation of the Pravda article, originally in the Russian language.
According to the 9/11 conspiracy theory websites currently circulating the Pravda article, the Illuminati are "freaking out in fear" over the news that Putin is about to expose them.
The problem, however, is that Pravda is a pro-Kremlin conspiracy theory website that buys into everything anti-U.S. with non-satirical conviction.
But Pravda is not the first to claim that Putin's government is preparing to release information showing that 9/11 was an "inside job." The far left-leaning website Global Research has also consistently championed an alternative narrative with regard to 9/11, which involves Putin's government exposing the U.S. government's role in staging the Sept. 11 attacks.
In an article published by Global Research in August, 2014, titled "The Walls Are Crumbling Down Around the 'official 9/11 Story,'" the website claimed that "an absolutely monumental shift" in the official 9/11 narrative was imminent.
Although Graham's statement that a "huge breakthrough was coming with classified 9/11 information" was in reference to beliefs that members of the Saudi ruling elite were involved in the Sept. 11 attacks, 9/11 truthers generally believe that the move by the Bush administration to classify 28 pages of the 9/11 intelligence report was motivated by fears that the pages could also reveal the role of the U.S. government and intelligence services in concert with the Saudis.
Global Research linked Graham's promise of "huge breakthrough" in declassifying the 28 pages of the intelligence reports with allegations that in May 2014, "Putin threatened … that he had evidence that 9/11 was an 'inside job' and [that he] was going to release it" in retaliation against NATO and the U.S. over the crisis in Ukraine.
Strangely, although 9/11 truthers have claimed for years that Putin is planning to release evidence that 9/11 was a false flag operation orchestrated by the U.S. intelligence services, the only time that Putin spoke publicly about the matter, he denied the allegations.
Despite Putin's public disavowal of 9/11 conspiracy theories, truthers continue to claim that Putin is planning to release evidence that 9/11 was a "false flag" operation.
According to 9/11 truthers, the U.S. has devised a plan to counter move by Putin to expose its role in the Sept. 11 attacks. The plan, according to Global Research, involves the U.S. government "front-running" Putin by releasing an alternative version of events that will implicate the Saudis and thus distract the attention of Americans from Putin's version of events.
The U.S. government will attempt to use the Saudis and a few expendable individuals in the Bush administration as "fall guys" to take the blame for an ostensibly unauthorized "inside job." According to 9/11 truthers, those officials in the Bush administration chosen to take the blame will be branded as rogue elements within the Bush administration who worked with the Saudis without the knowledge of the president and key officials of his administration.
The noise generated by the ensuing "scandal," according to 9/11 truthers, will thus serve to distract the public from the full blast of the implication of Putin's disclosure that 9/11 was a false flag operation orchestrated by the government.
The engineered "scandal" would also provide convenient casus belli for the U.S. government to begin pushing for a regime change in Saudi Arabia like it did against Saddam Hussein after he ceased to serve U.S. geopolitical interests. | {
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Erinije (gr. Ερινύες; erinyes v pomenu grmeče) so v grški mitologiji tri boginje maščevanja, ki imajo kače namesto las, ter strupene krvave solze.
Boginje maščevanja, spadajo med božanstva podzemlja. Bile so služabnice Hada in Perzefone. Njihova naloga je bila kaznovati zločince in vse, ki so motili javni red. Pravičnim se jih ni bilo treba bati, vendar so bile kljub temu med bogovi in ljudmi nepriljubljene.
Njihove rimske sorodnice so bile furije.
Viri
Zunanje povezave
Grške boginje
Mitološke osebe, po katerih so poimenovali asteroid
Trojna božanstva
Boginje maščevanja | {
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puppet-etc_puppetlabs
=====================
Base Directory for Puppet 4.2 and R10K Environment Configuration
Added Files
-----------
- r10k/r10k.yaml
- code/hiera.yaml
- code/Puppetfile
Contributors
------------
* Peter Pouliot <peter@pouliot.net>
Copyright and License
---------------------
Copyright (C) 2013 Peter J. Pouliot
Peter Pouliot can be contacted at: peter@pouliot.net
Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at
http://www.apache.org/licenses/LICENSE-2.0
Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.
| {
"redpajama_set_name": "RedPajamaGithub"
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The Paleogenetics group focuses on the evolution of human health with specific interests in the evolution of pathogens and microbiomes over time.
We recover DNA from a wide range of ancient, historic and modern samples to better understand how humans adapted to changes in environment, diet, and pathogens. State-of-the-art techniques in ancient DNA (link: clean room facilities) in combination with Next Generation Sequencing (link: Functional Genomic Centre Zürich) are used in this context. Our major goals are to obtain new insights into host-pathogen interactions in the past as well as to trace back the evolutionary history of pathogens. Using ancient pathogen genomics, we can identify causative agents of past pandemics and contribute to the reconstruction the origins and evolution of major human infectious diseases, such as leprosy or syphilis. | {
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Rhinorex is a genus of kritosaurin hadrosaur from the Late Cretaceous Neslen Formation, in central Utah. Its exact placement in time is uncertain, though it probably dates to 75 million years ago and was discovered in estuarine sediments.
It is likely a close relative of Gryposaurus, and indeed some have suggested that it falls within the genus Gryposaurus, as the phylogenetic analysis indicates.
It is found in a very similar time and place as Gryposaurus monumentensis and Gryposaurus sp. This would challenge the idea of provincialism in Late Cretaceous Laramidian faunas. It may be that Rhinorex lived in more coastal environments than Gryposaurus. The holotype specimen BYU 13258 is composed of a partial but mostly articulated skeleton, which includes the skull, the vertebral column and a partial pelvis.
Etymology
The type and only valid species known is Rhinorex condrupus. The genus name is derived from the Greek word "rhinos" which means "nose", and the Latin word "rex" which means "king." The specific name is derived from the Latin word "condo" which means "bury" and refers to the specimen being buried in rock and the Latin word "rupes" which means "cliffs" and is a reference to the fossil being discovered in the Book Cliffs of Utah.
See also
Timeline of hadrosaur research
References
Saurolophines
Late Cretaceous dinosaurs of North America
Fossil taxa described in 2014
Paleontology in Utah
Campanian genus first appearances
Campanian genus extinctions
Ornithischian genera | {
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> Index "L"
Race or Ethnicity:
Won't wake up for less than $10K/day
5' 9½" (1.77 m)
86-61-89 (EU); 34-24-35 (US)
36-38 (EU); 6 (US)
40 (EU); 9½ (US); 7 (UK); 25½ (JP)
Linda Evangelista - Pictures
Additional Free Pictures of Linda Evangelista
Linda Evangelista - Biography
It was a haircut that did it for Linda Evangelista – or rather, several haircuts and, later, various changes of color. The Canadian model had been modelling for several years, picking up around $600 per shoot for "mediocre" work until she went for the chop and parted with her long, light brown hair.
When British Vogue picked up on some shots Arthur Elgort had taken of her, she went stellar. Years later, after retiring from the modelling scene, she would make a comeback on the cover of the US edition of the magazine that first made her famous.
Born on May 10, 1965, into a large Italian-Canadian family in St Catherines, a small town near Niagara Falls, Linda developed striking good looks and was first spotted while just a teenager at a local beauty contest. An early modelling experience almost made her turn her back on the business, however. While in Japan on a work assignment, she contacted the Canadian embassy and was flown home after being asked to do nude photos. Later, nudity was not something Linda minded.
Evangelista later moved to New York City and signed with Elite Model Management there. She then moved to Paris, France, to further her career. She also appeared in music videos with George Michael.
Evangelista uttered the quote, "We don't wake up for less than $10,000 a day" (often misquoted as: "We don't get out of bed for less than..." or "I don't get out of bed for less than...") Spoken in Vogue (1990) to Jonathan van Meter, talking about money and how she and a few other models were calling the shots and changing the game.
In the late 1980s, she cut all her hair off to brandish a little-boy-style haircut and was promptly cancelled from all the important runway shows that season. This was the first of many dramatic hairstyle changes. Within months, she appeared on magazine covers everywhere and many women emulated the hairstyle.
In 2007, she signed a multiple-year exclusive contract with the cosmetics company L'Oreal Paris. It was announced in early 2008 that she would be featured in the Prada Fall 2008 campaign seen in magazines internationally.
She is signed to DNA Model Management in New York City, and Models 1 in London.
Evangelista was married to Elite executive Gerald Marie from 1987 to 1993. She has also dated actor Kyle MacLachlan (1992-1998), French football player Fabien Barthez (1998-2000), Italian oil mogul Ugo Brachetti Peretti (2003-2004), and former Formula One driver Paolo Barilla (2005).
Throughout her career, Evangelista has expressed her desire to have children.
To her delight, on October 11, 2006, she gave birth to a boy, Augustin James. While pregnant, Evangelista appeared on the August 2006 issue of Vogue magazine. She was the first non-actress/singer model to appear on Vogue in over a year.
Linda Evangelista - Personal Quotes
"It was God who made me so beautiful. If I weren't, then I'd be a teacher."
"I don't get out of bed for less than $10,000 a day."
"I don't diet. I just don't eat as much as I'd like to."
"People think modeling's mindless, that you just stand there and pose, but it doesn't have to be that way. I like to have a lot of input. I know how to wear a dress, whether it should be shot with me standing or sitting."
[In 2005, retracting her statement about not getting out of bed for less than $10,000 a day.] I can't stand the way this follows me around. I said it a long time ago, and I hope today I am a different person. Now I get out of bed for a much better reason. I'm part of a team that raises millions of dollars and raises awareness of HIV and AIDS all over the world.
Linda Evangelista - Filmography
"Sex and the City" .... Saleswoman (1 episode, 2003)
- A Woman's Right to Shoes (2003) TV Episode .... Saleswoman
The Loss of Sexual Innocence (1999) (uncredited) .... Italian Woman
"Primero izquierda" (1 episode, 1992)
- Episode dated 6 February 1992 (1992) TV Episode
Linda Evangelista - Related Links
Wikipedia: Linda Evangelista
YouTube: Linda Evangelista
Linda Evangelista at Babemania.com
Webmasters - LinkXchange
Copyrights are the property of their respective owners. The images displayed on this site are for newsworthy purposes only. All of the images on this site are either the property of CelebStar.net, used with permission of their respective copyright owners, or believed to have been granted into the public domain.
All original content Copyright ©
CelebStar™ All Rights Reserved. | {
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namespace DawidPerdekLab2
{
partial class FormMain
{
/// <summary>
/// Required designer variable.
/// </summary>
private System.ComponentModel.IContainer components = null;
/// <summary>
/// Clean up any resources being used.
/// </summary>
/// <param name="disposing">true if managed resources should be disposed; otherwise, false.</param>
protected override void Dispose(bool disposing)
{
if (disposing && (components != null))
{
components.Dispose();
}
base.Dispose(disposing);
}
#region Windows Form Designer generated code
/// <summary>
/// Required method for Designer support - do not modify
/// the contents of this method with the code editor.
/// </summary>
private void InitializeComponent()
{
this.labelMyNameAndSurname = new System.Windows.Forms.Label();
this.buttonShowPerson = new System.Windows.Forms.Button();
this.buttonShowHiperPerson = new System.Windows.Forms.Button();
this.dataGridViewListOfPeople = new System.Windows.Forms.DataGridView();
this.textBoxNewPersonName = new System.Windows.Forms.TextBox();
this.textBoxNewPersonSurname = new System.Windows.Forms.TextBox();
this.textBoxNewPersonAge = new System.Windows.Forms.TextBox();
this.buttonShow = new System.Windows.Forms.Button();
this.buttonAddPerson = new System.Windows.Forms.Button();
this.labelListOfPeople = new System.Windows.Forms.Label();
this.dataGridViewGrades = new System.Windows.Forms.DataGridView();
this.labelGrades = new System.Windows.Forms.Label();
this.textBoxGrade = new System.Windows.Forms.TextBox();
this.buttonAddGrade = new System.Windows.Forms.Button();
this.pictureBoxGrades = new System.Windows.Forms.PictureBox();
((System.ComponentModel.ISupportInitialize)(this.dataGridViewListOfPeople)).BeginInit();
((System.ComponentModel.ISupportInitialize)(this.dataGridViewGrades)).BeginInit();
((System.ComponentModel.ISupportInitialize)(this.pictureBoxGrades)).BeginInit();
this.SuspendLayout();
//
// labelMyNameAndSurname
//
this.labelMyNameAndSurname.AutoSize = true;
this.labelMyNameAndSurname.Font = new System.Drawing.Font("Microsoft Sans Serif", 24F, System.Drawing.FontStyle.Bold, System.Drawing.GraphicsUnit.Point, ((byte)(238)));
this.labelMyNameAndSurname.ForeColor = System.Drawing.SystemColors.HotTrack;
this.labelMyNameAndSurname.Location = new System.Drawing.Point(21, 9);
this.labelMyNameAndSurname.Name = "labelMyNameAndSurname";
this.labelMyNameAndSurname.Size = new System.Drawing.Size(227, 37);
this.labelMyNameAndSurname.TabIndex = 0;
this.labelMyNameAndSurname.Text = "Dawid Perdek";
//
// buttonShowPerson
//
this.buttonShowPerson.Location = new System.Drawing.Point(400, 6);
this.buttonShowPerson.Name = "buttonShowPerson";
this.buttonShowPerson.Size = new System.Drawing.Size(150, 40);
this.buttonShowPerson.TabIndex = 1;
this.buttonShowPerson.Text = "Pokaż dane osoby";
this.buttonShowPerson.UseVisualStyleBackColor = true;
this.buttonShowPerson.Click += new System.EventHandler(this.buttonShowPerson_Click);
//
// buttonShowHiperPerson
//
this.buttonShowHiperPerson.Location = new System.Drawing.Point(592, 6);
this.buttonShowHiperPerson.Name = "buttonShowHiperPerson";
this.buttonShowHiperPerson.Size = new System.Drawing.Size(150, 40);
this.buttonShowHiperPerson.TabIndex = 2;
this.buttonShowHiperPerson.Text = "Pokaż dane hiperosoby";
this.buttonShowHiperPerson.UseVisualStyleBackColor = true;
this.buttonShowHiperPerson.Click += new System.EventHandler(this.buttonShowHiperPerson_Click);
//
// dataGridViewListOfPeople
//
this.dataGridViewListOfPeople.ColumnHeadersHeightSizeMode = System.Windows.Forms.DataGridViewColumnHeadersHeightSizeMode.AutoSize;
this.dataGridViewListOfPeople.Location = new System.Drawing.Point(28, 110);
this.dataGridViewListOfPeople.Name = "dataGridViewListOfPeople";
this.dataGridViewListOfPeople.Size = new System.Drawing.Size(347, 267);
this.dataGridViewListOfPeople.TabIndex = 3;
//
// textBoxNewPersonName
//
this.textBoxNewPersonName.Location = new System.Drawing.Point(400, 73);
this.textBoxNewPersonName.Name = "textBoxNewPersonName";
this.textBoxNewPersonName.Size = new System.Drawing.Size(150, 20);
this.textBoxNewPersonName.TabIndex = 4;
//
// textBoxNewPersonSurname
//
this.textBoxNewPersonSurname.Location = new System.Drawing.Point(400, 110);
this.textBoxNewPersonSurname.Name = "textBoxNewPersonSurname";
this.textBoxNewPersonSurname.Size = new System.Drawing.Size(150, 20);
this.textBoxNewPersonSurname.TabIndex = 5;
//
// textBoxNewPersonAge
//
this.textBoxNewPersonAge.Location = new System.Drawing.Point(400, 146);
this.textBoxNewPersonAge.Name = "textBoxNewPersonAge";
this.textBoxNewPersonAge.Size = new System.Drawing.Size(150, 20);
this.textBoxNewPersonAge.TabIndex = 6;
//
// buttonShow
//
this.buttonShow.Location = new System.Drawing.Point(592, 62);
this.buttonShow.Name = "buttonShow";
this.buttonShow.Size = new System.Drawing.Size(150, 40);
this.buttonShow.TabIndex = 7;
this.buttonShow.Text = "Wyświetl";
this.buttonShow.UseVisualStyleBackColor = true;
this.buttonShow.Click += new System.EventHandler(this.buttonShow_Click);
//
// buttonAddPerson
//
this.buttonAddPerson.Location = new System.Drawing.Point(592, 126);
this.buttonAddPerson.Name = "buttonAddPerson";
this.buttonAddPerson.Size = new System.Drawing.Size(150, 40);
this.buttonAddPerson.TabIndex = 8;
this.buttonAddPerson.Text = "Dodaj osobę";
this.buttonAddPerson.UseVisualStyleBackColor = true;
this.buttonAddPerson.Click += new System.EventHandler(this.buttonAddPerson_Click);
//
// labelListOfPeople
//
this.labelListOfPeople.AutoSize = true;
this.labelListOfPeople.Location = new System.Drawing.Point(25, 80);
this.labelListOfPeople.Name = "labelListOfPeople";
this.labelListOfPeople.Size = new System.Drawing.Size(56, 13);
this.labelListOfPeople.TabIndex = 9;
this.labelListOfPeople.Text = "Lista ludzi:";
//
// dataGridViewGrades
//
this.dataGridViewGrades.ColumnHeadersHeightSizeMode = System.Windows.Forms.DataGridViewColumnHeadersHeightSizeMode.AutoSize;
this.dataGridViewGrades.Location = new System.Drawing.Point(400, 215);
this.dataGridViewGrades.Name = "dataGridViewGrades";
this.dataGridViewGrades.Size = new System.Drawing.Size(238, 162);
this.dataGridViewGrades.TabIndex = 10;
//
// labelGrades
//
this.labelGrades.AutoSize = true;
this.labelGrades.Location = new System.Drawing.Point(397, 197);
this.labelGrades.Name = "labelGrades";
this.labelGrades.Size = new System.Drawing.Size(41, 13);
this.labelGrades.TabIndex = 11;
this.labelGrades.Text = "Oceny:";
//
// textBoxGrade
//
this.textBoxGrade.Location = new System.Drawing.Point(400, 403);
this.textBoxGrade.Name = "textBoxGrade";
this.textBoxGrade.Size = new System.Drawing.Size(150, 20);
this.textBoxGrade.TabIndex = 12;
//
// buttonAddGrade
//
this.buttonAddGrade.Location = new System.Drawing.Point(592, 392);
this.buttonAddGrade.Name = "buttonAddGrade";
this.buttonAddGrade.Size = new System.Drawing.Size(150, 40);
this.buttonAddGrade.TabIndex = 13;
this.buttonAddGrade.Text = "Dodaj ocenę";
this.buttonAddGrade.UseVisualStyleBackColor = true;
this.buttonAddGrade.Click += new System.EventHandler(this.buttonAddGrade_Click);
//
// pictureBoxGrades
//
this.pictureBoxGrades.Location = new System.Drawing.Point(644, 215);
this.pictureBoxGrades.Name = "pictureBoxGrades";
this.pictureBoxGrades.Size = new System.Drawing.Size(128, 109);
this.pictureBoxGrades.TabIndex = 14;
this.pictureBoxGrades.TabStop = false;
//
// FormMain
//
this.AutoScaleDimensions = new System.Drawing.SizeF(6F, 13F);
this.AutoScaleMode = System.Windows.Forms.AutoScaleMode.Font;
this.ClientSize = new System.Drawing.Size(784, 462);
this.Controls.Add(this.pictureBoxGrades);
this.Controls.Add(this.buttonAddGrade);
this.Controls.Add(this.textBoxGrade);
this.Controls.Add(this.labelGrades);
this.Controls.Add(this.dataGridViewGrades);
this.Controls.Add(this.labelListOfPeople);
this.Controls.Add(this.buttonAddPerson);
this.Controls.Add(this.buttonShow);
this.Controls.Add(this.textBoxNewPersonAge);
this.Controls.Add(this.textBoxNewPersonSurname);
this.Controls.Add(this.textBoxNewPersonName);
this.Controls.Add(this.dataGridViewListOfPeople);
this.Controls.Add(this.buttonShowHiperPerson);
this.Controls.Add(this.buttonShowPerson);
this.Controls.Add(this.labelMyNameAndSurname);
this.Name = "FormMain";
this.Text = "FormMain";
((System.ComponentModel.ISupportInitialize)(this.dataGridViewListOfPeople)).EndInit();
((System.ComponentModel.ISupportInitialize)(this.dataGridViewGrades)).EndInit();
((System.ComponentModel.ISupportInitialize)(this.pictureBoxGrades)).EndInit();
this.ResumeLayout(false);
this.PerformLayout();
}
#endregion
private System.Windows.Forms.Label labelMyNameAndSurname;
private System.Windows.Forms.Button buttonShowPerson;
private System.Windows.Forms.Button buttonShowHiperPerson;
private System.Windows.Forms.DataGridView dataGridViewListOfPeople;
private System.Windows.Forms.TextBox textBoxNewPersonName;
private System.Windows.Forms.TextBox textBoxNewPersonSurname;
private System.Windows.Forms.TextBox textBoxNewPersonAge;
private System.Windows.Forms.Button buttonShow;
private System.Windows.Forms.Button buttonAddPerson;
private System.Windows.Forms.Label labelListOfPeople;
private System.Windows.Forms.DataGridView dataGridViewGrades;
private System.Windows.Forms.Label labelGrades;
private System.Windows.Forms.TextBox textBoxGrade;
private System.Windows.Forms.Button buttonAddGrade;
private System.Windows.Forms.PictureBox pictureBoxGrades;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
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BabesNetwork | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 119 |
<?php
require_once 'swift_required.php';
require_once 'Swift/Mime/EmbeddedFileAcceptanceTest.php';
class Swift_EmbeddedFileAcceptanceTest
extends Swift_Mime_EmbeddedFileAcceptanceTest
{
protected function _createEmbeddedFile()
{
return Swift_EmbeddedFile::newInstance();
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,104 |
{"url":"https:\/\/energyeducation.ca\/encyclopedia\/Thermal_efficiency","text":"# Thermal efficiency\n\nFigure 1: The amount of work output for a given amount of heat gives a system its thermal efficiency.[1]\n\nHeat engines turn heat into work. The thermal efficiency expresses the fraction of heat that becomes useful work. The thermal efficiency is represented by the symbol , and can be calculated using the equation:\n\nWhere:\n\nis the useful work and\n\nis the total heat energy input from the hot source.[2]\n\nHeat engines often operate at around 30% to 50% efficiency, due to practical limitations. It is impossible for heat engines to achieve 100% thermal efficiency () according to the Second law of thermodynamics. This is impossible because some waste heat is always produced produced in a heat engine, shown in Figure 1 by the term. Although complete efficiency in a heat engine is impossible, there are many ways to increase a system's overall efficiency.\n\n## Contents\n\n### An Example\n\nIf 200 joules of thermal energy as heat is input (), and the engine does 80 J of work (), then the efficiency is 80J\/200J, which is 40% efficient.\n\nThis same result can be gained by measuring the waste heat of the engine. For example, if 200 J is put into the engine, and observe 120 J of waste heat, then 80 J of work must have been done, giving 40% efficiency.\n\n## Carnot Efficiency\n\nmain article\n\nThere is a maximum attainable efficiency of a heat engine which was derived by physicist Sadi Carnot. Following laws of thermodynamics the equation for this turns out to be\n\nWhere\n\nis the temperature of the cold 'sink' and\n\nis the temperature of the heat reservoir.\n\nThis describes the efficiency of an idealized engine, which in reality is impossible to achieve.[3] From this equation, the lower the sink temperature or the higher the source temperature , the more work is available from the heat engine. The energy for work comes from a decrease in the total energy of the fluid used in the system. Therefore the greater the temperature change, the greater this decrease in the fluid and thus the greater energy available to do work is.[4]\n\nFor further information please see the related pages below:\n\n## References\n\n1. This picture was made by the Energy Education team.\n2. TPUB Engine Mechanics. (April 4, 2015). Thermal Efficiency [Online]. Available: http:\/\/enginemechanics.tpub.com\/14075\/css\/14075_141.htm\n3. Hyperphysics, Carnot Cycle [Online], Available: http:\/\/hyperphysics.phy-astr.gsu.edu\/hbase\/thermo\/carnot.html\n4. R. A. Hinrichs and M. Kleinbach, \"Heat and Work,\" in Energy: Its Use and the Environment, 4th ed. Toronto, Ont. Canada: Thomson Brooks\/Cole, 2006, ch.4, sec.E, pp.115\n\n## Authors and Editors\n\nBethel Afework, Jordan Hanania, Braden Heffernan, James Jenden, Kailyn Stenhouse, Jason Donev\nLast updated: May 18, 2018\nGet Citation","date":"2021-01-19 05:41:15","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8801149129867554, \"perplexity\": 1480.1844218848855}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-04\/segments\/1610703517966.39\/warc\/CC-MAIN-20210119042046-20210119072046-00461.warc.gz\"}"} | null | null |
{"url":"https:\/\/www.transtutors.com\/questions\/the-following-standards-for-variable-manufacturing-overhead-have-been-established-fo-5334186.htm","text":"# The following standards for variable manufacturing overhead have been established for a company that\n\nThe following standards for variable manufacturing overhead have been established for a company that makes only one product: Standard hours per unit of output ........ 8.0 hours Standard variable overhead rate.......... $11.55 per hour The following data pertain to operations for the last month: Actual hours. 7,000 hours Actual total variable overhead cost...........$79,100 Actual output ..... 600 units What is the variable overhead spending variance for the month? O $23,660 U$1,750 F $24,860 U$1,200 U The following materials standards have been established for a particular product: Standard quantity per unit of output .......... 3.2 meters Standard price.. ........ $13.40 per meter The following data pertain to operations concerning the product for the last month: Actual materials purchased. 4,600 meters Actual cost of materials purchased. .........$59,800 Actual materials used in production.......... 4,400 meters Actual output. 1,300 units Reference: 10-14 PUL.............. What is the materials price variance for the month? O $1,840 U 0$1,760 U 0 $1,760 F$1,840 F\n\n## Plagiarism Checker\n\nSubmit your documents and get free Plagiarism report\n\nFree Plagiarism Checker","date":"2020-11-27 11:54:33","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.23773953318595886, \"perplexity\": 11291.136599618665}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-50\/segments\/1606141191692.20\/warc\/CC-MAIN-20201127103102-20201127133102-00300.warc.gz\"}"} | null | null |
Discrepancies is een Amerikaanse experimentele raprockband, die invloeden uit rock, metal en rap combineert, afkomstig uit Saint Louis, Missouri.
Personele bezetting
Antonio Metcalf - vocals
Garrett Weakley - bas
Addison Bracher - gitaar, vocals
Zach Allard - drums
Discografie
Albums
2018 - The Awakening
EP's
2014 - Discrepancies
Singles
2018 - Art of War
2018 - Wake Up
2018 - Rock the Show
2018 - Raising the Bar
2018 - Prevail
2018 - Let it Go
2020 - Control
Amerikaanse metalband | {
"redpajama_set_name": "RedPajamaWikipedia"
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SAFETY GATE COUNTERWEIGHT CONNECTION CHAIN. SOLD BY FOOT VOLTAGE DECAL 460V 3PH 60hz Schneider Electric ZB4BZ009 Mounting Base 22mm for ZB4 Series.
The body has double-pole 1 normally closed (NC) + 1 normally open (NO) snap action contacts. PRESSURE SWITCH LIMIT SWITCH COMPLETE WITHOUT ACTUATOR ARM. | {
"redpajama_set_name": "RedPajamaC4"
} | 2,119 |
West Hampstead's 'celebrity road' bemoans loss of neighbourly love
By Paul Wright
Published: 6:02 PM November 10, 2014 Updated: 1:41 PM October 14, 2020
Parsifal Road residents association pictured with Cllr Flick Rae. Picture: Nigel Sutton - Credit: Nigel Sutton
It's been home to some of the most celebrated Hollywood stars and high-society bigwigs for over 100 years.
My Fair Lady star Rex Harrison, his wife, the actress Lilli Palmer, and Harrison's lover, actress Kay Kendall, all played roles in the extraordinary history of Parsifal Road in West Hampstead.
Named in 1883 after Wagner's opera Parsifal, the road of Victorian houses even held prestige in its early years with residents so distinguished it was a regularly haunt of former Prime Minister William Gladstone.
And it's still pulling in the big names today, with its most recent addition one of the most prominent men in the country.
Always in the background has been one of Camden's oldest residents' associations ensuring the street has never lost its community spirit.
Celebrating its 21st year, the Parsifal Road Association (PRA) has brought residents together, fought the council, and made sure neighbours look out for one another – no matter how high their profile.
But with the area becoming highly desirable for property investors and young professionals, the association fears West Hampstead is becoming full of "transient types".
Rosalie Miles, chairman of PRA, moved into the road in 1961 when there were "only two shops in West End Lane".
"I've lived here from teen to pensioner so have noticed so many changes," she said.
"At our first association meeting we held two sessions because so many people turned up. There was a strong feeling of community back then.
"We want to make sure this kind of interest continues, because I fear the work of groups like ours is being forgotten."
Built on the open fields of West End Farm and the Woodbine Cottage estate, the road has been part of, and bore witness to, many changes in West Hampstead. Some 15 nationalities are said to live in Parsifal Road now.
NGO fundraiser Rosy Hytner has been a resident and association member for 18 years.
"What's happening in this road is what's happening in West Hampstead as a whole," she said. "It's becoming more gentrified and the big challenge for the area is to retain the feel of community."
NW6 is known to have a strong online community but it's something that can be missed by those less familiar with technology.
"At a time when West Hampstead is going through so many changes, we're just urging people not to forget how important organisations like ours can be for keeping the community strong," Mrs Miles said. | {
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{"url":"https:\/\/mathematica.stackexchange.com\/tags\/units\/hot","text":"# Tag Info\n\n35\n\nIf we pre-evaluate the expression: x1 B[Quantity[ll, \"Micrometers\"], Quantity[1000, \"Kelvins\"]] then the run time can be reduced by about factor 10. We can do this by hoisting the expression out of the loop and pre-evaluating using With: p = Module[{ll} , With[{v = x1 B[Quantity[ll, \"Micrometers\"], Quantity[1000, \"Kelvins\"]]} , Table[{Quantity[...\n\n34\n\nThis is certainly the optimal way of obtaining the list you are looking for Quantity;QuantityUnitsPrivate$UnitReplacementRules[[1, All, 1]] EDIT for v10 (thanks @DavidCreech) In v10 this undocumented variable format has been changed into an association, whose keys are the units. Quantity; Keys[QuantityUnitsPrivate$UnitReplacementRules]\n\n28\n\nThere are system options available that should restore the old behavior for most of the currated data paclet: SetSystemOptions[SystemOptions[\"DataOptions\"] \/. True -> False] {\"DataOptions\" -> {\"ReturnEntities\" -> False, \"ReturnQuantities\" -> False, \"UseDataWrappers\" -> False}} Note that this prevents these paclets from returning Entity,...\n\n27\n\nI believe you can use \"DimensionlessUnit\" to get the desired result: Quantity[3, \"DimensionlessUnit\"] 3 (note this is the unit produced by QuantityUnit on a dimensionless value): QuantityUnit[3] \"DimensionlessUnit\"\n\n23\n\nHere is a cheap way which does not involve WA, but will only be as good as you make it to be (so that you'd have to customize it yourself): create a dynamic environment: ClearAll[withUnits]; SetAttributes[withUnits, HoldAll]; withUnits[code_] := Function[Null, Block[{Quantity}, SetAttributes[Quantity, HoldRest]; Quantity \/: UnitConvert[...\n\n23\n\nThis should list you all available units in Mathematica. Needs[\"QuantityUnits\"] Keys[QuantityUnitsPrivate$UnitReplacementRules] Inspired by eldo I made a little dynamic interface: Needs[\"QuantityUnits\"] table = Keys[QuantityUnitsPrivate$UnitReplacementRules]; Panel[DynamicModule[{f = \"\"}, Column[{Text[Style[\"Mathematica Unit Search:\", Bold]],...\n\n21\n\nIn physics, the Planck constant may be used as a natural unit. If you want to switch to another unit system, use UnitConvert[]. For example, you can switch to standard SI units this way: UnitConvert[Quantity[1, \"PlanckConstant\"], \"SIBase\"] which will give you: Quantity[6.626070*10^-34, (\"Kilograms\" (\"Meters\")^2)\/(\"Seconds\")] This can be done at the ...\n\n20\n\nOops - found it! QuantityMagnitude[quantity] does the job. For example, In[1]:= QuantityMagnitude[Quantity[1, \"Feet\"]] Out[1]= 1\n\n18\n\nThis appears to be a bug. The dimensions of the Boltzmann constant are incorrect. In fact, all the physical constants I checked have TemperatureUnit where they should have TemperatureDifferenceUnit. You should only have to make a substitution when making calls to physical constants in Quantity: q = UnitConvert[Quantity[\"elementary charge\"]]; k = (...\n\n18\n\nThis is a long comment for Nick Lariviere's answer. You can use Trace to see how lengthy the entity and quantity logic is. Version 9: Tuples@{Range@112, {\"Symbol\", \"Group\"}} \/\/ First ElementData @@ % \/\/ Trace; % \/\/ ByteCount 78336 TreeForm[%%, VertexLabeling -> False, ImageSize -> 800, AspectRatio -> 2] Version 10: ... % \/\/ ByteCount ...\n\n17\n\nJust some analysis to try to find where the slow down. On my PC, it took 25 seconds to build the table. ps. I never used Units before. Your main loop: x = UnitConvert@Quantity[\"PlanckConstant\" \"SpeedOfLight\"\/\"BoltzmannConstant\"] x1 = UnitConvert@Quantity[2, \"PlanckConstant\" (\"SpeedOfLight\")^2] B[L_, T_] := (L^(-5))\/(Exp[x\/(L T)] - 1) c = Quantity[1000, \"...\n\n17\n\nx is not in mL..it is just a pure number. Quantity[x, \"ml\"] is the 'thing' that is in mL. To get what you want, you need to recast your Solve command as Solve[x Quantity[5, \"mol\"] + (Quantity[250, \"ml\"] - x) Quantity[7, \"mol\"] == Quantity[250, \"ml\"] Quantity[6, \"mol\"], x] However, the result is a bit strange looking. {{x -> Quantity[1\/8000, (\"...\n\n16\n\nYou need to use 'better' units: UnitConvert[Quantity[1, \"GiB\"], \"MiB\"] Quantity[1024, \"Mebibytes\"] See explanation and definitions in https:\/\/en.wikipedia.org\/wiki\/Gigabyte\n\n15\n\nWhile it would've been nice if the package handled it automatically, it can be fixed with a simple overloading of Quantity: Unprotect@Quantity; Quantity \/: (0 | 0.) Quantity[_, unit_] := Quantity[0, unit] Protect@Quantity; You can add this to your init.m, so that you don't have to define it each time. You can test your examples with this: 0. Quantity[1, \"...\n\n15\n\nI have mined the Units package for the names of all units defined therein and correlated them to the built-in strings recognized by Quantity(referenced here). I then define a new function Quantify to convert the old school units into Quantity objects. unitRules = Dispatch[{Abampere -> Quantity[1, \"ABAmperes\"], Abcoulomb -> Quantity[1, \"...\n\n15\n\nAt the beginning of your notebook set the Metric system as default. $UnitSystem = \"Metric\" This works for me. If not try below suggestion Remember you are reading the data from wolfram alpha! it's a regional thing so if the upper solution didn't work, you can try something like (I don't remember exactly though) SetOptions[WolframAlpha, PodStates -> {\"... 14 The CGS units are available. Out of the need to ensure dimensional consistency, the different things which are all called ESUs must be carefully distinguished. In[59]:= Quantity[1, \"ESU of charge\"] Out[59]= Quantity[1, \"ESUsOfCharge\"] In[60]:= Quantity[1, \"ESU\"] Out[60]= Quantity[1, \"ESUOfDielectricDisplacement\"] Also, when a unit has a special name, ... 14 You can use the relative uncertainty 1.1*10^-5 from the CODATA website you also referenced, and use Around to construct a value for the Planck length with uncertainty which has a much nicer formatting for showing the relevant digits of the number: Around[UnitConvert[Quantity[\"PlanckLength\"], \"m\"], Scaled[1.1*^-5]] (1.616255\u00b10.000018) x 10^-35 m Since ... 13 It is generally better to work in dimensionless units to start with. In your case, your a and b have units of$L^{-2}$, while eta is dimensionless. Taking fx''[x] + a*fx[x] == -I*eta*fy''[x] - b*fy[x] as an example, you could use$\\xi=x\/L$,$\\alpha=a L^2$,$\\beta=b L^2$(all of which are now dimensionless), whereupon your differential equation becomes$L^{-2}...\n\n13\n\nThat code should work and it does work on my machine. The problem could be the following. You have only one part that requires Wolfram|Alpha interpretation: Quantity[24, \"1\/Seconds\"] It is not built in unit - so it goes to Wolfram|Alpha for interpretation (how cool is that? ;-) ). This works almost always - unless something is wrong with internet ...\n\n12\n\nUnitConvert[Quantity[179., \"Centimeters\"], MixedRadix[\"Feet\", \"Inches\"]] returns Quantity[MixedRadix[5, 10.472440944881885], MixedRadix[\"Feet\", \"Inches\"]] which formats as 5'10.4724\"\n\n11\n\nBelow is some code I use to work with units. I am aware that the unit system I am calling \"CGS\" is only semi-CGS, since I am keeping the SI electromagnetic units, but this is the flavor of consistent unit system we sometimes use in our lab. Really, though, this is a recipe for choosing your own set of base units. The method works by applying ...\n\n11\n\nAs noted in the documentation for Quantity, you can use ctrl-= to input units. This uses Wolfram|Alpha, so needs an internet connection. Quantity will also use Wolfram|Alpha to try to interpret strings, so you could also use: In[8]:= UnitConvert[Quantity[\"1 m\/s^2*(1 min)^2\"], Quantity[\"km\"]] Out[8]= Quantity[18\/5, \"Kilometers\"]\n\n11\n\nYou could set an input alias such as With[{rules = {\"m\" -> \"Meters\", \"km\" -> \"Kilometers\"}}, AppendTo[CurrentValue[InputNotebook[], InputAliases], \"qu\" -> TemplateBox[{\"\\[SelectionPlaceholder]\", \"\\[Placeholder]\"}, \"QuantityUnit\", DisplayFunction -> (PanelBox[RowBox[{##}], FrameMargins -> 2] &), InterpretationFunction -&...\n\n11\n\nThe difference between 0 \u00b0C and 100 \u00b0C in Fahrenheit is 180 \u00b0F. The difference between 0 K and 100 K in Fahrenheit is 180 \u00b0F. But 100 \u00b0C is 212 \u00b0F, while 100 K is 279.688 \u00b0F. The underlying reason are, of course, the differing scale origins (zero points); the conversions are therefore not direct proportions (not homogeneous functions). Instead they are ...\n\n11\n\nA bit too long for a comment... My best guess is that WolframAlpha[\"earth's gravity\", \"MathematicaResult\"] is simply rounding to too few digits. It returns 32.2 ft\/s^2, which seems pretty round for what it represents. Now maybe that's ok to do because gravity varies around the globe: data = GeogravityModelData[{{-90, -180.}, {90, 180.}}, \"Magnitude\", ...\n\n11\n\nThe 4.652... at the end of 1.616255000000000066841324.652207380644164*^-35 should tell you that Mathematica knows this constant only up to 4.65 decimal digits. You have to enforce first to treat the number as a higher precision number first: NumberForm[ SetPrecision[ QuantityForm[UnitConvert[Quantity[\"PlanckLength\"], \"Meters\"], \"LongForm\"], 7], {7,...\n\n10\n\nExpanding a little bit on paw's nice discovery: Needs[\"QuantityUnits\"] table = Keys[QuantityUnitsPrivate`\\$UnitReplacementRules]; Since this table is very long one can restrict the output, f.e. with Union @ Flatten[StringCases[#, \"Feet\" ~~ ___] & \/@ table] \/\/ TableForm UPDATE A similar question could arise with the more than 1000 inbuilt ...\n\n10\n\nThe following works for V10. First we define some abbreviation rules: rule = {\"Newtons\" :> N, \"Meters\" :> m, \"Pascals\" :> Pa, \"Farads\" :> F}; (* add more rules here *) Then: unit = TextString[QuantityUnit[Quantity[1, \"Newtons\/Meters^2\"]] \/. rule] \"N\/m^2\" StringQ[unit] True TextString[QuantityUnit[Quantity[1, \"Meters*Pascals\/Farads^2\"]] \/....\n\nOnly top voted, non community-wiki answers of a minimum length are eligible","date":"2019-12-09 10:54:27","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.45620566606521606, \"perplexity\": 6324.432655259143}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-51\/segments\/1575540518627.72\/warc\/CC-MAIN-20191209093227-20191209121227-00034.warc.gz\"}"} | null | null |
\section{Introduction}
Classical pulsating stars such as Cepheids and Miras are
useful tracers of the Galaxy. Period-luminosity relations of
those objects allow us to determine the distances to individual stars
(see \cite[Matsunaga 2017]{Matsunaga-2017b}, and references therein).
Each group of pulsating stars has a typical range of ages,
so that we can learn ages of stellar populations in galaxies
based on pulsating stars. Classical Cepheids are young,
Miras are intermediate to old, and type II Cepheids and RR Lyrs are
all old. Especially, ages of classical Cepheids are anti-correlated
with pulsation periods, and that makes it possible to estimate
their accurate ages (\cite[Bono \etal\ 2005]{Bono-2005}).
Furthermore, when appropriate datasets available, we can combine
kinetic and chemical parameters of these objects to reconstruct
chemodynamical information for tracing the structure
and evolution of the Galaxy.
There are some complications which should be borne in mind when
pulsating stars are used as tracers of the Galaxy.
Firstly, classification of the variability type is not always easy
and should be based on good data.
For example, it is possible to get confused between pulsating stars,
eclipsing binaries, and objects with bright or dim stellar spots;
see, e.g., a controversy about faint objects towards the Galactic disc found in
\cite[Chakrabarti \etal\ (2015, 2017)]{Chakrabarti-2015,Chakrabarti-2017}
and \cite{Pietrukowicz-2015b}.
Secondly, stellar rotation may affect ages of Cepheids
and maybe other stars as discussed by
\cite[Anderson \etal\ (2014, 2016)]{Anderson-2014,Anderson-2016}.
Thirdly, evolution in binary systems complicates the implication of
pulsating stars concerning stellar populations. The ages of such objects
can be totally different from those of isolated stars but with
similar variable characteristics. For example,
\cite{Feast-2013} found a carbon-rich Mira in an old-aged
globular clusters. Such old stars are not expected to evolve into
carbon stars but a merged star can become a carbon star. Such objects
must be relatively small in number, but when we discuss rare objects
we need to take such evolutionary paths into consideration.
We'll come back to this point later.
Another interesting case of unusual pulsating stars evolved in a binary system
is found in \cite{Pietrzynski-2012}.
While these complications require further investigations
to address their impacts and necessary corrections, if possible,
on studies on the Galaxy, at least a large fraction of pulsating stars
are useful for studying stellar populations and their distributions
in the Galaxy.
While pulsating stars have been actively used for studying the Galactic halo
(e.g.~\cite[Huxor \& Grevel 2015]{Huxor-2015};
\cite[Fiorentino \etal\ 2017]{Fiorentino-2017}),
our focus in this review is on the inner part of the Galaxy.
The use of pulsating stars as tracers of the inner part
and the disc of the Galaxy has been
limited by the incompleteness of surveys (e.g.\ see figure~1 of
\cite[Matsunaga 2012]{Matsunaga-2012}).
The main reasons for the incompleteness are twofold:
(1) early surveys were optical and relatively shallow
and thus were largely affected by the interstellar extinction
in the disc region,
and (2) sky coverage of sufficiently deep recent surveys, in particular
those in the infrared, has been limited.
In terms of the sky coverage, \textit{Gaia} will produce
an unprecedented catalogue of pulsating stars in the entire range of
the Galaxy, although the interstellar extinction limits its reach across
the Galactic plane. \cite{Windmark-2011} predicted
that the total number of Cepheids in the Galaxy is roughly 20,000 based on
a simple exponential-disc model and about half of them
would be detected by \textit{Gaia}.
There will be many regions
where Cepheids are too much obscured for \textit{Gaia};
in such obscured regions, one cannot expect to get stellar distribution
based on \textit{Gaia}'s parallaxes, and pulsating
stars found in the infrared will be crucial even in
the \textit{Gaia} era.
In this review, we summarize the currently known samples
of Cepheids and Miras in the inner Galaxy
in the sections \ref{sec:Cep} and \ref{sec:Mira}, respectively.
Fig.\,\ref{fig:LB} summarizes currently known Cepheids and Miras
towards the Galactic bulge.
Unless otherwise mentioned, our discussions are limited to this range
of the sky. At the distance to the Galactic centre,
{$\sim$}8.3~kpc (\cite[de Grijs \& Bono 2016]{deGrijs-2016}),
the horizontal stretch of Fig.\,\ref{fig:LB}, 27~degrees, corresponds to
approximately 4~kpc. The dominant component in this range, especially
in terms of stellar mass ($2\times 10^{10}$~M$_\odot$,
\cite[Valenti \etal\ 2016]{Valenti-2016}),
is the bulge with the bar structure inclined by
25--30~degrees to the direction of the Sun
(\cite[Bland-Hawthorn \& Gerhard 2016]{BlandHawthorn-2016}).
However, several systems with different characteristics are overlapped
in this range, which sometimes complicates interpretation of
observation data on objects in this direction.
At the very centre exists the Nuclear Stellar Cluster surrounding
the supermassive blackhole, and this cluster is within
the Nuclear Stellar Disc (or also known as the Central Molecular Zone).
The former has ${\sim}2\times 10^7$~M$_\odot$ in stars
within the radius of 10~pc,
and the latter has ${\sim}10^9$~M$_\odot$, within the radius of 200~pc,
of which a few percent are found in interstellar gas and dust
(\cite[Launhardt, Zylka, \& Mezger]{Launhardt-2002}).
The size of the NSD is illustrated by the ellipse
in Fig.\,\ref{fig:LB}.
In the foreground and background extended is the Galactic disc.
These different systems host different stellar populations
and thus different sets of pulsating stars as we see below.
\begin{figure}[hbtp]
\begin{center}
\includegraphics[width=0.98\hsize]{LB.png}
\caption{
Distributions of known Cepheids (panel a) and Miras (panel b).
The dashed ellipse around $(l,b)=(0,0)$ suggests the approximate
range of the Nuclear Stellar Disk
(\cite[Launhardt \etal\ 2002]{Launhardt-2002}).
The objects included in each panel are as follows:
(a)~Grey dots indicate classical Cepheids from OGLE
(\cite[Soszy\'nski \etal\ 2011]{Soszynski-2011}),
among which five indicated by triangles are discussed in \cite{Feast-2014a}.
Open and filled circles indicate those found by
\cite[D\'ek\'any \etal\ (2015ab)]{Dekany-2015a,Dekany-2015b}
and by \cite{Matsunaga-2016}, respectively.
Within the dashed ellipse for the NSD located are four Cepheids discussed in
\cite[Matsunaga \etal\ (2011, 2015)]{Matsunaga-2011,Matsunaga-2015}.
Cepheids in this region in the catalogue of \cite{Fernie-1995}
are indicated by crosses, but they are all located within 3~kpc of the Sun
and are foreground objects.
(b)~Grey dots indicate Miras from OGLE
(\cite[Soszy\'nski \etal\ 2013]{Soszynski-2013})
and those discussed in \cite{Catchpole-2016} whose main targets are
located within the two rectangles.
Four filled circles indicate the carbon-rich Miras reported in
\cite{Matsunaga-2017a}, while the cross and triangle symbols indicate
those which are probably foreground and background of the bulge
as discussed in the same paper.
}
\label{fig:LB}
\end{center}
\end{figure}
RR Lyrs and type II Cepheids are useful tracers of old stellar population
and have been found in the bulge, although they are outside the scope of
this review.
Readers are referred to recent observational results in the following papers:
\cite[Pietrukowicz \etal\ (2012,2015)]{Pietrukowicz-2012,Pietrukowicz-2015a},
\cite{Dekany-2013}, \cite[Gran \etal\ (2015, 2016)]{Gran-2015,Gran-2016},
\cite{Minniti-2016}, and \cite{Dong-2017} concerning RR Lyrs, and
\cite{Soszynski-2011}, \cite{Matsunaga-2013a}, and \cite{Bhardwaj-2017}
concerning type II Cepheids.
The surveys are rather incomplete for these important types
of pulsating stars (see, e.g.\ figure~2
in \cite[Pietrukowicz \etal\ 2015]{Pietrukowicz-2015}
and figure~6 in \cite[Soszy\'nski \etal\ 2011]{Soszynski-2011}).
\section{Cepheids towards the inner Galaxy}
\label{sec:Cep}
Classical Cepheids are pulsating supergiants with pulsation periods
ranging from one day to {$\sim$}80~days or even longer.
They are 10--300~Myr old and their initial masses are 4--10~M$_\odot$
(\cite[Bono \etal\ 2005]{Bono-2005}).
Their period--luminosity relation is one of the most
important steps of the cosmic distance ladder
(\cite[Freedman \etal\ 2001; Riess \etal\ 2016]{Freedman-2001,Riess-2016}).
Since the pioneering work by Baade
(summarized in \cite[Baade 1956]{Baade-1956}), classical Cepheids are
distinguished from type II Cepheids which are evolved low-mass stars
aged {$\sim$}10~Gyr. We'll consider only classical Cepheids hereinafter.
Classical Cepheids, as representatives of young stars, have been used
for studying various characteristics of the Galactic disc (e.g.~\cite[Feast \& Whitelock 1997; Majaess, Turner, \& Lane 2009]{Feast-1997,Majaess-2009}).
In particular, Cepheids have been the most successful tracers of
the metallicity gradient, i.e.~how the metallicity changes as
a function of the distance from the Galactic centre
(\cite[Genovali \etal\ 2014, 2015; da Silva \etal\ 2016]{Genovali-2014,Genovali-2015,daSilva-2016}).
However, the previous survey of classical Cepheids is incomplete as we
discussed above. Fig.\,\ref{fig:LB} plots about a dozen of previously
known Cepheids in the DDO database (\cite[Fernie \etal\ 1995]{Fernie-1995}),
but they are all within 3~kpc of the Sun. It is considered that
the bulge is dominated by old stellar populations even if a small fraction
of young stars exist (\cite[Zoccali \etal\ 2003; Clarkson \etal\ 2011; Benzby \etal\ 2013]{Zoccali-2003,Clarkson-2011,Bensby-2013}),
and therefore classical Cepheids are not expected
and haven't been found in the bulge. In contrast, the NSD
is known to host stars with a wide range of ages
(\cite[Serabyn, \& Morris 1996; Figer \etal\ 2004]{Serabyn-1996,Figer-2004}).
\cite[Matsunaga \etal\ (2011, 2015)]{Matsunaga-2011,Matsunaga-2015}
in fact found four classical Cepheids in this system.
They are located within 0.35~degrees of the Galactic centre,
or within 50~pc (projected distance), and their radial velocities
are consistent with the rotation of the NSD
(\cite[Matsunaga \etal\ 2015]{Matsunaga-2015}).
Interestingly, all of them have periods around 20~days (18.8--23.5~days),
which is significantly longer than the median period, {$\sim$}5~days,
of Galactic Cepheids, and thus are similarly aged {$\sim$}25~Myr,
which puts an important constraint on the star formation history
in this system (\cite[Matsunaga \etal\ 2011]{Matsunaga-2011}).
The foreground interstellar extinction for these Cepheids are
{$\sim$}2.5~mag or more at around 2~{$\mu$}m, corresponding to
{$\sim$}30~mag in the optical, so that infrared observations
are required for both surveys and follow-up studies.
Later, infrared surveys by
\cite[Dekany \etal\ (2015ab)]{Dekany-2015a,Dekany-2015b}
and \cite{Matsunaga-2016} found dozens of Cepheids in the mid plane of
the Galactic disc, i.e.~nearly 0~degree in Galactic latitude
(Fig.\,\ref{fig:LB}).
While \cite{Dekany-2015b} suggested the presence of an inner thin disc of
young stars surrounding the NSD, \cite{Matsunaga-2016}
reported the lack of young stars in the inner 2.5~kpc except the NSD.
As discussed in \cite{Matsunaga-2016}, the distances to these reddened
Cepheids ($E_{H-K_{\rm s}}=$1.2--2.5~mag) have large errors
due to the uncertainty in the extinction law, i.e.~the wavelength dependency
of the interstellar extinction.
The four classical Cepheids in the NSD give an important
constraint on the extinction law because they should be located
at the distance of the Galactic centre regardless of the extinction.
This supports the extinction law of \cite{Nishiyama-2006}
rather than that of \cite{Nishiyama-2009} which was used by
\cite[Dekany \etal\ (2015ab)]{Dekany-2015a,Dekany-2015b}.
More detailed discussions on the problem of the extinction law
are found in the original paper of \cite{Matsunaga-2016} and
a recent review (\cite[Matsunaga 2017]{Matsunaga-2017b}).
The lack of Cepheids and young stars in the innermost part of
the Galactic disc is supported by the distribution of
observed H~II regions and massive star-forming regions
(e.g.~\cite[Jones \etal\ 2013; Sanna \etal\ 2014]{Jones-2013,Sanna-2014}).
This suggests that there is no simple exponential disc extending
into the centre in contrast to the model used by \cite{Windmark-2011},
if we consider young stars, and we need further infrared surveys to
map this region including the entire NSD and the interface between
the discs and the bulge.
Another interesting sample of classical Cepheids towards
the region of our interest but a few degrees away from the Galactic mid plane
came from a large-scale optical survey,
Optical Gravitational Lensing Experiment (OGLE).
\cite{Soszynski-2011} discovered 32 classical Cepheids towards the bulge
(Fig.\,\ref{fig:LB}) besides hundreds of type II Cepheids.
Although most of the type II Cepheids belong to the bulge
considering their positions on the period--Wesenheit diagram
(figure~7 in \cite[Soszy\'nski \etal\ 2011]{Soszynski-2011}),
most of the classical Cepheids seem to be located further than the bulge
(their magnitudes are similar to those of type II Cepheids in spite of
the large difference between intrinsic magnitudes for these two types).
Five of the 32 classical Cepheids,
triangles in the panel (a) of Fig.\,\ref{fig:LB}, were investigated by
\cite{Feast-2014a} in more detail, and their distances are
larger than 20~kpc, indeed further than the bulge. Moreover, they are
separated from the Galactic plane by {$\sim$}1~kpc or more and
belong to the flared part of the disc outskirts.
Other Cepheids in \cite{Soszynski-2011} are also located
at similarly large distances and exotic objects like these five,
but their nature needs to be revealed by further investigations.
\section{Miras towards the inner Galaxy}
\label{sec:Mira}
Miras are pulsating giants with periods longer than 100~days.
Their initial masses and ages are relatively wide:
1--9~M$_\odot$ corresponding to between {$\sim$}10~Gyr and 30~Myr
(\cite[Iben 1983]{Iben-1983}).
There is an anti-correlation between periods and ages
(\cite[Feast, Whitelock, \& Menzies 2006]{Feast-2006}),
although its relation is not so tight or established as
the period--age relation of classical Cepheids.
Nevertheless, a wide range of the ages of Miras is useful
for tracing old to intermediate-age stellar populations.
Miras also have the period--luminosity relation and serves as
distance indicator, although one can find useful relations only
in the infrared or the bolometric magnitude but not in the optical
(\cite[Glass \& Lloyd Evans 1981; Feast \etal\ 1989; Whitelock \etal\ 2008]{Glass-1981,Feast-1989,Whitelock-2008}).
As has been discussed in \cite{Feast-2014b} and \cite{Whitelock-2014},
Miras are very bright in the infrared and can be a good alternative
to Cepheids as distance indicators to galaxies at large distances
for infrared facilities like
\textit{James Webb Space Telescope (JWST)} in the future.
Because of the high luminosities of Miras, surveys of Miras in the bulge,
especially towards low-extinction windows, have been done from the early days
(see the review given by \cite[Catchpole \etal\ 2016]{Catchpole-2016}).
Recently, \cite{Soszynski-2013} found more than 6500 Miras using
the OGLE dataset (Fig.\,\ref{fig:LB}), which doesn't include
the low-latitude region ($|b|<1^\circ$). Such central parts require
infrared surveys even if they are relatively shallow, and in fact
early surveys in the near infrared already detected dozens or hundreds
of Miras (\cite[Glass \etal\ 2001]{Glass-2001}, and references therein)
within the NSD. \cite{Matsunaga-2009} detected over 500 Miras
(with periods determined) and obtained the distance to the Galactic centre,
8.24~kpc, making use of multi-band photometry in $JHK_{\rm s}$ bands
which was necessary to make correction of the interstellar extinction.
This estimate or any other estimate based on standard candles
(i.e.~luminosity based distance indicators) is affected by
the uncertainty in the extinction law described above
(\cite[Matsunaga 2013]{Matsunaga-2013b}). Even with this uncertainty
in the distances, it is clear that Miras show a concentration
towards the Galactic centre (\cite[Matsunaga \etal\ 2009]{Matsunaga-2009};
see also the large-scale density gradient in Fig.\,\ref{fig:LB}).
It is, however, unclear how Miras in the central 100~pc should be
separated into the NSD and the extended bar-like bulge.
An important group of objects related to Miras, with at least some overlaps,
is sources with maser emissions with SiO, H$_2$O, and OH molecules.
These emissions are produced in evolved stars with thick circumstellar shell,
and those objects are often found to be Miras or related pulsating stars
like semi-regulars (e.g.~see the review by \cite[Habing 1996]{Habing-1996}).
In particular, OH/IR stars characterized by OH maser emission and
high infrared luminosity have played an important role in studying
intermediate-age populations in the inner Galaxy
(\cite[Lindqvist, Habing, \& Winnberg 1992; Wood, Habing, \& McGregor 1998]{Lindqvist-1992,Wood-1998}).
Their kinematics supports that these intermediate-age objects belong
to the NSD.
Later, \cite{Deguchi-2004} detected SiO maser in 180 Miras of
the catalogue published by \cite{Glass-2001},
and also found the rotation comparable to that of OH/IR stars.
This seems to suggest that at least a part of the Miras near the centre
belong to the NSD, but the more detailed studies on
distributions of these Miras and maser sources should be done
with more comprehensive datasets including kinematic information
from both radial velocities and proper motions.
The recent discovery by \cite{Matsunaga-2017a} has revealed
a rare group of objects, carbon-rich Miras, among the Miras in the bulge.
They selected several candidates of carbon-rich Miras,
among the Miras in \cite{Soszynski-2013} and \cite{Catchpole-2016}
based on the $(J-K_{\rm s})$-$([9]-[18])$ diagram in which
one can distinguish objects with carbon-rich dust shell from
those with oxygen-rich dust shell (\cite[Ishihara \etal\ 2011]{Ishihara-2011}).
Then, with spectroscopic follow-up observations,
they confirmed 8 carbon-rich Miras of which 3 are foreground objects.
These carbon-rich Miras are considered to represent
relatively (or totally) unexplored stellar population(s).
Both age and metallicity affect whether an AGB star is evolved into
carbon-rich stars or not, and intermediate-age stars (a few Gyr)
tend to finish the AGB phase as carbon stars (\cite[Marigo 2013]{Marigo-2013}).
For the Galactic bulge, no carbon-rich AGB stars had been confirmed before.
This was naturally understood because the dominant population
of the bulge is old and metal-rich. Moreover, many stars are expected
to be oxygen enhanced which makes it even harder to produce carbon-rich stars.
Unfortunately, the ages and the origin of the carbon-rich Miras found in
the bulge are still unclear because mass-augmented stars may
evolve into carbon-rich stars even if they are old
(\cite[Feast \etal\ 2013]{Feast-2013}).
Kinematics and chemical information may help uncover their origins.
\section{Summary}
In the last decade, large-scale surveys like OGLE and
some near-infrared ones have revealed
large collections of new Cepheids and Miras.
The same is true for RR Lyrs and other kinds of variable stars.
Nevertheless, Fig.\,\ref{fig:LB} clearly shows
that we still miss a large numbers of these objects
particularly in the low-latitude region, in which
every system of the inner Galaxy is overlapped
(i.e.~the Galactic bulge, the Nuclear Stellar Disk,
and the Nuclear Stellar Clusters) together with the Galactic disc
in the foreground and background.
Several large-scale surveys including
OGLE, VISTA Via Lactea (\cite[Minniti \etal\ 2010]{Minniti-2010}), and
\textit{Gaia}
are expected to provide us with more complete samples of pulsating stars
which will be crucial for studying stellar populations in the future.
Various parameters are readily accessible
using the period--luminosity and period--age relations, for example,
and other observables for the pulsating stars, and they will remain to be
useful tracers of stellar populations in the Galaxy.
\acknowledgement{
The author appreciate financial support from the Japan Society for the Promotion of Science (JSPS) through the Grant-in-Aid, No.~26287028.
}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,859 |
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To get the cheapest car insurance deal, it is crucial to compare using online resources that aggregate information from multiple insurers. This is a convenient and fast way to find out how much your insurance will cost.
To find the car insurance policy that suits you, you must decide on the type of cover that you need. It is important to note that third-party insurance may not always be the cheapest option as many competitive offers in the market can give you more value for your money. For instance, you may be presented with deals at a very similar price range, but you may opt for a policy that also included breakdown cover.
The excess you are required to pay is another aspect you must consider. This is the amount that you are expected to pay toward an insurance claim. Typically, you are required to pay a small amount while your insurer provided the rest of the funds needed to cover the repairs for damage. If you need to make a claim, having a high excess can place a lot of financial stress on you so you may wish to opt for a policy that allows you to pay a lower amount as excess. Finally, it is essential to be honest with your insurer as any discrepancies may cause your policy to be invalidated.
The average cost of car insurance in the UK is £485 a year according to the ABI. This concerns comprehensive coverage and equates roughly to £40 a month. Drivers under 25, however, pay on average over £1,600 annually compared to just £410 for the over 50s.
Which type of car insurance should I opt for?
Deciding on the type of cover that you need is effectively a matter of balance between the contents of the policy and its cost. Since the premiums you would be asked to pay depend very much on a variety of factors including your driving experience, line of work, you age, and the car that you drive, it is best that you compare different types of cover to help you decide what each policy is worth to you and how much you can afford. Luckily, comparing different insurance policies is free, and there are fast and convenient ways for you to make comparison online and help you reach a decision.
The best way to get cheap car insurance is to use a car insurance comparison site. Furthermore, you may wish to consider what the most affordable cars are to insure, opting to increase your voluntary excess, using a telematics policy (also known as 'black box insurance'), and limiting your mileage. Finally, after building your no-claims discount over a period, you would be entitled to cheaper car insurance.
Is it cheaper to pay for car insurance monthly or annually?
Car insurers offer you a discount when you choose to pay a lump sum to cover 12 months of car insurance. Paying on a monthly basis may be a more convenient option for many, but it could end up costing significantly more because the interest charged on the monthly instalments.
Usually, you can cancel a car insurance policy and be refunded for the rest of the year if you haven't made a claim. Some insurers may charge a cancellation fee, which is typically around £50. Cancelling or switching your car insurance policy would mean that your no-claims bonus would be lost. Therefore, it is essential to take into account these costs and whether it is worth switching insurers to make a saving.
Do I need a credit check for car insurance?
A credit check would only be necessary when getting car insurance if you choose to pay in monthly payments. The credit check is needed for the insurer to assert that you can make the payments and enter into a credit agreement with them. Be advised that a credit check will remain on your credit file and that your credit rating will determine the APR that your insurer offers you for your monthly payments. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,282 |
\section{INTRODUCTION}\label{sec:introduction}
The ocean is the largest ecosystem on earth facing dramatic changes such as deoxygenation, warming, acidification, and contamination by industrial pollution, to name a few. The involved physical, biological and biogeochemical processes are highly diverse and form a complex network of causal interactions that act on variable temporal and spatial scales. Some of them are fast (less than a day) and locally confined (several \SI{100}{m} to km) such as the response of coastal ecosystems to the passage of storms, input of increased nutrient loads from the coastal drainage area subsequent to heavy rainfall, or locally enhanced primary production effecting the local carbon cycling.
To understand major drivers and ecosystem response in space and time conventional ship based observation programs are not sufficient as ship based operations involve long planning periods and follow a fixed time schedule, which often do not allow to capture sporadic environmental events that are difficult to forecast.
Despite the availability of suitable carrier platforms (e.g. landers, floats, gliders, waveglider) and the increasing numbers of physical and biogeochemical sensors for marine observation, still many areas in the ocean and along the coasts are undersampled and processes are not quantified adequately.
To simultaneously monitor changes in the water column and at the seafloor an underwater (UW) – robotic sensing network has been developed within the framework of the Helmholtz innovation project ARCHES involving a consortium of partners from AWI (Alfred-Wegener-Institute Helmholtz Centre for Polar and Marine Research, Bremerhaven, Germany), DLR (German Aerospace Center, Oberpfaffenhofen, Germany) and the GEOMAR (Helmholtz Centre for Ocean Research Kiel, Germany). The major aim was to implement a robotic sensing network, which is able to autonomously respond to changes in the environment by adopting its measurement strategy.
During the research cruise AL547 with RV ALKOR (20.10. – 31.10.2020) the functionality of the network has been demonstrated. It was aimed to establish a mobile ad hoc network (MANET) \cite{manets_survey_2018} of heterogeneous, autonomous and interconnected robotic systems. The network consists of five different stationary and mobile sensing platforms that exchange environmental data with each other and the research vessel.
Despite the technological focus as a science case, the research cruise addressed the oxygen dynamics in coastal ecosystems, which are increasingly impacted by eutrophication and the progressing loss of oxygen (hypoxia) \cite{breitburg2018declining}. Particularly the deeper part of Eckernförder Bay experiences seasonal hypoxia, yet strong fluctuations of oxygen in the bottom water also occur at various frequencies (for reference see web page of the Boknis Eck time series station: https://www.bokniseck.de/de/literature). Hence, beside other sensors implemented in the different sensing platforms all of them were equipped with at least one oxygen sensor and only data of the oxygen sensors were transferred among the various network participants.
Here, we report on the experience of employing Digital Twins in the above described field experiment. \autoref{sec:context}, briefly describes the set-up of the network, the Digital Twin Prototype approach is introduced in \autoref{sec-dtp}. In \autoref{sec-field} we report on the field experiment. Conclusions and future work are discussed in \autoref{sec:conclusion}.
\begin{figure*}[tb]
\centering
\begin{subfigure}{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{images/BigoLander.jpg}
\caption{BIGO Lander}
\label{fig:bigo}
\end{subfigure}
\hspace{0.1cm}
\begin{subfigure}{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{images/CrawlerSim_big.jpg}
\caption{CRAWLERSIM}
\label{fig:crawlersim}
\end{subfigure}
\hspace{0.1cm}
\begin{subfigure}{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{images/AWILander.jpg}
\caption{FLUX Lander}
\label{fig:flux}
\end{subfigure}
\\
\vspace{0.5cm}
\centering
\begin{subfigure}{0.47\textwidth}
\centering
\includegraphics[width=1.0\textwidth]{images/EH_AWI_Robex-13_klein.jpg}
\caption{MANSIO \& VIATOR}
\label{fig:mansioviator}
\end{subfigure}
\caption{The ocean observation systems participating in the ARCHES network.}
\label{fig:multiplelander}
\end{figure*}
\section{PROJECT CONTEXT}\label{sec:context}
Since each field test of ocean observation systems always requires a research vessel and thus, a crew, it is quite expensive to test networks of observation systems this way. Furthermore, the placement or retrieval of an observation system from the seafloor, which would happen every time a software error has to be fixed, takes up to several hours, depending on the deployment depth and weather conditions. The advancement in hardware and software technology and especially in the embedded software domain, enables the usage of "Digital Twins" (DT) to reduce all these time-consuming activities and costs around a field test. In a previous study, the concept of how the Digital Twins in this network had been developed was presented in \cite{MFI2020}, see also \autoref{sec-dtp}.
The actual set up of the network and the geographical setting will be published elsewhere. Briefly, the network was established at two sites in the Eckernförder Bay (western Baltic Sea) close to Kiel. Major focus was on the region at the entrance of the Eckernförde Bay close to the Boknis Eck time series Station located approximately \SI{1}{km} from the coast at the position \ang{54;31.77;} N, \ang{010;02.36;} E in water depths of $17$ to $24$ meters.
The sensing carriers of our network included two lander type platforms (BIGO Lander, FLUX Lander), see Fig.~\ref{fig:multiplelander}, which are used outside of the ARCHES project to measure the exchange of solutes between the seafloor and the water column as well as to monitor changes in the bottom water \cite{sommer2009seabed,bigo2003, wenzhofer2016robex}, see Fig.~\ref{fig:mansioviator}. The CRAWLER SIM system was used to implement the crawler-control software and scientific payload of long-term benthic crawler systems \cite{lemburg2018benthic}. The MANSIO-VIATOR system comprises a stationary lander system serving as hangar (MANSIO) and a mobile deep-sea crawler (VIATOR), Fig.~\ref{fig:mansioviator}. The MANSIO lander serves as a hangar, which is used for transport to the site of investigation and for recovery at the ocean surface as well as to recharge the lithium polymer (LiPo) accumulators on the crawler.
The transfer of data and commands as well as positioning from all network partners was performed hydroacoustically using Evologics S2C R 7/17 USBL acoustic modems \cite{evologicsmodem717} on each observation system. A shipboard modem was mounted in the moon pool of RV ALKOR. Still the establishment of a reliable and robust hydroacoustic communication is a challenge as summarized by Akyildiz et al. \cite{underwaterchallenge}. In contrast to data transfer using optical systems or electromagnetic waves hydoacoustic communication is slow, it has a high energy demand, small bandwidths, and time-varying multi-path propagation. Furthermore, reliable transmission depends on physical properties of the water column including presence and position of a pycnocline, turbidity, temperature affecting sound velocity.
\section{DIGITAL TWIN PROTOTYPES}\label{sec-dtp}
In Barbie et al. \cite{MFI2020}, a detailed definition of the ``Digital Twin Prototype'' approach was given based on the Digital Twin definition by Saracco \cite{dtdefinition}. A Digital Twin Prototype is the software/model prototype of a real entity, the Physical Twin. It uses existing recordings of sensing and actuation data over time as digital shadow, to simulate the Physical Twin. In ARCHES each ocean observation system has a corresponding Digital Twin Prototype and Digital Twin. The software on the Physical Twin is identical to the software that is used for the Digital Twin Prototype. The difference is that the Physical Twin has real hardware connected and the Digital Twin Prototype uses emulated hardware components, yet both types synchronize their sensing/actuation data with the corresponding Digital Twin. The Digital Twin differs from the Digital Twin Prototype only by an environment flag that defines it is a Physical Twin or Digital Twin. If the flag is set \emph{true}, the Digital Twin Prototype is a Digital Twin and does not synchronize its sensing/actuation data, instead it receives sensing/actuation data and synchronizes only the incoming commands to its physical counterpart. This approach enables us to start a Digital Twin Prototype and a Digital Twin at the same time in the same development environment and hence, to develop new modules without the need of a connection to the real hardware \cite{MFI2020}.
The software framework was developed in Python using the middleware Robot Operating System (ROS) \cite{rospaper}. Each microservice is encapsulated in a Docker container. In Fig.~\ref{fig:architecture}, an overview of the software architecture of the Digital Twins is shown. The Physical Twin side carries real hardware such as sensors/actuators, while on the Digital Twin side the same hardware components are emulated. Even a mix of real and emulated hardware at the same time is possible.
\begin{figure*}[tb]
\centering
\includegraphics[width=1.0\textwidth]{images/architecture.pdf}
\caption{The Software Architecture of the Digital Twins \cite{MFI2020}.}
\label{fig:architecture}
\end{figure*}
Due to our publish-subscribe architecture, we face some challenges in the synchronization of the Physical Twins and Digital Twins. Messages published by a node XYZ in topic A/B/C on one Twin, have to be synchronized to the corresponding Twin in the same node and the same topic. Nodes that contain publishers or subscribers that are able to synchronize their messages with the Physical or Digital Twin are referred to as \emph{Skill}. Services and drivers can be part of a Skill. Which topics are synchronized with a twin are defined in a list of topics saved on the ROS Parameter Server. This list can contain absolute, relative, and topics mixed with wildcards. There are two types of synchronization. The first, on the Physical Twin, is the transmission of sensor/actuator data and statuses to the Digital Twin. The second, on the Digital Twin, is the transmission of commands to the Physical Twin. The Physical Twin does receive commands only via the corresponding Digital Twin. If a command causes a software failure on the Digital Twin, one can be quite sure that it causes an error on the Physical Twin, too.
The most advanced pair of Physical and Digital Twin in the project ARCHES is the MANSIO-VIATOR system. There are CAD models of both systems in a Gazebo \cite{gazebopaper} simulation. The ROS integration in the Gazebo simulation enables us to connect the models of MANSIO and VIATOR with the ROS software we developed, which is a fundamental idea of Digital Twins. Besides a 3D dynamic multi-robot environment, Gazebo also provides modules that help to simulate more complex hardware such as cameras and lasers. However, this simulation cannot only be used to simulate and visualize the current behavior, which is executed on the Physical Twin, in particular it can be used to develop and test advanced algorithms, e.g., for obstacle avoidance, docking (MANSIO-VIATOR specific), or train AI systems. In combination with our Digital Twin Prototype approach, Gazebo is also part of the automated tests in the continuous delivery pipeline used in ARCHES \cite{MFI2020}.
\section{THE DIGITAL THREAD}
The acoustic modems by Evologics offer different modes for data transmission. Instead of the \emph{Burst Mode} (BM), which has the highest bandwidth with up to \SI{6.9}{kbit/s}, \emph{Instant Messages} (IM) are used. In previous tests they proved to be the most reliable mode for data transmission. Additionally, instant messages can be broadcast to all network participants, which is not possible with the burst mode. The trade off is that the bandwidth for application data is limited to \SI{64}{B/s} with instant messages.
\begin{figure}[tb]
\centering
\includegraphics[clip, trim=8cm 0cm 8cm 0cm, width=0.5\textwidth]{images/basestation.pdf}
\caption{The Network Setup. Refer to Fig.~\ref{fig:multiplelander} for larger pictures of the ocean observation system components.}
\label{fig:setup}
\end{figure}
Fig.~\ref{fig:setup} shows that each Digital Twin is connected to the \emph{Basestation}, which consists of the ROS Master and one node that distributes the messages of the Physical Twins to the corresponding Digital Twins and vice versa. Connecting the Physical Twin with its digital counterpart on the research vessel, requires a vertical connection from the seafloor to the sea surface and vice versa. Vertical networks, where observation systems on the seafloor send signals to a buoy or research vessel on the sea surface, e.g. the tsunami warning systems GITEWS and DART, were already introduced two decades ago \cite{GITEWS, dart}. However, these networks did only transmit data in a predefined control language that limits the kind of data that can be sent to other platforms. To allow for a more dynamic message exchange, Schneider et al. \cite{dccl} developed a dynamic compact control language (DCCL). The DCCL also addresses another challenge in the underwater domain: the limited bandwidth and thus, the limitation of data that can be transmitted. Serializations in standard data formats such as YAML, which is the format of ROS messages, cause significant overhead. Schneider et al. utilize the binary serialization of Google Protocol Buffers (Protobuf) to serialize object messages and reduce unnecessary overhead. However, in combination with ROS the usage of Protobuf has one disadvantage, Protobuf also uses pre-compiled messages. Hence, two different message definitions by mistake or a missing Protobuf message on one side, leads to a faulty serialization and as a consequence, to corrupted data. To avoid such errors, a new control language was developed in ARCHES utilizing the open-source tool Apache Avro. Avro creates binary de-/serializations without requiring a pre-compile step for code generation, since message schemes can be defined inline. Thus, ROS messages are serialized on the sending observation system to an Avro message and are deserialized on the receiving observation system to a ROS message at runtime.
To synchronize the Physical Twins and Digital Twins, all ROS messages contain a header that indicates from which Skill and topic a message was published/subscribed. We used ROSBag as storage mechanism during the research cruise. With RQt there is already a Qt-based framework, to inspect all topics in that ROSBag files and visualize the contents in a GUI. In later missions, the main storage will be a MongoDB.
\section{Field Experiment}\label{sec-field}
The goal of this ARCHES Demonstration Mission was to establish a network of heterogeneous and interconnected ocean observation systems and to test its software components. Digital Twins are already used for robotic system in other extreme environments, such as space \cite{nasadt2012}. Hence, one of the research question for this Demo Mission was, whether the Digital Twin approach represents a suitable approach for MANETs in under water applications.
With the Digital Twin Prototype approach we were able to evaluate the scenarios described below in \autoref{subsec:scenarios} in a virtual environment before applying them during the field tests. The Physical Twins were simulated by the Digital Twin Prototypes. Instead of the real acoustic modems, we used the S2C D-MAC Emulator also provided by Evologics. This Demonstration Mission is the first time we evaluated the following scenarios under real conditions. All the tests of the scenarios were performed during the research cruise AL547. The Digital Twins were running on a server on the research vessel ALKOR and shared a common ROS Master provided by the Basestation. The Basestation was connected to the Evologics modem that was located in the moon pool of the research vessel.
\subsection{METHODOLOGY AND SCENARIOS}\label{subsec:scenarios}
\begin{figure*}[tb]
\centering
\begin{subfigure}{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{images/scenariosBoknisEckBilder1.pdf}
\caption{Synchronize the environmental data with the Digital Twins.}
\label{fig:scenarioa}
\end{subfigure}
\hspace{1cm}
\begin{subfigure}{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{images/scenariosBoknisEckBilder2.pdf}
\caption{Synchronize behaviors with Physical Twins.}
\label{fig:scenariob}
\end{subfigure}
\\
\begin{subfigure}{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{images/scenariosBoknisEckBilder3.pdf}
\caption{broadcast an event from the research vessel.}
\label{fig:scenarioc}
\end{subfigure}
\hspace{1cm}
\begin{subfigure}{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{images/scenariosBoknisEckBilder4.pdf}
\caption{One Digital Twins detects an event and its Physical Twin broadcasts this event to all other platforms.}
\label{fig:scenariod}
\end{subfigure}
\caption{Scenarios that are evaluated in the ARCHES Demo Mission. Refer to Fig.~\ref{fig:multiplelander} for larger pictures of the ocean observation system components.}
\label{fig:scenarios}
\end{figure*}
To investigate the suitability of the Digital Twin approach for under water networks, the scenarios displayed in Fig.~\ref{fig:scenarios} were evaluated. The approach was considered feasible, if all the scenarios can be repeated independently and the ocean observation systems synchronize environmental data and statuses reliably. While Scenarios \ref{fig:scenarioa} - \ref{fig:scenarioc} evaluate different aspects of the twin synchronization, Scenario \ref{fig:scenariod} evaluates (i.) the collaboration between a Physical Twin and its corresponding Digital Twin and (ii.) the communication between two Physical Twins. As a common standard between all ocean observation systems, four basic messages are exchanged: StandardO2, StandardStatus, SetBehavior, and O2Event. StandardO2 messages carry a timestamp, the oxygen value, oxygen saturation and the temperature. StandardStatus messages carry a timestamp, the current behavior ID and the status of the behavior such as running, finished, or failure. SetBehavior messages carry the ID of the behavior to execute. O2Event messages carry a string that indicates a detected environmental event. In this Demonstration Mission two event-scenarios were tested, ``Oxia'', chosen for well ventilated conditions in the ambient water body of a respective platform and ``Hypoxia'', indicating adverse environmental conditions with reduced oxygen levels.
In Scenario \ref{fig:scenarioa} all ocean observation systems were placed at the seafloor and synchronize their measurements to the corresponding Digital Twins on the research vessel. The Physical Twins are programmed to different measurement cycles from every five seconds to every five minutes, which were automatically started during the startup.
In Scenario \ref{fig:scenariob} commands were manually sent to the Digital Twin on the research vessel and from there synchronized to the corresponding Physical Twin. The Physical Twin respond with a status message of a changed behavior. MANSIO, VIATOR, and BIGO also respond with a status message periodically or after a given behavior has finished.
In Scenario \ref{fig:scenarioc} oxygen data measured by the platforms were used to identify changes in ambient oxygen levels, which were categorized either as oxic (fully ventilated) or hypoxic conditions (oxygen level declines below a certain threshold). Via broadcast an oxygen event such as Oxia or Hypoxia, was send to the other platforms, which changed their measurement strategy according to a predefined protocol. In this experiment the broadcast was manually triggered on the Basestation, without any automated decision algorithm.
Scenario \ref{fig:scenariod} combines vertical and horizontal communication from Physical Twin to Physical Twin. Similar to Scenario Fig.~\ref{fig:scenarioc} oxygen measurements were used to recognize oxic or hypoxic conditions and to trigger the corresponding platform specific measurement strategies Oxia or Hypoxia. However, the events were not broadcast from the research vessel, instead, the decision was triggered one of the Digital Twins. The Digital Twin then synchronizes its behavior to its corresponding Physical Twin and this twin broadcasts this event to all other ocean observation systems. Each observation system that receives this event, switches into the Oxia or Hypoxia behavior and synchronizes its status to the corresponding Digital Twin on the research vessel.
\subsection{RESULTS AND DISCUSSION}
It's beyond the scope of this field report to present the detailed behavior of the different platforms during the experiments. Instead, we will focus on the main aspects with regard to the application of Digital Twins in underwater networks.
For each ocean observation system Scenario \ref{fig:scenarioa} was evaluated in the moment it was placed on the seafloor. Since each Physical Twin starts measuring on start-up, we know that the synchronization with the corresponding Digital Twin is working, if StandardO2 messages arrive in corresponding Digital Twin.
\begin{figure}[tb]
\centering
\includegraphics[width=0.5\textwidth]{images/mansioo2data2110.eps}
\caption{Incorrect oxygen measurements taken by MANSIO.}
\label{fig:mansioo2}
\end{figure}
Scenario \ref{fig:scenariob} was executed one by one for each ocean observation system. To change the current behavior, we sent SetBehavior messages with different behaviorIDs. This command can only be send to the Digital Twins, which then synchronizes the message to the same topic on the corresponding Physical Twin. The Physical Twin solely reacts to SetBehavior messages sent by its Digital Twin. In this mission all SetBehavior commands sent to a Digital Twin were automatically synchronized to its corresponding Physical Twin. Nevertheless, this architecture allows to add an extra layer that checks, if the behavior was changed on the Digital Twin successfully and only after a successful behavior change the SetBehavior message will be synchronized to the Physical Twin. A behavior change on the Physical Twin leads to a StandardStatus message that is synchronized to its corresponding Digital Twin. In this scenario, we received this StandardStatus messages from all ocean observation systems.
Fig.~\ref{fig:mansioo2} shows a part of the results for Scenario \ref{fig:scenarioa} and \ref{fig:scenariob} for the ocean observation system MANSIO. In Fig.~\ref{fig:mansioo2} the measurements taken by MANSIO over a period of $9$ hours during the first deployment are visualized. The two gaps indicate where the Digital Twin successfully stopped and started the measurement on its physical counterpart. Notice, this measurements ($y\geq750 \, \si{\mu M}$) are contrary to measurements from the actual oxygen value during that period. The other oxygen sensors measured an oxygen concentration in the air around $\SI{230}{\mu M}$ and much less underwater. When MANSIO was deployed the first time, we forgot to remove the protective cap of its oxygen sensor, which caused this incorrect measurements. In a traditional approach, where the ocean observation system is deployed on the seafloor for a couple of weeks, this error would have been noticed only after the mission.
In Scenario \ref{fig:scenarioc} an O2Event message was broadcast to all ocean observation systems. Upon successful reception of the message, all of them switched to their platform specific behavior, i.e. to the behavior, which has been defined for the environmental condition Oxia or Hypoxia. For example, MANSIO turned on its lights and VIATOR moved backwards\footnote{We uploaded a video with this submission}. Nevertheless, the message was not received by all ocean observation systems during all broadcasting attempts. Problems of data transmission were caused by increased distances between the RV ALKOR or unfavorable positions of the research vessel within the network area. In some cases turbulence in the water induced by the propeller of the ship might have further contributed to disturb data transmission.
Instead of broadcasting a message from the research vessel to all the ocean observation system, an O2Event message was published to a decision node on one of the Digital Twins to evaluate the Scenario \ref{fig:scenariod}. In this scenario, the O2Event was sent to the Digital Twin of the FLUX Lander and its Physical Twin broadcast the event to all other platforms. The behaviors to be executed and results were the same as in Scenario \autoref{fig:scenarioc}. Again, not all ocean observation systems received always all broadcast messages. In horizontal communication the positioning of the ocean observation systems to each other is important. Nevertheless, this scenario showed a cooperation between a Digital Twin and its physical counterpart (Stage IV Digital Twin \cite{dtdefinition}). Since the battery capacity on ocean observation system is limited, see \autoref{sec:context}, tasks with extensive power consumption could be sourced out to the Digital Twin that is running on a server, e.g. placed on the research vessel. Power consuming tasks that are envisioned on the ocean observation systems in the context of the project ARCHES are machine learning algorithms to predict upcoming environmental changes. Velasco-Montero et al. analyzed different frameworks used for machine learning algorithms and showed the increased power consumption on a RaspberryPi Model B \cite{kipowercon}.
\section{CONCLUSION AND FUTURE WORK}\label{sec:conclusion}
In conclusion, the presented scenarios were evaluated successfully. Hence, Digital Twins Prototypes are capable to be used in underwater networks. Especially in 2020, during the travel restrictions caused by the COVID19 pandemic, the advantage of this approach was the independence of a permanent physical connection to the ocean observation systems from AWI and GEOMAR to develop and test the network. It would not have been possible to evaluate the network without this approach.
Monitoring and operating an ocean observation system from a research vessel is possible without Digital Twins. The advantage of Digital Twins is the visual response we get when synchronizing commands from the Digital Twin to the Physical Twin. If a command causes an error on the Digital Twin, it also causes an error on the Physical Twin. Hence, if an error is thrown, we immediately know where and maybe why it occurred. Adding a layer that prevents the synchronization of commands that cause errors on the Digital Twin, reduces the impact of mistakes done by an operator of that ocean observation system. Combined with a simulation tool like Gazebo, see also \autoref{sec-dtp}, different commands and scenarios can be simulated, before executed on the ocean observation system.
Furthermore, the importance of extensive software and hardware testing in the embedded domain became apparent. With the increasing complexity of the software, software tests become a prerequisite in the development process of ocean observation systems. The Digital Twin Prototype approach developed in this project allows state of the art software testing in the embedded software domain, including continuous integration~\cite{shahin_continuous_2017}. This is a generic approach, suitable for all kinds of Digital Twins, not only in the underwater domain.
The presented results for MANSIO in Scenario \ref{fig:scenarioa} hint another use case for Digital Twins: predictive maintenance. By monitoring the performance and condition of a Physical Twin's hardware and equipped with algorithms to analyze anomalies or deviations from monitored data in previous missions, a Digital Twin helps to detect failing hardware. The ocean observation system can be maintained and the failing hardware replaced before the gathered data during the entire research cruise is corrupted. Kapteyn et al. demonstrate a Digital Twin for predictive maintenance for unmanned aerial vehicles \cite{dtwillcox}.
Digital Twins do not have to run on a server on a research vessel near the ocean observation systems. Instead of expensive research cruises to retrieve the collected data from the ocean observation systems, a buoy at the ocean surface can be the gateway station between the ocean observation systems at the seafloor and a satellite. Scientists and technicians would be able to retrieve the data and operate ocean observation systems from any place in the world via the Internet.
Continuous long-term and interconnected ocean observation will be become increasingly important when facing the climate changes. A possible context for our approach could be the already existing FRAM Ocean Observing System \cite{fram} infrastructure operated by AWI, which targets the gateway between the North Atlantic and the Central Arctic.
\section*{Acknowledgment}
The project is supported through the HGF-Alliance ARCHES -- Autonomous Robotic Networks to Help Modern Societies and the Helmholtz Association.
\bibliographystyle{IEEEtran
| {
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Q: Auto renewable IN-App Subscriptions returns multiple transactions and "Sign-In required" dialogs I am implementing Auto renewable Subscriptions in my App. Whenever I launch my app, " func paymentQueue(_ queue: SKPaymentQueue, updatedTransactions transactions: [SKPaymentTransaction])" is called with multiple transactions. I am not sure why is this happening. May be because of "restore" functionality. I have implemented "RestoreCompletedTransactions" when user hits the Restore Button. I am running into issue either I get multiple dialogs for "Sign-In required"and/or I get so many transactions without calling anything. I am making sure I finish transaction. Please help!!
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,556 |
# Oracle PL/SQL Language
Pocket Reference
Fifth Edition
Steven Feuerstein, Bill Pribyl, and Chip Dawes
# Oracle PL/SQL Language Pocket Reference
by Steven Feuerstein, Bill Pribyl, and Chip Dawes
Copyright © 2015 Steven Feuerstein, Bill Pribyl, Chip Dawes. All rights reserved.
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978-1-491-92000-8
[LSI]
# Oracle PL/SQL Language Pocket Reference
# Introduction
The _Oracle PL/SQL Language Pocket Reference_ is a quick reference guide to the PL/SQL programming language, which provides procedural extensions to the SQL relational database language.
The purpose of this pocket reference is to help PL/SQL users find the syntax of specific language elements. It is not a self-contained user guide; basic knowledge of the PL/SQL programming language is assumed. For more information, see the following O'Reilly books:
* _Oracle PL/SQL Programming_ , Sixth Edition, by Steven Feuerstein with Bill Pribyl
* _Oracle PL/SQL Best Practices_ , Second Edition, by Steven Feuerstein
* _Oracle Essentials_ , Fifth Edition, by Rick Greenwald, Robert Stackowiak, and Jonathan Stern
Wherever a package, program, or function is supported only for a particular version of the Oracle database (e.g., Oracle Database 12 _c_ ), we indicate this in the text.
## Acknowledgments
We are grateful to all those who helped in the preparation of this book. In particular, thanks to Patrick Barel, Indu Janardanan, and Prachi Sharma for their technical reviews (though all mistakes and omissions remain the responsibility of the authors). Bryn Llewellyn gave us thorough and thoroughly helpful feedback on previous editions. Thanks as well to first-edition reviewers Eric J. Givler and Stephen Nelson and to second- and third-edition reviewer Jonathan Gennick. In addition, we appreciate all the good work by the O'Reilly crew, especially editor Tim McGovern, in editing and producing this book.
## Conventions
UPPERCASE indicates PL/SQL keywords, as well as certain identifiers used by Oracle Corporation as built-in function and package names.
_Italic_ indicates filenames and directories, as well as the first use of a term.
`Constant width` is used for code examples, literals, and identifiers.
**`Constant width bold`** indicates user input in examples showing an interaction.
`[]` enclose optional items in syntax descriptions.
`{}` enclose a list of items in syntax descriptions; you must choose one item from the list.
`|` separates bracketed list items in syntax descriptions.
# PL/SQL Language Fundamentals
This section summarizes the fundamental components of the PL/SQL language: characters, identifiers, literals, delimiters, use of comments and pragmas, and construction of statements and blocks.
## PL/SQL Character Set
The PL/SQL language is constructed from letters, digits, symbols, and whitespace, as defined in the following table:
Type | Characters
---|---
Letters | `A`–`Z`, `a`–`z`
Digits | `0`—`9`
Symbols | `~!@#$%*( )_−+=|:;"'< >,^.?/`
Whitespace | Space, tab, newline, carriage return
Characters are grouped together into four lexical units: identifiers, literals, delimiters, and comments.
## Identifiers
Identifiers are names for PL/SQL objects, such as constants, variables, exceptions, procedures, cursors, and reserved words. Identifiers have the following characteristics:
* Can be up to 30 characters in length
* Cannot include whitespace (space, tab, carriage return)
* Must start with a letter
* Can include a dollar sign (`$`), an underscore (`_`), and a pound sign (`#`)
* Are not case-sensitive
Using PL/SQL's reserved words as identifiers in your programs is not a good idea and can result in compilation or runtime errors that are difficult to troubleshoot.
###### Tip
Earlier editions of this book included a list of reserved words. However, Oracle Database 12 _c_ has more than 2,080 reserved words as listed in the V$RESERVED_WORDS data dictionary view. Although you can in fact use many of these words as your own identifiers without any complaints from the compiler, doing so can be confusing. We suggest you write PL/SQL using an editor that highlights reserved words automatically and avoid using them to name your own variables or subprograms.
If you enclose an identifier within double quotes, all but the first of these rules are ignored. For example, the following declaration is valid (although not exactly sensible):
DECLARE
"1 ^abc" VARCHAR2(100);
BEGIN
IF "1 ^abc" IS NULL THEN ...
END;
## Boolean, Numeric, and String Literals
_Literals_ are specific values not represented by identifiers. For example, `TRUE`, `3.14159`, `6.63E-34`, `'Moby Dick'`, and `NULL` are all literals of type Boolean, number, or string. There are no complex datatype literals because their values are internal representations; complex types receive values through direct assignment or via constructors. Unlike the rest of PL/SQL, literals are case-sensitive. To embed single quotes within a string literal, place two single quotes next to each other.
You can define your own quoting mechanism for string literals in both your SQL and PL/SQL statements. Use the characters `q'` (q followed by a straight single quote) to designate the programmer-defined delimiter for your string literal. Terminate the literal string with the programmer-defined delimiter followed by a trailing single quote—for example, `q'!my string!'`. NCHAR and NVARCHAR delimiters are preceded by the letters `nq`, as in `nq'^nchar string^'`. This technique can simplify your code when consecutive single quotes appear within a string, such as the literals in a SQL statement. If you define your delimiter with one of the four bracketing characters `( [ {<`, you must use the righthand version of the bracketing character as the closing delimiter. For example, `q'[` must be closed with `]'`.
See the following table for examples:
Literal | Actual value
---|---
`'That''s Entertainment!'` | That's Entertainment!
`q'#That's Entertainment!#'` | That's Entertainment!
`'"The Raven"'` | "The Raven"
`'TZ=''CDT6CST'''` | TZ='CDT6CST'
`q'$TZ='CDT6CST'$'` | TZ='CDT6CST'
`q'[TZ='CDT6CST']'` | TZ='CDT6CST'
`''''` | '
`'''hello world'''` | 'hello world'
`q'!'hello world'!'` | 'hello world'
`''''''` | ''
`q'['']'` | ''
`nq'<Price='£'>'` | Price='£'
`nq'-WHERE name LIKE 'ñ'-'` | WHERE name LIKE 'ñ'
## Numeric Literals
Oracle may improve runtime performance if you make explicit the datatype of numeric literals. You can do so by including or excluding a decimal point or by using a trailing `f` or `d`, as shown in the following table:
Literal | Datatype
---|---
`3.14159` | NUMBER
`42` | INTEGER
`0.0` | NUMBER
`3.14159f` | BINARY_FLOAT
`3.14159d` | BINARY_DOUBLE
You can also use the named constants:
* `BINARY_FLOAT_NAN` (not a number)
* `BINARY_FLOAT_INFINITY`
* `BINARY_FLOAT_MAX_NORMAL`
* `BINARY_FLOAT_MIN_NORMAL`
* `BINARY_FLOAT_MAX_SUBNORMAL`
* `BINARY_FLOAT_MIN_SUBNORMAL`
as well as the BINARY_DOUBLE versions of these constants.
## Datetime Interval Literals
The datetime interval datatypes represent a chronological interval expressed in terms of either years and months or days, hours, minutes, seconds, and fractional seconds. Literals of these datatypes require the keyword INTERVAL followed by the literal and format string(s). The interval must go from a larger field to a smaller one, so YEAR TO MONTH is valid, but MONTH TO YEAR is not. See the following table for examples:
Literal | Actual value
---|---
`INTERVAL '1–3' YEAR TO MONTH` | 1 year and 3 months later
`INTERVAL '125–11' YEAR(3) TO MONTH` | 125 years and 11 months later
`INTERVAL '-18' MONTH` | 18 months earlier
`INTERVAL '-48' HOUR` | 48 hours earlier
`INTERVAL '7 23:15' DAY TO MINUTE` | 7 days, 23 hours, 15 minutes later
`INTERVAL '1 12:30:10.2' DAY TO SECOND` | 1 day, 12 hours, 30 minutes, 10.2 seconds later
`INTERVAL '12:30:10.2' HOUR TO SECOND` | 12 hours, 30 minutes, 10.2 seconds later
## Delimiters
_Delimiters_ are symbols with special meaning, such as `:=` (assignment operator), `||` (concatenation operator), and `;` (statement delimiter). The following table lists the PL/SQL delimiters:
Delimiter | Description
---|---
`;` | Terminator (for statements and declarations)
`+` | Addition operator
`−` | Subtraction operator
`*` | Multiplication operator
`/` | Division operator
`**` | Exponentiation operator
`||` | Concatenation operator
`:=` | Assignment operator
`=` | Equality operator
`<>` and `!=` | Inequality operators
`^=` and `˜=` | Inequality operators
`<` | Less-than operator
`<=` | Less-than-or-equal-tooperator
`>` | Greater-than operator
`>=` | Greater-than-or-equal-to operator
`(` and `)` | Expression or list delimiters
`<<` and `>>` | Label delimiters
`,` | Item separator
`'` | Literal delimiter
`q'` and `'` | Programmer-defined string literal delimiter
`nq'` and `'` | Programmer-defined NCHAR string literal delimiter
`"` | Quoted literal delimiter
`:` | Host variable indicator
`%` | Attribute indicator
`.` | Component indicator (as in _record.field_ or _package.element_ )
`@` | Remote database indicator (database link)
`=>` | Association operator (named notation)
`..` | Range operator (used in the FOR loop)
`--` | Single-line comment indicator
`/*` and `*/` | Multiline comment delimiters
## Comments
_Comments_ are sections of code that exist to aid readability. The compiler ignores them.
A single-line comment begins with a double hyphen (--) and terminates at the end-of-line (newline). The compiler ignores all characters between the -- and the newline.
A multiline comment begins with slash asterisk (/*) and ends with asterisk slash (*/). The /* */ comment delimiters also can be used for a single-line comment. The following block demonstrates both kinds of comments:
DECLARE
-- Two dashes comment out remainder of line.
/* Everything is a comment until the compiler
encounters the following symbol */
You cannot embed multiline comments within a multiline comment, so be careful during development if you comment out portions of code that include comments. The following code demonstrates this issue:
DECLARE
/* Everything is a comment until the compiler
/* This comment inside another WON'T work!*/
encounters the following symbol. */
/* Everything is a comment until the compiler
-- This comment inside another WILL work!
encounters the following symbol. */
## Pragmas
The PRAGMA keyword is used to give instructions to the compiler. There are six types of pragmas in PL/SQL:
_AUTONOMOUS_TRANSACTION_
Tells the compiler that the function, procedure, top-level anonymous PL/SQL block, object method, or database trigger executes in its own transaction space. See "Database Interaction" for more information on this pragma.
_EXCEPTION_INIT_
Tells the compiler to associate the specified error number with an identifier that has been declared an EXCEPTION in your current program or an accessible package. See "Exception Handling" for more information on this pragma.
_INLINE_
Tells the compiler whether calls to a subprogram should be replaced with a copy of the subprogram. See "Optimizing Compiler" for more information on inline optimization.
_RESTRICT_REFERENCES_
This pragma is deprecated; use DETERMINISTIC and PARALLEL_ENABLE instead.
_SERIALLY_REUSABLE_
Tells the runtime engine that package data should not persist between references. This is used to reduce per-user memory requirements when the package data is needed only for the duration of the call and not for the duration of the session. See "Packages" for more information on this pragma.
_UDF_ (Oracle Database 12 _c_ and higher)
Tells the compiler that you intend for your user-defined function to be invoked primarily from SQL rather than PL/SQL. This allows Oracle to optimize for performance inside SQL (at the possible expense of its performance in PL/SQL).
## Statements
A PL/SQL program is composed of one or more logical statements. A _statement_ is terminated by a semicolon delimiter. The physical end-of-line marker in a PL/SQL program is ignored by the compiler, except to terminate a single-line comment (initiated by the -- symbol).
## Block Structure
Each PL/SQL program is a _block_ consisting of a standard set of elements, identified by keywords (see Figure 1). The block determines the scope of declared elements and how exceptions are handled and propagated. A block can be _anonymous_ or _named_. Named blocks include functions, procedures, packages, and triggers.
###### Figure 1. The PL/SQL block structure
Here is an example of an anonymous block:
DECLARE
today DATE DEFAULT SYSDATE;
BEGIN
-- Display the date.
DBMS_OUTPUT.PUT_LINE ('Today is ' || today);
END;
Here is a named block that performs the same action:
CREATE OR REPLACE PROCEDURE show_the_date
IS
today DATE DEFAULT SYSDATE;
BEGIN
-- Display the date.
DBMS_OUTPUT.PUT_LINE ('Today is ' || today);
END show_the_date;
The following table summarizes the sections of a PL/SQL block:
Section | Description
---|---
Header | Required for named blocks. Specifies the way the program is called by other PL/SQL blocks. Anonymous blocks do not have a header. They start with the DECLARE keyword if there is a declaration section, or with the BEGIN keyword if there are no declarations.
Declaration | Optional; declares variables, cursors, types, and local programs that are used in the block's execution and exception sections.
Execution | Contains statements that are executed when the block is run; optional in package specifications and type specifications.
Exception | Optional; describes error-handling behavior for exceptions raised in the executable section.
# Variables and Program Data
PL/SQL programs normally are used to manipulate database information. You commonly do this by declaring variables and data structures in your programs, and then working with that PL/SQL-specific data.
A _variable_ is a named instantiation of a data structure declared in a PL/SQL block (either locally or in a package). Unless you declare a variable as a CONSTANT, its value can be changed at any time in your program.
The following table summarizes the different types of program data:
Type | Description
---|---
Scalar | Variables made up of a single value, such as a number, date, or Boolean.
Composite | Variables made up of multiple values, such as a record, collection, or instance of a user-defined object type. See the sections "Records in PL/SQL", "Collections in PL/SQL", and "Object-Oriented Features".
Reference | Logical pointers to values or cursors.
LOB | Variables containing large object (LOB) locators.
## Scalar Datatypes
Scalar datatypes divide into four families: number, character, datetime, and Boolean. Subtypes further define a base datatype by restricting the values or size of the base datatype.
### Numeric datatypes
Numeric datatypes represent real numbers, integers, and floating-point numbers. They are stored as NUMBER, PLS_INTEGER, and IEEE floating-point storage types.
Decimal numeric datatypes store fixed and floating-point numbers of just about any size. They include the subtypes NUMBER, DEC, DECIMAL, NUMERIC, FLOAT, REAL, and DOUBLE PRECISION. The maximum precision of a variable with type NUMBER is 38 digits, which yields a range of values from `1.0E-129` through `9.999E125`.
Variables of type NUMBER can be declared with precision and scale, as follows:
NUMBER( _precision, scale_ )
where _precision_ is the number of digits, and _scale_ is the number of digits to the right (positive scale) or left (negative scale) of the decimal point at which rounding occurs. Legal values for _scale_ range from −84 to 127. The following table shows examples of precision and scale:
Declaration | Assigned value | Stored value
---|---|---
NUMBER | 6.02 | 6.02
NUMBER(4) | 8675 | 8675
NUMBER(4) | 8675309 | Error
NUMBER(12,5) | 3.14159265 | 3.14159
NUMBER(12,−5) | 8675309 | 8700000
Oracle provides a variety of datatypes to store whole numbers: BINARY_INTEGER, INTEGER, INT, SMALLINT, NATURAL, NATURALN, POSITIVE, POSITIVEN, SIGNTYPE, and PLS_INTEGER.
Binary integer datatypes store signed integers in the range of −231 \+ 1 to 231 − 1. The subtypes include NATURAL (0 through 231 − 1) and POSITIVE (1 through 231 − 1) together with the NOT NULL variations NATURALN and POSITIVEN. SIGNTYPE is restricted to three values (−1, 0, 1). PLS_INTEGER is an unconstrained subtype (alias) of BINARY_INTEGER.
SIMPLE_INTEGER has the same range as BINARY_INTEGER except that it does not allow for null values and does not raise an exception if an overflow occurs. For example, 2147483647 + 1 = −2147483648 (note the negative value!). SIMPLE_INTEGER datatypes can result in significantly faster execution speeds when the PL/SQL code is compiled to native machine code.
IEEE 754–compliant floating-point numbers are available in both SQL and PL/SQL. These subtypes are the single-precision BINARY_FLOAT and the double-precision BINARY_DOUBLE. Because these datatypes require less memory and use native machine arithmetic, they perform much better for scientific or engineering applications that are computer-intensive or that require comparison to infinity or not a number (NaN). These two datatypes have binary precision instead of the decimal precision used in the NUMBER family. So, if you are developing financial applications that are concerned with rounding errors or require decimal precision, you probably should not use these floating-point datatypes.
The following table lists the PL/SQL numeric datatypes with ANSI and IBM compatibility. In this table:
* _precision_ is the precision for the subtype.
* _scale_ is the scale of the subtype.
* _binary_ is the binary precision of the subtype.
PL/SQL datatype | Compatibility | Oracle database datatype
---|---|---
DEC( _precision_ , _scale_ ) | ANSI | NUMBER( _precision_ , _scale_ )
DECIMAL( _precision_ , _scale_ ) | IBM | NUMBER( _precision_ , _scale_ )
DOUBLE PRECISION | ANSI | NUMBER
FLOAT( _binary_ ) | ANSI, IBM | NUMBER
INT | ANSI | NUMBER(38)
INTEGER | ANSI, IBM | NUMBER(38)
NUMERIC( _precision_ , _scale_ ) | ANSI | NUMBER( _precision_ , _scale_ )
REAL | ANSI | NUMBER
SMALLINT | ANSI, IBM | NUMBER(38)
BINARY_FLOAT | IEEE 754 | BINARY_FLOAT
BINARY_ DOUBLE | IEEE 754 | BINARY_ DOUBLE
### Character datatypes
Character datatypes store alphanumeric text and are manipulated by character functions. As with the numeric family, there are several subtypes in the character family, shown in the following table:
Family | Description
---|---
CHAR | Fixed-length alphanumeric strings. Valid sizes are 1 to 32,767 bytes.
VARCHAR2 | Variable-length alphanumeric strings. Valid sizes are 1 to 32,767 bytes.
LONG | Variable-length alphanumeric strings. Valid sizes are 1 to 32,760 bytes. LONG is included primarily for backward compatibility; instead use VARCHAR2(32767), CLOB, BLOB, or NCLOB for large character strings.
RAW | Variable-length binary strings. Valid sizes are 1 to 32,767 bytes (which is larger than the database limit of 2,000). RAW data does not undergo character set conversion when selected from a remote database.
LONG RAW | Variable-length binary strings. Valid sizes are 1 to 32,760 bytes. LONG RAW is included primarily for backward compatibility; BLOB and BFILE are the preferred datatypes for large binary data.
ROWID | Fixed-length binary data. Every row in a database has a physical address or ROWID. A ROWID has four parts in base 64:
_OOOOOOFFFBBBBBBRRR_
where:
* _OOOOOO_ is the object number.
* _FFFF_ is the absolute or relative file number.
* _BBBBBB_ is the block number within the file.
* _RRR_ is the row number within the block.
UROWID | Universal ROWID. Variable-length hexadecimal string depicting a logical, physical, or non-Oracle row identifier. Valid sizes are up to 4,000 bytes.
### Unicode character datatypes
The standard WE8MSWIN1252 or WE8ISO8859P2 character set does not support some languages, such as Chinese and Greek. To support multiple languages, the database allows two character sets—the _database character set_ and a _Unicode character set_ , sometimes called the _national character set_ (NLS).
The two NLS datatypes, NCHAR and NVARCHAR2, are used to represent data in the Unicode character set. NCHAR values are fixed-length character data; the maximum length is 32,767 bytes. NVARCHAR2 values are variable-length character data; the maximum length also is 32,767 bytes.
### Datetime datatypes
The datetime datatypes are DATE, TIMESTAMP, TIMESTAMP WITH TIME ZONE, and TIMESTAMP WITH LOCAL TIME ZONE. There are also two interval datatypes, INTERVAL YEAR TO MONTH and INTERVAL DAY TO SECOND.
DATE values are fixed-length, date-plus-time values. The DATE datatype can store dates from January 1, 4712 BC to December 31, 9999 AD. Each DATE includes the century, year, month, day, hour, minute, and second. Subsecond granularity is not supported via the DATE datatype; use one of the TIMESTAMP datatypes instead. The time portion of a DATE defaults to midnight (12:00:00 a.m.) if it is not included explicitly.
TIMESTAMP values store date and time to subsecond granularity. The subsecond precision (the number of digits to the right of the decimal) either defaults to 6 or is set to 0 through 9 digits by declaration, as in:
DECLARE
mytime_declared TIMESTAMP(9); -- max precision
mytime_default TIMESTAMP; -- default 6 digits precision
mytime_to_seconds TIMESTAMP(0); -- no fractional seconds
TIMESTAMP WITH TIME ZONE values store date and time values like a TIMESTAMP but also store the hourly offset from Coordinated Universal Time (UTC, which is essentially equivalent to Greenwich Mean Time). As with TIMESTAMP, the subsecond precision is 0 to 9 digits, either declared or inherited from the default 6 digits of precision:
DECLARE
mytime_declared TIMESTAMP(9) WITH TIME ZONE;
mytime_default TIMESTAMP WITH TIME ZONE;
TIMESTAMP WITH LOCAL TIME ZONE values store date and time values together with the UTC offset, like a TIMESTAMP WITH TIME ZONE. The principal difference between these timestamp datatypes occurs when values are saved to or retrieved from a database table. TIMESTAMP WITH LOCAL TIME ZONE values are converted to the database time zone and saved without an offset. The values retrieved from the database table are converted from the database time zone to the session's time zone.
The offset from UTC for both TIMESTAMP WITH TIME ZONE and TIMESTAMP WITH LOCAL TIME ZONE can be hours and minutes or a time zone region (found in the V$TIMEZONE_NAMES data dictionary view) with the optional daylight savings time name (also found in V$TIMEZONE_NAMES). For example:
ALTER SESSION SET NLS_TIMESTAMP_TZ_FORMAT=
'DD-Mon-YYYY HH24:MI:SS.FF TZR';
DECLARE
my_tswtz TIMESTAMP(4) WITH TIME ZONE;
BEGIN
my_tswtz := '31-MAR-2016 07:32:45.1234 US/Pacific';
INTERVAL YEAR TO MONTH values store a period of time in years and months:
DECLARE
myy2m INTERVAL YEAR TO MONTH;
BEGIN
myy2m := INTERVAL '1–6' YEAR TO MONTH; -- interval of 18 months
INTERVAL DAY TO SECOND values store a period of time in days, hours, minutes, seconds, and fractional seconds:
DECLARE
myd2s INTERVAL DAY TO SECOND;
BEGIN
myd2s := INTERVAL '2 10:32:15.678' DAY TO SECOND;
### BOOLEAN datatype
The BOOLEAN datatype can store one of _three_ values: TRUE, FALSE, or NULL. BOOLEAN variables usually are used in logical control structures such as IF . . . THEN or LOOP statements.
The following truth tables show the results of logical AND, OR, and NOT operations with PL/SQL's three-value Boolean model:
AND | TRUE | FALSE | NULL
---|---|---|---
TRUE | TRUE | FALSE | NULL
FALSE | FALSE | FALSE | FALSE
NULL | NULL | FALSE | NULL
OR | TRUE | FALSE | NULL
---|---|---|---
TRUE | TRUE | TRUE | TRUE
FALSE | TRUE | FALSE | NULL
NULL | TRUE | NULL | NULL
NOT (TRUE) | NOT (FALSE) | NOT (NULL)
---|---|---
FALSE | TRUE | NULL
## LOB Datatypes
PL/SQL supports a number of large object (LOB) datatypes, which can store objects up to 8 terabytes in size, or even up to 128 TB, depending on database block size. Unlike the scalar datatypes, variables declared for LOBs use _locators_ , or pointers to the actual data. LOBs are manipulated in PL/SQL using the built-in package DBMS_LOB. The LOB datatypes are:
_BFILE_
File locators pointing to read-only large binary objects in operating system files. With BFILEs, the large objects are outside the database, and the maximum size is system-dependent.
_BLOB_
LOB locators that point to large binary objects inside the database.
_CLOB_
LOB locators that point to large character (alphanumeric) objects inside the database.
_NCLOB_
LOB locators that point to large Unicode character objects inside the database.
## Implicit Datatype Conversions
Whenever PL/SQL detects that a datatype conversion is necessary, it attempts to change the values as required to perform the operation. Not all values in each datatype can be converted to another datatype. For example, an attempt to convert BINARY_FLOAT_NAN to a number datatype will raise an INVALID NUMBER exception.
## NULLs in PL/SQL
PL/SQL represents unknown or inapplicable values as NULL values. Because a NULL is unknown, a NULL is never equal or not equal to anything (including another NULL value). In addition, most functions and operators return a NULL when passed a NULL argument. You cannot check for equality or inequality to NULL; therefore, you must use the IS NULL or IS NOT NULL syntax to check for NULL values.
Here is an example of the IS NULL syntax used to check the value of a variable:
BEGIN
/* Correct usage */
IF myvar IS NULL THEN ...
/* Dangerous! Always evaluates to NULL */
IF myvar = NULL THEN ...
There are several ways in SQL and PL/SQL to substitute a non-null value for a null variable or expression. NVL is commonly used:
NVL ( _exprn1_ , _exprn2_ )
With NVL, Oracle will always evaluate both arguments, even if _`exprn1` _is not null. A potentially more efficient function is COALESCE:
COALESCE( _exprn1_ , _exprn2_ [, _exprn3_ , ...] )
COALESCE has two distinct characteristics: first, it accepts more than two arguments; second, it uses short-circuit evaluation, meaning it stops looking for a non-null value when it finds one (and, therefore, does not evaluate every expression in the list, which NVL does).
## Declaring Variables
Before you can use a variable, you must first declare it in the declaration section of your PL/SQL block or in a package as a global. When you declare a variable, PL/SQL allocates memory for the variable's value and names the storage location so that the value can be retrieved and changed. The syntax for a variable declaration is:
_variable_name_ [CONSTANT] _datatype_ [NOT NULL]
[{ := | DEFAULT } _initial_value_ ];
### Constrained declarations
A programmer may constrain certain datatypes at declaration so that variables have a size, scale, or precision that is less than the maximum allowed. Some examples:
total_sales NUMBER(15,2); -- Constrained
emp_id VARCHAR2(9); -- Constrained
company_number NUMBER; -- Unconstrained
book_title VARCHAR2; -- Not valid
Constrained declarations can require less memory than unconstrained declarations. Not all datatypes can be declared unconstrained; VARCHAR2 is a notable exception.
### Constants
The CONSTANT keyword in a declaration requires an initial value and does not allow that value to be changed in the program. For example:
min_order_qty NUMBER(1) CONSTANT := 5;
### Default values
The PL/SQL runtime engine assigns a default value of NULL to each variable you explicitly declare. If you want to initialize a variable to a value other than NULL, you can do so in the declaration with either the assignment operator (:=) or the DEFAULT keyword:
counter BINARY_INTEGER := 0;
priority VARCHAR2(8) DEFAULT 'LOW';
A NOT NULL constraint can be appended to the variable's datatype declaration to indicate that NULL is not a valid value. If you add the NOT NULL constraint, you must explicitly assign an initial value for that variable.
## Anchored Declarations
Use the %TYPE attribute to _anchor_ the datatype of a scalar variable to either another variable or to a column in a database table or view. Use %ROWTYPE to anchor a record's declaration to a cursor or table (see "Records in PL/SQL" for more details on the %ROWTYPE attribute).
The following block shows several variations of anchored declarations:
DECLARE
tot_sales NUMBER(20,2);
-- Anchor to a local variable
monthly_sales tot_sales%TYPE;
-- Anchor to a database column
v_ename employee.last_name%TYPE;
-- Anchor to a cursor
CURSOR mycur IS SELECT * FROM employee;
myrec mycur%ROWTYPE; -
The NOT NULL clause on a variable declaration (but not on a database column definition) follows the %TYPE anchoring and requires anchored declarations to have a default in their declaration. The default value for an anchored declaration can be different from that for the base declaration:
tot_sales NUMBER(20,2) NOT NULL DEFAULT 0;
monthly_sales tot_sales%TYPE DEFAULT 10;
## Programmer-Defined Subtypes
PL/SQL allows you to define unconstrained scalar subtypes, which you might want to do for greater "self-documentation" of your code or for future-proofing. An unconstrained subtype provides an alias to the original underlying datatype; for example:
CREATE OR REPLACE PACKAGE std_types
IS
-- Declare reusable app-specific type
SUBTYPE dollar_amt_t IS NUMBER;
END std_types;
CREATE OR REPLACE PROCEDURE process_money
IS
-- Use standardized type declared in package
credit std_types.dollar_amt_t;
A constrained subtype limits or constrains the new datatype to a subset of the original datatype. For example, POSITIVE is a constrained subtype of BINARY_INTEGER. The declaration for POSITIVE in the STANDARD package is:
SUBTYPE POSITIVE IS BINARY_INTEGER RANGE 1..2147483647;
You can define your own constrained subtypes in your programs:
PACKAGE std_types
IS
SUBTYPE currency_t IS NUMBER (15, 2);
END;
# Conditional and Sequential Control
PL/SQL includes conditional (IF, CASE) structures as well as sequential control (GOTO, NULL) constructs.
## Conditional Control Statements
There are several varieties of IF-THEN-ELSE and CASE structures.
### IF-THEN combination
IF _condition_ THEN
_executable statement(s)_
END IF;
For example:
IF caller_type = 'VIP' THEN
generate_response('GOLD');
END IF;
### IF-THEN-ELSE combination
IF _condition_ THEN
_TRUE sequence_of_executable_statement(s)_
ELSE
_FALSE (or NULL) sequence_of_statement(s)_
END IF;
For example:
IF caller_type = 'VIP' THEN
generate_response('GOLD');
ELSE
generate_response('BRONZE');
END IF;
Note that a condition that evaluates to NULL at runtime will cause control to branch to the ELSE clause.
### IF-THEN-ELSIF combination
Oracle's else-if condition allows testing a second conditional expression. Note that the second letter "e" is missing in the ELSIF keyword:
IF _condition-1_ THEN
_statements-1_
ELSIF _condition-N_ THEN
_statements-N_
[ELSE
ELSE statements]
END IF;
For example:
IF caller_type = 'VIP' THEN
generate_response('GOLD');
ELSIF priority_client THEN
generate_response('SILVER');
ELSE
generate_response('BRONZE');
END IF;
### CASE statement
There are two types of CASE statements: simple and searched.
A simple CASE statement is similar to an IF-THEN-ELSIF structure. The statement has a switch expression immediately after the keyword CASE. The expression is evaluated and compared to the value in each WHEN clause. The first WHEN clause with a matching value is executed, and then control passes to the next statement following the END CASE. For example:
CASE region_id
WHEN 'NE' THEN
mgr_name := 'MINER';
WHEN 'SE' THEN
mgr_name := 'KOOI';
ELSE
mgr_name := 'LANE';
END CASE;
If a switch expression evaluates to NULL, the ELSE case is the only one that can possibly match; WHEN NULL will never match because the database performs an equality comparison on the expressions.
Both the CASE statement and the CASE expression (see next section) should include an ELSE clause that will execute statements if no WHEN clause evaluates to TRUE, because PL/SQL's runtime engine will raise an exception if it finds no matching expression.
The searched CASE statement does not have a switch; instead, each WHEN clause has a complete Boolean expression. The first matching WHEN clause is executed, and control passes to the next statement following the END CASE; for example:
CASE
WHEN region_id = 'EAME' THEN
mgr_name := 'SCHMIDT';
WHEN division = 'SALES' THEN
mgr_name := 'KENNEDY';
ELSE mgr_name := 'GUPTA';
END CASE;
### CASE expression
There are also two types of CASE expressions: simple and searched. You can use CASE expressions anywhere that expressions are valid in your SQL or PL/SQL programs.
A simple CASE expression lets you choose an expression to evaluate based on a scalar value that you provide as input. The following example shows a simple CASE expression being used with the built-in DBMS_OUTPUT package to output the value of a Boolean variable. DBMS_OUTPUT.PUT_LINE is not overloaded to handle Boolean types, so in this example, the CASE expression converts the Boolean value in a character string, which PUT_LINE can then handle:
DECLARE
boolean_true BOOLEAN := TRUE;
boolean_false BOOLEAN := FALSE;
boolean_null BOOLEAN;
FUNCTION b2vc (flag IN BOOLEAN)
RETURN VARCHAR2 IS
BEGIN
RETURN
CASE flag
WHEN TRUE THEN 'True'
WHEN FALSE THEN 'False'
ELSE 'Null'
END;
END;
BEGIN
DBMS_OUTPUT.PUT_LINE (b2vc(boolean_true));
DBMS_OUTPUT.PUT_LINE (b2vc(boolean_false));
DBMS_OUTPUT.PUT_LINE( b2vc(boolean_null));
END;
A searched CASE expression evaluates a list of expressions to find the first one that evaluates to TRUE, and then returns the results of an associated expression. In the following example, a searched CASE expression returns the proper bonus value for any given salary:
DECLARE
salary NUMBER := 20000;
employee_id NUMBER := 36325;
PROCEDURE give_bonus
(emp_id IN NUMBER, bonus_amt IN NUMBER) IS
BEGIN
DBMS_OUTPUT.PUT_LINE(emp_id);
DBMS_OUTPUT.PUT_LINE(bonus_amt);
END;
BEGIN
give_bonus(employee_id,
CASE
WHEN salary >= 10000
AND salary <=20000 THEN 1500
WHEN salary > 20000
AND salary <= 40000 THEN 1000
WHEN salary > 40000 THEN 500
ELSE 0
END);
END;
## Sequential Control Statements
PL/SQL provides a GOTO statement and a NULL statement to aid in sequential control operations.
### GOTO
Although rarely used, the GOTO statement performs unconditional branching to a named label. At least one executable statement must follow the label (the NULL statement can be this necessary executable statement). The format of a GOTO statement is:
GOTO _label_name_ ;
For example:
BEGIN
GOTO second_output;
DBMS_OUTPUT.PUT_LINE('Will never execute.');
<<second_output>>
DBMS_OUPUT.PUT_LINE('We are here!);
END
There are several scope restrictions on where a GOTO can branch control. A GOTO:
* Can branch out of an IF statement, LOOP, or subblock.
* Cannot branch into an IF statement, LOOP, or subblock.
* Cannot branch from one section of an IF statement to another (from the IF-THEN section to the ELSE section is illegal).
* Cannot branch into or out of a subprogram.
* Cannot branch from the exception section to the executable section of a PL/SQL block.
* Cannot branch from the executable section to the exception section of a PL/SQL block, although a RAISE does this.
### NULL
The NULL statement is an executable statement that does nothing. It is useful when an executable statement must follow a GOTO label or to aid readability in an IF-THEN-ELSE structure. For example:
IF :report.selection = 'DETAIL' THEN
exec_detail_report;
ELSE
NULL;
END IF;
# Loops
The LOOP construct allows you to execute a sequence of statements repeatedly. There are three types of loops: simple (infinite), FOR, and WHILE.
You can use the EXIT statement to break out of the LOOP and pass control to the statement following the END LOOP. Use the CONTINUE statement, described later, to break out of the current loop iteration and pass control to the next loop iteration.
## Simple Loop
LOOP
_executable_statement(s)_
END LOOP;
The simple loop should contain an EXIT or EXIT WHEN unless you want it to execute infinitely. Use the simple loop when you want the body of the loop to execute at least once. For example:
LOOP
FETCH company_cur INTO company_rec;
EXIT WHEN company_cur%ROWCOUNT > 5 OR
company_cur%NOTFOUND;
process_company(company_cur);
END LOOP;
## Numeric FOR Loop
FOR _loop_index_ IN [REVERSE]
_lowest_number..highest_number_
LOOP
_executable_statement(s)_
END LOOP;
The PL/SQL runtime engine automatically declares the loop index a PLS_INTEGER variable, and you don't want or need to declare a variable with that name yourself. The _lowest_number_ and _highest_number_ ranges can be variables but are evaluated only once—on initial entry into the loop. The REVERSE keyword causes PL/SQL to start with the _highest_number_ and decrement down to the _lowest_number_. For example, this code:
BEGIN
FOR counter IN 1 .. 4
LOOP
DBMS_OUTPUT.PUT(counter);
END LOOP;
DBMS_OUTPUT.NEW_LINE;
FOR counter IN REVERSE 1 .. 4
LOOP
DBMS_OUTPUT.PUT(counter);
END LOOP;
DBMS_OUTPUT.NEW_LINE;
END;
yields the following output:
1234
4321
## Cursor FOR Loop
FOR _loop_index_ IN
{ _cursor_name_ | _(SELECT statement)_ }
LOOP
_executable_statement(s)_
END LOOP;
The PL/SQL runtime engine implicitly declares the loop index as a record of _cursor_name_ %ROWTYPE; do not explicitly declare a variable with that name.
The cursor FOR loop automatically opens the cursor, fetches all rows identified by the cursor, and then closes the cursor. You can embed the SELECT statement directly in the cursor FOR loop or use a previously declared cursor; for example:
FOR emp_rec IN emp_cur
LOOP
IF emp_rec.title = 'Oracle Programmer'
THEN
give_raise(emp_rec.emp_id,30)
END IF;
END LOOP;
The cursor FOR loop is an elegant, _declarative_ construct (you tell the database to fetch every row in the cursor without specifying _how_ to do it). Internally, Oracle will attempt to optimize such a fetch loop by retrieving 100 rows at a time, but you may still want to explicitly use BULK COLLECT and FORALL. See "Bulk Binds" for information on these statements.
## WHILE Loop
WHILE _condition_
LOOP
_executable_statement(s)_
END LOOP;
Use the WHILE loop in cases where you may not want the loop body to execute even once:
WHILE NOT end_of_analysis
LOOP
perform_analysis;
get_next_record;
IF analysis_cursor%NOTFOUND
AND next_step IS NULL
THEN
end_of_analysis := TRUE;
END IF;
END LOOP;
## "Repeat Until" Loop Emulation
PL/SQL does not directly support a REPEAT UNTIL construct, but a modified simple loop can emulate one. The syntax for this emulated REPEAT UNTIL loop is:
LOOP
_executable_statement(s)_
EXIT WHEN _boolean_condition_ ;
END LOOP;
Use such a loop when executing iterations indefinitely before conditionally terminating the loop.
## EXIT Statement
EXIT [WHEN _condition_ ];
If you do not include a WHEN clause, EXIT will terminate the loop unconditionally. Otherwise, the loop terminates only if the Boolean _condition_ evaluates to TRUE. The EXIT statement is optional and can appear anywhere in the loop.
## CONTINUE Statement
The CONTINUE statement terminates the current iteration of a loop, passing control to the next iteration. The format of a CONTINUE statement is:
CONTINUE [ _label_name_ ][WHEN _boolean_expression_ ];
The optional _label_name_ identifies which loop to terminate. If no _label_name_ is specified, the innermost loop's current iteration is terminated.
Like the EXIT statement, the CONTINUE statement is optional and can appear anywhere in the loop. The CONTINUE statement can be used to pass control out of a loop, optionally crossing loop boundaries, but cannot pass control out across a procedure, function, or method boundary.
The following example uses CONTINUE to branch out of an inner loop as well an outer loop:
DECLARE
TYPE dow_tab_t IS TABLE OF VARCHAR2(10);
dow_tab dow_tab_t := dow_tab_t('Sunday'
,'Monday','Tuesday','Wednesday','Thursday'
,'Friday','Saturday');
BEGIN
<<day_loop>>
FOR counter IN 2 .. 6 LOOP
--Skip Wednesdays
CONTINUE day_loop
WHEN dow_tab(counter)='Wednesday';
DBMS_OUTPUT.PUT_LINE (dow_tab(counter));
END LOOP;
END;
This yields the following output:
Monday
Tuesday
Thursday
Friday
This example shows how control may be passed from within an inner loop to the next iteration of an outer loop:
BEGIN
<<outer_loop>>
FOR outer_counter IN 1 .. 3 LOOP
DBMS_OUTPUT.PUT_LINE(outer_counter);
<<inner_loop>>
FOR inner_counter IN 10 .. 15 LOOP
CONTINUE outer_loop
WHEN outer_counter > 1
AND inner_counter = 12;
DBMS_OUTPUT.PUT_LINE('...'||inner_counter);
END LOOP;
END LOOP;
END;
This yields the following output:
1
...10
...11
...12
...13
...14
...15
2
...10
...11
3
...10
...11
## Loop Labels
Loops can be optionally labeled to improve readability and execution control, as shown in the preceding code. The label must appear immediately in front of the statement that initiates the loop.
The following example demonstrates the use of loop labels to qualify variables within a loop and also to terminate nested and outer loops:
<<year_loop>>
FOR yearind IN 1 .. 20
LOOP
<<month_loop>>
LOOP
...
IF year_loop.yearind > 10
THEN
EXIT year_loop;
END IF;
END LOOP month_loop;
END LOOP year_loop;
# Database Interaction
PL/SQL is tightly integrated with the underlying SQL layer of the Oracle database. You can execute SQL statements (SELECT, INSERT, UPDATE, DELETE, MERGE, LOCK TABLE) directly in PL/SQL programs. You also can execute Data Definition Language (DDL) statements through the use of dynamic SQL. In addition, you can manage transactions with COMMIT, ROLLBACK, and other Data Control Language (DCL) statements.
## Sequences in PL/SQL
Sequences are frequently used to generate keys in an Oracle database. In code that predates Oracle Database 11 _g_ , it was customary to select from DUAL to obtain a sequence's NEXTVAL or CURRVAL as follows:
/* No longer necessary! */
SELECT my_sequence.NEXTVAL
INTO my_variable FROM dual;
However, NEXTVAL and CURRVAL are now available anywhere in your program in which a number expression can appear. For example:
my_variable := my_sequence.NEXTVAL;
## Transaction Management
The Oracle database provides a transaction model based on a unit of work. A transaction begins with the first change to data and ends with either a COMMIT or a ROLLBACK. Transactions can span multiple PL/SQL blocks, or multiple transactions can be in a single PL/SQL block. The PL/SQL-supported transaction statements include COMMIT, ROLLBACK, SAVEPOINT, SET TRANSACTION, and LOCK TABLE, described in the following sections.
### COMMIT
COMMIT [WORK] [ _comment_ _ _text_ ];
COMMIT makes the database changes permanent and visible to other database sessions. The WORK keyword is optional and aids only readability; it is rarely used. The COMMENT text is optional and can be up to 50 characters in length. It is germane only to in-doubt distributed (two-phase commit) transactions. The database statement COMMIT FORCE, also relevant for distributed transactions, is not directly supported in PL/SQL, but can be invoked using the built-in procedure DBMS_TRANSACTION.COMMIT_FORCE.
### ROLLBACK
ROLLBACK [WORK] [TO [SAVEPOINT] _savepoint_name_ ];
ROLLBACK undoes the changes made in the current transaction either to the beginning of the transaction or to a _savepoint_. A savepoint is a named processing point in a transaction, created with the SAVEPOINT statement. Rolling back to a savepoint is a partial rollback of a transaction, wiping out all changes (and savepoints) that occurred later than the named savepoint.
### SAVEPOINT
SAVEPOINT _savepoint_name_ ;
SAVEPOINT establishes a savepoint in the current transaction. _savepoint_name_ is an undeclared identifier. More than one savepoint can be established within a transaction. If you reuse a savepoint name, that savepoint is moved to the later position, and you will not be able to roll back to the initial savepoint position.
### SET TRANSACTION
SET TRANSACTION READ ONLY;
SET TRANSACTION ISOLATION LEVEL SERIALIZABLE;
SET TRANSACTION USE ROLLBACK SEGMENT _rbseg_name_ ;
SET TRANSACTION has three transaction control functions:
_READ ONLY_
This marks the beginning of a read-only transaction. It indicates to the Oracle database that a read-consistent view of the database is to be enforced for the transaction (the default is for the statement). This read-consistent view means that only changes committed before the transaction begins are visible for the duration of the transaction. The transaction is ended with either a COMMIT or a ROLLBACK. Only LOCK TABLE, SELECT, SELECT INTO, OPEN, FETCH, CLOSE, COMMIT, and ROLLBACK statements are permitted during a read-only transaction. Issuing other statements, such as INSERT or UPDATE, in a read-only transaction results in an ORA-1456 error.
_ISOLATION LEVEL SERIALIZABLE_
Similar to a READ ONLY transaction in that transaction-level read consistency is enforced rather than the default statement-level read consistency. Serializable transactions do allow changes to data, however.
_USE ROLLBACK SEGMENT_
If your database is using rollback segments for undo management, as opposed to Oracle's recommended automatic undo management, this statement tells the database to use the specifically named rollback segment _rbseg_name_. For example, if we know that our large rollback segment is named `rbs_large`, we can tell the database to use it by issuing the following statement before our first change to data:
SET TRANSACTION USE ROLLBACK SEGMENT rbs_large;
### LOCK TABLE
LOCK TABLE _table_list_ IN _lock_mode_ MODE [NOWAIT];
This statement bypasses the implicit database row-level locks by explicitly locking one or more tables in the specified mode. The _table_list_ is a comma-delimited list of tables. The _lock_mode_ is one of the following: ROW SHARE, ROW EXCLUSIVE, SHARE UPDATE, SHARE, SHARE ROW EXCLUSIVE, or EXCLUSIVE. The NOWAIT keyword specifies that the Oracle database should not wait for a lock to be released. If there is a lock when NOWAIT is specified, the database raises the exception "ORA-00054: resource busy and acquire with NOWAIT specified." The default database locking behavior is to wait indefinitely.
## Autonomous Transactions
Autonomous transactions, often used for recording application log messages, execute within a block of code as separate transactions from the outer (main) transaction. Changes can be committed or rolled back in an autonomous transaction without committing or rolling back the main transaction. Changes committed in an autonomous transaction are visible to the main transaction, even though they occur after the start of the main transaction. Those changes committed in an autonomous transaction are visible to other transactions as well. The database suspends the main transaction while the autonomous transaction executes:
PROCEDURE main IS
BEGIN
UPDATE ... -- Main transaction begins here
DELETE ...
**at_proc; -- Call the autonomous transaction**
SELECT ...
INSERT ...
COMMIT; -- Main transaction ends here
END;
PROCEDURE at_proc IS
**PRAGMA AUTONOMOUS_TRANSACTION;**
BEGIN -- Main transaction suspends here
SELECT ...
INSERT ... -- Autonomous transaction begins here
UPDATE ...
DELETE ...
COMMIT;
/* Always commit or rollback at end of
autonomous transaction. */
END; -- Main transaction resumes after closing
Changes made in the main transaction are not visible to the autonomous transaction, and if the main transaction holds any locks that the autonomous transaction waits for, a deadlock occurs. Using the NOWAIT option on UPDATE statements in autonomous transactions can help to minimize this kind of deadlock. Functions and procedures (local program, standalone, or packaged), database triggers, top-level anonymous PL/SQL blocks, and object methods can be declared autonomous via the compiler directive PRAGMA AUTONOMOUS_TRANSACTION. In addition, there must be a COMMIT or a ROLLBACK at each exit point in the autonomous program.
# Cursors in PL/SQL
Every SQL statement executed by the Oracle database has a private SQL area that contains information about the SQL statement and the set of data returned. In PL/SQL, a _cursor_ is a name assigned to a specific private SQL area for a specific SQL statement. There can be either _static cursors_ , whose SQL statement is determined at compile time, or _dynamic cursors_ , whose SQL statement is determined at runtime.
Static cursors are used only for DML statements (SELECT, INSERT, UPDATE, DELETE, MERGE, or SELECT FOR UPDATE). These static cursors may be explicitly declared and named or may appear inline as implicit cursors.
Dynamic cursors are used for any type of valid SQL statement, including DDL (CREATE, TRUNCATE, ALTER) and DCL (GRANT, REVOKE). Dynamic cursors are typically implemented with the EXECUTE IMMEDIATE statement.
## Explicit Cursors
Explicit cursors are SELECT statements that are declared explicitly in the declaration section of the current block or in a package specification. Use OPEN, FETCH, and CLOSE in the execution or exception sections of your programs.
### Declaring explicit cursors
To use an explicit cursor, you must first declare it in the declaration section of a block or package. There are three types of explicit cursor declarations:
* A cursor without parameters; for example:
CURSOR company_cur IS
SELECT company_id FROM company;
* A cursor that accepts arguments through a parameter list; for example:
CURSOR company_cur (id_in IN NUMBER) IS
SELECT name FROM company
WHERE company_id = id_in;
* A cursor header that contains a RETURN clause in place of the SELECT statement; for example:
CURSOR company_cur (id_in IN NUMBER)
RETURN company%ROWTYPE;
This last example shows that the "header" of a cursor can be declared separately from its implementation—which will then be found in the package _body_ , as in:
PACKAGE BODY my_pkg
IS
CURSOR company_cur (id_in IN NUMBER) IS
SELECT * FROM company WHERE id = id_in;
END;
See "Packages" for more information on package structure.
### Opening explicit cursors
To open a cursor, you can use the following syntax:
OPEN _cursor_name_ [( _argument_ [, _argument_ ...])];
where _cursor_name_ is the name of the cursor as declared in the declaration section. The arguments are required if the definition of the cursor contains a parameter list, in which case you may use either positional notation, as above, and/or named notation:
OPEN _cursor_name_
( _parameter_name_ => _argument_ [, ...]);
You must open an explicit cursor before you can fetch rows from that cursor. When the cursor is opened, the processing actually includes the parse, bind, open, and execute phases of SQL statement execution. This OPEN processing includes determining an execution plan, associating host variables and cursor parameters with the placeholders in the SQL statement, determining the result set, and, finally, setting the current row pointer to the first row in the result set.
When using a cursor FOR loop, the OPEN is implicit in the FOR statement. If you try to open a cursor that is already open, PL/SQL will raise an "ORA-06511: PL/SQL: cursor already open" exception.
### Fetching from explicit cursors
The FETCH statement places the contents of the current row into local variables. To retrieve all rows in a result set, each row needs to be fetched. The syntax for a FETCH statement is:
FETCH _cursor_name_ INTO
{ _record_ | _variable_ [, _variable_... ] };
where _cursor_name_ is the name of the cursor as declared and opened.
### Closing explicit cursors
After all rows have been fetched, a cursor needs to be closed. Closing a cursor enables the PL/SQL memory optimization algorithm to release the associated memory at an appropriate time. You can close an explicit cursor by specifying a CLOSE statement as follows:
CLOSE _cursor_name_ ;
where _cursor_name_ is the name of the cursor declared and opened.
If you declare a cursor in a local anonymous, procedure, or function block, that cursor will close automatically when the block terminates. Package-based cursors must be closed explicitly, or they stay open for the duration of your session. Closing a cursor that is not open raises an INVALID_CURSOR exception.
### Explicit cursor attributes
There are four attributes associated with cursors: ISOPEN, FOUND, NOTFOUND, and ROWCOUNT. These attributes can be accessed with the % delimiter to obtain information about the state of the cursor. The syntax for a cursor attribute is:
_cursor_name_ % _attribute_
where _cursor_name_ is the name of the explicit cursor.
The behaviors of the explicit cursor attributes are described in the following table:
Attribute | Description
---|---
%ISOPEN | TRUE if cursor is open
FALSE if cursor is not open
%FOUND | NULL before the first fetch
TRUE if record was fetched successfully
FALSE if no row was returned
INVALID_CURSOR if cursor isn't open
%NOTFOUND | NULL before the first fetch
FALSE if record was fetched successfully
TRUE if no row was returned
INVALID_CURSOR if cursor isn't open
%ROWCOUNT | The number of rows fetched from the cursor
INVALID_CURSOR if cursor isn't open
Frequently, a cursor attribute is checked as part of a loop that fetches rows from a cursor, as shown here:
DECLARE
caller_rec caller_pkg.caller_cur%ROWTYPE;
BEGIN
OPEN caller_pkg.caller_cur;
LOOP
FETCH caller_pkg.caller_cur into caller_rec;
EXIT WHEN caller_pkg.caller_cur%NOTFOUND
OR
caller_pkg.caller_cur%ROWCOUNT > 10;
UPDATE call
SET caller_id = caller_rec.caller_id
WHERE call_timestamp < SYSDATE;
END LOOP;
CLOSE caller_pkg.caller_cur;
END;
## Implicit Cursors
Whenever a SQL statement is directly in the execution or exception section of a PL/SQL block, you are working with implicit cursors. SQL statements handled this way include INSERT, UPDATE, DELETE, MERGE, and SELECT INTO. Unlike explicit cursors, implicit cursors do not require separate statements to perform declaration, open, fetch, or close operations.
SELECT statements handle the %FOUND and %NOTFOUND attributes differently from the way that explicit cursors do. When an implicit SELECT statement does not return any rows, PL/SQL immediately raises the NO_DATA_FOUND exception, and control passes to the exception section. When an implicit SELECT returns more than one row, PL/SQL immediately raises the TOO_MANY_ROWS exception, and control passes to the exception section.
Implicit cursor attributes are referenced via the SQL cursor. For example:
BEGIN
UPDATE activity SET last_accessed := SYSDATE
WHERE UID = user_id;
IF SQL%NOTFOUND THEN
INSERT INTO activity_log (uid, last_accessed)
VALUES (user_id, SYSDATE);
END IF
END;
The following table lists the implicit cursor attributes:
Attributes | Description
---|---
SQL%ISOPEN | Always FALSE because the cursor is opened implicitly and closed immediately after the statement is executed.
SQL%FOUND | NULL before the statement.
TRUE if one or more rows were inserted, merged, updated, or deleted, or if only one row was selected.
FALSE if no row was selected, merged, updated, inserted, or deleted.
SQL%NOTFOUND | NULL before the statement.
TRUE if no row was selected, merged, updated, inserted, or deleted.
FALSE if one or more rows were inserted, merged, updated, or deleted.
SQL%ROWCOUNT | Number of rows affected by the cursor.
SQL%BULK_ ROWCOUNT | Pseudoassociative array (index-by table) containing the number of rows affected by the statements executed in bulk bind operations. See "Bulk Binds" for more information.
Use the RETURNING clause in INSERT, UPDATE, and DELETE statements to obtain data modified by the associated DML statement. This clause allows you to avoid an additional SELECT statement to query the affected row after applying the DML statement. Here is an example for the case that your DML affects exactly one row:
BEGIN
UPDATE activity SET last_accessed := SYSDATE
WHERE UID = user_id
RETURNING last_accessed, cost_center
INTO timestamp, chargeback_acct;
If your DML affects more than one row, you can use RETURNING BULK COLLECT to return all rows into collection variables. For example:
DECLARE
TYPE date_array IS TABLE OF DATE;
TYPE number_array IS TABLE OF NUMBER;
dates date_array;
nums number_array;
BEGIN
UPDATE activity SET last_accessed := SYSDATE
RETURNING last_accessed, cost_center
BULK COLLECT INTO dates, nums;
### SELECT FOR UPDATE clause
By default, the Oracle database locks rows as they are changed. To lock all rows in a result set, use the FOR UPDATE clause in your SELECT statement when you OPEN the cursor, instead of when you change the data. Using the FOR UPDATE clause does not require you to actually make changes to the data; it only locks the rows when opening the cursor. These locks are released on the next COMMIT or ROLLBACK. As always, these row locks do not affect other SELECT statements unless they, too, are FOR UPDATE. The FOR UPDATE clause is appended to the end of the SELECT statement and has the following syntax:
SELECT ...
FROM ...
FOR UPDATE [OF _column_reference_ ] [NOWAIT];
where _column_reference_ is a comma-delimited list of columns that appear in the SELECT clause. The optional NOWAIT keyword means that if the program cannot obtain the locks immediately, Oracle will raise an ORA-00054 exception. In the absence of NOWAIT, the program will wait forever for any locks to be released.
In the following example, only columns from the `inventory` table are referenced FOR UPDATE, so no rows in the `dog_breeds` table are locked when `hounds_in_stock_cur` is opened:
DECLARE
CURSOR hounds_in_stock_cur IS
SELECT i.stock_no, i.breeder, d.size
FROM dog_breeds d, inventory i
WHERE d.breed = i.breed
AND d.class = 'HOUND'
FOR UPDATE OF i.stock_no, i.breeder;
### WHERE CURRENT OF clause
UPDATE and DELETE statements in PL/SQL can use a WHERE CURRENT OF clause if they reference a cursor declared FOR UPDATE. This syntax indicates that the UPDATE or DELETE statement should modify the current row identified by the FOR UPDATE cursor. The syntax is:
[UPDATE | DELETE ] ...
WHERE CURRENT OF _cursor_name_ ;
By using WHERE CURRENT OF, you do not have to repeat the WHERE clause in the SELECT statement. For example:
DECLARE
CURSOR account_cur IS
SELECT acct_no, enter_date FROM accounts
WHERE enter_date < SYSDATE - 7
FOR UPDATE;
BEGIN
FOR account_rec IN wip_cur
LOOP
INSERT INTO acct_log (acct_no, order_date)
VALUES (account_rec.acct_no
, account_rec.enter_date);
DELETE FROM accounts
WHERE CURRENT OF account_cur;
END LOOP;
END;
## Native Dynamic SQL
Dynamic SQL is commonly implemented "natively" with the EXECUTE IMMEDIATE statement together with the OPEN FOR, FETCH, and CLOSE statements. The EXECUTE IMMEDIATE statement is typically used for DDL, single-row SELECTs, and other DML, while the OPEN FOR, FETCH, and CLOSE statements support dynamic multirow queries. Here is the syntax for these statements:
EXECUTE IMMEDIATE _sql_statement_
[INTO { _variable_ [, _variable_ ...] |
_record_ | _object_ }]
[USING [IN | OUT | IN OUT] _bind_argument_
[,[IN | OUT | IN OUT] _bind_argument_ ...] ]
[{RETURNING | RETURN}
INTO _bind_argument_ [, _bind_argument_ ]...];
The _sql_statement_ is passed as an expression in the character datatype family. Your _sql_statement_ itself can be in a literal string, VARCHAR2 variable, or even in a CLOB variable if the statement length is greater than 32,767 bytes. The EXECUTE IMMEDIATE statement parses and executes the SQL statement in a single step. The EXECUTE IMMEDIATE statement requires a terminating semicolon, but the _sql_statement_ must not have a trailing semicolon.
Here is an example of executing a DDL statement inside PL/SQL:
EXECUTE IMMEDIATE 'TRUNCATE TABLE foo';
When designing your programs, keep in mind that executing DDL will automatically commit any pending transactions.
Next, an example with an UPDATE, which introduces the use of bind variable placeholders (recognizable because their names begin with a colon):
DECLARE
new_sal NUMBER := 75000;
BEGIN
sql_stmt :=
'UPDATE emp SET salary = **:new_sal** WHERE emp_id = **:empno** ';
EXECUTE IMMEDIATE sql_stmt USING new_sal, 123;
At runtime, Oracle will positionally associate the values you supply in the USING clause with the bind variable placeholders. You can normally bind values only to variables in the dynamic PL/SQL block that have a SQL type. Beginning with Oracle Database 12 _c_ , however, you may also bind some PL/SQL-specific types such as record and collection types.
This bind variable syntax is needed only in the case that you are using dynamic SQL; when embedding static SQL into your program, any PL/SQL variables you reference in the statement automatically become bind variables. So the preceding example is for illustration purposes; it is the equivalent of the more straightforward:
DECLARE
new_sal NUMBER := 75000;
BEGIN
UPDATE emp SET salary = new_sal
WHERE emp_id = 123;
If you have a dynamic query that will return multiple rows, you can use the OPEN FOR statement with a weakly-typed cursor variable. The rows are then fetched and the cursor closed:
DECLARE
TYPE cv_typ IS REF CURSOR;
cv cv_typ;
laccount_no NUMBER;
lbalance NUMBER;
BEGIN
OPEN cv FOR
'SELECT account_no, balance
FROM accounts
WHERE balance < 500';
LOOP
FETCH cv INTO laccount_no, lbalance;
EXIT WHEN cv%NOTFOUND;
-- _Process the row..._
END LOOP;
CLOSE cv;
END;
## DBMS_SQL
For most dynamic SQL requirements, native dynamic SQL (NDS), illustrated in the preceding code block, is the easiest route to implementation and will likely perform the best. But Oracle's venerable DBMS_SQL package will be essential in the scenario where you have to deal with an unpredictable number of columns in the select-list or an unpredictable number of bind variables in the WHERE clause. This scenario is known in Oracle parlance as "method 4."
The DBMS_SQL functions TO_REFCURSOR and TO_CURSOR_NUMBER allow you to switch back and forth between DBMS_SQL and native dynamic SQL, taking advantage of the best of both dynamic SQL techniques.
Please refer to the Oracle documentation, or to O'Reilly's _Oracle PL/SQL Programming_ , for more details on using DBMS_SQL.
## SQL Injection and Bind Variables
SQL injection is a security vulnerability that can occur with dynamic SQL when unintended (malicious) code is injected into a program's dynamic SQL statement. One important defense against SQL injection is to use bind variables. This technique is generally recommended, in any case, because SQL statements usually execute repeatedly, and bind variables reduce the need for parsing. Declare your dynamic cursor with placeholders for bind variables and then pass the bind arguments to the Oracle database at runtime with a USING clause. The parsed form of the statement will be reused from the shared pool, improving performance.
Bind variables cannot be used in any arbitrary part of a SQL statement; you still need to be conscious of SQL injection when designing programs that use dynamically constructed column names, WHERE clauses, table names, and the like. To learn more about securing PL/SQL from injection attacks, read the white paper entitled "How to write SQL injection proof PL/SQL," available on the Oracle Technology Network.
## Cursor Variables
A cursor variable is a data structure that points to a cursor object, which in turn points to the cursor's result set. You can use cursor variables to more easily retrieve rows in a result set from client and server programs. You also can use cursor variables to hide minor variations in queries.
The pattern for declaring a REF_CURSOR type and associated cursor variable is:
DECLARE
TYPE _ref_cursor_type_name_ IS REF CURSOR
[RETURN _record_type_ ];
_cursor_variable_ _ref_cursor_type_name_ ;
If you do not include a RETURN type, you are declaring a _weakly-typed_ REF CURSOR; variables of this type can be associated with any query at runtime. If you include the RETURN type, you're declaring a _strongly-typed_ REF CURSOR; any cursor variable declared using that type can only FETCH INTO data structures that match the specified record type.
The advantage of using a strongly-typed REF CURSOR is that you get an early (compile-time) warning of any mismatch between the shape of the SQL statement and the destination variable. So use a weakly-typed REF CURSOR when you don't know the shape in advance.
The following code shows examples of declaration sections for weakly- and strongly-typed REF CURSORs:
1 DECLARE
2 -- Create type based on company table
3 TYPE company_ct IS REF CURSOR RETURN companies%ROWTYPE;
4 company_cur company_ct;
5
6 -- And now the weak cursor:
7 TYPE any_ct IS REF CURSOR;
8 generic_curvar any_ct;
For weak cursors, you can use the built-in type SYS_REFCURSOR as a shortcut; lines 7 and 8 could be replaced with the more succinct:
generic_curvar SYS_REFCURSOR;
The syntax to OPEN a cursor variable is:
OPEN _cursor_name_ FOR _SELECT_statement_ ;
You can FETCH and CLOSE a cursor variable using the same syntax as for explicit cursors.
There are several restrictions on cursor variables:
* You cannot declare package-level cursor variables because they do not have a persistent state. (You can declare them in packaged procedures and functions, however.)
* You cannot assign NULLs to a cursor variable nor can you use comparison operators to test for equality, inequality, or nullity.
* Neither database columns nor collections can store cursor variables.
* You cannot use remote procedure calls (RPCs) to pass cursor variables from one server to another.
## Cursor Expressions
A cursor expression provides a way to return a nested cursor from within a query. The syntax for a cursor expression is:
CURSOR ( _subquery_ )
Cursor expressions can reduce the amount of redundant data returned to a calling program over techniques that involve joining the tables together. The cursor expression is opened automatically when the parent row is fetched. Cursor expressions can be nested as well. These nested cursors are closed when one of the following occurs:
* The nested cursor is explicitly closed by the program.
* The parent cursor is closed.
* The parent cursor is re-executed.
* An exception is raised during the fetch of the parent row.
An example of a cursor expression follows:
DECLARE
CURSOR order_cur IS
SELECT o.order_date ,o.order_status
, **CURSOR(** SELECT p.translated_name
,i.unit_price
,i.quantity
FROM oe.order_items i
,oe.prod_descrips p
WHERE i.product_id = p.product_id
AND i.order_id = o.order_id **)**
FROM oe.orders o
WHERE order_date BETWEEN TO_DATE('01-Oct-15')
AND TO_DATE('31-Oct-15');
odate oe.orders.order_date%TYPE;
ostatus oe.orders.order_status%TYPE;
od_cur SYS_REFCURSOR;
tname oe.prod_descrips.translated_name%TYPE;
price oe.order_items.unit_price%TYPE;
qty oe.order_items.quantity%TYPE;
BEGIN
OPEN order_cur;
LOOP
FETCH order_cur INTO odate, ostatus, **od_cur** ;
EXIT WHEN order_cur%NOTFOUND;
LOOP
FETCH **od_cur** INTO tname, price, qty;
EXIT WHEN od_cur%NOTFOUND;
DBMS_OUTPUT.PUT_LINE(odate||','||ostatus
||','||tname||','||price||','||qty);
END LOOP;
END LOOP;
CLOSE order_cur;
END;
# Exception Handling
PL/SQL allows developers to raise and handle errors (exceptions) in a very flexible and powerful way. Each PL/SQL block can have its own exception section in which exceptions can be trapped and handled (resolved or passed on to the enclosing block). When an exception is raised in a PL/SQL block, its execution section immediately terminates. Control is passed to the exception section. Every exception in PL/SQL has an error number and error message; some exceptions also have names.
## Declaring Exceptions
Some exceptions have been predefined by Oracle in the STANDARD package or other built-in packages, such as UTL_FILE. See the following table for some of the most common predefined exceptions. You also can declare your own exceptions as follows:
_exception_name_ EXCEPTION;
Error | Named exception
---|---
ORA-00001 | DUP_VAL_ON_INDEX
ORA-00051 | TIMEOUT_ON_RESOURCE
ORA-01001 | INVALID_CURSOR
ORA-01012 | NOT_LOGGED_ON
ORA-01017 | LOGIN_DENIED
ORA-01403 | NO_DATA_FOUND
ORA-01410 | SYS_INVALID_ROWID
ORA-01422 | TOO_MANY_ROWS
ORA-01476 | ZERO_DIVIDE
ORA-01722 | INVALID_NUMBER
ORA-06500 | STORAGE_ERROR
ORA-06501 | PROGRAM_ERROR
ORA-06502 | VALUE_ERROR
ORA-06504 | ROWTYPE_MISMATCH
ORA-06511 | CURSOR_ALREADY_OPEN
ORA-06530 | ACCESS_INTO_NULL
ORA-06531 | COLLECTION_IS_NULL
ORA-06532 | SUBSCRIPT_OUTSIDE_LIMIT
ORA-06533 | SUBSCRIPT_BEYOND_COUNT
ORA-06548 | NO_DATA_NEEDED
ORA-06592 | CASE_NOT_FOUND
ORA-30625 | SELF_IS_NULL
An exception can be declared only once in a block, but nested blocks can declare an exception with the same name as an outer block. If this multiple declaration occurs, scope takes precedence over name when handling the exception. The inner block's declaration takes precedence over a global declaration.
When you declare your own exception, you must RAISE it explicitly. All programmer-declared exceptions have an error code of 1 and the error message "User-defined exception," unless you use the EXCEPTION_INIT pragma.
You can associate an error number with a declared exception with the PRAGMA EXCEPTION_INIT statement using the following syntax:
DECLARE
_exception_name_ EXCEPTION;
PRAGMA EXCEPTION_INIT
( _exception_name_ , _error_number_ );
where _error_number_ is a literal value (variable references are not allowed). This number can be an Oracle error, such as -955 (object exists), or an error in the user-definable range 20000 to -20999. For example, to execute the dynamic SQL in the variable `sql_stmt`, ignoring any ORA-00955 errors, run the following:
DECLARE
ObjectExists EXCEPTION;
PRAGMA EXCEPTION_INIT (ObjectExists, -955);
sql_stmt VARCHAR2(100) :=
'CREATE TABLE mydual AS SELECT * FROM dual';
BEGIN
EXECUTE IMMEDIATE sql_stmt;
-- Ignore ORA-955 errors (object already exists)
EXCEPTION
WHEN ObjectExists THEN NULL;
END;
## Raising Exceptions
An exception can be raised in three ways:
* By the PL/SQL runtime engine
* By an explicit RAISE statement in your code
* By a call to the built-in function RAISE_APPLICATION_ ERROR
The syntax for the RAISE statement is:
RAISE [ _exception_name_ ];
where _exception_name_ is the name of an exception that you have declared, or an exception that is declared in the STANDARD package. If you use the RAISE statement inside an exception handler, you can omit the exception name to re-raise the current exception:
EXCEPTION
WHEN _exception_name_
THEN
...do_something;
RAISE;
This syntax is not valid outside the exception section.
The RAISE_APPLICATION_ERROR built-in function has the following header:
RAISE_APPLICATION_ERROR (
_num_ BINARY_INTEGER,
_msg_ VARCHAR2,
_keeperrorstack_ BOOLEAN DEFAULT FALSE);
where _num_ is the error number (an integer between -20999 and -20000), _msg_ is the associated error message, and _keeperrorstack_ defines whether your error adds to (TRUE), or replaces (FALSE), the existing errors on the stack.
## Scope
The _scope_ of an exception section is that portion of the code that is "covered" by the exception section. An exception handler will handle, or attempt to handle, only those exceptions that are raised in the executable section of its PL/SQL block. Exceptions raised in the declaration or exception sections are passed to the outer block automatically. Any line or set of PL/SQL code can be placed inside a BEGIN-END to define its own block and have its own exception section. This allows you to limit the propagation of an exception.
## Propagation
Exceptions raised in a PL/SQL block propagate to an outer block if they are unhandled or re-raised in the exception section. When an exception occurs, PL/SQL looks for an exception handler that checks for the exception (or the WHEN OTHERS clause) in the current block. If a match is not found, PL/SQL propagates the exception to the enclosing block or calling program. This propagation continues until the exception is handled or propagated out of the outermost block, back to the calling program. In this case, the exception is "unhandled" and (1) stops the calling program, and (2) the host environment usually issues an automatic rollback of any outstanding transactions in that session.
Once an exception is handled, it will not propagate upward. If you want to trap an exception, display a meaningful error message, and have the exception propagate upward as an error, you must re-raise the exception. The RAISE statement can re-raise the current exception or raise a new exception, as shown here:
PROCEDURE delete_dept(deptno_in IN NUMBER)
IS
still_have_employees EXCEPTION;
PRAGMA EXCEPTION_INIT
(still_have_employees, −2292);
BEGIN
DELETE FROM dept
WHERE deptno = deptno_in;
EXCEPTION
WHEN still_have_employees
THEN
DBMS_OUTPUT.PUT_LINE
('Please delete employees in dept first');
RAISE; --Re-raise the current exception.
END;
### WHEN OTHERS clause
Use the WHEN OTHERS clause in the exception handler as a catch-all to trap any exceptions that are not handled by specific WHEN clauses in the exception section. If present, this clause must be the last exception handler in the exception section. Specify this clause as follows:
EXCEPTION
WHEN OTHERS
THEN
...
Using WHEN OTHERS THEN NULL is generally considered poor programming practice; if you must employ a general-purpose exception handler, it should do something reasonable (e.g., log the error, alert operators, etc.) to avoid an application that silently ends with a failure.
### SQLCODE and DBMS_UTILITY formatters for exception handlers
Inside any WHEN OTHERS section, you can use Oracle's built-ins, such as the SQLCODE function, and the various error and stack formatting tools. The former will enable your program to branch appropriately, while the latter will allow you to record the needed information about what went wrong. Prior to Oracle Database 12.1, you could use the functions in the DBMS_UTILITY package to return strings containing information about the execution call stack, error stack, and error backtrace, respectively:
CREATE TABLE err_test
(widget_name VARCHAR2(100)
,widget_count NUMBER
,CONSTRAINT no_small_numbers CHECK
(widget_count > 1000));
BEGIN
INSERT INTO err_test (widget_name, widget_count)
VALUES ('Athena',2);
EXCEPTION
WHEN OTHERS THEN
IF SQLCODE = −2290
AND **DBMS_UTILITY.FORMAT_ERROR_STACK** LIKE
'%NO_SMALL_NUMBERS%'
THEN
DBMS_OUTPUT.PUT_LINE (
'Widget_count is too small.' );
ELSE
DBMS_OUTPUT.PUT_LINE(
'Unhandled exception ' || SQLCODE);
DBMS_OUTPUT.PUT_LINE(
**DBMS_UTILITY.FORMAT_CALL_STACK** );
DBMS_OUTPUT.PUT_LINE(
**DBMS_UTILITY.FORMAT_ERROR_BACKTRACE** );
END IF;
END;
This produces the following output:
Widget_count is too small.
In Oracle Database 12.1, a single package, UTL_CALL_STACK, provides that same information, plus much more fine-grained access to the contents of these formatted strings. Here, for example, is a block that utilizes the UTL_CALL_STACK API to obtain the fully qualified name of the _nested_ subprogram that was executed:
FUNCTION call_stack_string RETURN VARCHAR2
IS
l_subprogram VARCHAR2 (32767);
l_return VARCHAR2 (32767);
BEGIN
FOR indx IN REVERSE 2 ..
utl_call_stack.dynamic_depth
LOOP
l_subprogram :=
utl_call_stack.concatenate_subprogram (
utl_call_stack.subprogram (indx));
l_return :=
l_return || l_subprogram
|| ' ('
|| TO_CHAR
(utl_call_stack.unit_line (indx))
|| ')';
END LOOP;
RETURN l_return;
END;
Please refer to Oracle documentation and O'Reilly's _Oracle PL/SQL Programming_ for more details of this package.
There is another built-in function that provides error string information: SQLERRM. However, Oracle Corporation recommends that you limit use of this function to the SAVE EXCEPTIONS clause in FORALL statements.
### Exceptions and transactions
When an exception is raised in a PL/SQL block, it does _not_ instantly roll back your current transaction, even if the block issued an INSERT, UPDATE, or DELETE. If your exception handler catches the exception, you have control over what happens; for example, you may want to issue your own ROLLBACK statement.
If your exception goes unhandled (propagates out of the outermost block), however, most host environments, including SQL*Plus, will then force an automatic, unqualified rollback of any outstanding changes in your session.
# Records in PL/SQL
A PL/SQL record is a data structure composed of multiple pieces of information called _fields_. To use a record, you must first define it and declare a variable of this type. There are three types of records: table-based, cursor-based, and programmer-defined.
## Declaring Records
Define and declare records either in the declaration section of a PL/SQL block or globally, via a package specification.
You do not have to explicitly define table-based or cursor-based records, as they are implicitly defined with the same structure as a table or a cursor. Variables of these types are declared via the %ROWTYPE attribute. The record's fields correspond to the table's columns or the columns in the SELECT list.
DECLARE
comp_rec company%ROWTYPE;
CURSOR comp_summary_cur IS
SELECT c.company_id,SUM(s.gross_sales) gross
FROM company c, sales s
WHERE c.company_id = s.company_id;
-- Declare a cursor-based record.
comp_summary_rec comp_summary_cur%ROWTYPE;
Programmer-defined records must be explicitly defined with the TYPE statement in the PL/SQL declaration section or in a package specification. Variables of this type then can be declared as shown here:
DECLARE
TYPE name_rectype IS RECORD(
prefix VARCHAR2(15)
,first_name VARCHAR2(30)
,middle_name VARCHAR2(30)
,sur_name VARCHAR2(30)
,suffix VARCHAR2(10) );
TYPE employee_rectype IS RECORD (
emp_id NUMBER(10) NOT NULL
,mgr_id NUMBER(10)
,dept_no dept.deptno%TYPE
,title VARCHAR2(20)
,name name_rectype
,hire_date DATE := SYSDATE
,fresh_out BOOLEAN );
-- Declare a variable of this type.
new_emp_rec employee_rectype;
BEGIN
## Referencing a Record's Fields
Individual fields are referenced via dot notation:
_record_name.field_name_
For example:
employee.first_name
Individual fields within a record can be read from or written to. They can appear on either the left or right side of the assignment operator:
BEGIN
start_date := new_emp_rec.hire_date + 30;
new_emp_rec.fresh_out := FALSE;
...
## Assigning Records
An entire record can be assigned to another record of the same type, but one record cannot be compared to another record via Boolean operators. This is a valid assignment:
shipto_address_rec := customer_address_rec
But this is not a valid comparison:
IF shipto_addr_rec = cust_addr_rec -- illegal
THEN
...
END IF;
The individual fields of the records need to be compared instead.
Values can be assigned to records or to the fields within a record in four different ways:
* You can use the assignment operator to assign a value to a field:
new_emp_rec.hire_date := SYSDATE;
* You can SELECT INTO a whole record or the individual fields:
SELECT emp_id, dept_id, hire_date
INTO new_emp_rec
FROM emp
WHERE surname = 'LI'
* You can FETCH INTO a whole record or the individual fields:
FETCH emp_cur INTO new_emp_rec;
FETCH emp_cur
INTO new_emp_rec.emp_id, new_emp_rec.name;
* You can assign all the fields of one record variable to another record variable of the same type:
IF rehire THEN
new_emp_rec := former_emp_rec;
ENDIF;
This aggregate assignment technique works only for records declared with the same TYPE statement.
## Records and DML
You insert into or update a database table using a %ROWTYPE record without having to specify each field individually in the following ways:
* Insert into a database table, using a %ROWTYPE record and the VALUES keyword:
DECLARE
emp_rec emp%ROWTYPE;
BEGIN
emp_rec.empno := employees_seq.NEXTVAL;
INSERT INTO emp VALUES (emp_rec);
END;
* Update a database table using a %ROWTYPE record and the SET ROW keywords:
FOR emp_rec IN emp_cur
LOOP
change_record_values(emp_rec);
UPDATE emp SET ROW = emp_rec
WHERE empno = emp_rec.empno;
END LOOP;
These techniques allow you to write more compact code to interact with a database. If you are using Oracle Database 12 _c_ or higher, this approach also works with dynamic SQL, because bind variables can now be of a record datatype.
## Nested Records
Nested records are records contained in fields that are records themselves. Nesting records is a powerful way to normalize data structures and hide complexity within PL/SQL programs. For example:
DECLARE
-- Define a record.
TYPE phone_rectype IS RECORD (
area_code VARCHAR2(3),
exchange VARCHAR2(3),
phn_number VARCHAR2(4),
extension VARCHAR2(4));
-- Define a record composed of records.
TYPE contact_rectype IS RECORD (
day_phone# phone_rectype,
eve_phone# phone_rectype,
cell_phone# phone_rectype);
-- Declare a variable for the nested record.
auth_rep_info_rec contact_rectype;
BEGIN
# Collections in PL/SQL
There are three types of collections in PL/SQL: associative arrays (formerly known as index-by tables or PL/SQL tables), nested tables, and VARRAYs:
_Associative arrays_
Single-dimension, unbounded collections of homogeneous elements available only in PL/SQL, not in the Oracle database. Associative arrays are initially sparse; they have nonconsecutive subscripts. You can index by both integer and string, while nested tables and varrays can only be indexed by integer.
_Nested tables_
Single-dimension, unbounded collections of homogeneous elements that can be declared within a PL/SQL block and also stored in the column of a table. Nested tables initially are dense (they have consecutive subscripts), but they can become sparse through deletions.
_VARRAYs_
Variable-size arrays. Single-dimension, bounded collections of homogeneous elements that can be declared within a PL/SQL block and also stored in the column of a table. VARRAYs are never sparse. Unlike nested tables, their element order is preserved when you store and retrieve them from the database.
The following table compares these similar collection types:
Characteristic | Associative array | Nested table | VARRAY
---|---|---|---
Dimensionality | Single | Single | Single
Usable in SQL? | Yes, as of Oracle Database 12 _c_ a | Yes | Yes
Usable as a column datatype in a table? | No | Yes; data stored "out of line" (in a separate table) | Yes; data typically stored "inline" (in the same table)
Uninitialized state | Empty (cannot be null); elements are undefined | Atomically null; illegal to reference elements | Atomically null; illegal to reference elements
Initialization | Automatic, when declared | Via constructor, fetch, assignment | Via constructor, fetch, assignment
In PL/SQL, elements referenced by | BINARY_INTEGER (−2,147,483,647 .. 2,147,483,647) or character string (VARCHAR2); maximum length of VARCHAR2 is 32,767, minimum length is 1 | Positive integer between 1 and 2,147,483,647; elements cannot be referenced unless they have been initialized via the constructor or with the EXTEND function | Positive integer between 1 and 2,147,483,647; elements cannot be referenced unless they have been initialized via the constructor or with the EXTEND function
Sparse? | Yes | Initially no; after deletions, yes | No
Bounded? | No | Can be extended | Yes
Growing the number of elements in the collection | Assign a value to the element using a new subscript | Elements are added via the constructor or later via the EXTEND method | Elements are added via the constructor or later via the EXTEND method; however, you cannot EXTEND past the upper bound
Shrinking the size of the collection | Automatic | TRIM function | TRIM function
Can be compared for equality? | No | Yes (in Oracle Database 10 _g_ and later) | No
Elements retain ordinal position and subscript when stored and retrieved from the database | N/A; can't be stored in database | No | Yes
a SQL statements embedded in PL/SQL can use associative arrays via the TABLE operator (Oracle Database 12 _c_ and later).
## Declaring a Collection
Collections are implemented as TYPEs. As with any programmer-defined type, you must first define the type; then you can declare instances of that type. The TYPE definition can be stored in the database or declared in the PL/SQL program. Each instance of the TYPE is a collection.
The syntax for declaring an associative array is:
TYPE _type_name_ IS TABLE OF _element_type_ [NOT NULL]
INDEX BY {BINARY_INTEGER | VARCHAR2( _size_limit_ )};
The syntax for a nested table is:
[CREATE [OR REPLACE]] TYPE _type_name_ { IS | AS } TABLE OF
_element_type_ [NOT NULL];
The syntax for a VARRAY is:
[CREATE [OR REPLACE]] TYPE _type_name_ { IS | AS } { VARRAY |
VARYING ARRAY } ( _max_elements_ ) OF _element_type_
[NOT NULL];
The CREATE keyword defines the statement to be DDL and indicates that this type will exist in the database. The optional OR REPLACE keywords are used to rebuild an existing type, preserving the privileges. _type_name_ is any valid identifier that will be used later to declare the collection. _max_elements_ is the maximum size of the VARRAY. _element_type_ is the type of the collection's elements. All elements are of a single type, which can be most scalar datatypes, an object type, or a REF object type. If the elements are objects, the object type itself cannot have an attribute that is a collection. Explicitly disallowed collection datatypes are BOOLEAN, NCHAR, NCLOB, NVARCHAR2, REF CURSOR, TABLE, and VARRAY.
NOT NULL indicates that a collection of this type cannot have any null elements. However, the collection can be atomically null (uninitialized).
## Initializing a Collection
Initializing an associative array is not necessary—simply declaring it also initializes it. Initializing a nested table or a VARRAY can be done in any of three ways: explicitly with a constructor, implicitly with a fetch from the database, or implicitly with a direct assignment of another collection variable.
The constructor is a built-in function with the same name as the collection. It constructs the collection from the elements passed to it. The following example shows how you can create a nested table of colors and explicitly initialize it to three elements with a constructor:
DECLARE
TYPE colors_tab_t IS TABLE OF VARCHAR2(30);
colors_tab colors_tab_t :=
colors_tab_t('RED','GREEN','BLUE');
BEGIN
The next example shows how you can create the nested table of colors and implicitly initialize it with a fetch from the database:
-- Create nested table in the database.
CREATE TYPE colors_tab_t IS TABLE OF VARCHAR2(32);
-- Create table with nested table type as column.
CREATE TABLE color_models
(model_type VARCHAR2(12)
,colors colors_tab_t)
NESTED TABLE colors STORE AS
color_model_colors_tab;
-- Add some data to the table.
INSERT INTO color_models
VALUES('RGB',colors_tab_t('RED','GREEN','BLUE'));
INSERT INTO color_models
VALUES('CYMK'
,colors_tab_t('CYAN','YELLOW'
,'MAGENTA' 'BLACK'));
-- Initialize collection of colors from table.
DECLARE
basic_colors colors_tab_t;
BEGIN
SELECT colors INTO basic_colors
FROM color_models
WHERE model_type = 'RGB';
END;
The third example shows how you can implicitly initialize the table via an assignment from an existing collection:
DECLARE
basic_colors Color_tab_t := Color_tab_t
('RED','GREEN','BLUE');
my_colors Color_tab_t;
BEGIN
my_colors := basic_colors;
my_colors(2) := 'MUSTARD';
## Adding and Removing Elements
Elements in an associative array can be added simply by referencing new subscripts. To add elements to nested tables or VARRAYs, you must first enlarge the collection with the EXTEND function, and then you can assign a value to a new element using one of the methods described in the previous section.
Use the DELETE function to remove an element in a nested table regardless of its position. The TRIM function also can be used to remove elements, but only from the end of a collection. To avoid unexpected results, do not use both DELETE and TRIM on the same collection.
## Nested Table Functions
Several collection (multiset) functions can be used to manipulate collections. These are summarized in the following table. Note that COLLECT is valid only in a SQL statement; it cannot be used, for example, in a PL/SQL assignment.
Function | Return value | Description
---|---|---
= | BOOLEAN | Compares two nested tables and returns TRUE if they have the same named type and cardinality, and if the elements are equal.
<> | BOOLEAN | Compares two nested tables and returns FALSE if they differ in named type, cardinality, or equality of elements.
[NOT] IN ( ) | BOOLEAN | Returns TRUE [FALSE] if the nested table to the left of IN exists in the list of nested tables specified in the parentheses.
CARDINALITY( _x_ ) | NUMBER | Returns the number of elements in VARRAY or nested table _x_. Returns NULL if the collection is atomically NULL (not initialized).
CAST( _k_ AS _t_ ) | TYPE _t_ | Changes the datatype of _k_ to type _t_ ; used in conjunction with COLLECT or MULTISET.
COLLECT | NESTED TABLE | Used in conjunction with CAST to map a column to a collection.
MULTISET | NESTED TABLE | Used in conjunction with CAST to map a subquery to a collection.
_x_ MULTISET EXCEPT [DISTINCT] _y_ | NESTED TABLE | Performs a MINUS set operation on nested tables _x_ and _y_ , returning a nested table whose elements are in _x_ , but not in _y_. The returned nested table, _x_ , and _y_ must all be of the same type. The DISTINCT keyword forces the elimination of duplicates from the returned nested table.
_x_ MULTISET INTERSECT [DISTINCT] _y_ | NESTED TABLE | Performs an INTERSECT set operation on nested tables _x_ and _y_ , returning a nested table whose elements are in both _x_ and _y_. The returned nested table, _x_ , and _y_ must all be of the same type. The DISTINCT keyword forces the elimination of duplicates from the returned nested table.
_x_ MULTISET UNION [DISTINCT] _y_ | NESTED TABLE | Performs a UNION set operation on nested tables _x_ and _y_ , returning a nested table whose elements include all those in _x_ as well as those in _y_. The returned nested table, _x_ , and _y_ must all be of the same type. The DISTINCT keyword forces the elimination of duplicates from the returned nested table.
SET( _x_ ) | NESTED TABLE | Returns nested table _x_ without duplicate elements.
_x_ IS [NOT] A SET | BOOLEAN | Returns TRUE [FALSE] if the nested table _x_ is composed of unique elements.
_x_ IS [NOT] EMPTY | BOOLEAN | Returns TRUE [FALSE] if the nested table _x_ is empty.
_e_ [NOT] MEMBER [OF] _x_ | BOOLEAN | Returns TRUE [FALSE] if an expression _e_ is a member of the nested table _x_.
_y_ [NOT] SUBMULTISET [OF] _x_ | BOOLEAN | Returns TRUE [FALSE] if the nested table _y_ contains only elements that are also in the nested table _x_.
The CAST function works together with the COLLECT and MULTISET functions. MULTISET operates on a subquery. COLLECT operates on a column in a SQL statement:
CREATE TYPE email_list_t AS TABLE OF VARCHAR2(64);
-- COLLECT operates on a column
SELECT CAST(COLLECT(cust_email) AS email_list_t)
FROM oe.customers;
-- which is equivalent to
SELECT CAST(
MULTISET(
SELECT cust_email
FROM oe.customers)
AS email_list_t)
FROM dual;
The following provides examples of the other nested table functions, operators, and expressions:
DECLARE
TYPE nested_type IS TABLE OF NUMBER;
nt1 nested_type := nested_type(1,2,3);
nt2 nested_type := nested_type(3,2,1);
nt3 nested_type := nested_type(2,3,1,3);
nt4 nested_type := nested_type(1,2,4);
answer nested_type;
BEGIN
/* (1,2,3,1,2,4) */
answer := nt1 MULTISET UNION nt4;
/* (1,2,3,2,3,1,3) */
answer := nt1 MULTISET UNION nt3;
/* (1,2,3) */
answer := nt1 MULTISET UNION DISTINCT nt3;
/* (3,2,1) */
answer := nt2 MULTISET INTERSECT nt3;
/* (3,2,1) */
answer := nt2 MULTISET INTERSECT DISTINCT nt3;
/* (3) */
answer := nt3 MULTISET EXCEPT nt2;
/* ( ) */
answer := nt3 MULTISET EXCEPT DISTINCT nt2;
/* (2,3,1) */
answer := SET(nt3);
IF (nt1 IS A SET) AND (nt3 IS NOT A SET) THEN
DBMS_OUTPUT.PUT_LINE(
'nt1 has unique elements,
but nt3 does not');
END IF;
IF (nt3 MULTISET EXCEPT DISTINCT nt2)
IS EMPTY THEN
DBMS_OUTPUT.PUT_LINE('empty set');
END IF;
IF 3 MEMBER OF (nt3 MULTISET EXCEPT nt2) THEN
DBMS_OUTPUT.PUT_LINE('3 in answer set');
END IF;
IF nt1 SUBMULTISET nt3 THEN
DBMS_OUTPUT.PUT_LINE('nt1 subset of nt3');
END IF;
IF SET(nt3) IN (nt1,nt2,nt3) THEN
DBMS_OUTPUT.PUT_LINE (
'expression in list of nested tables');
END IF;
END;
## Collection Methods
Several built-in functions (methods) are defined for all collections. These methods are called with dot notation:
_collection_name.method_name_ [( _parameters_ )]
The methods are listed in the following table:
Collection method | Description
---|---
COUNT | A function. Returns the current number of elements in the collection. If a collection is atomically NULL, this method raises an exception. Consider the CARDINALITY( ) function instead.
DELETE [( _i_ [, _j_ ])] | A procedure. Removes element _i_ or elements _i_ through _j_ from a nested table or associative array. When called with no parameters, removes all elements in the collection. Reduces the COUNT if the element is not already DELETEd. Does not apply to VARRAYs.
EXISTS ( _i_ ) | A function. Returns TRUE or FALSE to indicate whether element _i_ exists. If the collection is an uninitialized nested table or VARRAY, returns FALSE.
EXTEND [( _n_ [, _i_ ])] | A procedure. Appends _n_ elements to a nested table or VARRAY, initializing them to the value of element _i_. Both _i_ and _n_ are optional, and _n_ defaults to 1.
FIRST | A function. Returns the lowest index in use. Returns NULL when applied to empty initialized collections.
LAST | A function. Returns the greatest index in use. Returns NULL when applied to empty initialized collections.
LIMIT | A function. Returns the maximum number of allowed elements in a VARRAY. Returns NULL for associative arrays and nested tables.
PRIOR ( _i_ ) | A function. Returns the index immediately before element _i_. Returns NULL if _i_ is less than or equal to FIRST.
NEXT ( _i_ ) | A function. Returns the index immediately after element _i_. Returns NULL if _i_ is greater than or equal to LAST.
TRIM [( _n_ )] | A procedure. Removes _n_ elements at the end of the collection with the largest index. _n_ is optional and defaults to 1. If _n_ is NULL, TRIM does nothing. Associative arrays cannot be TRIMmed.
The EXISTS function returns a BOOLEAN, and all other functions and procedures return BINARY_INTEGER except for collections indexed by VARCHAR2, which can return character strings. All parameters are of the BINARY_INTEGER type. Only EXISTS can be used on uninitialized nested tables or VARRAYs. Other methods applied to these atomically null collections will raise the COLLECTION_IS_NULL exception.
DELETE and TRIM both remove elements from a nested table, but TRIM also removes the placeholder, while DELETE does not. This behavior may be confusing because TRIM can remove previously DELETEd elements.
Here are examples of some collection methods in use with an associative array:
DECLARE
TYPE population_type IS
TABLE OF NUMBER INDEX BY VARCHAR2(64);
continent_population population_type;
howmany NUMBER;
continent_name VARCHAR2(64);
BEGIN
continent_population('Australia') := 30000000;
-- Create new entry
continent_population('Antarctica') := 1000;
-- Replace old value
continent_population('Antarctica') := 1001;
continent_name := continent_population.FIRST;
DBMS_OUTPUT.PUT_LINE (continent_name);
DBMS_OUTPUT.PUT_LINE (
continent_population(continent_name));
continent_name := continent_population.LAST;
DBMS_OUTPUT.PUT_LINE (continent_name);
DBMS_OUTPUT.PUT_LINE
(continent_population(continent_name));
END;
This example produces the following output:
Antarctica
1001
Australia
30000000
Here are examples of some collection methods in use with a nested table:
DECLARE
TYPE colors_tab_t IS TABLE OF VARCHAR2(30);
my_list colors_tab_t :=
colors_tab_t('RED','GREEN','BLUE');
element BINARY_INTEGER;
BEGIN
DBMS_OUTPUT.PUT_LINE('my_list has '
||my_list.COUNT||' elements');
my_list.DELETE(2); -- delete element two
DBMS_OUTPUT.PUT_LINE('my_list has '
||my_list.COUNT||' elements');
FOR element IN my_list.FIRST..my_list.LAST
LOOP
IF my_list.EXISTS(element)
THEN
DBMS_OUTPUT.PUT_LINE(my_list(element)
|| ' Prior= '||my_list.PRIOR(element)
|| ' Next= ' ||my_list.NEXT(element));
ELSE
DBMS_OUTPUT.PUT_LINE('Element '|| element
||' deleted. Prior=
'||my_list.PRIOR(element)
|| ' Next= '||my_list.NEXT(element));
END IF;
END LOOP;
END;
This example produces the following output:
my_list has 3 elements
my_list has 2 elements
RED Prior= Next= 3
Element 2 deleted. Prior= 1 Next= 3
BLUE Prior= 1 Next=
## Collections and Privileges
As with other TYPEs in the database, you need the EXECUTE privilege on that TYPE to use a collection type created by another schema (user account) in the database. You can use synonyms for user-defined TYPE names.
## Nested Collections
Nested collections are collections contained in members that are collections themselves. Nesting collections is a powerful way to implement object-oriented programming constructs within PL/SQL programs. For example:
CREATE TYPE books IS TABLE OF VARCHAR2(64);
CREATE TYPE our_books IS TABLE OF books;
## Bulk Binds
You can use collections to improve the performance of SQL operations executed iteratively by using _bulk binds_. Bulk binds reduce the number of context switches between the PL/SQL engine and the SQL engine. Two PL/SQL language constructs implement bulk binds: FORALL and BULK COLLECT INTO.
The syntax for the FORALL statement is:
FORALL _bulk_index_ IN [ _lower_bound..upper_bound_
| INDICES OF _collection_variable_ [BETWEEN _lower_bound_ AND _upper_bound_ ]
| VALUES OF _collection_variable_ ]
[SAVE EXCEPTIONS]
_sql_statement_ ;
_bulk_index_ can be used only in the _sql_statement_ and only as a collection index (subscript). When PL/SQL processes this statement, the whole collection—instead of each individual collection element—is sent to the database server for processing. To delete all the accounts in the collection named `inactives` from the table ledger, do this:
FORALL i IN inactives.FIRST..inactives.LAST
DELETE FROM ledger WHERE acct_no = inactives(i);
If nonconsecutive index values result from deletions, you will need to use the INDICES OF syntax to skip over the deleted elements:
FORALL i IN INDICES OF inactives
DELETE FROM ledger WHERE acct_no = inactives(i);
If you are interested in the values of a sparse collection of integers instead of the indices, you will need to use the VALUES OF syntax:
FORALL i IN VALUES OF inactives_list
DELETE FROM ledger WHERE acct_no = inactives(i);
The INDICES OF and VALUES OF keywords allow you to specify a subset of rows in a driving collection that will be used in the FORALL statement. To match the row numbers in the data collection with the row numbers in the driving collection, use the INDICES OF clause. To match the row numbers in the data collection with the values found in the defined rows of the driving collection, use the VALUES OF clause.
The default is for the database to stop after the first exception encountered. Specify SAVE EXCEPTIONS to indicate that processing should continue after encountering exceptions. The cursor attribute %BULK_EXCEPTIONS stores a collection of records containing the errors. These records have two fields, EXCEPTION_INDEX and EXCEPTION_CODE, which contain the FORALL iteration (during which the exception was raised) as well as the SQLCODE for the exception. If no exceptions are raised, the SQL%BULK_EXCEPTIONS.COUNT method returns 0. For example:
DECLARE
TYPE NameList IS TABLE OF VARCHAR2(32);
name_tab NameList := NameList(
'Dawes','Feuerstein','Gennick'
,'Pribyl','Beresniewicz','Dawes','Dye');
error_count NUMBER;
bulk_errors EXCEPTION;
PRAGMA exception_init(bulk_errors, -24381);
BEGIN
FORALL indx IN name_tab.FIRST..name_tab.LAST
SAVE EXCEPTIONS
INSERT INTO authors (name) VALUES
(name_tab(indx));
EXCEPTION
WHEN OTHERS
THEN
error_count := SQL%BULK_EXCEPTIONS.COUNT;
DBMS_OUTPUT.PUT_LINE
('Number of errors is ' || error_count);
FOR indx IN 1..error_count
LOOP
DBMS_OUTPUT.PUT_LINE('Error ' || indx ||
' occurred during iteration ' ||
SQL%BULK_EXCEPTIONS(indx).ERROR_INDEX);
DBMS_OUTPUT.PUT_LINE('Error is ' ||
SQLERRM (
-SQL%BULK_EXCEPTIONS(indx).
ERROR_CODE));
END LOOP;
END;
Number of errors is 2
Error 1 occurred during iteration 5
Error is ORA-00001: unique constraint (.) violated
Error 2 occurred during iteration 7
Error is ORA-00001: unique constraint (.) violated
Moving along to the second bulk approach, BULK COLLECT INTO:
BULK COLLECT INTO _collection_name_list_ ;
where _collection_name_list_ is a comma-delimited list of collections, one for each column in the SELECT.
The BULK COLLECT INTO clause can be used in SELECT INTO, FETCH INTO, or RETURNING INTO statements. For example:
DECLARE
TYPE name_tab IS TABLE OF vendors.name%TYPE;
TYPE term_tab IS TABLE OF vendors.terms%TYPE;
v_names name_tab;
v_terms term_tab;
BEGIN
SELECT name, terms
BULK COLLECT INTO v_names, v_terms
FROM vendors
WHERE terms < 30;
...
END;
If you use Oracle Database 12 _c_ or later, you can use SQL's FETCH FIRST clause to get top-N results; the preceding SELECT statement could become:
SELECT name, terms
BULK COLLECT INTO v_names, v_terms
FROM vendors
ORDER BY terms, name
FETCH FIRST 50 ROWS ONLY;
And, if your application needs to "paginate" the results, combine FETCH FIRST with the OFFSET clause:
PROCEDURE fetch_search_results
(p_page# IN INTEGER DEFAULT 1)
IS
c_lines CONSTANT INTEGER := 50;
v_offset INTEGER := c_lines * (p_page# - 1);
BEGIN
SELECT name, terms
BULK COLLECT INTO v_names, v_terms
FROM vendors
ORDER BY terms, name
OFFSET v_offset ROWS
FETCH NEXT c_lines ROWS ONLY;
...
The next example deletes products in an input list of categories, and the SQL RETURNING clause returns a list of deleted products:
FUNCTION cascade_category_delete (
categorylist clist_t) RETURN prodlist_t
IS
prodlist prodlist_t;
BEGIN
FORALL aprod IN categorylist.FIRST ..
categorylist.LAST
DELETE FROM product
WHERE product_id IN categorylist(aprod)
RETURNING product_id
BULK COLLECT INTO prodlist;
RETURN prodlist;
END;
You can use the SQL%BULK_ROWCOUNT cursor attribute for bulk bind operations. It is like an associative array containing the number of rows affected by the executions of the bulk bound statements. The _n_ th element of SQL%BULK_ROWCOUNT contains the number of rows affected by the _n_ th execution of the SQL statement. For example:
FORALL i IN inactives.FIRST..inactives.LAST
DELETE FROM ledger WHERE acct_no = inactives(i);
FOR counter IN inactives.FIRST..inactives.LAST
LOOP
IF SQL%BULK_ROWCOUNT(counter) = 0
THEN
DBMS_OUTPUT.PUT_LINE(
'No rows deleted for '|| counter);
END IF;
END LOOP;
You cannot pass SQL%BULK_ROWCOUNT as a parameter to another program or use an aggregate assignment to another collection. The value of %ROWCOUNT is the sum of all %BULK_ROWCOUNT elements. Attributes %FOUND and %NOTFOUND reflect only the last execution of the SQL statement.
# Built-in Functions and Packages
Oracle Corporation supplies many built-in functions and packages. The following sections describe some of the more commonly used built-ins.
## Built-in Functions
The Oracle database provides dozens of built-in functions. Although most of these functions are available to both SQL and PL/SQL, there are some notable exceptions. The DECODE function, for example, is not available to PL/SQL except within SQL statements included in your PL/SQL programs. The _Oracle Database SQL Reference_ contains implementation details for the many functions of PL/SQL with the package STANDARD. You can view this package specification within the file _stdspec.sql_ located in the _ORACLE_HOME/rdbms/admin_ directory on the database server.
### Conversion functions
Built-in function | Description
---|---
ASCIISTR( _string_ ) | Converts _string_ in any character set to the ASCII version of _string_ in the database character set.
CAST( _expression_ AS _datatype_ ) | Converts _expression_ to type _datatype_. (Use as a powerful substitute for TO_DATE.)
CHARTOROWID( _string_ ) | Converts _string_ to a ROWID datatype.
CONVERT( _string_ , _destination_character_set, source_character_set_ ) | Converts _string_ from the source character set to the destination character set. (The default source character set is the database character set.)
FROM_TZ( _timestamp_value, time_zone_ ) | Adds time zone information to a TIMESTAMP value (converting it to a TIMESTAMP WITH TIME ZONE value).
HEXTORAW( _string_ ) | Converts _string_ containing hexadecimal characters to its raw equivalent.
MULTISET | Maps a database table to a collection.
NUMTODSINTERVAL( _n, interval_unit_ ) | Converts numeric expression _n_ to an INTERVAL DAY TO SECOND literal; _interval_unit_ can be DAY, HOUR, MINUTE, or SECOND.
NUMTOYMINTERVAL( _n, interval_unit_ ) | Converts numeric expression _n_ to an INTERVAL YEAR TO MONTH literal; _interval_unit_ can be YEAR or MONTH.
RAWTOHEX( _raw_ ) | Converts _raw_ value to its hexadecimal equivalent.
REFTOHEX( _ref_ ) | Converts _ref_ expression into a string containing its hexadecimal equivalent.
ROWIDTOCHAR( _rowid_ ) | Converts _rowid_ to a VARCHAR2(18) string equivalent.
ROWIDTONCHAR( _rowid_ ) | Converts _rowid_ to an NVARCHAR2(18) string equivalent.
TABLE | Maps a collection to a database table (inverse of MULTISET).
TO_BINARY_FLOAT( _expression_ [, _fmt_ ]) | Converts number or string _expression_ to a BINARY_FLOAT; optionally use format model _fmt_.
TO_BINARY_DOUBLE( _expression_ [, _fmt_ ]) | Converts number or string _expression_ to a BINARY_DOUBLE; optionally use format model _fmt_.
TO_CHAR, TO_NCHAR( _expression_ [, _fmt_ ]) | Converts _expression_ to a string (VARCHAR2 or NVARCHAR2, respectively); optionally use format model _fmt_.
TO_CLOB, TO_NCLOB( _c_ ) | Converts _c_ from a VARCHAR2, NVARCHAR2, or NCLOB value to a CLOB (or NCLOB).
TO_DATE( _expression_ [, _fmt_ ]) | Converts string _expression_ to a DATE datatype; optionally use format model _fmt_.
TO_DSINTERVAL( _string_ ) | Converts character _string_ of a CHAR, VARCHAR2, NCHAR, or NVARCHAR2 datatype to an INTERVAL DAY TO SECOND type.
TO_LOB( _c_ ) | Converts _c_ from a LONG or LONG RAW to a LOB.
TO_MULTI_BYTE( _string_ ) | Where possible, converts single-byte characters in the input _string_ to their multibyte equivalents.
TO_NUMBER( _expression_ [, _fmt_ ]) | Converts string or numeric (such as a BINARY_FLOAT) _expression_ to a NUMBER; optionally use format model _fmt_.
TO_RAW( _b_ ) | Converts _b_ from a BLOB to a RAW.
TO_SINGLE_BYTE( _string_ ) | Converts multibyte characters in _string_ to their corresponding single-byte characters.
TO_TIMESTAMP( _expression_ [, _fmt_ ]) | Converts string _expression_ to a value of type TIMESTAMP; optionally use format model _fmt_.
TO_TIMESTAMP_TZ( _expression_ [, _fmt_ ]) | Converts string _expression_ to a value of type TIMESTAMP WITH TIMEZONE; optionally use format model _fmt_.
TO_YMINTERVAL( _string_ ) | Converts character _string_ of a CHAR, VARCHAR2, NCHAR, or NVARCHAR2 datatype to an INTERVAL YEAR TO MONTH type.
TRANSLATE ... USING( _string_ USING CHAR_CS) | Converts _string_ between the database character set and the national character set. Used for ANSI compatibility; mimics TO_CHAR or TO_NCHAR functionality.
UNISTR( _string_ ) | Translates _string_ containing Unicode encoding values (e.g., x00e5) to a Unicode string literal in the database Unicode character set.
### String functions
Built-in function | Description
---|---
ASCII( _string_ ) | Returns the numeric code in the database character set representing the first character in _string_.
ASCIISTR( _string_ ) | Takes a _string_ in any character set and converts it into a string of ASCII characters. Any non-ASCII characters are represented using \XXXX Unicode notation.
CHR( _code_location_ ) | Inverse of ASCII function; returns a VARCHAR2 character of length 1 that corresponds to the location in the collating sequence provided as a parameter. A variation of CHR is useful when working with national character set data.
CHR( _code_location_ USING NCHAR_CS) | Returns an NVARCHAR2 character from the national character set.
COMPOSE( _string_ ) | Takes a Unicode _string_ as input and returns that string in its fully normalized form.
CONCAT( _string1, string2_ ) | Appends _string2_ to the end of _string1_.
CONVERT( _string, target_char_set_ [, _source_char_set_ ]) | Converts a _string_ from one character set to another. Default source is the database character set.
DECOMPOSE( _string_ [CANONICAL]) | Takes a Unicode _string_ as input and returns that string with any precomposed characters decomposed into their separate elements. CANONICAL optionally gives a result that may be reversed using COMPOSE.
GREATEST( _string1, string2_ , ...) | Takes one or more _string_ s as input and returns the string that would come last (i.e., sorts highest) if the inputs were sorted in ascending order. Compare to LEAST.
INITCAP( _string_ ) | Reformats the case of the _string_ argument, setting the first letter of each word to uppercase and the remainder of the letters to lowercase. A word is a set of characters separated by a space or nonalphanumeric character (such as # or _ ). For example, INITCAP('this is lower') gives 'This Is Lower'.
INSTR( _string1, string2_ [ [ _, start_position_ ] _, nth_ ]) | Returns the position at which _string2_ is found within _string1_ ; otherwise, returns 0. _start_position_ defines the starting position and defaults to 1 if not present. With a negative _start_position_ , searching begins at the end of _string1_ and works backward. _n_ th defines the ordinal occurrence (1st, 2nd, 3rd, etc.) of _string2_ in _string1_ , and defaults to 1 if not present.
LEAST( _string1, string2_ , ...) | Takes one or more _string_ s as input and returns the string that would come first (i.e., the least) if the inputs were sorted in ascending order. Compare to GREATEST.
LENGTH( _string_ ) | Returns the number of characters in a _string_. The variations LENGTHB, LENGTHC, LENGTH2, and LENGTH4 return the number of bytes, the number of Unicode characters, the number of USC2 code points, and the number of USC4 code points, respectively. LENGTH returns NULL when passed a NULL, and zero when passed a zero length (but non-NULL) CLOB.
LOWER( _string_ ) | Converts all letters in the specified _string_ to lowercase (the inverse of UPPER).
LPAD( _string, padded_length_ [ _,pad_string_ ]) | Returns the value from _string_ , but padded on the left with enough _pad_string_ characters to make the result _padded_length_ characters long. _pad_string_ defaults to a space if not present.
LTRIM( _string_ [ _,trim_string_ ]) | Removes, or trims, any characters found in _trim_string_ from the leading edge of _string. trim_string_ defaults to a space if not present. (See also TRIM [ISO standard] and RTRIM.)
NCHR( _code_location_ ) | Returns an NVARCHAR2 character (length 1) that corresponds to the location in the national character set collating sequence specified by _code_location_. (Provides the same functionality as the CHR function's USING NCHAR_CS clause.)
NLS_INITCAP( _string_ ) | Returns a version of _string_ (which should be of type NVARCHAR2 or NCHAR), setting the first letter of each word to uppercase and the remainder to lowercase. The return value is a VARCHAR2. A word is a set of characters separated by a space or nonalphanumeric character.
NLS_INITCAP( _string_ , 'NLS_SORT= _sort_sequence_name_ ') | You may specify a linguistic sorting sequence that affects the definition of "first letter," as in the preceding built-in function. For _sort_sequence_name_ , specify a linguistic sort name as described in the _Oracle Database Globalization Support Guide_.
NLS_LOWER( _string_ ) and NLS_LOWER( _string_ , 'NLS_SORT= _sort_sequence_name_ ') | Lowercases a _string_ in accordance with language-specific rules. (See NLS_INITCAP for how NLS_SORT can affect the results.)
NLS_UPPER( _string_ ) and NLS_UPPER( _string_ , 'NLS_SORT= _sort_sequence_name_ ') | Uppercases a _string_ in accordance with language-specific rules. (See NLS_INITCAP for how NLS_SORT can affect the results.)
NLSSORT( _string_ ) and NLSSORT( _string_ , 'NLS_SORT= _sort_sequence_name_ ') | Returns a _string_ of bytes that can be used to sort a string value in accordance with language-specific rules. The string returned is of the RAW datatype.
REGEXP_INSTR, REGEXP_LIKE, REGEXP_REPLACE, REGEXP_SUBSTR | See "Built-in Regular Expression Functions" for descriptions.
REPLACE( _string, match_string, replace_string_ ) | Returns a _string_ in which all occurrences of _match_string_ in _string_ are replaced by _replace_string_. Use REPLACE to search for a pattern of characters and then change all instances of that pattern in a single function call.
RPAD( _string, padded_length_ [ _,pad_string_ ]) | Returns _string_ padded on the right with enough _pad_string_ occurrences to make the result _padded_length_ characters long. _pad_string_ defaults to a space.
RTRIM( _string_ [ _,trim_string_ ]) | Removes, or trims, any _trim_string_ characters from the right, or trailing edge, of _string_. (See also TRIM [ISO standard] and LTRIM.) _trim_string_ defaults to a space.
SOUNDEX( _string_ ) | Returns a string that is the "phonetic representation" of _string_ (via algorithm defined by Donald E. Knuth).
SUBSTR( _string, start_ , [ _length_ ]) | Returns a substring from _string_ , beginning with the character at position _start_ and going for _length_ characters. If _start_ is negative, the beginning position is counted from the end of the string rather than the beginning. _length_ defaults to the remainder of _string_.
TO_CHAR( _national_character_data_ ) | Converts data in the national character set to its equivalent representation in the database character set. (See also TO_NCHAR.) You can also use TO_CHAR to convert date and time values, as well as numbers, into strings.
TO_MULTI_BYTE( _string_ ) | Translates single-byte characters to their multibyte equivalents (the inverse of TO_SINGLE_BYTE).
TO_NCHAR( _database_character_data_ ) | Converts data in the database character set to its equivalent representation in the national character set. (See also TO_CHAR and TRANSLATE...USING.)
TO_SINGLE_BYTE( _string_ ) | Translates multibyte characters to their single-byte equivalents (the inverse of TO_MULTI_BYTE).
TRANSLATE ( _string_ , _search_set, replace_set_ ) | Replaces every instance in _string_ of a character from _search_set_ with the corresponding character from _replace_set_.
TRANSLATE( _text_ USING CHAR_CS) and TRANSLATE( _text_ USING NCHAR_CS) | Translates character data to either the database character set (CHAR_CS) or the national character set (NCHAR_CS). The output datatype will be either VARCHAR2 or NVARCHAR2, depending on whether you are converting to the database or the national character set, respectively.
TRIM ([ [LEADING | TRAILING | BOTH] _trim_character_ FROM ] _string_ ) | Returns a version of _string_ that omits any leading and trailing spaces. The optional keywords LEADING FROM, TRAILING FROM, and BOTH FROM cause the trimming of only leading, trailing, or both (the default) leading and trailing _trim_character_ s. _trim_character_ defaults to a space.
UNISTR( _string_ ) | Returns _string_ converted into Unicode (the inverse of ASCISTR). You can represent nonprintable characters in the input _string_ using \XXXX Unicode notation.
UPPER( _string_ ) | Returns a version of _string_ with all letters made uppercase.
### Numeric functions
Built-in function | Description
---|---
ABS( _n_ ) | Returns the absolute value of _n_.
ACOS( _n_ ) | Returns the arc cosine of _n_ , where _n_ must be between π and 1. The returned value is between 0 and π.
ASIN( _n_ ) | Returns the arc sine, where _n_ must be between −1 and 1. The returned value is between −π/2 and π /2.
ATAN( _n_ ) | Returns the arc tangent, where the number _n_ must be between −infinity and infinity. The returned value is between −π/2 and π/2.
ATAN2( _n_ , _m_ ) | Returns the arc tangent of _n/m_ , where the numbers _n_ and _m_ must be between −infinity and infinity. The returned value is between −π and π. The result of ATAN2( _n,m_ ) is defined to be identical to ATAN( _n/m_ ).
BIN_TO_NUM( _b1, b2,...bn_ ) | Converts the bit vector represented by _b1_ through _bn_ into a number. Each of _b1_ through _bn_ must evaluate to either 0 or 1.
BITAND( _n_ , _m_ ) | Performs a logical AND between _n_ and _m_.
CEIL( _n_ ) | Returns the smallest integer greater than or equal to _n_.
COS( _n_ ) | Returns the cosine of the angle _n_ , which must be expressed in radians.
COSH( _n_ ) | Returns the hyperbolic cosine of _n_. If _n_ is a real number, and _i_ is the imaginary square root of −1, then the relationship between COS and COSH can be expressed as follows: COS ( _i * n_ ) = COSH ( _n_ ).
EXP( _n_ ) | Returns the value _e_ raised to the _n_ th power, where _n_ is the input argument. The number _e_ (approximately equal to 2.71828) is the base of the system of natural logarithms.
FLOOR( _n_ ) | Returns the largest integer that is less than or equal to _n_.
GREATEST _(n1, n2,...n3_ ) | Returns the highest ordinal element from a list of input numbers.
LEAST( _n1, n2,...n3_ ) | Returns the lowest ordinal element from a list of input numbers.
LN( _n_ ) | Returns the natural logarithm of _n_. The argument _n_ must be greater than or equal to 0.
LOG( _b, n_ ) | Returns the base _b_ logarithm of _n_. The argument _n_ must be greater than or equal to 0. The base _b_ must be greater than 1.
MOD( _n, m_ ) | Returns the remainder of _n_ divided by _m_. The remainder is computed using a formula equivalent to _n_ −( _m_ *FLOOR( _n/m_ )) when _n_ and _m_ are both positive or both negative, and _n_ −( _m_ *CEIL( _n/m_ )) when the signs of _n_ and _m_ differ.
NANVL( _n, m_ ) | Returns _m_ if _n_ is NaN (not a number); otherwise, returns _n_. The value returned will be in the type of the argument with the highest numeric precedence: BINARY_DOUBLE, BINARY_FLOAT, or NUMBER, in that order.
POWER( _n, m_ ) | Raises _n_ to the power _m_. If _n_ is negative, then _m_ must be an integer.
REMAINDER( _n, m_ ) | Returns the remainder of _n_ divided by _m_. The remainder is defined as follows: _n_ −( _m_ *ROUND( _n/m_ )).
ROUND(n) | Returns _n_ rounded to the nearest integer.
ROUND( _n, m_ ) | Returns _n_ rounded to _m_ decimal places. The value of _m_ can be less than zero. A negative value for _m_ directs ROUND to round digits to the left of the decimal point rather than to the right.
SIGN( _n_ ) | Returns −1, 0, or +1, depending on whether _n_ is less than zero, equal to zero, or greater than zero.
SIN( _n_ ) | Returns the sine of the specified angle, which must be expressed in radians.
SINH( _n_ ) | Returns the hyperbolic sine of _n_. If _n_ is a real number, and _i_ is the imaginary square root of −1, then the relationship between SIN and SINH can be expressed as follows: SIN ( _i * n_ ) = _i_ * SINH ( _n_ ).
SQRT( _n_ ) | Returns the square root _n_ , which must be greater than or equal to 0.
TAN( _n_ ) | Returns the tangent of the angle _n_ , which must be expressed in radians.
TANH( _n_ ) | Returns the hyperbolic tangent of _n_. If _n_ is a real number, and _i_ is the imaginary square root of −1, then the relationship between TAN and TANH can be expressed as follows: TAN ( _i * n_ ) = _i_ * TANH ( _n_ ).
TRUNC( _n_ [, _p_ ]) | Truncates _n_ to _m_ decimal places. The optional precision _p_ defaults to 0 and, if negative, truncates (zeros out) _p_ places to left of the decimal.
### Date and time functions
Built-in function | Description
---|---
ADD_MONTHS( _date, n_ ) | Adds _n_ months to _date_ , returning a DATE.
CAST( { _expression_ | MULTISET( _subquery_ ) } AS _type_ ) | Converts a value from one datatype or collection type to another. Use the MULTISET keyword when casting to a collection type.
CURRENT_DATE | Returns the current date and time as a DATE value in the session time zone.
CURRENT_TIMESTAMP( _p_ ) | Returns the current date and time as a TIMESTAMP WITH TIME ZONE value in the session time zone. The optional precision _p_ specifies the subsecond number of digits to the right of the decimal and defaults to 6.
DBTIMEZONE | Returns the time zone offset (from UTC) of the database time zone in the form of a character string in format [+|-]HH24:MI; for example, `-05:00`.
EXTRACT( _element_ FROM _expression_ ) | Returns the value of a specific datetime element from the datetime expression. The element can be one of YEAR, MONTH, DAY, HOUR, MINUTE, SECOND, TIMEZONE_HOUR, TIMEZONE_MINUTE, TIMEZONE_REGION, or TIMEZONE_ABBR.
FROM_TZ( _ts, tz_ ) | Adds time zone _tz_ to TIMESTAMP _ts_ , converting it to a TIMESTAMP WITH TIME ZONE.
LAST_DAY( _expression_ ) | Returns the last day in the month containing the DATE _expression_.
LOCALTIMESTAMP( _p_ ) | Returns the current date and time as a TIMESTAMP value in the local time zone. The optional precision _p_ specifies the subsecond number of digits to the right of the decimal.
MONTHS_BETWEEN( _end_date, start_date_ ) | Calculates the number of months between _start_date_ and _end_date_.
NEW_TIME( _date,tz1,tz2_ ) | Translates the date value from time zone _tz1_ to _tz2_. Included for backward compatibility; consider using a TIMESTAMP WITH TIMEZONE datatype instead.
NEXT_DAY( _date,dow_ ) | Returns the DATE of the first _dow_ weekday that is later than _date_.
NUMTODSINTERVAL( _n, unit_ ) | Converts number _n_ representing _unit_ number to a value of type INTERVAL DAY TO SECOND. _unit_ can be one of DAY, HOUR, MINUTE, or SECOND.
NUMTOYMINTERAL( _n, unit_ ) | Converts number _n_ representing _unit_ number to a value of type INTERVAL YEAR TO MONTH. _unit_ can be one of YEAR or MONTH.
ROUND( _date, fmt_ ) | Returns _date_ rounded to the optional format model _fmt_ level of granularity. If _fmt_ is not specified, _date_ is rounded to the nearest day.
SESSIONTIMEZONE | Returns the time zone offset (from UTC) of the session time zone in the form of a character string.
SYSDATE | Returns the current date and time from the Oracle database server as a DATE value.
SYS_EXTRACT_UTC( _dt_ ) | Converts the TIMESTAMP WITH TIME ZONE value _dt_ to a TIMESTAMP having the same date and time, but normalized to UTC.
SYSTIMESTAMP | Returns the current date and time from the Oracle database server as a TIMESTAMP WITH TIME ZONE value.
TO_CHAR( _dt, fmt_ ) | Converts the datetime _dt_ to a string using optional format model _fmt_ , which defaults to the session NLS_DATE_FORMAT.
TO_DATE( _string, fmt_ ) | Converts _string_ to a DATE; optionally use format model _fmt_ , which defaults to the session NLS_DATE_FORMAT.
TO_DSINTERVAL( _string_ ) | Converts the character string representation of an interval expressed in days, hours, minutes, and seconds to a value of INTERVAL DAY TO SECOND.
TO_TIMESTAMP( _string, fmt_ ) | Converts the character string representation of a date and time to a value of type TIMESTAMP; optionally use format model _fmt_ , which defaults to the session NLS_DATE_FORMAT.
TO_TIMESTAMP_TZ( _string, fmt_ ) | Converts the character string representation of a date and time to a value of type TIMESTAMP WITH TIME ZONE; optionally use format model _fmt_ , which defaults to the session NLS_TIMESTAMP_FORMAT.
TO_YMINTERVAL( _string_ ) | Converts the character string representation of an interval expressed in years and months to a value of INTERVAL YEAR TO MONTH.
TRUNC( _date,fmt_ ) | Truncates the _date_ value to format model _fmt_ level of granularity. The default granularity is day.
TZ_OFFSET( _tz_ ) | Returns the time zone offset from UTC for _tz_ , where _tz_ is a time zone name, a time zone offset, or the keywords SESSIONTIMEZONE or DBTIMEZONE.
## Built-in Regular Expression Functions
The Oracle database supports the use of regular expressions via five built-in functions: REGEXP_COUNT, REGEXP_INSTR, REGEXP_LIKE, REGEXP_REPLACE, and REGEXP_SUBSTR.
### Metacharacters
Regular expressions are found in Unix utilities, such as _grep_ , _sed_ , and the _ex_ editor; in the Perl scripting language; and in many other tools. Regular expressions are a powerful and popular means of processing text, mainly because they use _metacharacters_ to facilitate searching for strings. The metacharacters supported by the database are shown in the following table:
Pattern metacharacter | Description
---|---
* | Asterisk. Matches zero or more occurrences.
+ | Plus sign. Matches one or more occurrences.
? | Question mark. Matches zero or one occurrence.
^ | Caret. Matches beginning of line.
$ | Dollar sign. Matches end of line.
. | Period. Matches any single character.
\ | Backslash. Treats the following metacharacter as a nonspecial character.
{ _m_ } | Curly braces. Matches exactly _m_ times.
{ _m_ ,} | Curly braces. Matches at least _m_ times.
{ _m_ , _n_ } | Curly braces. Matches at least _m_ times, but no more than _n_ times.
[ ] | Square brackets. Matches any of the characters in the square brackets.
| | Vertical bar. Alternation operator for specifying alternative matches.
( ) | Parentheses. Grouping expression.
\ _n_ | Backslash. Backreference expression (\1 through \9). Used in conjunction with ( ) to identify the _n_ th occurrence of the backreferenced expression. (REGEXP_ REPLACE allows up to 500 backreferences in _replacement_string_.)
[`::`] | Character class. Examples are [:digit:] for numeric digits or [:alnum:] for alphanumeric characters.
[`..`] | Collation element. Encloses multiple characters treated as one character (e.g., 'ch' in Spanish).
[`==`] | Equivalence class. Matches accented and unaccented versions of a letter.
### REGEXP_COUNT
The REGEXP_COUNT function returns a number containing the tally of the occurences of a regular expression in a specific column, variable, or text literal. The syntax is:
REGEXP_COUNT ( _source_string_ , _pattern_ [,postion
[, _match_modifier_ ]])
where _source_string_ is the character string to be searched, _pattern_ is the regular expression pattern to search for in the _source_string_ , and _match_modifier_ is one or more modifiers that apply to the search (see "Match modifiers"). For example:
/* Count #of phone numbers in contact info */
phone_pattern :=
'\(?\d{3}\)?[\s.-]?\d{3}[\s-.]?\d{4}';
phone_count :=
regep_count (contact_clob, phone_pattern);
### REGEXP_LIKE
The REGEXP_LIKE function determines whether a specific column, variable, or text literal contains text matching a regular expression. It returns Boolean TRUE if the regular expression is found in the _source_string_ and FALSE if the regular expression is not found. The syntax is:
REGEXP_LIKE ( _source_string_ , _pattern_ [, _match_modifier_ ])
where _source_string_ is the character string to be searched, _pattern_ is the regular expression pattern to search for in _source_string_ , and _match_modifier_ is one or more modifiers that apply to the search. For example:
IF REGEXP_LIKE(phone_number,'^\(?212\)?'
THEN
-- phone number begins with 212
-- optionally enclosed by parentheses
apply_nyc_surcharge;
END IF;
### REGEXP_INSTR
The REGEXP_INSTR function locates, by character position, an occurrence of text matching a regular expression pattern. It returns the beginning or ending position of the regular expression within a string. The syntax is:
REGEXP_INSTR ( _source_string_ , _pattern_
[, _beginning_position_ [, _occurrence_ [, _return_option_
[, _match_modifier_ [, _subexp_ ]]]]])
where _source_string_ is a character string to be searched, _pattern_ is a regular expression pattern to search for in _source_string_ , _beginning_position_ is the character position at which to begin the search, _occurrence_ is the ordinal occurrence desired (1 = first, 2 = second, etc.), _return_option_ is either 0 for the beginning position or 1 for the ending position, and _match_modifier_ is one or more modifiers that apply to the search. You can also specify _subexp_ ; if the pattern uses subexpressions, this parameter tells the database which subexpression to return the position of from the pattern found in the source string. Subexpressions are used to parse out the interesting pieces. You define a subexpression by enclosing it in parentheses. For example:
witty_saying :=
'Man fears time, but time fears the Pyramids';
-- Display the witty_saying
-- starting w/ 2nd occurence of the word 'time'
DBMS_OUTPUT.PUT_LINE(
SUBSTR(witty_saying
,REGEXP_INSTR(witty_saying,'time',1,2)));
The output is:
time fears the Pyramids
### REGEXP_SUBSTR
The REGEXP_SUBSTR function extracts text matching a regular expression from a character column, variable, or text literal. It returns as many matching substrings as it finds (which might be zero). The syntax is:
REGEXP_SUBSTR ( _source_string_ , _pattern_ [, _position_
[, _occurrence_ [, _match_modifier_ [, _subexp_ ]]]])
where _source_string_ is the character string to be searched, _pattern_ is the regular expression pattern to search for in _source_string_ , _position_ is the character position at which to begin the search, _occurrence_ is the ordinal occurrence desired (1 = first, 2 = second, etc.), and _match_modifier_ is one or more modifiers that apply to the search. You can also specify _subexp_ ; if the pattern uses subexpressions, this parameter tells the database which subexpression to return from the pattern found in the source string. For example:
-- get the leading number part of the address
-- (up to a whitespace character)
street_num := REGEXP_SUBSTR (
address_line1,'[[:digit:]]+[:space:]');
In the following example, we parse out the exchange (second group of three digits) from the first telephone number found in the variable `c_rec.c_clob`. The regular expression pattern is defined as three digits optionally enclosed by parentheses; followed by an optional dot, dash, or whitespace character; followed by three digits; followed by an optional dot, dash, or whitespace character; followed by four digits. The whole pattern must match for the substring to be recognized as a matching pattern (telephone number). We then parse out the interesting part—the middle three digits—and assign it to the variable exchange. Here is the example data:
SELECT * FROM contacts WHERE contact_id=26;
CONTACT_ID CONTACT_NAME CONTACT_CLOB
---------- ------------ ------------
26 Elwood Blues Brother of
"Joliet" Jake
address:
1060 W Addison St
Chicago, IL 60613
home 773-555-5253
club 312-555-2468
Next is the subexpression parsing example:
DECLARE
ptrn VARCHAR2(45);
exchange VARCHAR2(3);
CURSOR c_cur IS
SELECT contact_clob c_clob
FROM contacts WHERE contact_id=26;
BEGIN
ptrn :=
'\(?(\d{3})\)?[\s.-]?(\d{3})[\s.-]?(\d{4})';
-- Get the second subexpression from the first
-- occurrence of the pattern
FOR c_rec in c_cur LOOP
exchange :=
REGEXP_SUBSTR(c_rec.c_clob,ptrn,1,1,'i',2);
DBMS_OUTPUT.PUT_LINE(
'exchange='||exchange);
END LOOP;
END;
This displays:
exchange=555
### REGEXP_REPLACE
The REGEXP_REPLACE function replaces a regular expression with new text that you specify. Your replacement text may include back references to values in the regular expression. The syntax is:
REGEXP_REPLACE ( _source_string_ , _pattern_
[, _replacement_string_ [, _position_ [, _occurrence_
[, _match_modifier_ ]]])
where _source_string_ is the character string to be searched, _pattern_ is the regular expression pattern to search for in _source_string_ , _replacement_string_ is the replacement text for _pattern_ , _position_ is the character position at which to begin the search, _occurrence_ is the ordinal occurrence desired (0 = all occurrences, 1 = first, 2 = second, etc.), and _match_modifier_ is one or more modifiers that apply to the search. For example:
-- Change the domain part of the email addresses
-- Replace all between the @ and the '.com' with
-- the new domain name
REGEXP_REPLACE(email_address
, '@.*\.com', '@new_domain.com'));
### Match modifiers
The _match_modifiers_ available to the regular expression condition and functions are shown in the following table:
match_ modifier | Description
---|---
i | Uses a case-insensitive search; the default behavior is based on NLS_SORT.
c | Uses a case-sensitive search; the default behavior is based on NLS_SORT.
n | Enables the dot metacharacter to match newlines.
m | Treats the _source_string_ as multiple lines for purposes of the beginning and end-of-line metacharacters ^ and $.
For more details, see the _Oracle Regular Expressions Pocket Reference_, by Jonathan Gennick and Peter Linsley (O'Reilly).
# Stored Procedures and Functions
PL/SQL allows you to create a variety of named program units, or containers, for your code. These include:
_Procedure_
A program that executes one or more statements.
_Function_
A program that executes one or more statements and returns a value.
_Trigger_
A program that executes in response to database changes.
_Package_
A container for procedures, functions, and data structures.
_Object type_
Oracle's version of an object-oriented class; object types can contain member procedures and functions.
The following sections describe stored procedures and functions. Later sections describe triggers, packages, and object types.
## Procedures
Procedures are program units that execute one or more statements and can receive or return zero or more values through their parameter lists. The syntax of a procedure is:
CREATE [OR REPLACE] PROCEDURE _name_
[ ( _parameter_ [, _parameter_ ]) ]
[AUTHID { CURRENT_USER | DEFINER } ]
[ACCESSIBLE BY ( _program_unit_list_ )]
{ IS | AS }
_declaration_section_
BEGIN
_executable_section_
[EXCEPTION
_exception_section_ ]
END [ _name_ ];
where ACCESSIBLE BY is only available for 12.1 and higher.
Inside a PL/SQL executable section, a procedure is called as a standalone executable statement:
apply_discount(new_company_id, 0.15);
Many execution environments, such as SQL*Plus, also support ANSI SQL's CALL syntax:
CALL apply_discount(new_company_id, 0.15);
However, SQL*Plus programmers commonly invoke procedures with the SQL*Plus EXEC command (short for EXECUTE):
EXEC apply_discount(new_company_id, 0.15);
or the equivalent anonymous block:
BEGIN
apply_discount(new_company_id, 0.15);
END;
## Functions
Functions are program units that execute zero or more statements and return a value through the RETURN clause. Functions also can receive or return zero or more values through their parameter lists. The syntax of a function is:
CREATE [OR REPLACE] FUNCTION _name_
[ ( _parameter_ [, _parameter_ ]) ]
RETURN _return_datatype_
[AUTHID { CURRENT_USER | DEFINER } ]
[DETERMINISTIC]
[PARALLEL_ENABLE [ _partition_clause_ ]]
[PIPELINED [USING _implementation_type_ ]]
[RESULT_CACHE]
[ACCESSIBLE BY ( _program_unit_list_ )]
[AGGREGATE USING _implementation_type_ ]
{ IS | AS }
[ _declaration_section_ ]
BEGIN
_executable_section_
[EXCEPTION
_exception_section_ ]
END [ _name_ ];
where ACCESSIBLE BY is only available for 12.1 and higher.
A function must have at least one RETURN statement in the execution section. The RETURN clause in the function header specifies the datatype of the returned value.
See "Compiling Stored PL/SQL Programs" for information on the keywords OR REPLACE, AUTHID, DETERMINISTIC, PARALLEL_ENABLE, PIPELINED, and AGGREGATE USING. See "Privileges and Stored PL/SQL" for AUTHID. See "Function Result Cache" for RESULT_CACHE.
A function can be called anywhere that an expression of the same type can be used. You can call a function:
* In an assignment statement:
sales07 := tot_sales(2007,'C');
* To set a default value:
DECLARE
sales07 NUMBER DEFAULT tot_sales(2007,'C');
BEGIN
* In a Boolean expression:
IF tot_sales(2007,'C') > 10000
THEN ...
* In a SQL statement (note that there are special rules that apply to calling a user-defined function inside SQL; see "Calling PL/SQL Functions in SQL"):
SELECT first_name, surname
FROM sellers
WHERE tot_sales(2007,'C') > 1000;
* As an argument in another program unit's parameter list
Here, for example, `max_discount` is a programmer-defined function, and SYSDATE is a built-in function:
apply_discount(company_id, max_discount(SYSDATE));
## Parameters
Procedures, functions, and cursors may have a parameter list. This list contains one or more parameters that allow you to pass information back and forth between the subprogram and the calling program. Each parameter is defined by its name, datatype, mode, and optional default value. The syntax for a parameter is:
_parameter_name_ [ _mode_ ] [NOCOPY] _datatype_ [ { := | DEFAULT } _value_ ]
### Datatype
The datatype can be any PL/SQL or programmer-defined datatype but cannot be constrained by a size (NUMBER is valid, NUMBER(10) is not valid). The actual size of the parameter is determined from the calling program:
CREATE OR REPLACE PROCEDURE empid_to_name
(in_id emp.emp_id%TYPE -- Compiles OK
,last_name VARCHAR2 -- Compiles OK
,first_name VARCHAR2(10) -- Won't compile
)
The lengths of `out_last_name` and `out_first_name` are determined by the calling program:
DECLARE
surname VARCHAR2(10);
first_name VARCHAR2(10);
BEGIN
empid_to_name(10, surname, first_name);
END;
### Mode
The _mode_ of a parameter specifies whether the parameter can be read from or written to, as shown in the following table:
Mode | Description | Parameter usage
---|---|---
IN | Read-only | The value of the actual parameter can be referenced inside the program, but the parameter cannot be changed.
IN OUT | Read/write | The program can both reference (read) and modify (write) the parameter.
OUT | Write | The value of any argument passed in is ignored; thereafter, the program can modify the parameter and read its value, passing the final value out.
If the mode is not explicitly defined, it defaults to IN.
OUT parameters are not the same as IN OUT parameters. When running the called program, the runtime engine ignores (sets to NULL) any argument value you supply for an OUT parameter; it preserves the value provided for an IN OUT. If an exception is raised during execution of a procedure or function, assignments made to OUT or IN OUT parameters get rolled back unless the parameter includes the NOCOPY option.
NOCOPY is a request to the compiler to make the parameter a call by reference instead of a call by value. Normally, PL/SQL passes IN/OUT parameters by value—a copy of the parameter is created for the subprogram. When parameter items are large (as may be the case with CLOBs, objects, and collections), the copy can eat memory and slow down processing. NOCOPY asks PL/SQL to pass the parameter by reference, using a pointer to the single copy of the parameter.
The main disadvantage of NOCOPY is that when an exception is raised during execution of a program that has modified an OUT or IN OUT parameter, the changes to the actual parameters are not "rolled back" because the parameters were passed by reference instead of being copied. Also, NOCOPY does not always apply; see _Oracle PL/SQL Programming_ for a list of cases in which the compiler ignores the NOCOPY request.
### Default values
IN parameters can be given default values (usually either the DEFAULT keyword or the := operator). If an IN parameter has a default value, you do not need to supply an argument for that parameter when you call the program unit. It automatically uses the default value. For example:
CREATE OR REPLACE PROCEDURE hire_employee
(emp_id IN VARCHAR2
,hire_date IN DATE := SYSDATE
,company_id IN NUMBER DEFAULT 1
)
IS
...
Here are some example calls to the preceding procedure:
-- Use two default values.
hire_employee(new_empno);
-- Use one default value.
hire_employee(new_empno,'12-Jan-2007');
-- Use nontrailing default value, named notation.
hire_employee(emp_id => new_empno, company_id => 12);
### Parameter-passing notations
_Formal parameters_ are the names that are declared in the header of a procedure or function. _Actual parameters_ ( _arguments_ ) are the values or expressions placed in the parameter list when a procedure or function is called. In the `empid_to_name` example shown earlier in "Datatype", the formal parameters to the procedure are `in_id`, `out_last_name`, and `out_first_name`. The actual parameters used in the call to this procedure are `10`, `surname`, and `first_name`.
PL/SQL lets you use either of two styles for passing arguments in parameter lists:
_Positional notation_
The default. Each value in the list of arguments supplied in the program call is associated with the parameter in the corresponding position.
_Named notation_
Explicitly associates the argument value with its parameter by name (not position). When you use named notation, you can supply the arguments in any order, and you can omit IN arguments that have default values.
The call to the `empid_to_name` procedure is shown here with both notations:
BEGIN
-- Implicit positional notation.
empid_to_name(10, surname, first_name);
-- Explicit named notation.
empid_to_name(in_id => 10
,out_last_name => surname
,out_first_name => first_name);
END;
You may combine positional and named notation, as long as positional arguments appear to the left of any named notation arguments; for example:
empid_to_name (
10, surname
, out_first_name => first_name);
As of Oracle Database 11 _g_ , you can use both positional and named notation when calling stored functions inside a SQL statement.
## Local Programs
A local program is a procedure or function that is defined in the declaration section of a PL/SQL block. The declaration of a local program must appear at the end of the declaration section, after the declarations of any types, records, cursors, variables, and exceptions. A program defined in a declaration section may be referenced only within that block's execution and exception sections. It is not defined outside that block.
The following program defines a local procedure and function:
PROCEDURE track_revenue
IS
l_total NUMBER;
PROCEDURE calc_total (year_in IN INTEGER) IS
BEGIN
calculations here ...
END;
FUNCTION below_minimum (comp_id IN INTEGER)
RETURN BOOLEAN
IS
BEGIN
...
END;
BEGIN
... _main procedure logic here_
END;
Local programs may be overloaded with the same restrictions as overloaded packaged programs.
## Program Overloading
PL/SQL allows you to define two or more programs with the same name within any declaration section, when declared in a package specification or body. This is called _overloading_. If two or more programs have the same name, they must be different in some other way so that the compiler can determine which program should be used.
Here is an example of overloaded programs in a built-in package specification:
PACKAGE DBMS_OUTPUT
IS
PROCEDURE PUT_LINE (a VARCHAR2);
PROCEDURE PUT_LINE (a NUMBER);
PROCEDURE PUT_LINE (a DATE);
END;
Each PUT_LINE procedure is identical, except for the datatype of the parameter. That is enough difference for the compiler.
To overload programs successfully, one or more of the following conditions must be true:
* Parameters must differ by datatype family (number, character, datetime, or Boolean).
* The program type must be different (you can overload a function and a procedure of the same name and identical parameter list).
* The numbers of parameters must be different.
In general, you _cannot_ overload programs if:
* Only the datatypes of the functions' RETURN clauses are different.
* Parameter datatypes are within the same family (CHAR and VARCHAR2, NUMBER and INTEGER, etc.).
* Only the modes of the parameters are different (IN versus IN OUT, for example).
* The programs are standalone functions or procedures.
You can overload programs whose parameters differ only in numeric datatypes, as long as they are in different datatype "families." The runtime environment will search first for a matching program with a PLS_INTEGER (or the equivalent BINARY_INTEGER) parameter; then it will try to match NUMBER, then, BINARY_FLOAT, and finally BINARY_DOUBLE, in that order. If you want to force the use of the faster IEEE 754 datatypes, you may need to use the TO_BINARY_FLOAT or TO_BINARY_DOUBLE functions on the input argument, or for literals, append with `f` or `d`, as discussed in "Numeric Literals".
## Forward Declarations
Programs must be declared before they can be used. PL/SQL supports _mutual recursion_ , in which program A calls program B, whereupon program B calls program A. To implement this mutual recursion, you must use a _forward declaration_ of the programs. This technique declares a program in advance of the program definition, thus making it available for other programs to use. The forward declaration is the program header up to the IS/AS keyword:
PROCEDURE perform_calc(year_in IN NUMBER)
IS
**/* Forward decl for total_cost function. */**
FUNCTION total_cost (year_in IN NUMBER)
RETURN NUMBER;
/* net_profit function can use total_cost. */
FUNCTION net_profit(year_in IN NUMBER)
RETURN NUMBER
IS
BEGIN
RETURN total_sales(year_in) -
total_cost(year_in);
END;
**/* Implementation of total_cost function */**
FUNCTION total_cost (year_in IN NUMBER)
RETURN NUMBER
IS
BEGIN
IF net_profit(year_in) < 0 THEN
RETURN 0;
ELSE
RETURN...;
END IF;
END;
BEGIN
...
END perform_calc;
## Table Functions
Table functions are functions that can be called within the FROM clause of a query, as if they were relational tables. To act as a table function, a function must have a header that is SQL-compatible (no Boolean arguments, for example), and the function must return a supported collection type: either a nested table or VARRAY whose type is declared at the schema level. As of Oracle Database 12 _c_ , a SQL statement embedded in PL/SQL _can_ use a table function typed as an associative array that you have declared in a package spec.
Pipelined table functions are special cases of table functions that allow you to "pipe" data out of the function back to the calling query while the function is still executing. Here is a very simple example of a pipelined table function:
CREATE TYPE num_tab_typ AS TABLE OF NUMBER;
/
CREATE OR REPLACE FUNCTION piped_func(
factor IN NUMBER)
RETURN num_tab_typ PIPELINED AS
BEGIN
FOR counter IN 1..1000
LOOP
PIPE ROW (counter*factor);
END LOOP;
RETURN;
END piped_func;
/
SELECT COLUMN_VALUE FROM TABLE (piped_func (2))
WHERE rownum < 5;
COLUMN_VALUE
------------
2
4
6
8
## Function Result Cache
With the PL/SQL function result cache, you can tell the database to retain the results of your function in a cache, located in the System Global Area (SGA), and available to all sessions that invoke the function. The RESULT_CACHE feature is best suited for functions that are executed relatively often (think every few seconds or minutes) against data that changes relatively slowly (think hourly or daily). Oracle will automatically ensure that the function results are flushed whenever there is DML on any table or view used by the function. In a RAC environment, each instance has its own result cache, which may differ in which items are cached, but common items in different instances will never disagree with each other.
Whenever a result-cached function is called with new parameters, both the parameters and the return value are saved in the cache. When the result-cached function is called with cached parameters, whether from your session or from a different one, the results are returned from the cache instead of being calculated anew. These cached entries can be monitored with the V$RESULT_CACHE% series of data dictionary views. Tune the size and usage of the result cache with the RESULT_CACHE_SIZE and RESULT_CACHE_MODE initialization parameters and the DBMS_RESULT_CACHE built-in package.
To enable result caching for your function, it must _not_ be:
* In an anonymous block
* Defined with invoker's rights
* A pipelined table function
* Defined with any OUT or IN OUT parameters
* Defined with any IN parameters of type BLOB, CLOB, NCLOB, REF CURSOR, object, or record
* Defined with a RETURN type of BLOB, CLOB, NCLOB, REF CURSOR, object, or compound datatype (record or collection) containing any of these unsupported types
* Dependent on session-specific settings or application contexts
Oracle recommends that result-cached functions not modify the database state or external state. For example, result-cached functions should not call DBMS_OUTPUT or UTL_FILE or send email because these external operations will not execute consistently between result cache hits and misses. Recursive fuctions are good candidates for result caching. The factorial function is an example:
CREATE OR REPLACE FUNCTION factorial(n NUMBER)
RETURN NUMBER RESULT_CACHE IS
BEGIN
IF n <= 1 THEN
RETURN 1;
ELSE
RETURN n*factorial(n-1);
END IF;
END;
For packaged functions, use the keyword RESULT_CACHE in both the package specification and the package body:
CREATE OR REPLACE PACKAGE bi_sales IS
FUNCTION avg_sales(cust_id IN NUMBER)
RETURN NUMBER **RESULT_CACHE** ;
END bi_sales;
CREATE OR REPLACE PACKAGE BODY bi_sales IS
FUNCTION avg_sales(cust_id IN NUMBER)
RETURN NUMBER **RESULT_CACHE** IS
BEGIN
...
END;
END bi_sales;
## Privileges and Stored PL/SQL
An Oracle user who creates a stored program will be able to execute that program unless nondefault privileges are in place. For other Oracle users to run the program, they would, at a minimum, need to be granted EXECUTE permission on it. This permission can be received via direct grant to the user, or via a grant to a database role that has been granted to the user.
When a user executes a stored procedure, Oracle has to evaluate whether the user has permissions on the underlying objects (tables, views, procedures, etc.) that the procedure uses internally. To make this decision, Oracle supports two different models for evaluating names and privileges at runtime.
The default model is _definer rights_ , which executes the stored program with the privileges of the _owner_ of the program. This model works well in most cases. If the program refers to objects that the program owner does not also own, the grants on those must be directly to him—he cannot inherit these privileges from a role.
The other model, known as _invoker rights_ , causes Oracle to resolve program identifiers and execute the program, with the permissions of the user who is _running_ the program. Stored programs that include the keywords AUTHID CURRENT_USER will run with invoker rights; anonymous PL/SQL blocks always execute with invoker rights.
Be aware that Oracle performs additional runtime checks, and incurs a slight performance overhead, when running an invoker rights program that uses any database objects.
By default, an invoker rights program can use the full range of the invoker's database permissions at runtime, even if those privileges exceed the programmer's. In other words, an invoker rights program "inherits" the privileges of the invoker, and a programmer could (either intentionally or accidentally) write code that would perform an operation beyond his or her pay grade.
To help guard against such misuse, Oracle Database 12 _c_ introduced the INHERIT [ ANY ] PRIVILEGES feature. An end user or DBA could prevent such inheritance at the user level by issuing the following REVOKE statement:
REVOKE INHERIT PRIVILEGE ON _invoking_user_ FROM PUBLIC;
or:
REVOKE INHERIT PRIVILEGE ON _invoking_user_ FROM _unit_owner_ ;
Any subsequent privilege escalation attempt by an invoker rights program that is run by _invoking_user_ will result in a run-time failure ORA-06598: insufficient INHERIT PRIVILEGES privilege. Significantly, this failure cannot be trapped as an exception in the program.
For backward compatibility, Oracle automatically runs GRANT INHERIT PRIVILEGE ON username TO PUBLIC when a database user is migrated or created.
Even with EXECUTE privilege and an appropriate runtime rights model, there is another mechanism that may limit one program's ability to invoke another: the ACCESSIBLE BY feature. Using this clause in a program specification restricts which programs can invoke it to those explicitly listed in the clause. This is a program-level, rather than a user-level, restriction, so it applies even when the same user owns both the called program and its calling program.
# Triggers
Triggers are programs that execute in response to changes in table data or certain database events. A predefined set of events can be "hooked" with a trigger, enabling you to integrate your own processing with that of the database. A triggering event _fires_ or executes the trigger.
There are three types of triggering events:
_DML events_
Fire when an INSERT, UPDATE, or DELETE statement executes
_DDL events_
Fire when a CREATE, ALTER, or DROP statement executes
_Database events_
Fire when one of the predefined database-level events occurs
Complete lists of these events are included in later sections.
## Creating Triggers
The syntax for creating a trigger on a DML event is:
CREATE [OR REPLACE] TRIGGER _trigger_name_
{ BEFORE | AFTER | INSTEAD OF | FOR } _trigger_event_
ON { _table_or_view_reference_ |
NESTED TABLE _nested_table_column_ OF _view_ }
[REFERENCING [OLD AS _old_ ] [NEW AS _new_ ]
[PARENT AS _parent_ ]]
[FOR EACH ROW ]
[FOLLOWS _other_trigger_ ] [DISABLE]
[COMPOUND TRIGGER]
[WHEN _trigger_condition_ ]
_trigger_body_ ;
The syntax for creating a trigger on a DDL or database event is:
CREATE [OR REPLACE] TRIGGER _trigger_name_
{ BEFORE | AFTER } _trigger_event_
ON [ DATABASE | _schema_ ]
[FOLLOWS other_trigger][DISABLE]
[WHEN _trigger_condition_ ]
_trigger_body_ ;
Trigger events are listed in the following table:
Trigger event | Description
---|---
INSERT | Fires whenever a row is added to the _table_ or_view_reference_.
UPDATE | Fires whenever an UPDATE changes the _table_or_view_reference_. UPDATE triggers can additionally specify an OF clause to restrict firing to updates of certain columns.
DELETE | Fires whenever a row is deleted from the _table_or_view_reference_. Does not fire on a TRUNCATE of the table.
ALTER | Fires whenever an ALTER statement changes a database object. In this context, objects are things such as tables or packages (found in ALL_OBJECTS). Can apply to a single schema or the entire database.
ANALYZE | Fires whenever the database collects or deletes statistics or validates the structure of a database object.
ASSOCIATE STATISTICS | Fires whenever the database associates a statistic type with a database object.
AUDIT | Fires whenever the database records an audit operation.
COMMENT | Fires whenever a comment on a database object is modified.
CREATE | Fires whenever a database object is created. Does not fire on CREATE CONTROLFILE statements.
DB_ROLE_CHANGE | In a Data Guard configuration, fires whenever a role change from primary to standby or standby to primary occurs. Only AFTER DB_ROLE_CHANGE triggers on the DATABASE are allowed.
DDL | Fires whenever one of the following events occurs: ALTER, ANALYZE, ASSOCIATE STATISTICS, AUDIT, COMMENT, CREATE, DISASSOCIATE, DROP, GRANT, NOAUDIT, RENAME, REVOKE, or TRUNCATE.
DISASSOCIATE STATISTICS | Fires whenever the database disassociates a statistic type from a database object.
DROP | Fires whenever a DROP statement removes an object from the database. In this context, objects are things such as tables or packages (found in ALL_OBJECTS). Can apply to a single schema or the entire database.
GRANT | Fires whenever a system, role, or object privilege is assigned.
NOAUDIT | Fires whenever the database processes a NOAUDIT statement to stop auditing database operations.
RENAME | Fires whenever a RENAME statement changes a database object name.
REVOKE | Fires whenever a system, role, or object privilege is rescinded.
TRUNCATE | Fires whenever a TRUNCATE statement is processed to purge a table or cluster.
SERVERERROR | Fires whenever a server error message is logged. Only AFTER triggers are allowed in this context.
LOGON | Fires whenever a session is created (a user connects to the database). Only AFTER triggers are allowed in this context.
LOGOFF | Fires whenever a session is terminated (a user disconnects from the database). Only BEFORE triggers are allowed in this context.
STARTUP | Fires when the database is opened. Only AFTER triggers are allowed in this context.
SHUTDOWN | Fires when the database is closed. Only BEFORE triggers are allowed in this context.
SUSPEND | Fires whenever a server error causes a transaction to be suspended.
Triggers can fire BEFORE or AFTER the triggering event. AFTER DML triggers are slightly more efficient than BEFORE triggers.
The REFERENCING clause is allowed only for the data events INSERT, UPDATE, and DELETE. It lets you give a nondefault name to the old and new pseudorecords. These pseudo-records give the program visibility to the pre- and postchange values in row-level triggers. These records are defined like %ROWTYPE records, except that columns of type LONG or LONG RAW cannot be referenced. They are prefixed with a colon in the trigger body and referenced with dot notation. Unlike other records, these fields can only be assigned individually—aggregate assignment is not allowed. All old fields are NULL within INSERT triggers, and all new fields are NULL within DELETE triggers. Parent fields are valid only in triggers on nested tables and refer to the current row in the parent table.
FOR EACH ROW defines the trigger to be a row-level trigger. Row-level triggers fire once for each row affected. The default is a statement-level trigger, which fires only once for each triggering statement.
If you have multiple triggers on the same event, you can use the FOLLOWS keyword to define the order in which they fire.
If you specify the DISABLE keyword, the database creates the trigger in a disabled state. You can then issue ALTER TRIGGER ENABLE or ALTER TABLE ENABLE ALL TRIGGERS to enable the trigger. Creating a trigger in a disabled state allows you to verify that it will compile and helps you avoid "ORA-04098: trigger NAME is invalid and failed re-validation" errors.
The WHEN _trigger_condition_ specifies the conditions that must be met for the trigger to fire. Stored functions and object methods are not allowed in the trigger condition.
The trigger body is a standard PL/SQL block. For example:
CREATE OR REPLACE TRIGGER add_tstamp
BEFORE INSERT ON emp
REFERENCING NEW as new_row
FOR EACH ROW
FOLLOWS audit_emp
BEGIN
-- Automatically timestamp the entry.
SELECT CURRENT_TIMESTAMP
INTO :new_row.entry_timestamp
FROM dual;
END add_tstamp;
Triggers are enabled by default on creation and can be disabled (so that they do not fire) with an ALTER statement, issued with the following syntax:
ALTER TRIGGER _trigger_name_ { ENABLE | DISABLE };
ALTER TABLE _table_name_ { ENABLE | DISABLE } ALL TRIGGERS;
## Trigger Predicates
When using a single trigger for multiple events, use the trigger predicates INSERTING, UPDATING, and DELETING in the trigger condition to identify the triggering event, as shown in this example:
CREATE OR REPLACE TRIGGER emp_log_t
AFTER INSERT OR UPDATE OR DELETE ON emp
FOR EACH ROW
DECLARE
dmltype CHAR(1);
BEGIN
IF INSERTING THEN
dmltype := 'I';
INSERT INTO emp_log (emp_no, who, operation)
VALUES (:new.empno, USER, dmltype);
ELSIF UPDATING THEN
dmltype := 'U';
INSERT INTO emp_log (emp_no, who, operation)
VALUES (:new.empno, USER, dmltype);
END IF;
END;
## DML Events
The DML events include INSERT, UPDATE, and DELETE statements on a table. An INSTEAD OF trigger is associated with a view and fires in lieu of DML to that view. Triggers on these events can be statement-level triggers (table only) or row-level triggers, and can fire BEFORE or AFTER the triggering event. BEFORE triggers can modify the data in affected rows, but perform an additional logical read. AFTER triggers do not perform this additional logical read and therefore perform slightly better, but are not able to change the : _new_ values. AFTER triggers are thus better suited for data-validation functionality. Triggers cannot be created on SYS-owned objects. The order in which these triggers fire, if present, is as follows:
1. BEFORE statement-level trigger
2. BEFORE row-level trigger for each row affected by statement
3. The triggering statement
4. AFTER row-level trigger for each row affected by statement
5. AFTER statement-level trigger
## Compound DML Triggers
Compound triggers allow you to combine up to four DML triggering events into a single coordinated program. These compound triggers let you share common elements (subprograms and state data) among the different triggering events; for example, you can use bulk binds in your DML trigger and achieve significantly better performance when several rows are affected by a statement.
A compound trigger has as many as four sections: a BEFORE STATEMENT section, a BEFORE EACH ROW section, an AFTER EACH ROW section, and an AFTER STATEMENT section. The FOR and COMPOUND TRIGGER keywords tell the database that the trigger is a compound trigger. In the compound trigger's declaration section, you declare the structures that are to be shared by all sections. These structures include the collections needed for bulk binds, variables, local programs, etc. This trigger data is created when the trigger fires and is automatically destroyed (cleaned up) when the triggering statement completes.
Here is an example of using a compound trigger to record audit information on changes to the `salary` column in the `employee` table. The example shows how this is done with the old FOR EACH ROW technique as well as the newer, more efficient bulk binds:
-- First row-by-row way
CREATE OR REPLACE TRIGGER old_trg
AFTER UPDATE OF salary ON employees
FOR EACH ROW
BEGIN
INSERT INTO employee_audit
VALUES (:new.employee_id
,:old.salary
,:new.salary ,SYSTIMESTAMP);
END old_trg;
/* This next trigger is created disabled and
must be enabled for use
Here is the bulk bind approach: */
CREATE OR REPLACE TRIGGER new_trg
FOR UPDATE OF salary ON employees DISABLE
COMPOUND TRIGGER
-- General declarations here
TYPE emp_aud_t IS TABLE OF
employee_audit%ROWTYPE
INDEX BY BINARY_INTEGER;
emps emp_aud_t;
cntr PLS_INTEGER := 0;
batch_size CONSTANT PLS_INTEGER := 100;
-- local procedure
PROCEDURE bulk_flush IS
BEGIN
FORALL idx IN 1..emps.count
INSERT INTO employee_audit
VALUES emps(idx);
emps.delete;
cntr := 0;
END ;
-- Each section is defined like this:
AFTER EACH ROW IS
BEGIN
cntr := cntr+1;
emps(cntr).employee_id := :new.employee_id;
emps(cntr).old_salary := :old.salary;
emps(cntr).new_salary := :new.salary;
emps(cntr).change_ts := systimestamp;
IF cntr >= batch_size THEN
bulk_flush;
END IF;
END AFTER EACH ROW;
-- Final flush in after statement section
AFTER STATEMENT IS
BEGIN
bulk_flush;
END AFTER STATEMENT;
END new_trg;
## DDL Events
The DDL events are ALTER, ANALYZE, ASSOCIATE STATISTICS, AUDIT, COMMENT, CREATE, DISASSOCIATE, DROP, GRANT, NOAUDIT, RENAME, REVOKE, and TRUNCATE. These triggers fire whenever the respective DDL statement is executed. DDL triggers can apply to either a single schema or the entire database.
## Database Events
The database events are SERVERERROR, LOGON, LOGOFF, STARTUP, SHUTDOWN, and SUSPEND. Only BEFORE triggers are allowed for LOGOFF and SHUTDOWN events. Only AFTER triggers are allowed for LOGON, STARTUP, and SERVERERROR events. A SHUTDOWN trigger will fire on a SHUTDOWN NORMAL and a SHUTDOWN IMMEDIATE, but not on a SHUTDOWN ABORT.
# Packages
A package is a collection of PL/SQL elements that are grouped together. There are several benefits to using packages, including information hiding, object-oriented design, top-down design, object persistence across transactions, and improved performance.
Elements that can be placed in a package include procedures, functions, constants, variables, cursors, exception names, and TYPE statements (for associative arrays, records, REF CURSORs, etc.).
## Package Structure
A package can have two parts: the specification and the body. The _package specification_ is required and lists all the objects that are "publicly" available (i.e., may be referenced from outside the package) for use in applications. It also provides all the information a developer needs to use objects in the package; essentially, it is the package's API.
The _package body_ contains all the code needed to implement procedures, functions, and cursors listed in the specification, as well as any private objects (accessible only to other elements defined in that package) and an optional initialization section.
If a package specification does not contain any procedures or functions, and no private code is needed, that package does not need to have a package body.
The syntax for the package specification is:
CREATE [OR REPLACE] PACKAGE _package_name_
[AUTHID { CURRENT_USER | DEFINER } ]
[ACCESSIBLE BY ( _program_unit_list_ )]
{ IS | AS }
[ _definitions of public TYPEs
,declarations of public variables, types, and objects
,declarations of exceptions
,pragmas
,declarations of cursors, procedures, and
functions
,headers of procedures and functions_ ]
END [ _package_name_ ];
The syntax for the package body is:
CREATE [OR REPLACE] PACKAGE BODY _package_name_
{ IS | AS }
[ _definitions of private TYPEs_
, _declarations of private variables, types, and objects_
, _full definitions of cursors_
, _full definitions of procedures and functions_ ]
[BEGIN
_executable_statements_
[EXCEPTION
_exception_handlers_ ]]
END [ _package_name_ ];
Specify the optional OR REPLACE to rebuild an existing package, preserving any EXECUTE privileges previously granted to other accounts. The declarations in the specifications cannot be repeated in the body. Both the execution section and the exception section are optional in a package body. If the execution section is present, it is called the _initialization section_ and executes only once—the first time any package element is referenced during a session.
You must compile the package specification before the body specification. When you grant EXECUTE authority on a package to another schema or to PUBLIC, you are giving access only to the specification; the body remains hidden.
Here's an example of a package:
CREATE OR REPLACE PACKAGE time_pkg
AS
FUNCTION GetTimestamp RETURN DATE;
PROCEDURE ResetTimestamp(
new_time DATE DEFAULT SYSDATE);
END time_pkg;
-- StartTimeStamp is package-level data.
CREATE OR REPLACE PACKAGE BODY time_pkg
AS
StartTimeStamp DATE := SYSDATE;
FUNCTION GetTimestamp RETURN DATE IS
BEGIN
RETURN StartTimeStamp;
END GetTimestamp;
PROCEDURE ResetTimestamp(
new_time DATE DEFAULT SYSDATE)
IS
BEGIN
StartTimeStamp := new_time;
END ResetTimestamp;
END time_pkg;
## Referencing Package Elements
The elements declared in the specification are referenced from the calling application via dot notation:
_package_name.package_element_
For example, the built-in package DBMS_OUTPUT has a procedure PUT_LINE, so a call to this package would look like this:
DBMS_OUTPUT.PUT_LINE('This is parameter data');
## Package Data
Data structures declared within a package specification or body, but outside any procedure or function in the package, are _package data_. The default lifetime of package data is your entire session, spanning transaction boundaries and acting as globals for your programs.
Keep the following guidelines in mind as you work with package data:
* The state of your package variables is not affected by COMMITs and ROLLBACKs.
* A cursor declared in a package has global scope. It remains OPEN until you close it explicitly or until your session ends.
* A good practice is to _hide_ your data structures in the package body and provide "get and set" programs to read and write that data. This technique can help protect your data.
## SERIALLY_REUSABLE Pragma
If you need package data to exist only during a call to the packaged functions or procedures and not between calls of the current session, you can potentially save runtime memory by using the pragma SERIALLY_REUSABLE. After each call, PL/SQL closes the cursors and releases the memory used in the package. This technique is applicable only to large user communities executing the same routine. Normally, the database server's memory requirements grow linearly with the number of users; with SERIALLY_REUSABLE, this growth can be less than linear because work areas for package states are kept in a pool in the database's SGA and are shared among all users. This pragma must appear in both the specification and the body, as shown here:
CREATE OR REPLACE PACKAGE my_pkg IS
PRAGMA SERIALLY_REUSABLE;
PROCEDURE gig_em;
END my_pkg;
CREATE OR REPLACE PACKAGE BODY my_pkg IS
PRAGMA SERIALLY_REUSABLE;
PROCEDURE gig_em IS
...
END my_pkg;
## Package Initialization
The first time a user references a package element, the entire package is loaded into the SGA of the database instance to which the user is connected. That code is then shared by all sessions that have EXECUTE authority on the package.
Any package data is then instantiated into the session's User Global Area (UGA), a private area in either the SGA or the Process Global Area (PGA). If the package body contains an initialization section, that code will be executed. The initialization section is optional and appears at the end of the package body, beginning with a BEGIN statement and ending with the EXCEPTION section (if present) or the END of the package.
The following package initialization section runs a query to transfer the user's minimum balance into a global package variable. Programs can then reference the packaged variable (via the function) to retrieve the balance, rather than execute the query repeatedly:
CREATE OR REPLACE PACKAGE usrinfo
IS
FUNCTION minbal RETURN VARCHAR2;
END usrinfo;
CREATE OR REPLACE PACKAGE BODY usrinfo
IS
g_minbal NUMBER; -- Package-level data
FUNCTION minbal RETURN VARCHAR2
IS
BEGIN
RETURN g_minbal;
END;
BEGIN -- Initialization section
SELECT minimum_balance
INTO g_minbal
FROM user_configuration
WHERE username = USER;
EXCEPTION
WHEN NO_DATA_FOUND
THEN g_minbal := NULL;
END usrinfo;
# Calling PL/SQL Functions in SQL
Stored functions can be called from SQL statements in a manner similar to built-in functions, such as DECODE, NVL, or RTRIM. This is a powerful technique for incorporating business rules into SQL in a simple and elegant way, but there are several caveats and restrictions.
The most notable caveat is that stored functions executed from SQL are not by default guaranteed to return results from the database that are read-consistent with respect to the parent SQL statement. Unless the SQL statement and any stored functions in that statement are in the same read-consistent transaction (even if they are read-only), each execution of the stored function may look at a different time-consistent set of data. To avoid this potential problem, you have two choices:
1. Restrict the stored function to perform only computations that do not involve using data from database tables.
2. Ensure read consistency programmatically by issuing the SET TRANSACTION READ ONLY or SET TRANSACTION ISOLATION LEVEL SERIALIZABLE statement before executing your SQL statement containing the stored function. A COMMIT or ROLLBACK then needs to follow the SQL statement to end this read-consistent transaction.
## Calling a Function
The syntax for calling a stored function from SQL is the same as that used to reference it from PL/SQL:
[ _schema_name_.][ _pkg_name_.] _func_name_ [@ _db_link_ ][ _parm_list_ ]
_schema_name_ is optional and refers to the user/owner of the function or package. _pkg_name_ is optional and refers to the package containing the called function. _func_name_ is required and is the function name. _db_link_ is optional and refers to the database link name to the remote database containing the function. _parm_list_ is optional, as are the parameters passed to the function.
The following are example calls to the `GetTimestamp` function in the `time_pkg` example seen earlier in "Package Structure":
-- Capture system events.
INSERT INTO v_sys_event
(timestamp, event, qty_waits)
SELECT time_pkg.GetTimestamp, event, total_waits
FROM v$system_event
-- Capture system statistics.
INSERT INTO v_sys_stat (timestamp, stat#, value)
SELECT time_pkg.GetTimestamp, statistic#, value
FROM v$sysstat;
There are several requirements for calling stored functions in SQL:
* All parameters must be IN; no IN OUT or OUT parameters are allowed.
* The datatypes of the function's parameters and RETURN must be compatible with RDBMS datatypes. You cannot have arguments or RETURN types, such as BOOLEAN, programmer-defined record, and associative array.
* The function must be a schema-level object in the database or defined in the specification of a package.
## Improving Performance of Calling PL/SQL Functions from SQL
Oracle Database 12 _c_ introduced the UDF pragma, which signals to the compiler that you would like your subprogram to be optimized for invocation from SQL statements. The `GetTimestamp` function mentioned previously would become:
...
FUNCTION GetTimestamp RETURN DATE IS
**PRAGMA UDF;**
BEGIN
RETURN StartTimeStamp;
END GetTimestamp;
...
After compiling with this pragma, Oracle's runtime engine may be able to reduce the overhead associated with context switching, resulting in better performance. However, performance when called from inside PL/SQL is likely to be worse.
Another feature added in Oracle Database 12 _c_ is the ability to declare a function inside the WITH clause of a SELECT statement. If the UDF pragma is not helpful, you could explore this feature as a possible performance optimization. An example:
WITH
FUNCTION betwnstr (str IN VARCHAR2,
p1 IN INTEGER, p2 IN INTEGER)
RETURN VARCHAR2
IS
BEGIN
RETURN SUBSTR(str, p1, p2 - p1 +1);
END;
SELECT betwnstr(last_name, 3, 5)
FROM employees;
## Column Name Versus Function Name Precedence
If your function has the same name as a table column in your SELECT statement, and the function has no parameter, the column takes precedence over the function. To force the Oracle database to resolve the name to your function, prepend the schema name to it:
CREATE TABLE emp(new_sal NUMBER ...);
CREATE FUNCTION new_sal RETURN NUMBER IS ...;
-- Resolves to column
SELECT new_sal FROM emp;
-- Resolves to function
SELECT scott.new_sal FROM emp;
# Object-Oriented Features
In the Oracle database, an _object type_ combines attributes (data structures) and methods (functions and procedures) into a single programming construct. The object type construct allows programmers to define their own reusable datatypes for use in PL/SQL programs and table and column definitions. An object type must be created in a database before it can be used in a PL/SQL program.
An instance of an object type is an _object_ in the same way that a variable is an instance of a scalar type. As with scalars, objects are either _persistent_ (stored in the database) or _transient_ (stored only in PL/SQL variables). Objects can be stored in a database as a row in a table (a row object) or as a column in a table. A table of row objects can be created with syntax such as this:
CREATE TABLE _table_name_ OF _object_type_ ;
When stored in such a table, the object (row) has a system-generated object identifier (OID) that is unique throughout the database.
## Object Types
An object type has two parts: a specification and a body. The specification is required and contains the attributes and method specifications. The syntax for creating the object type specification is:
CREATE [OR REPLACE] TYPE _obj_type_name_
[AUTHID { CURRENT_USER | DEFINER } ]
[ACCESSIBLE BY ( _program_unit_list_ )]
{ { IS | AS } OBJECT | UNDER _parent_type_name_ }
(
_attribute_name datatype_ ,...,
[ [ [NOT] OVERRIDING ] [ [NOT] FINAL ] [ [NOT]
INSTANTIABLE ]
_method_spec_ ,...,
]
)
[ [NOT] FINAL ]
[ [NOT] INSTANTIABLE ];
where _method_spec_ is one of the following:
MEMBER { PROCEDURE | FUNCTION } _program_spec_
or:
STATIC { PROCEDURE | FUNCTION } _program_spec_
or:
{ ORDER | MAP } MEMBER FUNCTION _comparison_function_spec_
or:
[ FINAL ] [ INSTANTIABLE ] CONSTRUCTOR FUNCTION
RETURNING SELF AS RESULT _constructor_function_spec_
Attribute specifications must appear before method specifications. Object attributes, like table columns, are defined with a name and a datatype. The name can be any legal identifier, and the datatype can be almost any datatype known to SQL other than LONG, LONG RAW, ROWID, and UROWID. Attributes can be declared using other programmer-defined object types or collection types, but not of the special types ANYTYPE, ANYDATA, or ANYDATASET. Attributes cannot be of datatypes unique to PL/SQL, such as BOOLEAN.
Method headers appear in the object type specification in a comma-delimited list. Unlike in a package specification, commas (not semicolons) terminate the object type program specifications. To support object comparisons and sorting, the type optionally can include one comparison method—either ORDER or MAP. Member methods can be overloaded in object types following the same rules as function and procedure overloading in packages.
Method "specs" that appear earlier in the syntax actually can be call specs for Java classes in the database or for external procedures written in C.
The syntax for creating the object type body is:
CREATE [OR REPLACE] TYPE BODY obj_type_name
{ IS | AS }
_method_implementation_ ;
[ _method_implementation_ ... ]
;
Where _`method_implementation`_ is one of:
MEMBER { PROCEDURE | FUNCTION } _function_body_
or:
STATIC { PROCEDURE | FUNCTION } _function_body_
or:
{ ORDER | MAP } MEMBER FUNCTION
_comparison_function_body_
or:
[ FINAL ] [ INSTANTIABLE ] CONSTRUCTOR FUNCTION
RETURNING SELF AS RESULT
_constructor_function_body_
Again, the program bodies can be call specs to Java or C programs.
## Type Inheritance
You can define subtypes of object types following a single-inheritance model. The database does not have a master root-level object; instead, each type is "standalone" unless declared otherwise.
The UNDER keyword specifies that the type exists as a subtype in a hierarchy. When you are using UNDER, the parent type must be marked NOT FINAL. By default, types are FINAL, meaning that you cannot declare a subtype of that type.
A subtype contains all the attributes and methods of its parent (supertype) and may contain additional attributes and methods. Methods can override corresponding methods from the parent. Changes to the supertype—such as the addition of attributes or methods—are reflected in the subtypes automatically.
By default, object types are INSTANTIABLE—that is, an invoking program may create an object of that type. The phrase NOT INSTANTIABLE indicates that you don't want any objects of the type, in which case the database will not create a constructor for it. This variation generally makes sense only with types that will serve as parents of other types.
## Methods
There are four kinds of methods: _member_ , _static_ , _constructor_ , and _comparison_.
### Member methods
A member method is a procedure or function designated with the keyword MEMBER. Calling programs may invoke such a method only on objects that have been instantiated.
### Static methods
A static method has no access to a current (SELF) object. Such a method is declared using the keyword STATIC and can be invoked at any time using _type.method_ syntax.
### Constructor methods
Even if you don't declare any methods, every instantiable object has a default constructor method that allows a calling program to create new objects of that type. This built-in method:
* Has the same name as the object type.
* Is a function that returns an object of that type.
* Accepts attributes in named or positional notation.
* Must be called with a value (or NULL) for every attribute—there is no DEFAULT clause for object attributes.
* Cannot be modified.
You can replace this default constructor with your own using the CONSTRUCTOR FUNCTION syntax. This method must have the same name as the object type, but there are no restrictions on its parameter list. The RETURN clause of the constructor's header must be RETURN SELF AS RESULT. The database supports overloading of programmer-defined constructors. All nonstatic methods have the implied parameter SELF, which refers to the current instance of the object. The default mode for the SELF parameter is IN for functions and IN OUT for procedures. A programmer can alter the mode by explicitly including SELF in the formal parameter list. An example of a programmer-defined default constructor follows:
CREATE OR REPLACE TYPE book_t AS OBJECT (
isbn VARCHAR2(13),
pages INTEGER,
CONSTRUCTOR FUNCTION book_t
(id IN INTEGER DEFAULT NULL
,title IN VARCHAR2 DEFAULT NULL
,isbn IN VARCHAR2 DEFAULT NULL
,pages IN INTEGER DEFAULT NULL
)
RETURN SELF AS RESULT,
OVERRIDING MEMBER FUNCTION ck_digit_okay
RETURN BOOLEAN,
OVERRIDING MEMBER FUNCTION print
RETURN VARCHAR2
);
### Comparison methods
The comparison methods, ORDER and MAP, establish ordinal positions of objects for comparisons such as "<" or "between" and for sorting (ORDER BY, GROUP BY, DISTINCT). The database invokes a comparison method automatically whenever it needs to perform such an operation.
MAP and ORDER methods are actually special types of member methods—that is, they execute only in the context of an existing object. An ORDER function accepts two parameters: SELF and another object of the same type. It must return an INTEGER value as explained in the following table:
Return value | Object comparison
---|---
Any negative integer (commonly −1) | SELF < second object
0 | SELF = second object
Any positive integer (commonly 1) | SELF > second object
NULL | Undefined comparison: attributes needed for the comparison are NULL
For example, the Senate ranks majority party members higher than nonmajority party members and within the majority (or nonmajority) by years of service. Here is an example ORDER function incorporating these rules:
CREATE TYPE senator_t AS OBJECT
(
majority INTEGER,
yrs_service NUMBER,
ORDER MEMBER FUNCTION ranking (
other IN senator_t)
RETURN INTEGER
);
CREATE OR REPLACE TYPE BODY senator_t
AS
ORDER MEMBER FUNCTION ranking (
other IN senator_t)
RETURN INTEGER
IS
BEGIN
IF SELF.majority = 'Y' AND
other.majority.istrue = 'Y'
THEN
RETURN SIGN(
SELF.yrs_service -
other.yrs_service);
ELSIF SELF.majority = 'Y' AND
other.majority ='N'
THEN
RETURN 1;
ELSIF SELF.majority = 'N' AND
other.majority.istrue = 'Y'
THEN
RETURN −1;
ELSIF SELF.majority = 'N' AND
other.majority.istrue = 'N'
THEN
RETURN SIGN(
SELF.yrs_service -
other.yrs_service);
END IF;
END ranking;
END;
A MAP function accepts no parameters and returns a scalar datatype such as DATE, NUMBER, or VARCHAR2 for which the database already knows a collating sequence. The MAP function translates, or _maps_ , each object into this scalar datatype space.
If no ORDER or MAP function exists for an object type, then SQL, but not PL/SQL, supports only limited equality comparisons of objects. Objects are equal if they are of the same object type and if each attribute is equal.
Use MAP if possible when frequently sorting or comparing a large number of objects, as in a SQL statement; an internal optimization reduces the number of function calls. With ORDER, the function must run once for every comparison.
## Methods in Subtypes
When defining methods in a subtype, you have two options: you can _inherit_ a supertype's method, or you can _override_ a supertype's method by defining your own subtype method of the same name and parameter list. If you choose to inherit, you do not need to write any code in the subtype.
To override a supertype, you must use the OVERRIDING keyword in the header of the program, as shown here:
CREATE TYPE food_t AS OBJECT (
name VARCHAR2(100),
food_group VARCHAR2 (100),
MEMBER FUNCTION price RETURN NUMBER
)
NOT FINAL;
CREATE TYPE dessert_t UNDER food_t (
contains_chocolate CHAR(1),
OVERRIDING MEMBER FUNCTION price RETURN NUMBER
);
This example also shows that if you want to allow a method to be overridden, you must specify that this method be NOT FINAL. By default, methods are FINAL and cannot be overridden.
You also can define a method to be NOT INSTANTIABLE, which means that you specify only the header of the method, but you do not need to provide an implementation in the OBJECT TYPE body for that method. For example:
CREATE TYPE food_t AS OBJECT (
name VARCHAR2(100),
food_group VARCHAR2 (100),
NOT INSTANTIABLE MEMBER
FUNCTION price RETURN NUMBER
)
NOT FINAL
NOT INSTANTIABLE;
The consequences of a NOT INSTANTIABLE method are as follows:
* The entire object type must be defined as NOT INSTANTIABLE, which means that you cannot instantiate an instance from this type. You can use it only as a supertype in an object type hierarchy.
* Any subtype of `food_t` must provide an implementation of the `price` function or in turn also be defined as a NOT INSTANTIABLE object type.
The database supports _dynamic method dispatch_ , also known as _dynamic polymorphism_ , to determine which overridden method to invoke at runtime. That is, it will choose the method in the most specific subtype associated with the currently instantiated object. However, it is also possible to directly invoke a parent type's method. For example, you can invoke the `food_t` version of a dessert's `price` method using the following syntax:
DECLARE
my_dessert dessert_t := dessert_t (
'tres leches', 'sugar', 'N');
BEGIN
DBMS_OUTPUT.PUT_LINE(
( **my_dessert AS food_t).price**);
END;
This is also possible inside the implementation sections of subtype methods using SELF:
(SELF AS _parent_type_ ). _method_invocation_ ;
## Manipulating Objects in PL/SQL and SQL
Variables declared as objects begin their life _atomically null_ , meaning that the expression:
_object_ IS NULL
evaluates to TRUE. Attempting to assign values to the attributes of an atomically null object will return an ACCESS_ INTO_NULL exception. Instead, you must initialize the object, in one of these ways:
* Use either the default constructor method or a user-defined constructor
* Assign to it the value of an existing object
* Use SELECT INTO or FETCH INTO
Here is an example using each initialization technique:
DECLARE
project_boiler_plate project_t;
build_web_site project_t;
-- Initialize via constructor.
new_web_mgr proj_mgr_t :=
proj_mgr_t('Ruth', 'Home Office');
-- Initialize via user-defined constructor
-- that provides defaults
new_web_mgr proj_mgr_t := NEW proj_mgr_t();
CURSOR template_cur IS
SELECT VALUE(proj)
FROM projects
WHERE project_type = 'TEMPLATE'
AND sub_type = 'WEB SITE';
BEGIN
OPEN template_cur;
-- Initialize via FETCH INTO.
FETCH template_cur INTO project_boiler_plate;
-- Initialize via assignment.
build_web_site := project_boiler_plate;
...
After an object is initialized, it can be stored in the database, and you can then locate and use that object with the REF, VALUE, and DEREF operators.
## Upcasting and Downcasting
The Oracle database supports implicit _upcasting_ (widening) of a subtype and provides the TREAT operator to _downcast_ (narrow) a supertype. TREAT also can explicitly upcast a subtype.
The following example returns to the food-dessert hierarchy to demonstrate upcasting and downcasting:
CREATE TYPE food_t AS OBJECT (
name VARCHAR2(100),
food_group VARCHAR2 (100)
)
NOT FINAL;
CREATE TYPE dessert_t UNDER food_t (
contains_chocolate CHAR(1)
);
DECLARE
marzipan dessert_t :=
NEW dessert_t('marzipan', 'sweets', 'N');
ice_cream_sundae dessert_t;
tasty_treat food_t;
BEGIN
/* An implied upcast */
tasty_treat := marzipan;
/* An explicit downcast */
ice_cream_sundae :=
TREAT(tasty_treat AS dessert_t);
END;
The syntax of TREAT is:
TREAT ( _object_instance_ AS [REF] _type_ )
where _object_instance_ is a value that is of a particular supertype in an object hierarchy, and _type_ is the name of the subtype (or supertype) in the same hierarchy. The TREAT expression won't compile if you attempt to cast a type to another from a different type hierarchy. If you supply an object from the correct type hierarchy, TREAT will return either the casted object or NULL—but not an error.
You also can use dot notation to obtain access to the casted object's attributes and methods:
TREAT ( _object_instance_ AS _type_ ).{ _attribute_ | _method_ ( _args_...) } ]
SQL also supports TREAT and implied upcasting.
### REF operator
REF, short for REFerence, designates a datatype modifier or an operator to retrieve a logical pointer to an object. This pointer encapsulates the OID and can simplify navigation among related database objects. The syntax for a REF operator is:
REF( _table_alias_name_ )
For example:
SELECT REF(p) FROM pets p WHERE ...
A PL/SQL variable can hold a reference to a particular object type:
DECLARE
petref REF Pet_t;
BEGIN
SELECT REF(p) INTO petref FROM pets p WHERE ...
Through deletions, REFs can reference a nonexistent object, called a _dangling REF_ , resulting in a state that can be detected with the IS DANGLING predicate. For example:
UPDATE pets
SET owner_ref = NULL
WHERE owner_ref IS DANGLING;
The built-in package UTL_REF provides programmatic access to stored objects via their REF.
### VALUE operator
Use the VALUE operator to retrieve a row object as a single object rather than as multiple columns. The syntax for the VALUE operator is:
VALUE( _table_alias_name_ )
For example:
SELECT VALUE(p) FROM pets p WHERE ...
### DEREF operator
Use the DEREF operator to retrieve the value of an object for which you have a REF. The syntax for DEREF is:
DEREF( _table_alias_name_ )
For example:
DECLARE
person_ref REF person_t;
author person_t;
BEGIN
-- Get the ref.
SELECT REF(p) INTO person_ref
FROM persons WHERE p.last_name ='Pribyl';
-- Dereference the pointer back to the value.
SELECT DEREF(person_ref) INTO author FROM dual;
In addition, the database uses an OID internally as a unique identifier for each object. As with a ROWID, you don't typically use an OID directly. The following table shows ways of referencing persistent objects:
Scheme | Description | Applications
---|---|---
OID | An opaque, globally unique handle, produced when the object is stored in the database as a table (row) object. | The persistent object's handle; it is what REFs point to. Your program never uses it directly.
VALUE | An operator. In SQL, it acts on an object in an object table and returns the object's _contents_. Different from the VALUES keyword found in some INSERT statements. | Used when fetching a table (row) object into a variable, or when you need to refer to an object table as an object instead of a list of columns.
REF | A pointer to an object. May be used within a SQL statement as an operator or in a declaration as a type modifier. | Allows quasi-normalizing of object-relational databases and joining of object tables using dot navigation. In PL/SQL, REFs serve as input/output variables.
DEREF | Reverse pointer lookup for REFs. | Used for retrieving the contents of an object when all you know is its REF.
## Changing Object Types
You can add methods, but not attributes, to an object type stored in the database using the ALTER TYPE statement. There are several forms of this statement, with many options, some of which are in the following code block. Check Oracle documentation for the most up-to-date syntax diagrams:
ALTER TYPE _typename_
{ ADD | MODIFY | DROP } ATTRIBUTE _attribute_spec_
{ INVALIDATE | CASCADE
{ [ NOT ] INCLUDING TABLE DATA |
CONVERT TO SUBSTITUTABLE }
[ FORCE ] };
ALTER TYPE _typename_
[ NOT ] { INSTANTIABLE | FINAL }
{ INVALIDATE | CASCADE
{ [ NOT ] INCLUDING TABLE DATA |
CONVERT TO SUBSTITUTABLE }
[ FORCE ] };
ALTER TYPE _typename_
COMPILE [ DEBUG ] [ SPECIFICATION | BODY ]
[ REUSE SETTINGS ];
Because altering the structure of a type can have quite a few repercussions for database objects, the database requires that you either INVALIDATE the dependent objects or CASCADE the change.
When making a change from FINAL to NOT FINAL and cascading the change, you can cause existing table objects to be either NOT SUBSTITUTABLE (the default) or SUBSTITUTABLE. The following is an example of adding an attribute:
ALTER TYPE catalog_item_t
ADD ATTRIBUTE publication_date VARCHAR2(400)
CASCADE INCLUDING TABLE DATA;
The next example shows adding a method:
ALTER TYPE catalog_item_t
ADD MEMBER PROCEDURE save,
CASCADE;
After adding a method to a spec, you would use CREATE OR REPLACE TYPE BODY to implement it in the body (include all the other methods as well).
There are a variety of restrictions on modifying types; for example, you cannot change a type from INSTANTIABLE to NOT INSTANTIABLE if you have created tables that depend on the type.
The syntax for dropping an object type is:
DROP TYPE _typename_ [FORCE];
You can drop only an object type that has not been implemented in a table (or you can drop the tables first). The FORCE option will drop object types even if they have dependencies, but FORCE will irreversibly invalidate any dependent objects such as tables. FORCE does not do a DROP CASCADE.
If you are dropping a type whose parent type has table dependents, this form of the statement:
DROP TYPE _subtype_name_ VALIDATE;
will "validate" the safety of dropping the subtype before performing it. That is, the database will perform the drop only if there are no objects of the subtype in any substitutable columns of the parent type.
# Compilation
PL/SQL compilation is an area that has seen several improvements in recent database versions. These capabilities include conditional compilation, informational warnings, optimization, and compilation to native code.
## Compiling Stored PL/SQL Programs
The following keywords are available when creating stored programs:
_OR REPLACE_
Used with CREATE to rebuild an existing program unit, preserving privileges granted on it to users and roles.
_AUTHID_
Defines whether the program will execute with the privileges of, and resolve names like, the object owner (DEFINER), or as the user executing the function (CURRENT_USER). The default AUTHID is DEFINER. See "Privileges and Stored PL/SQL" for additional information.
_ACCESSIBLE BY_ (program unit list)
(Oracle Database 12 _c_ and later.) Restricts execution of the program to a white list of other programs. Useful when you want to hide some utility programs behind your stable API.
_DETERMINISTIC_
Required for function-based indexes. A function is DETERMINISTIC if it always returns the same value when called with the same parameters. Deterministic functions do not meaningfully reference package variables or the database. The built-in INITCAP is deterministic, but SYSDATE is not.
_PARALLEL_ENABLE..._
Informs the optimizer that a function is safe for parallel execution. Can improve runtime performance of pipelined table functions that are called in a parallel SQL query.
_PIPELINED_
Used with table functions. Specifies that the results of this table function should be returned iteratively via the PIPE ROW statement. A pipelined function can start to return data as it is generated instead of all at once after processing is complete.
_AGGREGATE USING_
Required for aggregate functions. Tells the database that the function evaluates a group of rows and returns a single result. For example, the built-in function AVG is an aggregate function.
The following compiler settings are established at program creation time based on the database or session configuration and can be changed or retained during recompilation. The database stores these compiler settings on a program-by-program basis, so you can recompile your program later using the REUSE SETTINGS option. If you do not reuse the stored settings, or if you explicitly define one or more settings, your current session settings are used:
_PLSQL_CCFLAGS_
Contains a comma-delimited list of name:value pairs controlling conditional compilation. See "Conditional Compilation" for more information.
_PLSQL_CODE_TYPE_
Controls whether interpreted or native code is created during compilation. Valid values are INTERPRETED or NATIVE. See "Performing Native Compilation of PL/SQL" for more information.
_PLSQL_DEBUG_
Controls whether or not the program will be instrumented for debugging during compilation. Valid values are TRUE or FALSE. When compiled for debugging, a program will always be INTERPRETED and never NATIVE.
_PLSQL_OPTIMIZE_LEVEL_
Controls the level of optimization employed by the compiler. Valid values are 0, 1, 2, or 3. See "Optimizing Compiler" for more information.
_PLSQL_WARNINGS_
Controls the level of warnings that the compiler will report. See "Compiler Warnings" for more information.
_NLS_LENGTH_SEMANTICS_
Controls whether VARCHAR2 and CHAR datatypes are defined with BYTE (default) or CHAR semantics. NVARCHAR2, NCHAR, CLOB, and NCLOB datatypes are always defined with CHAR semantics.
To recompile the procedure `my_proc`, explicitly setting the optimization level to 3, run the following:
ALTER PROCEDURE my_proc COMPILE PLSQL_OPTIMIZE_ LEVEL = 3;
Then to recompile it later with the saved settings, run the following:
ALTER PROCEDURE my_proc COMPILE REUSE SETTINGS;
To view all of the stored compiler settings for your programs, query the view USER_PLSQL_OBJECT_SETTINGS.
## Conditional Compilation
Conditional compilation allows your programs to decide at compile time which blocks of code will be implemented. You can conditionally include code in the compiled program based on the database version, environment, or other configurable settings. There are three types of compiler directives available for you to use:
_Selection directives_
Use the $IF directive to evaluate an expression and determine which code should be included. For example:
DECLARE
emp_rec employees%ROWTYPE;
BEGIN
$IF DBMS_DB_VERSION.VER_LE_10_2 $THEN
-- Legacy code
SELECT employees_seq.NEXTVAL
INTO emp_rec.employee_id FROM dual;
$ELSE
-- Oracle Database 11g and higher
emp_rec.employee_id := employees_seq.NEXTVAL;
$END
INSERT INTO emp VALUES (emp_rec);
END;
_Inquiry directives_
Use the $$identifier directive to refer to conditional compilation flags (PLSQL_CCFLAGS). These inquiry directives can be referenced in an $IF directive or independently. For example:
ALTER SESSION SET PLSQL_CCFLAGS =
'pl_debug:false, pl_trace_level:2';
CREATE OR REPLACE PROCEDURE extract_client_data
AS
BEGIN
$IF $$pl_debug OR $$pl_trace_level >= 2 $THEN
DBMS_SUPPORT.START_TRACE
(waits=>TRUE, binds=>TRUE);
$ELSIF $$pl_trace_level >= 1 $THEN
DBMS_SUPPORT.START_TRACE
(waits=>TRUE, binds=>FALSE);
$END
NULL; -- code goes here
END extract_client_data;
_Error directives_
Use the $ERROR directive to force compilation errors if your prerequisite conditions are not met. For example, I want to _make sure_ no one compiles a compute-intensive program with a level below 2:
CREATE OR REPLACE PROCEDURE compute_intensive IS
BEGIN
$IF $$plsql_optimize_level < 2
$THEN
$ERROR 'Must use full optimization!' $END
$END
NULL;
END;
The settings that are available for use in these directives include:
_Compiler settings_
PLSCOPE_SETTINGS, PLSQL_CCFLAGS, PLSQL_DEBUG, PLSQL_WARNINGS, PLSQL_OPTIMIZE_LEVEL, PLSQL_CODE_TYPE, and NLS_LENGTH_SEMANTICS.
_PLSQL_LINE (PLS_INTEGER literal)_
The line number within the program or can be explicitly defined with the PLSQL_CCFLAGS parameter.
_PLSQL_UNIT (VARCHAR2 literal)_
The name of the program being compiled. For anonymous blocks, it is null. PLSQL_UNIT can also be explicitly defined with the PLSQL_CCFLAGS parameter.
_Static expressions defined in package specifications_
These expressions cannot change when a package is recompiled.
The PL/SQL compiler reads and interprets these directives, generating the code to be implemented. To identify what has actually been deployed in the compiled program, use the DBMS_PREPROCESSOR package, as in the following example:
-- Compile with conditional compilation statements
CREATE OR REPLACE PROCEDURE my_cc_proc IS
BEGIN
DBMS_OUTPUT.PUT_LINE('DB Version is:'
||DBMS_DB_VERSION.VERSION||'r'
||DBMS_DB_VERSION.RELEASE);
$IF DBMS_DB_VERSION.VER_LE_11 $THEN
DBMS_OUTPUT.PUT_LINE('Version 11 code here');
$ELSIF DBMS_DB_VERSION.VER_LE_12 $THEN
DBMS_OUTPUT.PUT_LINE('Version 12 code here');
$END
END;
/
-- Display the deployed code
BEGIN
DBMS_PREPROCESSOR.PRINT_POST_PROCESSED_SOURCE
('PROCEDURE', USER, 'MY_CC_PROC');
END;
This displays the following text (note that whitespace is preserved--and intentionally displayed here--to ensure line numbers at runtime correspond to original source code):
PROCEDURE my_cc_proc IS
BEGIN
DBMS_OUTPUT.PUT_LINE('DB Version is:'
||DBMS_DB_VERSION.VERSION||'r'
||DBMS_DB_VERSION.RELEASE);
DBMS_OUTPUT.PUT_LINE('Version 12 code here');
END;
## Compiler Warnings
_Compile-time warnings_ can help make your programs more robust. These warnings highlight potential problems that are not severe enough to raise an exception, but may result in runtime errors or poor performance. You can configure the compiler to reject as an error any of these warnings. Warnings result in the program compiling to a VALID status, but errors result in an INVALID status.
To enable these warnings, you need to set the database initialization parameter PLSQL_WARNINGS. This parameter can be set globally in the _SPFILE_ initialization file, in your session via the ALTER SESSION statement, or with the built-in package DBMS_WARNING.
The PLSQL_WARNINGS parameter is a comma-delimited list of values, each of which has the syntax:
[ENABLE | DISABLE | ERROR] : [ALL | SEVERE | INFORMATIONAL
| PERFORMANCE | _warning_number_ ]
For example, to enable all warnings in your session, execute:
ALTER SESSION SET plsql_warnings = 'enable:all';
If you want to configure warning message number 06009 ("OTHERS handler does not end in RAISE or RAISE_APPLICATION_ERROR") as an error and enable all warnings in the performance category except warning number 07202 ("Parameter may benefit from use of the NOCOPY compiler hint"), execute:
ALTER SESSION SET plsql_warnings =
'error:06009'
,'enable:performance'
,'disable:07203';
To see what your current setting is, you can execute:
DBMS_OUTPUT.PUT_LINE
(DBMS_WARNING.get_warning_setting_string());
Some examples of warnings follow (edited for space reasons):
SQL>ALTER SESSION SET plsql_warnings ='ENABLE:ALL';
Session altered.
SQL> **CREATE OR REPLACE PROCEDURE bad_practice IS**
2 test_string VARCHAR2(32);
3 BEGIN
4 test_string := 'My demo program';
5 EXCEPTION
6 WHEN OTHERS THEN NULL;
7 END;
8 /
Warning: Procedure created with compilation errors.
SQL> **SHOW ERRORS**
LINE/COL ERROR
---- ------------------------------------------
4/1 PLW-07206: analysis suggests that the
assignment to 'TEST_STRING' may be
unnecessary
6/6 PLS-06009: procedure "BAD_PRACTICE"
OTHERS handler does not
end in RAISE or RAISE_APPLICATION_ERROR
SQL> **CREATE OR REPLACE PACKAGE create_policy IS**
2 PROCEDURE proc_dec_page (dec_page IN OUT CLOB);
3 END create_policy;
4
/
SP2-0808: Package created with compilation warnings
SQL> **SHOW ERRORS**
Errors for PACKAGE CREATE_POLICY:
LINE/COL ERROR
---- ------------------------------------------
2/32 PLW-07203: parameter 'DEC_PAGE' may benefit
from use of the NOCOPY compiler hint
SQL> **CREATE OR REPLACE PACKAGE BODY create_policy IS**
2 PROCEDURE process_dec_page (
3 dec_page IN OUT NOCOPY CLOB ) IS
4 BEGIN
5 default_dec(dec_page);
6 END process_dec_page;
7 END create_policy;
8 /
SP2-0810: Package Body created with compilation warnings
SQL> **SHOW ERRORS**
Errors for PACKAGE BODY CREATE_POLICY:
LINE/COL ERROR
--- -------------------------------------------
3/6 PLW-05000: mismatch in NOCOPY qualification
between specification and body
SQL> **CREATE OR REPLACE PROCEDURE dead_code IS**
2 x NUMBER := 10;
3 BEGIN
4 IF x = 10 THEN
5 x := 20;
6 ELSE
7 x := 100; -- dead code
8 END IF;
9 END dead_code;
10 /
SP2-0804: Procedure created with compilation warnings
SQL> **SHOW ERRORS**
Errors for PROCEDURE DEAD_CODE:
LINE/COL ERROR
---- --------------------------------------
7/7 PLW-06002: Unreachable code
## Optimizing Compiler
PL/SQL's optimizing compiler can improve runtime performance dramatically while imposing only a relatively slight overhead at compile time. Fortunately, the benefits of optimization apply to both interpreted and natively compiled PL/SQL because optimizations are applied by analyzing patterns in source code.
The optimizing compiler is enabled by default. However, you may want to alter its behavior, by either lowering its aggressiveness or disabling it entirely. For example, if, in the course of normal operations, your system must perform recompilation of many lines of code, or if an application generates many lines of dynamically executed PL/SQL, the overhead of optimization may be unacceptable. Note, however, that Oracle's tests show that the optimizer doubles the runtime performance of computationally intensive PL/SQL.
In some cases, the optimizer may even alter program behavior. One such case might occur in code written for Oracle9 _i_ Database, which depends on the relative timing of initialization sections in multiple packages. If your testing demonstrates such a problem, yet you want to enjoy the performance benefits of the optimizer, you may want to rewrite the offending code or introduce an initialization routine that ensures the desired order of execution.
The Orace database utilizes _intra-unit inline optimization_. This optimization technique replaces a call to a subprogram with a copy of the program, at compile time. The performance improvement occurs because the subprogram does not have to be loaded separately at runtime. This technique is especially useful for short utility helper programs.
To change the optimizer settings, set the initialization parameter PLSQL_OPTIMIZE_LEVEL, either for your session with an ALTER SESSION statement or for the database with an ALTER SYSTEM statement. Valid settings are:
_0_
No optimization
_1_
Moderate optimization, such as eliminating superfluous code or exceptions
_2 (default)_
Aggressive optimization beyond level 1, including rearranging source code
_3_
Include inline subprogram optimization
You can also modify these settings for the current session; for example:
ALTER SESSION SET PLSQL_OPTIMIZE_LEVEL = 0;
With optimization level 2, you can control inline optimization with the INLINE pragma (see "Pragmas"). The syntax of the INLINE pragma is:
PRAGMA INLINE ( _program_name_ ,'YES | NO');
YES requests the compiler to use inline optimization for calls to _program_name_ , while NO explicitly requests the compiler to not use inline optimization for such calls. In the following example, compiled with the default optimization level 2, calls to the procedure `P` are requested to be inlined:
CREATE OR REPLACE PACKAGE BODY bi_util IS
FUNCTION avg_sales(cust_id IN NUMBER)
RETURN NUMBER
IS
BEGIN
-- inline calls to program P
PRAGMA INLINE (P,'YES');
p('Inside simulation');
RETURN cust_id; -- simulation only
END;
END bi_util;
## Performing Native Compilation of PL/SQL
You can speed up many of your PL/SQL programs by compiling them into code native to the hardware rather than using the default, system-independent compiled form (known as DIANA). Using native compilation, you will realize the greatest performance gains with computer-intensive applications and the least from programs that contain only declarations, such as types and package specifications. Note, though, that if you are working in development mode and need to compile a program with debug information, the native compilation feature is not available. Follow these steps to compile a stored program natively:
ALTER SESSION SET PLSQL_CODE_TYPE = 'NATIVE';
1. Ensure that database (or session-level) parameter PLSQL_OPTIMIZE_LEVEL is set to 2 or higher.
2. Set the database parameter PLSQL_CODE_TYPE to NATIVE, or issue a session-level statement:
3. Recompile your stored program.
If you want to recompile all your stored programs natively, follow the step-by-step procedure in Oracle's PL/SQL documentation (search for "Compiling the Entire Database for PL/SQL Native or Interprested Compilation").
# Java Language Integration
Java programmers can write server-side classes that invoke SQL and PL/SQL using standard JDBC or SQLJ calls. PL/SQL programmers can call server-side Java methods by writing a PL/SQL cover or _call spec_ for Java using Oracle database DDL.
Server-side Java in the database may be faster than PL/SQL for computer-intensive programs, but not as nimble for database access. PL/SQL is much more efficient for database-intensive routines because, unlike Java, it doesn't have to pay the overhead for converting SQL datatypes for use inside the stored program. In my experience, the most common use of Java stored procedures is to supplement functionality not provided by Oracle built-ins (for example, to obtain a sorted listing of a directory in the host filesystem).
Follow these steps to create a Java stored procedure (JSP):
1. Write or otherwise obtain functional Java code. Having source code is not necessary, though, so you can use class libraries from third parties. The classes must meet two requirements: methods published to SQL and PL/SQL must be declared static, because PL/SQL has no mechanism for instantiating nonstatic Java classes; and, the classes must not issue any GUI calls (for example, to AWT) at runtime.
If you write your own JSP and it needs to connect to the database for access to tables or stored procedures, use standard JDBC and/or SQLJ calls in your code. Many JDBC and SQLJ reference materials are available to provide assistance in calling SQL or PL/SQL from Java, but be sure to review the product-specific documentation that ships with your tool.
2. Once you have the Java class in hand, either in source or _.class_ file format, load it into the database. The database's `loadjava` command-line utility is a convenient way to accomplish the load. Refer to the _Oracle Java Developer's Guide_ for further assistance with `loadjava`.
3. Create a call spec for the Java method, specifying the AS LANGUAGE JAVA clause of the CREATE statement (described in "Publishing Java to PL/SQL"). You may create a function or procedure cover as appropriate.
4. Grant EXECUTE privileges on the new JSP using GRANT EXECUTE; PL/SQL routines can now call the JSP as if it were another PL/SQL module.
## Example
Let's write a simple "Hello, World" JSP that will accept an argument:
package oreilly.plsquick.demos;
public class Hello {
public static String sayIt (String toWhom) {
return "Hello, " + toWhom + "!";
}
}
Saved in a file called _Hello.java_ , the source code can be loaded directly into the database. Doing so will compile the code automatically. Here is a simple form of the _loadjava_ command:
loadjava -user scott/tiger oreilly/plsquick/demos/Hello.java
The _Hello.java_ file follows the Java file placement convention for packages, and thus exists in a subdirectory named _oreilly/plsquick/demos_.
We can fire up our favorite SQL interpreter, connect as SCOTT/TIGER, and create the call spec for the `Hello.sayIt()` method:
CREATE FUNCTION hello_there (to_whom IN VARCHAR2)
RETURN VARCHAR2
AS LANGUAGE JAVA
NAME 'oreilly.plsquick.demos.Hello.sayIt
(java.lang.String) return java.lang.String';
Now we can call our function very easily:
BEGIN
DBMS_OUTPUT.PUT_LINE(hello_there('world'));
END;
And we get the following as the expected output:
Hello, world!
## Publishing Java to PL/SQL
To write a call spec, use the AS LANGUAGE JAVA clause in a CREATE statement. The syntax for this clause is:
{ IS | AS } LANGUAGE JAVA
NAME ' _method_fullname_ [ ( _type_fullname_ ,... ]
[ RETURN _type_fullname_ ]'
_method_fullname_ is the package-qualified name of the Java class and method. It is case-sensitive and uses dots to separate parts of the package's full name. _type_fullname_ is the package-qualified name of the Java datatype. Notice that a literal string, not a SQL identifier, follows the NAME keyword.
Type mapping follows most JDBC rules regarding the legal mapping of SQL types to Java types. JDBC extensions exist for Oracle-specific datatypes. Most datatype mappings are relatively straightforward, but passing database objects of a user-defined type is harder than one would think. Oracle provides a tool named JPublisher that generates the Java required to encapsulate a database object and its corresponding REF. Refer to Oracle's JPublisher documentation for guidelines on usage.
The AS LANGUAGE JAVA clause is the same regardless of whether you are using Java as a standalone JSP, the implementation of a packaged program, or the body of an object type method. For example, here is the complete syntax for creating JSPs as PL/SQL-callable functions or procedures:
CREATE [OR REPLACE]
{ PROCEDURE _procedure_name_ [( _param_ [, _param_ ]...)]
| FUNCTION _function_name_ [( _param_ [, _param_ ]...)]
RETURN _sql_type_
}
[AUTHID {DEFINER | CURRENT_USER}]
[PARALLEL_ENABLE]
[DETERMINISTIC]
{ IS | AS } LANGUAGE JAVA
NAME ' _method_fullname_ [ ( _type_fullname_ ,... ]
[ RETURN _type_fullname_ ]'
When using Java as the implementation of a packaged procedure or function, the database allows you to place the Java call spec either in the package specification (where the call spec substitutes for the subprogram specification) or in the package body (where the call spec substitutes for the subprogram body). Similarly, when using JSPs in object type methods, the Java call spec can substitute for either the object type method specification or its body.
Note that Java functions typically map to PL/SQL functions, but Java functions declared void map to PL/SQL procedures. Also, you will quickly learn that mistakes in mapping PL/SQL parameters to Java parameters become evident only at runtime.
## Data Dictionary
To learn what Java library units are available in your schema, look in the USER_OBJECTS data dictionary view where the _object_type_ is like "JAVA%". If you see a Java class with INVALID status, it has not yet been resolved successfully. Note that the names of the Java source library units need not match the names of the classes they produce.
There is one exception to this rule: associative arrays are _not_ null when declared, and there is no way to make them null. However, when declared, associative arrays have no elements, a state unsurprisingly known as "empty."
Well, most statements, anyway. CREATE OR REPLACE statements and anonymous blocks do require semicolons.
Oracle's reference documentation lists four different dynamic SQL scenarios, and labels them methods 1 through 4. Methods 1 through 3 are now commonly implemented with native dynamic SQL.
# Index
### Symbols
* != (inequality) operator, Delimiters
* " (double quotes), Identifiers, Delimiters
* # (pound sign), Identifiers
* $ (dollar sign)
* in identifiers, Identifiers
* metacharacter, Metacharacters
* % (attribute indicator), Delimiters, Explicit cursor attributes
* ' (single quote), Boolean, Numeric, and String Literals, Delimiters
* () (parentheses)
* as delimiters, Boolean, Numeric, and String Literals, Delimiters
* metacharacter, Metacharacters
* * (asterisk)
* metacharacter, Metacharacters
* multiplication operator, Delimiters
* ** (exponentiation) operator, Delimiters
* \+ (plus sign)
* addittion operator, Delimiters
* metacharacter, Metacharacters
* , (comma), Delimiters
* \- (subtraction) operator, Delimiters
* \-- (single-line comment indicator), Delimiters
* . (period)
* in collection methods, Collection Methods
* component indicator, Delimiters
* in Java names, Publishing Java to PL/SQL
* metacharacter, Metacharacters
* referencing record fields, Referencing a Record's Fields
* in TREAT expressions, Upcasting and Downcasting
* in triggers, Creating Triggers
* .. (range) operator, Delimiters
* / (division) operator, Delimiters
* /* and */, for multiline comments, Delimiters
* : (host variable indicator), Delimiters
* := (assignment) operator, Delimiters, Default values, Assigning Records
* ; (semicolon), statements and, Delimiters, Native Dynamic SQL
* < (less-than) operator, Delimiters
* <= (less-than-or-equal-to) operator, Delimiters
* <> (angle brackets)
* as delimiters, Boolean, Numeric, and String Literals, Delimiters
* inequality operator, Delimiters
* nested table functions and, Nested Table Functions
* = (equals sign)
* comparing nested tables, Nested Table Functions
* equality operator, Delimiters
* => (association) operator, Delimiters
* > (greater-than) operator, Delimiters
* >= (greater-than-or-equal-to) operator, Delimiters
* ? (question mark), metacharacter, Metacharacters
* @ (remote database indicator), Delimiters
* ..] (collation element), metacharacter, [Metacharacters
* ::] (character class), metacharacter, [Metacharacters
* ==] (equivalence class), metacharacter, [Metacharacters
* [] (square brackets)
* as delimiters, Boolean, Numeric, and String Literals
* metacharacter, Metacharacters
* \ (backslash), metacharacter, Metacharacters, Metacharacters
* ^ (caret), metacharacter, Metacharacters
* ^= (inequality) operator, Delimiters
* _ (underscore), Identifiers
* {} (curly braces)
* as delimiters, Boolean, Numeric, and String Literals
* metacharacter, Metacharacters
* || (concatenation) operator, Delimiters
* ~= (inequality) operator, Delimiters
* ~| (vertical bar), metacharacter, Metacharacters
### A
* ABS built-in function, Numeric functions
* ACCESSIBLE BY keyword, Compiling Stored PL/SQL Programs
* ACCESS_ INTO_NULL exception, Manipulating Objects in PL/SQL and SQL
* ACOS built-in function, Numeric functions
* actual parameters, Parameter-passing notations
* addition (+) operator, Delimiters
* ADD_MONTHS built-in function, Date and time functions
* AFTER triggers, Creating Triggers, Compound DML Triggers
* aggregate assignment technique , Assigning Records, Creating Triggers
* AGGREGATE USING keyword, Compiling Stored PL/SQL Programs
* ALTER SESSION statement, Compiler Warnings
* ALTER trigger event, Creating Triggers, DDL Events
* ALTER TYPE statement, Changing Object Types
* ANALYZE trigger event, Creating Triggers, DDL Events
* anchored declarations, Anchored Declarations
* AND operation and Boolean datatype, BOOLEAN datatype
* angle brackets <>
* as delimiters, Boolean, Numeric, and String Literals, Delimiters
* inequality operator, Delimiters
* nested table functions and, Nested Table Functions
* anonymous blocks, Block Structure
* arguments, passing in parameter lists, Mode
* AS LANGUAGE JAVA clause, Java Language Integration, Publishing Java to PL/SQL
* ASCII built-in function, String functions
* ASCIISTR built-in function, Conversion functions, String functions
* ASIN built-in function, Numeric functions
* assigning records, Assigning Records
* assignment (:=) operator, Delimiters, Default values, Assigning Records
* ASSOCIATE STATISTICS trigger event, Creating Triggers, DDL Events
* association (=>) operator, Delimiters
* associative arrays
* about, Collections in PL/SQL
* adding/removing elements, Adding and Removing Elements
* collection methods used with, Collection Methods
* comparing collection types, Collections in PL/SQL
* declaring, Declaring a Collection
* INDEX BY BINARY_INTEGER, Collections in PL/SQL
* INDEX BY VARCHAR2, Collections in PL/SQL
* initializing, Initializing a Collection
* asterisk (*)
* metacharacter, Metacharacters
* multiplication operator, Delimiters
* ATAN built-in function, Numeric functions
* ATAN2 built-in function, Numeric functions
* atomically null collections
* COLLECTION_IS_NULL exception, Collection Methods
* nested tables, Collections in PL/SQL
* VARRYs, Collections in PL/SQL
* atomically null objects, Manipulating Objects in PL/SQL and SQL
* attribute indicator (%), Delimiters, Explicit cursor attributes
* attributes
* explicit cursors, Explicit cursor attributes
* implicit cursors, Implicit Cursors
* object type specifications, Object Types
* AUDIT trigger event, Creating Triggers, DDL Events
* AUTHID CURRENT_USER keyword, Privileges and Stored PL/SQL
* AUTHID keyword, Compiling Stored PL/SQL Programs
* AUTONOMOUS_TRANSACTION pragma, Pragmas, Autonomous Transactions
### B
* backslash (\\), metacharacter, Metacharacters, Metacharacters
* BEFORE triggers, Creating Triggers, Compound DML Triggers
* BFILE datatype, LOB Datatypes
* BINARY_DOUBLE datatype, Numeric Literals, Numeric datatypes
* BINARY_FLOAT datatype, Numeric Literals, Numeric datatypes
* BINARY_INTEGER datatype, Numeric datatypes
* bind variables, SQL Injection and Bind Variables, Records and DML
* BIN_TO_NUM built-in function, Numeric functions
* BITAND built-in function, Numeric functions
* BLOB datatype, LOB Datatypes
* block structure (PL/SQL programs), Block Structure-Block Structure
* BOOLEAN datatype, BOOLEAN datatype
* Boolean literals, Boolean, Numeric, and String Literals
* bracketing characters, Boolean, Numeric, and String Literals
* built-in functions, Initializing a Collection, Built-in Functions-Match modifiers
* bulk binds and collections, Bulk Binds-Bulk Binds
* BULK COLLECT INTO clause , Bulk Binds
* BULK COLLECT statement, Cursor FOR Loop
* %BULK_EXCEPTIONS attribute, Bulk Binds
### C
* cache, function result, Function Result Cache-Function Result Cache
* CALL syntax (ANSI SQL), Procedures
* CARDINALITY function, Nested Table Functions
* caret (^), metacharacter, Metacharacters
* CASCADE keyword, Changing Object Types
* CASE expression, CASE expression
* case sensitivity for identifiers, Identifiers
* CASE statement, CASE statement
* CAST built-in function, Conversion functions, Date and time functions
* CAST function, Nested Table Functions
* CEIL built-in function, Numeric functions
* CHAR datatype, Character datatypes
* character datatypes, Character datatypes-Unicode character datatypes
* character set (PL/SQL), PL/SQL Character Set, Unicode character datatypes
* CHARTOROWID built-in function, Conversion functions
* CHR built-in function, String functions
* CLOB datatype, LOB Datatypes
* closing
* cursor variables, Cursor Variables
* explicit cursors, Closing explicit cursors
* COALESCE function, NULLs in PL/SQL
* COLLECT function, Nested Table Functions
* collections (PL/SQL)
* adding/removing elements, Adding and Removing Elements
* bulk binds and, Bulk Binds-Bulk Binds
* declaring, Declaring a Collection
* disallowed datatypes, Declaring a Collection
* initializing, Initializing a Collection-Initializing a Collection
* methods for, Collection Methods
* nested collections, Nested Collections
* nested table functions, Nested Table Functions-Nested Table Functions
* privileges and, Collections and Privileges
* types of, Collections in PL/SQL-Collections in PL/SQL
* COLLECTION_IS_NULL exception, Collection Methods
* column/function name precedence, Column Name Versus Function Name Precedence
* comma (,), Delimiters
* COMMENT trigger event, Creating Triggers, DDL Events
* comments (PL/SQL)
* about, Comments
* multiline indicator, Delimiters
* single-line indicator, Delimiters
* COMMIT FORCE statement, COMMIT
* COMMIT statement, COMMIT, Calling PL/SQL Functions in SQL
* comparison methods for object types, Comparison methods-Comparison methods
* compilation
* about, Compilation
* compiler warnings, Compiler Warnings-Optimizing Compiler
* conditional, Conditional Compilation-Conditional Compilation
* native, Performing Native Compilation of PL/SQL
* optimizing compiler, Optimizing Compiler-Optimizing Compiler
* pragmas and, Pragmas
* settings, Compiling Stored PL/SQL Programs, Conditional Compilation
* COMPOSE built-in function, String functions
* composite datatype, Variables and Program Data
* compound triggers, Compound DML Triggers
* CONCAT built-in function, String functions
* concatenation (||) operator, Delimiters
* conditional compilation, Conditional Compilation-Conditional Compilation
* conditional control statements, Conditional Control Statements-CASE expression
* CONSTANT keyword, Constants
* constants, numeric literals, Numeric Literals
* constrained declarations, Constrained declarations
* constrained subtypes, Programmer-Defined Subtypes
* CONSTRUCTOR FUNCTION syntax, Constructor methods
* constructor methods, Constructor methods
* constructors, Initializing a Collection
* CONTINUE statement, CONTINUE Statement-Loop Labels
* control statements
* BOOLEAN datatype and, BOOLEAN datatype
* conditional, Conditional Control Statements-CASE expression
* sequential, Sequential Control Statements-NULL
* conversion functions, Conversion functions
* CONVERT built-in function, Conversion functions, String functions
* Coordinated Universal Time (UTC), Datetime datatypes
* COS built-in function, Numeric functions
* COSH built-in function, Numeric functions
* COUNT function, Collection Methods
* CREATE keyword , Declaring a Collection
* CREATE OR REPLACE TYPE BODY statement, Changing Object Types
* CREATE trigger event, Creating Triggers, DDL Events
* curly braces {}
* as delimiters, Boolean, Numeric, and String Literals
* metacharacter, Metacharacters
* CURRENT_DATE built-in function, Date and time functions
* CURRENT_TIMESTAMP built-in function, Date and time functions
* cursor expressions, Cursor Expressions
* cursor FOR loop, Cursor FOR Loop, Opening explicit cursors
* cursor variables, Cursor Variables
* cursors (PL/SQL)
* about, Cursors in PL/SQL
* dynamic, Cursors in PL/SQL
* explicit, Explicit Cursors
* implicit, Implicit Cursors-WHERE CURRENT OF clause
* static, Cursors in PL/SQL
### D
* d BINARY_DOUBLE literal suffix, Numeric Literals
* dangling REFs and IS DANGLING predicate, REF operator
* Data Control Language (DCL), Database Interaction
* Data Definition Language (DDL)
* dynamic cursors and, Cursors in PL/SQL
* events and triggers, Triggers, Creating Triggers, DDL Events
* executing statements, Database Interaction
* data dictionary views
* USER_OBJECTS, Data Dictionary
* V$RESERVED_WORDS, Identifiers
* V$RESULT_CACHE, Function Result Cache
* V$TIMEZONE_NAMES, Datetime datatypes
* database character set, Unicode character datatypes
* database events and triggers, Triggers, Creating Triggers, Database Events
* database interaction, Database Interaction-Autonomous Transactions
* datatypes
* datetime interval, Datetime Interval Literals
* disallowed for collections, Declaring a Collection
* implicit conversions, Implicit Datatype Conversions
* LOB, LOB Datatypes
* mapping, Publishing Java to PL/SQL
* numeric literals, Numeric Literals
* of parameters, Datatype
* program data and, Variables and Program Data-LOB Datatypes
* date and time functions, Date and time functions-Date and time functions
* DATE datatype, Datetime datatypes
* datetime datatypes, Datetime datatypes-Datetime datatypes
* datetime interval literals, Datetime Interval Literals
* DBMS_LOB package, LOB Datatypes
* DBMS_OUTPUT package, CASE expression, Function Result Cache, Referencing Package Elements
* DBMS_PREPROCESSOR package, Conditional Compilation
* DBMS_RESULT_CACHE package, Function Result Cache
* DBMS_SQL package, DBMS_SQL
* TO_CURSOR_NUMBER function, DBMS_SQL
* TO_REFCURSOR function, DBMS_SQL
* DBMS_TRANSACTION.COMMIT_FORCE procedure, COMMIT
* DBMS_UTILITY package, SQLCODE and DBMS_UTILITY formatters for exception handlers-SQLCODE and DBMS_UTILITY formatters for exception handlers
* DBMS_WARNING package, Compiler Warnings
* DBTIMEZONE built-in function, Date and time functions
* DB_ROLE_CHANGE trigger event, Creating Triggers
* DCL (Data Control Language), Database Interaction
* DDL (Data Definition Language)
* dynamic cursors and, Cursors in PL/SQL
* events and triggers, Triggers, Creating Triggers, DDL Events
* executing statements, Database Interaction
* DDL trigger event, Creating Triggers
* DEC datatype, Numeric datatypes
* DECIMAL datatype, Numeric datatypes
* decimal numeric datatypes, Numeric datatypes
* declarations
* anchored, Anchored Declarations
* collections, Declaring a Collection
* constraining datatypes, Constrained declarations
* exceptions, Declaring Exceptions
* explicit cursors, Declaring explicit cursors
* forward, Forward Declarations
* local programs, Local Programs
* records, Declaring Records
* variables, Declaring Variables-Anchored Declarations
* DECODE function, Built-in Functions
* DECOMPOSE built-in function, String functions
* DEFAULT keyword, Default values
* default values
* functions setting, Functions
* for parameters, Default values
* for variables, Default values
* definer rights, Privileges and Stored PL/SQL
* DELETE function, Adding and Removing Elements
* DELETE procedure, Collection Methods
* DELETE trigger event, Creating Triggers
* DELETING trigger predicate, Trigger Predicates
* delimiters (PL/SQL), Delimiters
* DEREF operator, DEREF operator
* DETERMINISTIC keyword, Compiling Stored PL/SQL Programs
* DETERMINISTIC pragma, Pragmas
* DISABLE keyword, Creating Triggers
* DISASSOCIATE STATISTICS trigger event, Creating Triggers, DDL Events
* DISTINCT keyword , Nested Table Functions
* division (/) operator, Delimiters
* DML events and triggers, Triggers, Creating Triggers, DML Events-DDL Events
* DML, records and, Records and DML
* dollar sign ($)
* in identifiers, Identifiers
* metacharacter, Metacharacters
* dot notation (see entries under period (.))
* DOUBLE PRECISION datatype, Numeric datatypes
* double quotes ("), Identifiers
* downcasting supertypes, Upcasting and Downcasting-DEREF operator
* DROP trigger event, Creating Triggers, DDL Events
* dynamic cursors, Cursors in PL/SQL
* dynamic method dispatch (dynamic polymorphism), Methods in Subtypes
* dynamic SQL, Native Dynamic SQL-Native Dynamic SQL
### E
* equals sign (=)
* comparing nested tables, Nested Table Functions
* equality operator, Delimiters
* exception handling
* about, Exception Handling
* built-in list of exceptions, Declaring Exceptions
* bulk binds and collections, Bulk Binds
* declaring exceptions, Declaring Exceptions
* propagating exceptions, Propagation
* raising exceptions, Raising Exceptions
* scope of, Scope
* EXCEPTION_CODE field, Bulk Binds
* EXCEPTION_INDEX field, Bulk Binds
* EXCEPTION_INIT pragma, Pragmas, Declaring Exceptions
* EXEC command (SQL *Plus), Procedures
* EXECUTE IMMEDIATE statement, Native Dynamic SQL
* EXECUTE privileges
* for collections, Collections and Privileges
* for JSPs, Java Language Integration
* for packages, Package Structure, Package Initialization
* for programs, Privileges and Stored PL/SQL
* EXISTS function, Collection Methods
* EXIT statement, Simple Loop, EXIT Statement
* EXIT WHEN statement, Simple Loop
* EXP built-in function, Numeric functions
* explicit cursors
* about, Explicit Cursors
* attributes of, Explicit cursor attributes
* closing, Closing explicit cursors
* declaring, Declaring explicit cursors
* fetching from, Fetching from explicit cursors
* opening, Opening explicit cursors
* exponentiation (**) operator, Delimiters
* EXTEND function, Collections in PL/SQL, Adding and Removing Elements
* EXTEND procedure, Collection Methods
* EXTRACT built-in function, Date and time functions
### F
* f BINARY_FLOAT literal suffix, Numeric Literals
* FALSE value, BOOLEAN datatype
* FETCH FIRST clause, Bulk Binds
* FETCH INTO statement
* assigning records, Assigning Records
* BULK COLLECT INTO clause and, Bulk Binds
* manipulating objects, Manipulating Objects in PL/SQL and SQL
* FETCH statement, SET TRANSACTION, Fetching from explicit cursors, Cursor Variables
* fields of records
* about, Records in PL/SQL
* DML and, Records and DML
* referencing, Referencing a Record's Fields
* FINAL keyword, Object Types, Type Inheritance, Methods in Subtypes, Changing Object Types
* FIRST function, Collection Methods
* FLOAT datatype, Numeric datatypes
* FLOOR built-in function, Numeric functions
* FOLLOWS keyword, Creating Triggers
* FOR EACH ROW statement, Creating Triggers
* FOR loops, Numeric FOR Loop-Cursor FOR Loop, Opening explicit cursors
* FORALL statement, Cursor FOR Loop, Bulk Binds-Bulk Binds
* FORCE option, Changing Object Types
* formal parameters, Parameter-passing notations
* forward declarations, Forward Declarations
* %FOUND attribute, Explicit cursor attributes, Bulk Binds
* FROM_TZ built-in function, Conversion functions, Date and time functions
* function result cache, Function Result Cache-Function Result Cache
* functions
* about, Stored Procedures and Functions, Functions
* built-in, Initializing a Collection, Built-in Functions-Match modifiers
* column/function name precedence, Column Name Versus Function Name Precedence
* conversion functions, Conversion functions
* date and time funcitons, Date and time functions-Date and time functions
* improving performance of, Improving Performance of Calling PL/SQL Functions from SQL
* nested table functions, Nested Table Functions-Nested Table Functions
* numeric functions, Numeric functions-Numeric functions
* regular expression, Built-in Regular Expression Functions-REGEXP_REPLACE
* RETURN clause in, Functions
* stored functions, Calling PL/SQL Functions in SQL-Column Name Versus Function Name Precedence
* string functions, String functions
* table functions, Table Functions
### G
* Gennick, Jonathan, Match modifiers
* GOTO statement, GOTO
* GRANT EXECUTE statement, Java Language Integration
* GRANT INHERIT PRIVILEGE statement, Privileges and Stored PL/SQL
* GRANT trigger event, Creating Triggers, DDL Events
* greater-than (>) operator, Delimiters
* greater-than-or-equal-to (>=) operator, Delimiters
* GREATEST built-in function, String functions, Numeric functions
* Greenwich Mean Time, Datetime datatypes
### H
* HEXTODRAW built-in function, Conversion functions
* host variable indicator (:), Delimiters
### I
* identifiers (PL/SQL), Identifiers
* IEEE floating-point storage types, Numeric datatypes, Numeric datatypes, Program Overloading
* IF-THEN-ELSE statements, Conditional Control Statements
* implicit cursors, Implicit Cursors-WHERE CURRENT OF clause
* implicit datatype conversions, Implicit Datatype Conversions
* IN function, Nested Table Functions
* IN OUT parameters, Mode
* IN parameters, Mode, Calling a Function
* INDEX BY BINARY_INTEGER associative array, Collections in PL/SQL
* INDEX BY VARCHAR2 associative array, Collections in PL/SQL
* INDICES OF keyword, Bulk Binds
* inequality operators, Delimiters
* INHERIT PRIVILEGES feature, Privileges and Stored PL/SQL
* INITCAP built-in function, String functions
* initializing
* objects, Manipulating Objects in PL/SQL and SQL
* packages, Package Initialization
* initializing collections, Initializing a Collection-Initializing a Collection
* INLINE pragma, Pragmas, Optimizing Compiler
* INSERT trigger event, Creating Triggers
* INSERTING trigger predicate, Trigger Predicates
* INSTANTIABLE keyword, Object Types, Type Inheritance, Changing Object Types
* INSTR built-in function, String functions
* INT datatype, Numeric datatypes
* INTEGER datatype, Numeric Literals, Numeric datatypes
* INTERSECT set operation, Nested Table Functions
* INTERVAL DAY TO MINUTE datatype, Datetime Interval Literals
* INTERVAL DAY TO SECOND datatype, Datetime Interval Literals, Datetime datatypes
* INTERVAL HOUR datatype, Datetime Interval Literals
* INTERVAL HOUR TO SECOND datatype, Datetime Interval Literals
* INTERVAL keyword, Datetime Interval Literals
* INTERVAL MONTH datatype, Datetime Interval Literals
* INTERVAL YEAR TO MONTH datatype, Datetime Interval Literals, Datetime datatypes
* intra-unit inline optimization, Optimizing Compiler
* INVALID NUMBER exception, Implicit Datatype Conversions
* INVALIDATE keyword, Changing Object Types
* INVALID_CURSOR exception, Closing explicit cursors
* invoker rights model, Privileges and Stored PL/SQL
* IS A SET function, Nested Table Functions
* IS EMPTY function, Nested Table Functions
* IS NOT A SET function, Nested Table Functions
* IS NOT EMPTY function, Nested Table Functions
* IS NULL/IS NOT NULL syntax, NULLs in PL/SQL
* ISOLATION LEVEL SERIALIZABLE transaction control function, SET TRANSACTION, Calling PL/SQL Functions in SQL
* %ISOPEN attribute, Explicit cursor attributes
### J
* Java language
* call specs for , Object Types, Java Language Integration, Java Language Integration, Java Language Integration, Publishing Java to PL/SQL
* data dictionary views, Data Dictionary
* object types and, Object Types, Data Dictionary
* publishing Java to PL/SQL, Publishing Java to PL/SQL
* writing simple JSP, Example
* Java stored procedures (JSPs), Java Language Integration-Example
* JPublisher tool, Publishing Java to PL/SQL
* JSPs (Java stored procedures), Java Language Integration-Example
### K
* keywords
* datetime interval datatypes, Datetime Interval Literals
* DISABLE, Creating Triggers
* pragmas, Pragmas
* RESULT_CACHE, Function Result Cache
### L
* labels for loops, Loop Labels
* language fundamentals (PL/SQL), PL/SQL Language Fundamentals-Block Structure
* large object (LOB) datatypes, LOB Datatypes
* LAST function, Collection Methods
* LAST_DAY built-in function, Date and time functions
* LEAST built-in function, String functions, Numeric functions
* LENGTH built-in function, String functions
* less-than (<) operator, Delimiters
* less-than-or-equal-to (<=) operator, Delimiters
* LIMIT function, Collection Methods
* Linsley, Peter, Match modifiers
* lists, delimiters for, Delimiters
* literals (PL/SQL), Boolean, Numeric, and String Literals-Datetime Interval Literals
* LN built-in function, Numeric functions
* loadjava command-line utility, Java Language Integration
* LOB (large object) datatypes, LOB Datatypes
* local programs, Local Programs
* LOCALTIMESTAMP built-in function, Date and time functions
* LOCK TABLE statement, LOCK TABLE
* locking rows in result sets, SELECT FOR UPDATE clause
* LOG built-in function, Numeric functions
* LOGOFF trigger event, Creating Triggers, Database Events
* LOGON trigger event, Creating Triggers, Database Events
* LONG datatype, Character datatypes, Creating Triggers
* LONG RAW datatype, Character datatypes, Creating Triggers
* loops
* about, Loops-"Repeat Until" Loop Emulation
* CONTINUE statement, CONTINUE Statement-Loop Labels
* EXIT statement, EXIT Statement
* labels for, Loop Labels
* LOWER built-in function, String functions
* LPAD built-in function, String functions
* LTRIM built-in function, String functions
### M
* MAP method, Object Types, Comparison methods
* mapping datatypes, Publishing Java to PL/SQL
* match modifiers for regular expressions, Match modifiers
* MEMBER keyword, Member methods
* MEMBER OF function, Nested Table Functions
* metacharacters supported by Oracle, Metacharacters
* methods
* collection, Collection Methods
* comparison, Comparison methods
* constructor, Constructor methods
* in subtypes, Methods in Subtypes-Methods in Subtypes
* types of, Methods-Comparison methods
* MINUS set operation, Nested Table Functions
* MOD built-in function, Numeric functions
* modes of parameters, Mode
* MONTHS_BETWEEN built-in function, Date and time functions
* multiplication (*) operator, Delimiters
* MULTISET built-in function, Conversion functions
* MULTISET EXCEPT function, Nested Table Functions
* MULTISET function, Nested Table Functions
* MULTISET INTERSECT function, Nested Table Functions
* MULTISET UNION function, Nested Table Functions
* mutual recursion, Forward Declarations
### N
* NAME keyword, Publishing Java to PL/SQL
* named blocks, Block Structure
* named notation, Delimiters, Opening explicit cursors, Parameter-passing notations
* named program units, Stored Procedures and Functions
* NANY built-in function, Numeric functions
* national character set (NLS), Unicode character datatypes
* native dynamic SQL (NDS), Native Dynamic SQL-DBMS_SQL
* NATURAL datatype, Numeric datatypes
* NATURALN datatype, Numeric datatypes
* NCHAR datatype, Unicode character datatypes
* NCHAR delimiter, Boolean, Numeric, and String Literals, Delimiters
* NCHR built-in function, String functions
* NCLOB datatype, LOB Datatypes
* NDS (native dynamic SQL), Native Dynamic SQL-DBMS_SQL
* nested collections, Nested Collections
* nested cursors, Cursor Expressions
* nested tables
* about, Collections in PL/SQL
* adding/removing elements, Adding and Removing Elements
* collection methods and, Collection Methods
* comparing collection types, Collections in PL/SQL
* declaring, Declaring a Collection
* functions for, Nested Table Functions-Nested Table Functions
* initializing, Initializing a Collection
* NEW_TIME built-in function, Date and time functions
* NEXT function, Collection Methods
* NEXT_DAY built-in function, Date and time functions
* NLS (national character set), Unicode character datatypes
* NLSSORT built-in function, String functions
* NLS_INITCAP built-in function, String functions
* NLS_LENGTH_SEMANTICS setting, Compiling Stored PL/SQL Programs
* NLS_LOWER built-in function, String functions
* NLS_UPPER built-in function, String functions
* NOAUDIT trigger event, Creating Triggers, DDL Events
* NOCOPY option, Mode
* NOT A MEMBER OF function, Nested Table Functions
* NOT FINAL keyword, Type Inheritance, Methods in Subtypes, Changing Object Types
* NOT IN function, Nested Table Functions
* NOT INSTANTIABLE keyword, Type Inheritance, Methods in Subtypes, Changing Object Types
* NOT NULL constraint, Default values, Declaring a Collection
* NOT operation and Boolean datatype, BOOLEAN datatype
* NOT SUBMULTISET OF function, Nested Table Functions
* NOT SUBSTITUTABLE keyword, Changing Object Types
* %NOTFOUND attribute, Explicit cursor attributes, Bulk Binds
* NOWAIT keyword
* LOCK TABLE statement and, LOCK TABLE
* SELECT FOR UPDATE clause and, SELECT FOR UPDATE clause
* UPDATE statement, Autonomous Transactions
* NO_DATA_FOUND exception, Implicit Cursors
* nq' programmer-defined delimiter, Boolean, Numeric, and String Literals, Delimiters
* NULL statement, NULL
* NULLs in PL/SQL
* about, NULLs in PL/SQL
* atomically null collections, Collections in PL/SQL, Collection Methods
* atomically null objects, Manipulating Objects in PL/SQL and SQL
* BOOLEAN datatype and, BOOLEAN datatype
* NUMBER datatype, Numeric Literals, Numeric datatypes
* NUMERIC datatype, Numeric datatypes
* numeric datatypes, Numeric datatypes-Numeric datatypes
* numeric FOR loop, Numeric FOR Loop
* numeric functions, Numeric functions-Numeric functions
* numeric literals, Numeric Literals
* NUMTODSINTERVAL built-in function, Conversion functions, Date and time functions
* NUMTOYMINTERVAL built-in function, Conversion functions, Date and time functions
* NVARCHAR delimiter, Boolean, Numeric, and String Literals
* NVARCHAR2 datatype, Unicode character datatypes
* NVL function, NULLs in PL/SQL
### O
* object identifiers (OIDs)
* about, Object-Oriented Features
* encapsulated by REF operator, REF operator
* referencing persistent objects using, DEREF operator
* object types
* about, Stored Procedures and Functions, Object-Oriented Features
* changing, Changing Object Types-Changing Object Types
* comparison methods for, Comparison methods-Comparison methods
* initializing objects, Manipulating Objects in PL/SQL and SQL
* manipulating, Manipulating Objects in PL/SQL and SQL
* syntax for creating body, Object Types
* syntax for creating specification, Object Types-Object Types
* object-oriented features
* about, Object-Oriented Features
* changing object types, Changing Object Types-Changing Object Types
* manipulating objects, Manipulating Objects in PL/SQL and SQL
* methods and, Methods-Comparison methods
* methods in subtypes, Methods in Subtypes-Methods in Subtypes
* object types and, Object Types
* type inheritance, Type Inheritance
* upcasting and downcasting, Upcasting and Downcasting-DEREF operator
* OFFSET clause, Bulk Binds
* OIDs (object identifiers)
* about, Object-Oriented Features
* encapsulated by REF operator, REF operator
* referencing persistent objects using, DEREF operator
* OPEN FOR statement , Native Dynamic SQL
* operators (see delimiters (PL/SQL))
* optimizing compiler, Optimizing Compiler-Optimizing Compiler
* OR operation and Boolean datatype, BOOLEAN datatype
* OR REPLACE keyword
* creating stored programs, Compiling Stored PL/SQL Programs
* rebuilding existing collection types, Declaring a Collection
* rebuilding existing packages, Package Structure
* ORA-00054 exception, LOCK TABLE, SELECT FOR UPDATE clause
* ORA-04098 exception, Creating Triggers
* ORA-06511 exception, Opening explicit cursors
* ORA-1456 error, SET TRANSACTION
* ORA-nnnnn (list of built-in exceptions), Declaring Exceptions
* Oracle PL/SQL Programming, Mode
* Oracle Regular Expressions Pocket Reference, Match modifiers
* ORDER method, Object Types, Comparison methods
* OUT parameters, Mode
* overloading
* member methods, Object Types
* programmer-defined constructors, Constructor methods
* programs, Program Overloading
* OVERRIDING keyword, Methods in Subtypes
### P
* package body
* about, Package Structure
* placing Java call specs in, Publishing Java to PL/SQL
* syntax for, Package Structure
* package data, Package Data
* package specification
* about, Package Structure
* placing Java call specs in, Publishing Java to PL/SQL
* syntax for, Package Structure
* packaged functions, calling, SERIALLY_REUSABLE Pragma
* packages
* about, Stored Procedures and Functions, Packages
* initializing, Package Initialization
* referencing elements of, Referencing Package Elements
* structure of, Package Structure-Package Structure
* PARALLEL_ENABLE keyword, Compiling Stored PL/SQL Programs
* PARALLEL_ENABLE pragma, Pragmas
* parameters
* about, Parameters
* datatypes of, Datatype
* default values for, Default values
* modes of, Mode
* passing arguments, Mode
* program overloading, Program Overloading
* parentheses ()
* as delimiters, Boolean, Numeric, and String Literals, Delimiters
* metacharacter, Metacharacters
* passing parameters by reference, Mode
* period (.)
* in collection methods, Collection Methods
* component indicator, Delimiters
* in Java names, Publishing Java to PL/SQL
* metacharacter, Metacharacters
* in TREAT expressions, Upcasting and Downcasting
* in triggers, Creating Triggers
* persistent objects, Object-Oriented Features, DEREF operator
* PGA (Process Global Area), Package Initialization
* PIPELINED keyword, Compiling Stored PL/SQL Programs
* PLSQL_CCFLAGS setting, Compiling Stored PL/SQL Programs
* PLSQL_CODE_TYPE setting, Compiling Stored PL/SQL Programs, Performing Native Compilation of PL/SQL
* PLSQL_DEBUG setting, Compiling Stored PL/SQL Programs
* PLSQL_OPTIMIZE_LEVEL setting, Compiling Stored PL/SQL Programs, Optimizing Compiler
* PLSQL_WARNINGS setting, Compiling Stored PL/SQL Programs, Compiler Warnings
* PLS_INTEGER datatype, Numeric datatypes, Numeric datatypes
* plus sign (+)
* addition operator, Delimiters
* metacharacter, Metacharacters
* positional notation, Opening explicit cursors, Parameter-passing notations
* POSITIVEN datatype, Numeric datatypes
* POSITVE datatype, Numeric datatypes
* pound sign (#), Identifiers
* POWER built-in function, Numeric functions
* PRAGMA keyword, Pragmas
* precedence, column/function name, Column Name Versus Function Name Precedence
* predicates, trigger, Trigger Predicates
* PRIOR function, Collection Methods
* privileges and stored PL/SQL, Privileges and Stored PL/SQL-Privileges and Stored PL/SQL
* procedures (PL/SQL), Stored Procedures and Functions, Procedures
* Process Global Area (PGA), Package Initialization
* program data, types of, Variables and Program Data-BOOLEAN datatype
* program overloading, Program Overloading
* propagating exceptions, Propagation
* publishing Java to PL/SQL, Publishing Java to PL/SQL
* PUT_LINE procedure , Program Overloading
### Q
* q' programmer-defined delimiter, Boolean, Numeric, and String Literals, Delimiters
* queries, multirow, Native Dynamic SQL
* question mark (?), metacharacter, Metacharacters
### R
* RAISE statement, Raising Exceptions
* RAISE_APPLICATION_ ERROR function, Raising Exceptions
* raising exceptions, Raising Exceptions
* range (..) operator, Delimiters
* RAW datatype, Character datatypes
* RAWTOHEX built-in function, Conversion functions
* READ ONLY transaction control function, SET TRANSACTION, Calling PL/SQL Functions in SQL
* REAL datatype, Numeric datatypes
* records (PL/SQL)
* about, Records in PL/SQL
* assigning, Assigning Records
* declaring, Declaring Records
* DML and, Records and DML
* nested, Nested Records
* referencing fields of, Referencing a Record's Fields
* recursion, mutual, Forward Declarations
* REF object type, Declaring a Collection
* REF operator, REF operator
* reference datatype, Variables and Program Data
* referencing
* fields of records, Referencing a Record's Fields
* package elements, Referencing Package Elements
* persistent objects, DEREF operator
* REFERENCING clause , Creating Triggers
* REFTOHEX built-in function, Conversion functions
* REF_CURSOR type , Cursor Variables
* REGEXP_COUNT function, REGEXP_COUNT
* REGEXP_INSTR function, String functions, REGEXP_INSTR
* REGEXP_LIKE function, String functions, REGEXP_LIKE
* REGEXP_REPLACE function, String functions, String functions, REGEXP_REPLACE
* REGEXP_SUBSTR function , REGEXP_SUBSTR
* regular expressions, Built-in Regular Expression Functions-Match modifiers
* REMAINDER built-in function, Numeric functions
* remote database indicator (@), Delimiters
* RENAME trigger event, Creating Triggers, DDL Events
* REPEAT UNTIL loop emulation, "Repeat Until" Loop Emulation
* REPLACE built-in function, String functions
* reserved words, Identifiers
* RESTRICT_REFERENCES pragma, Pragmas
* RESULT_CACHE feature, Function Result Cache-Function Result Cache
* RETURN clause in functions, Functions
* RETURNING clause, Implicit Cursors, Bulk Binds
* REUSE SETTINGS option, Compiling Stored PL/SQL Programs
* REVERSE keyword , Numeric FOR Loop
* REVOKE statement, Privileges and Stored PL/SQL
* REVOKE trigger event, Creating Triggers, DDL Events
* ROLLBACK statement
* about, ROLLBACK
* exceptions and transactions, Exceptions and transactions
* executing stored functions, Calling PL/SQL Functions in SQL
* ROUND built-in function, Numeric functions, Date and time functions
* row objects, retrieving, VALUE operator
* row-level triggers, Creating Triggers
* %ROWCOUNT attribute, Explicit cursor attributes, Bulk Binds
* ROWDTOCHAR built-in function, Conversion functions
* ROWDTONCHAR built-in function, Conversion functions
* ROWID datatype, Character datatypes
* %ROWTYPE attribute
* anchoring declarations, Anchored Declarations
* declaring records, Declaring Records
* DML and, Records and DML
* RPAD built-in function, String functions
* RTRIM built-in function, String functions
### S
* SAVE EXCEPTIONS keyword, Bulk Binds
* SAVEPOINT statement, ROLLBACK
* scalar datatypes, Scalar Datatypes-BOOLEAN datatype, Anchored Declarations-Programmer-Defined Subtypes
* scope of exceptions, Scope
* searched CASE expression , CASE expression
* searched CASE statement, CASE statement
* SELECT FOR UPDATE clause, SELECT FOR UPDATE clause
* SELECT INTO statement
* assigning records, Assigning Records
* BULK COLLECT INTO clause and, Bulk Binds
* manipulating objects, Manipulating Objects in PL/SQL and SQL
* SELECT statement
* cursor FOR loop and, Cursor FOR Loop
* explicit cursors and, Explicit Cursors
* implicit cursors and, Implicit Cursors
* read-only transactions and, SET TRANSACTION
* static cursors and, Cursors in PL/SQL
* WHERE clause and, WHERE CURRENT OF clause
* WITH clause, Improving Performance of Calling PL/SQL Functions from SQL
* SELF parameter, Constructor methods
* semicolon (;), statements and, Delimiters, Native Dynamic SQL
* sequences (PL/SQL), Sequences in PL/SQL
* sequential control statements, Sequential Control Statements-NULL
* SERIALLY_REUSABLE pragma, Pragmas, SERIALLY_REUSABLE Pragma
* SERVERERROR trigger event, Creating Triggers, Database Events
* SESSIONTIMEZONE built-in function, Date and time functions
* SET function, Nested Table Functions
* SET ROW keyword, Records and DML
* SET TRANSACTION statement, SET TRANSACTION
* SGA (System Global Area), Function Result Cache, Package Initialization
* SHUTDOWN trigger event, Creating Triggers, Database Events
* SIGN built-in function, Numeric functions
* SIGNTYPE datatype, Numeric datatypes
* simple CASE expression, CASE expression
* SIMPLE_INTEGER datatype, Numeric datatypes
* SIN built-in function, Numeric functions
* single quote ('), Boolean, Numeric, and String Literals, Delimiters
* SINH built-in function, Numeric functions
* SMALLINT datatype, Numeric datatypes
* SOUNDEX built-in function, String functions
* SQL injection, SQL Injection and Bind Variables
* SQL%BULK_EXCEPTIONS.COUNT method, Bulk Binds
* SQL%BULK_ROWCOUNT attribute, Implicit Cursors, Bulk Binds
* SQL%FOUND attribute, Implicit Cursors
* SQL%ISOPEN attribute, Implicit Cursors
* SQL%NOTFOUND attribute, Implicit Cursors
* SQL%ROWCOUNT attribute, Implicit Cursors
* SQLCODE function, SQLCODE and DBMS_UTILITY formatters for exception handlers-SQLCODE and DBMS_UTILITY formatters for exception handlers, Bulk Binds
* SQLERRM function, SQLCODE and DBMS_UTILITY formatters for exception handlers
* SQRT built-in function, Numeric functions
* square brackets []
* as delimiters, Boolean, Numeric, and String Literals
* metacharacter, Metacharacters
* STARTUP trigger event, Creating Triggers, Database Events
* statements (PL/SQL)
* about, Statements
* bind variables and, SQL Injection and Bind Variables
* control, BOOLEAN datatype, Conditional and Sequential Control-NULL
* delimiters for, Delimiters
* loops and, Loops-Loop Labels
* static cursors, Cursors in PL/SQL
* STATIC keyword, Static methods
* stored functions
* calling from SQL statements, Calling PL/SQL Functions in SQL
* requirements for calling in SQL, Calling PL/SQL Functions in SQL
* syntax for calling from SQL, Calling a Function
* stored programs
* compiling, Compiling Stored PL/SQL Programs
* compiling natively, Performing Native Compilation of PL/SQL
* string functions, String functions
* string literals, Boolean, Numeric, and String Literals, Delimiters
* SUBMULTISET OF function, Nested Table Functions
* SUBSTITUTABLE keyword, Changing Object Types
* SUBSTR built-in function, String functions
* subtraction (-) operator, Delimiters
* subtypes (object), Type Inheritance, Methods in Subtypes-Methods in Subtypes, Upcasting and Downcasting-DEREF operator
* subtypes (scalar), Scalar Datatypes-Character datatypes, Programmer-Defined Subtypes
* supertypes
* downcasting, Upcasting and Downcasting-DEREF operator
* overriding, Methods in Subtypes
* SUSPEND trigger event, Creating Triggers, Database Events
* SYSDATE built-in function, Date and time functions
* System Global Area (SGA), Function Result Cache, Package Initialization
* SYSTIMESTAMP built-in function, Date and time functions
* SYS_EXTRACT_UTC built-in function, Date and time functions
* SYS_REFCURSOR type, Cursor Variables
### T
* TABLE built-in function, Conversion functions
* table functions, Table Functions
* TAN built-in function, Numeric functions
* TANH built-in function, Numeric functions
* terminator (;), Delimiters
* time functions (see date and time functions)
* TIMESTAMP datatype, Datetime datatypes
* TIMESTAMP WITH LOCAL TIME ZONE datatype, Datetime datatypes
* TIMESTAMP WITH TIME ZONE datatype, Datetime datatypes
* TOO_MANY_ROWS exception, Implicit Cursors
* TO_BINARY_DOUBLE built-in function, Conversion functions
* TO_BINARY_FLOAT built-in function, Conversion functions
* TO_CHAR/TO_NCHAR built-in functions, Conversion functions, String functions, Date and time functions
* TO_CLOB/TO_NCLOB built-in functions, Conversion functions
* TO_DATE built-in function, Conversion functions, Date and time functions
* TO_DSINTERVAL built-in function, Conversion functions, Date and time functions
* TO_LOB built-in function, Conversion functions
* TO_MULTI_BYTE built-in function, Conversion functions, String functions
* TO_NUMBER built-in function, Conversion functions
* TO_RAW built-in function, Conversion functions
* TO_SINGLE_BYTE built-in function, Conversion functions, String functions
* TO_TIMESTAMP built-in function, Conversion functions, Date and time functions
* TO_TIMESTAMP_TZ built-in function, Conversion functions, Date and time functions
* TO_YMINTERVAL built-in function, Conversion functions, Date and time functions
* transaction management
* about, Transaction Management-Autonomous Transactions
* autonomous transactions, Pragmas, Autonomous Transactions
* exceptions and, Exceptions and transactions
* transient objects, Object-Oriented Features
* TRANSLATE built-in function, String functions
* TRANSLATE_USING built-in function, Conversion functions, String functions
* TREAT operator, Upcasting and Downcasting
* triggers
* about, Stored Procedures and Functions, Triggers
* compound, Compound DML Triggers
* creating, Creating Triggers-Creating Triggers
* disabling/enabling, Creating Triggers
* predicates, Trigger Predicates
* row-level, Creating Triggers
* TRIM function, Collections in PL/SQL, Adding and Removing Elements, String functions
* TRIM procedure, Collection Methods
* TRUE value, BOOLEAN datatype
* TRUNC built-in function, Numeric functions, Date and time functions
* TRUNCATE trigger event, Creating Triggers, DDL Events
* truth tables, BOOLEAN datatype
* %TYPE attribute, Anchored Declarations
* TYPE statement, Declaring Records, Declaring a Collection
* TZ_OFFSET built-in function, Date and time functions
### U
* UDF pragma, Pragmas, Improving Performance of Calling PL/SQL Functions from SQL
* UGA (User Global Area), Package Initialization
* unconstrained subtypes, Programmer-Defined Subtypes
* underscore (_), Identifiers
* Unicode character datatypes, Unicode character datatypes
* UNION set operation, Nested Table Functions
* UNISTR built-in function, Conversion functions, String functions
* upcasting subtypes, Upcasting and Downcasting-DEREF operator
* UPDATE statement, Autonomous Transactions
* UPDATE trigger event, Creating Triggers
* UPDATING trigger predicate, Trigger Predicates
* UPPER built-in function, String functions
* UROWID datatype, Character datatypes
* USE ROLLBACK SEGMENT transaction control function, SET TRANSACTION
* User Global Area (UGA), Package Initialization
* USER_OBJECTS data dictionary view, Data Dictionary
* UTC (Coordinated Universal Time), Datetime datatypes
* UTL_CALL_STACK package, SQLCODE and DBMS_UTILITY formatters for exception handlers
* UTL_FILE package, Declaring Exceptions, Function Result Cache
* UTL_REF package, REF operator
### V
* V$RESERVED_WORDS data dictionary view, Identifiers
* V$RESULT_CACHE data dictionary views, Function Result Cache
* V$TIMEZONE_NAMES data dictionary view, Datetime datatypes
* VALUE operator, VALUE operator
* VALUES OF keyword, Bulk Binds
* VARCHAR2 datatype, Character datatypes, Constrained declarations
* variables
* about, Variables and Program Data
* assigning fields to, Assigning Records
* bind variables , SQL Injection and Bind Variables, Records and DML
* cursor variables, Cursor Variables
* declaring, Declaring Variables-Anchored Declarations
* default values of, Default values
* manipulating, Manipulating Objects in PL/SQL and SQL
* program data and, Variables and Program Data
* VARRAYs
* about, Collections in PL/SQL
* adding/removing elements, Adding and Removing Elements
* comparing collection types, Collections in PL/SQL
* declaring, Declaring a Collection
* initializing, Initializing a Collection
### W
* WHEN clause, CASE statement
* WHEN OTHERS clause, WHEN OTHERS clause
* WHERE CURRENT OF clause, WHERE CURRENT OF clause
* WHILE loop, WHILE Loop
* whitespace, Identifiers, Conditional Compilation
* WITH clause, Improving Performance of Calling PL/SQL Functions from SQL
* WORK keyword, COMMIT
1. Introduction
1. Acknowledgments
2. Conventions
2. PL/SQL Language Fundamentals
1. PL/SQL Character Set
2. Identifiers
3. Boolean, Numeric, and String Literals
4. Numeric Literals
5. Datetime Interval Literals
6. Delimiters
7. Comments
8. Pragmas
9. Statements
10. Block Structure
3. Variables and Program Data
1. Scalar Datatypes
2. LOB Datatypes
3. Implicit Datatype Conversions
4. NULLs in PL/SQL
5. Declaring Variables
6. Anchored Declarations
7. Programmer-Defined Subtypes
4. Conditional and Sequential Control
1. Conditional Control Statements
2. Sequential Control Statements
5. Loops
1. Simple Loop
2. Numeric FOR Loop
3. Cursor FOR Loop
4. WHILE Loop
5. "Repeat Until" Loop Emulation
6. EXIT Statement
7. CONTINUE Statement
8. Loop Labels
6. Database Interaction
1. Sequences in PL/SQL
2. Transaction Management
3. Autonomous Transactions
7. Cursors in PL/SQL
1. Explicit Cursors
2. Implicit Cursors
3. Native Dynamic SQL
4. DBMS_SQL
5. SQL Injection and Bind Variables
6. Cursor Variables
7. Cursor Expressions
8. Exception Handling
1. Declaring Exceptions
2. Raising Exceptions
3. Scope
4. Propagation
9. Records in PL/SQL
1. Declaring Records
2. Referencing a Record's Fields
3. Assigning Records
4. Records and DML
5. Nested Records
10. Collections in PL/SQL
1. Declaring a Collection
2. Initializing a Collection
3. Adding and Removing Elements
4. Nested Table Functions
5. Collection Methods
6. Collections and Privileges
7. Nested Collections
8. Bulk Binds
11. Built-in Functions and Packages
1. Built-in Functions
2. Built-in Regular Expression Functions
12. Stored Procedures and Functions
1. Procedures
2. Functions
3. Parameters
4. Local Programs
5. Program Overloading
6. Forward Declarations
7. Table Functions
8. Function Result Cache
9. Privileges and Stored PL/SQL
13. Triggers
1. Creating Triggers
2. Trigger Predicates
3. DML Events
4. Compound DML Triggers
5. DDL Events
6. Database Events
14. Packages
1. Package Structure
2. Referencing Package Elements
3. Package Data
4. SERIALLY_REUSABLE Pragma
5. Package Initialization
15. Calling PL/SQL Functions in SQL
1. Calling a Function
2. Improving Performance of Calling PL/SQL Functions from SQL
3. Column Name Versus Function Name Precedence
16. Object-Oriented Features
1. Object Types
2. Type Inheritance
3. Methods
4. Methods in Subtypes
5. Manipulating Objects in PL/SQL and SQL
6. Upcasting and Downcasting
7. Changing Object Types
17. Compilation
1. Compiling Stored PL/SQL Programs
2. Conditional Compilation
3. Compiler Warnings
4. Optimizing Compiler
5. Performing Native Compilation of PL/SQL
18. Java Language Integration
1. Example
2. Publishing Java to PL/SQL
3. Data Dictionary
19. Index
| {
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{"url":"https:\/\/www.r-bloggers.com\/2020\/10\/machine-learning-with-r-logistic-regression\/","text":"Want to share your content on R-bloggers? click here if you have a blog, or here if you don't.\n\nOur little journey to machine learning with R continues! Today\u2019s topic is logistic regression \u2013 as an introduction to machine learning classification tasks. We\u2019ll cover data preparation, modeling, and evaluation of the well-known Titanic dataset.\n\nIf you want to read the series from the beginning, here are the links to the previous articles:\n\n\u2022 Intro to logistic regression\n\u2022 Data preparation\n\u2022 Model training and evaluation\n\u2022 Conclusion\n\nYou can download the source code\u00a0here. That\u2019s it for the introduction section \u2013 we have many things to cover, so let\u2019s jump right to it.\n\n### Intro to logistic regression\n\nLogistic regression is a great introductory algorithm for binary classification (two class values) borrowed from the field of statistics. The algorithm got the name from its underlying mechanism \u2013 the logistic function (sometimes called the sigmoid function).\n\nThe logistic function is an S-shaped function developed in statistics, and it takes any real-valued number and maps it to a value between 0 and 1. That\u2019s just what we need for binary classification, as we can set the threshold at 0.5 and make predictions according to the output of the logistic function.\n\nHere\u2019s how the logistic function looks like:\n\nIn case you\u2019re interested, below is the equation for the logistic function. Remember \u2013 it takes any real-valued number and transforms it to a value between 0 and 1.\n\nAnd that\u2019s quite enough for the theory. I repeat \u2013 this article\u2019s aim isn\u2019t to cover the theory, as there\u2019s a plethora of theoretical articles\/books out there. It\u2019s a pure hands-on piece.\n\nOkay, now we have a basic logistic regression understanding under our belt, and we can begin with the coding portion. We\u2019ll use the Titanic dataset, as mentioned previously. You don\u2019t have to download it, as R does that for us.\n\nlibrary(dplyr)\nlibrary(stringr)\nlibrary(caTools)\nlibrary(caret)\n\ndf <- read.csv('https:\/\/raw.githubusercontent.com\/datasciencedojo\/datasets\/master\/titanic.csv')\n\nAnd here\u2019s how the first couple of rows look like:\n\nAwesome! The dataset requires a bit of preparation to get it to a ml-ready format, so that\u2019s what we\u2019ll do next.\n\n### Data preparation\n\nThere are a couple of essential things we have to do:\n\n\u2022 Extract titles from the Name attribute\n\u2022 Remap extracted titles as usual\/unusual\n\u2022 Convert Cabin attribute to binary \u2013 HasCabin\n\u2022 Remove unnecessary attributes\n\nThis\u00a0snippet from Kaggle\u00a0helped a lot with title extraction and remapping, with slight modifications. Other points are relatively straightforward, as the following snippet shows:\n\nmaleNobleTitles <- c('Capt', 'Col', 'Don', 'Dr', 'Jonkheer', 'Major', 'Rev', 'Sir')\nfemaleNobleTitles <- c('Lady', 'Mlle', 'Mme', 'Ms', 'the Countess')\n\ndf$Title <- str_sub(df$Name, str_locate(df$Name, ',')[ , 1] + 2, str_locate(df$Name, '\\\\.')[ , 1] - 1)\ndf$Title[df$Title %in% maleNobleTitles] <- 'MaleNoble'\ndf$Title[df$Title %in% femaleNobleTitles] <- 'FemaleNoble'\ndf$HasCabin <- ifelse(df$Cabin == '', 0, 1)\ndf <- df %>% select(-PassengerId, -Name, -Ticket, -Cabin)\n\nWe essentially created two arrays for noble titles, one for males and one for females, extracted the title to the Title column, and replaced noble titles with the expressions \u2018MaleNoble\u2019 and \u2018FemaleNoble\u2019.\n\nFurther, the ifelse function helped make the HasCabin attribute, which has a value of 1 if the value for Cabin is not empty and 0 otherwise. Finally, we\u2019ve kept only the features that are relevant for analysis.\n\nHere\u2019s how the dataset looks now:\n\nAwesome! Let\u2019s deal with missing values next.\n\n#### Handling missing data\n\nThe following line of code prints out how many missing values there are per attribute:\n\nlapply(df, function(x) { length(which(is.na(x))) })\n\nThe attribute Age is the only one that contains missing values. As this article covers machine learning and not data preparation, we\u2019ll perform the imputation with a simple mean. Here\u2019s the snippet:\n\ndf$Age <- ifelse(is.na(df$Age), mean(df$Age, na.rm=TRUE), df$Age)\n\nAnd that\u2019s it for the imputation. There\u2019s only one thing left to do, preparation-wise.\n\n#### Factor conversion\n\nWe have a bunch of categorical attributes in our dataset. R provides a simple factor() function that converts categorical attributes to an algorithm-understandable format.\n\nHere\u2019s the structure of our dataset before the transformation:\n\nAnd here\u2019s the code snippet to perform the transformation:\n\ndf$Survived <- factor(df$Survived)\ndf$Pclass <- factor(df$Pclass)\ndf$Sex <- factor(df$Sex)\ndf$SibSp <- factor(df$SibSp)\ndf$Parch <- factor(df$Parch)\ndf$Embarked <- factor(df$Embarked)\ndf$Title <- factor(df$Title)\ndf$HasCabin <- factor(df$HasCabin)\n\nThe data preparation part is finished, and we can now proceed with the modeling.\n\n### Model training and evaluation\n\nBefore the actual model training, we need to split our dataset on the training and testing subset. Doing so ensures we have a subset of data to evaluate on, and know how good the model is. Here\u2019s the code:\n\nset.seed(42)\n\nsampleSplit <- sample.split(Y=df$Survived, SplitRatio=0.7) trainSet <- subset(x=df, sampleSplit==TRUE) testSet <- subset(x=df, sampleSplit==FALSE) The above code divides the original dataset into 70:30 subsets. We\u2019ll train on the majority (70%), and evaluate on the rest. We can now train the model with the glm() function. We\u2019ll use all of the attributes, indicated by the dot, and the Survived column is the target variable. model <- glm(Survived ~ ., family=binomial(link='logit'), data=trainSet) And that\u2019s it \u2013 we have successfully trained the model. Let\u2019s see how it performed by calling the summary() function on it: summary(model) The most exciting thing here is the P-values, displayed in the Pr(>|t|) column. Those values indicate the probability of a variable not being important for prediction. It\u2019s common to use a 5% significance threshold, so if a P-value is 0.05 or below, we can say there\u2019s a low chance for it not being significant for the analysis. As we can see, the most significant attributes\/attribute subsets are Pclass3, Age, SibSp3, SibSp4, and HasCabin1 We now have some more info on our model \u2013 we know the most important factors to decide if a passenger survived the Titanic accident. Now we can move on the evaluation of previously unseen data \u2013 test set. We\u2019ve kept this subset untouched deliberately, just for model evaluation. To start, we\u2019ll need to calculate the prediction probabilities and predicted classes on top of those probabilities. We\u2019ll set 0.5 as a threshold \u2013 if the chance of surviving is less than 0.5, we\u2019ll say the passenger didn\u2019t survive the accident. Here\u2019s the code: probabs <- predict(model, testSet, type='response') preds <- ifelse(probabs > 0.5, 1, 0) It\u2019s now easy to build on top of that. The go-to approach for classification tasks is to make a confusion matrix \u2013 a 2\u00d72 matrix showing correct classification on the first and fourth element, and incorrect classification on the second and third element (reading left to right, top to bottom). Here\u2019s how to obtain it through code: confusionMatrix(factor(preds), factor(testSet$Survived))\n\nSo, overall, our model is correct in roughly 84% of the test cases \u2013 not too bad for a couple of minutes of work. Let\u2019s wrap things up in the next section.\n\n### Before you go\n\nWe\u2019ve covered the most basic regression and classification machine learning algorithms thus far. It was quite a tedious process, I know, but necessary to create foundations for what\u2019s coming later \u2013 more complex algorithms and optimization.\n\nThe next article in the series on KNN is coming in a couple of days, so stay tuned.","date":"2021-12-03 10:25:19","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.481426864862442, \"perplexity\": 1812.327876860904}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-49\/segments\/1637964362619.23\/warc\/CC-MAIN-20211203091120-20211203121120-00272.warc.gz\"}"} | null | null |
Sony RX0
Sony Smart Speaker
Acer Switch 7
IFA 2017: 5 noteworthy new product launches
Europe's largest tech show IFA 2017 is underway and like every year, it's the ideal place to see the latest wares from all the global tech giants. Here are a few products that were unveiledHitesh Raj Bhagat | ET Bureau | September 02, 2017, 10:38 IST
Europe's largest tech show IFA 2017 is underway and like every year, it's the ideal place to see the latest wares from all the global tech giants. Here are a few products that were unveiled:
From Fitbit, a company normally known for fitness bands, comes a brand new smartwatch called the Ionic. It's still not powered by Android Wear instead, it has Fitbit OS.
The reason we say smartwatch is because it supports third-party apps. It still does all the usual fitness features, has Bluetooth, GPS, WiFi, multiple watch faces and promises 4 days of battery life. The Ionic claims to have a different, more accurate way of measuring heartbeat.
Instead of the usual optical sensor + green LEDs, this uses a combination of red, green and infrared LEDs.
You can also use it for swim workouts and it shows notifications from your phone.
This is LG's latest flagship and it incorporates learnings from the current LG G6 and the V20. Gone is the gimmicky second screen, replaced with a 6-inch FullVision OLED display.
FullVision is the name LG gives to the ultra slim bezel design, one of the defining characteristics of its new flagships. Despite the large screen, it's about the size of a 5.5-inch phone.
Like the G6, it has a dual camera system with one ultra wide angle lens that helps capture interesting perspectives. There's also IP68 certification and for those who think that new phones are getting too delicate, military standard durability rating (MIL-STD 810G).
This latest update to the Acer Switch series has a couple of firsts: it's the first two-in-one hybrid Windows laptop with a fanless design and the first with a new kind of kickstand that can automatically deploy and auto retract once you're done.
Despite having no fan (and hence no noise, just like tablets), the Switch 7 includes a discrete graphics chip for better gaming performance. To prevent overheating, it uses liquid cooling technology impressive, considering the 1.13kg weight. It has a 13.5-inch touchscreen (2256 x 1504 pixels) and will be powered by Intel's latest 8th generation Core-I processors. Prices should start at $1699.
The market is soon going to be flooded with smart speakers and this Sony is one of them. Like Google Home, Sony's LF-S50G is powered by Google Assistant.
It's designed to sit unobtrusively on your desk or side table, play music (it has Bluetooth and NFC), tell the time, control smart home products, perform Google searches, check your calendar, tell the weather and all the usual things you'd expect. And if you're playing your music through it, you can also use gesture control (just wave your hand in the air above the speaker) to control playback. Price has been set at $199 and it starts shipping in October.
Sony offers a range of compact cameras with large, 1-inch CMOS sensors (RX100 series). They've taken the same imaging components and squeezed them into a rugged, waterproof shell that's roughly the size of a typical action camera.
It's waterproof (down to 33ft10m), shockproof (from drops of 6.5ft/2m), and crushproof (from forces up to 440lb/200kg).
Sony is essentially bringing high quality imaging to a more versatile size. Other features include the ability to link multiple RX0 cameras together for creative techniques like bullet time. Price is set at $700 each and it will be available to purchase from October onwards.
Tags : Mobile, IFA 2017, LG V30, Sony RX0, Fitbit Ionic, Sony Smart Speaker, Acer Switch 7
Trending in Mobile
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India proposes incentives to woo Apple, Samsung suppliers | {
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} | 108 |
I've mentioned a few times my bipolar has spiralled into a depression the past few weeks. Music is always really important to me but when I'm unwell especially so. I have been a huge fan of the Italian band Lacuna Coil from day one, and I literally couldn't tell you how many times I've seen them live but it's in the double figures by now.
Last year they released Delirium, an album completely centred around mental health. It's a solid and sensitive album but it doesn't hide from the brutal truth's of life with a severe mental illness either. It's my favourite of all their albums probably because I connected with it on such a personal level.
I've chosen the title track from the album to share with you, but I highly recommend this album.
I haven't listened to LC for ages!
I saw them live in 2009 and used to be a huge fan. Time has dragged me away!
She is rather fine! I have quite the girl crush on her!
Lacuna Coil for the win!
My brother, who is actually crazy into trance and such loves Lacuna Coil … ha… I love 'em too, they have som mean tunes and the vocals are to die for!
Yay for your brother liking them! Funnily enough Hubs likes them too and he's really into dance (or blippy bloppy as I call it) as well!
LOL at blippy bloppy! 😀 Good one! | {
"redpajama_set_name": "RedPajamaC4"
} | 2,377 |
Eero Emil Kössi Salo (4 July 1921, Hämeenlinna – 13 October 1975) was a Finnish politician. He was a Member of the Parliament of Finland from 1968 until his death in 1975, representing the Social Democratic Party of Finland (SDP).
References
1921 births
1975 deaths
People from Hämeenlinna
Social Democratic Party of Finland politicians
Members of the Parliament of Finland (1966–70)
Members of the Parliament of Finland (1970–72)
Members of the Parliament of Finland (1972–75)
Members of the Parliament of Finland (1975–79) | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,626 |
Sinjar est une ville du Gouvernorat d'Idleb au nord-ouest de la Syrie.
Sa population était de en 2004.
Notes et références
Liens externes
Ville en Syrie | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 623 |
Борок — деревня в Череповецком районе Вологодской области. Административный центр Дарвинского заповедника.
Входит в состав сельского поселения Уломское, с точки зрения административно-территориального деления — в Ягницкий сельсовет.
Расстояние по автодороге до районного центра Череповца — 160 км, до центра муниципального образования Ягницы — 50 км. Ближайший населённый пункт — Лоша.
По переписи 2002 года население — 109 человек (52 мужчины, 57 женщин). Преобладающая национальность — русские (94 %).
Примечания
Ссылки
в реестре населённых пунктов Вологодской области
Лист карты NO 37-4. Масштаб 1 : 250 000. Состояние местности на 1967 год.
Населённые пункты Череповецкого района | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 2,963 |
Preparing for college is not the same as it was for us.
You have unanswered questions…so much you don't know. You need information from a trusted resource and time to ask your questions. We have the solution to your stress with Tips for the College-Bound.
Join Beth Probst and other experts in all aspects of college preparation and support. Perfect for families with 8th-11th graders. We encourage parents and students to attend together.
Topics include: ACT/SAT exams, choosing high school courses (AP, IB, College Credit Plus, etc.), college applications, choosing a major or a college, how to pay for it, and much more!
Families will leave with a ton of information, a helpful workbook, and the feeling of confidence and knowledge they're looking for.
Click to read more about the program. | {
"redpajama_set_name": "RedPajamaC4"
} | 2,945 |
package com.xeiam.xchange.bitso.dto.trade;
import com.fasterxml.jackson.annotation.JsonIgnore;
import com.fasterxml.jackson.annotation.JsonProperty;
import com.fasterxml.jackson.databind.annotation.JsonDeserialize;
import com.xeiam.xchange.bitso.BitsoUtils;
import java.math.BigDecimal;
import java.util.Date;
/**
* @author Piotr Ładyżyński
*/
public final class BitsoOrder {
private final String id;
private final String datetime;
/** 0 - buy (bid); 1 - sell (ask) */
private final int type;
private final BigDecimal price;
private final BigDecimal amount;
private final String errorMessage;
/**
* Constructor
*
* @param id
* @param datetime
* @param type
* @param price
* @param amount
*/
public BitsoOrder(@JsonProperty("id") String id, @JsonProperty("datetime") String datetime, @JsonProperty("type") int type,
@JsonProperty("price") BigDecimal price, @JsonProperty("amount") BigDecimal amount, @JsonProperty("error") @JsonDeserialize(
using = BitsoErrorDeserializer.class) String errorMessage) {
this.id = id;
this.datetime = datetime;
this.type = type;
this.price = price;
this.amount = amount;
this.errorMessage = errorMessage;
}
public String getDatetime() {
return datetime;
}
public String getId() {
return id;
}
public int getType() {
return type;
}
public BigDecimal getPrice() {
return price;
}
public BigDecimal getAmount() {
return amount;
}
@JsonIgnore
public Date getTime() {
return BitsoUtils.parseDate(getDatetime());
}
@JsonIgnore
public String getErrorMessage() {
return errorMessage;
}
@Override
public String toString() {
return errorMessage != null ? errorMessage : String.format("Order{id=%s, datetime=%s, type=%s, price=%s, amount=%s}", id, datetime, type, price,
amount);
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,684 |
Definition - What does Areal Density mean?
Areal density is the measurement of storage units per square inch, or, more generally, the measurement of storage capacity relative to physical dimensions.
Areal density is sometimes used interchangeably with the terms area density and surface density.
Areal density is an important idea in IT. For an easy example of areal density, think of a small piece of storage media. Say the disk is one square inch in terms of its physical size. If that disk has 1 GB of storage capacity, the areal density measurement is 1 GB per square inch.
Areal density is a useful term in looking at the relative storage capacity of physical storage media like magnetic tapes or disks and optical disks. Reports from the IT industry over time have shown how areal density improves at a dramatic rate. The ability to pack more digital storage capacity into smaller disks and devices is a large part of what has driven improvement in the hardware industry and other parts of the tech business. | {
"redpajama_set_name": "RedPajamaC4"
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\section{Introduction}
Due to an increasing ability to handle massive data with extraordinary model accuracy, machine learning (ML) algorithms have achieved remarkable success in many applications, such as computer vision \cite{KA2014VGG,SLJS2015GOOGLE,HZRS15Res,SVI2016INC} and natural language processing \cite{SVQ2014seq2,VSPU2017Attention,DCL2018BERT,YDY2019XLNET}.
However, empirical studies have also revealed that ML algorithms may incorporate bias from training data into model predictions.
Due to historical biases, vulnerable groups are often under-represented in available data \cite{JJSL2016, ZWY2017, SEC2019}.
As a consequence, without fairness considerations, ML algorithms can be systematically biased against certain groups defined by protected attributes such as race and gender.
As algorithmic decision-making systems are now widely integrated in high-stakes decision-making processes, such as in healthcare \cite{MQ2017} and criminal prediction \cite{JJSL2016},
fair machine learning has grown rapidly over the last few years into a foundational area of trustworthy AI.
A main task in fair machine learning is to design efficient algorithms satisfying fairness constraints with a small sacrifice in model accuracy.
This field has made substantial progress in recent years, as many effective approaches have been proposed to mitigate algorithmic bias \cite{ZWKT2013,CKYMR2016,CFDV2017,XWYZW2019,LV2019,CJG2019,DETR2018,CMJW2019,JL2019,CHS2020,ZDC2022}.
An important fundamental benchmark for fair classification is provided by fair Bayes-optimal classifiers, which maximize accuracy subject to fairness \cite{MW2018,ZDC2022}.
A representative class of classifiers is group-wise thresholding rules (GWTRs) over the feature-conditional probabilities of the target label, for each protected group (e.g., probability of repaying a loan given income).
Intuitively, being a GWTR is a minimal requirement for within-group fairness, as it ensures that the most qualified individuals are selected in every group.
\cite{CSFG2017,MW2018,CCH02019,I2020,SC2021,ZDC2022} have studied fair Bayes-optimal classifiers under various fairness constraints and proved that,
for many fairness metrics, the optimal fair classifiers are GWTRs. Moreover, the associated thresholds can be learned efficiently \cite{MW2018,ZDC2022}.
Current literature on Bayes-optimality focuses mainly on the independence- and separation-based fairness measures (e.g., demographic parity, equality of opportunity, equalized odds; see Section \ref{fm} for definitions and a review).
However, sufficiency-based measures such as predictive parity are less commonly considered, possibly due to the complexity of their constraints.
Sufficiency-based measures are often applied to assess recidivism prediction instruments \cite{FBC2016,DMB2016,C2017fair}.
\cite{LSMH2019} shows that a particular sufficiency-based measure, group calibration,
is implicitly favored by unconstrained optimization:
calibration error is bounded by the excess risk over the unconstrained Bayes-optimal classifier.
For selective classification, \cite{LBRSP2021} finds that sufficiency-based representation learning leads to fairness.
Despite these findings, little is known about (1) what the optimal fair classifiers under sufficiency-based measures are, and (2) how to learn them efficiently.
In this paper, we aim to answer these two questions by focusing on predictive parity, which requires that the positive predictive value (probability of a successful outcome given a positive prediction) be equalized among protected groups.
For example, in credit lending, for example, predictive parity requires that, for individuals who receive the loans, the repayment rates are the same in different protected groups.
We study fair Bayes-optimal classifiers under predictive parity.
Perhaps surprisingly, our theoretical results reveal that the \emph{optimal fair classifiers are often, but not always group-wise thresholding rules}, depending on the data distribution.
We identify a sufficient condition under which all fair Bayes-optimal classifier are GWTRs.
Without this condition,
we show that fair Bayes-optimal classifiers may not be a GWTR when the minority group is more qualified than the majority group.
In these cases, predictive parity may have limitations as a fairness measure, as it can lead to within-group unfairness for the minority group.
Our findings are a reminder that the improper use of fairness measures may result in severe unintended consequences.
Careful analysis before applying fairness measures is necessary.
We then develop an algorithm, FairBayes-DPP, aiming for predictive parity.
Our method is a three-stage plug-in method.
In the first step, we use standard learning algorithms to estimate the group-wise conditional probabilities of the labels.
In the second step, we check our sufficient condition. If the sufficient condition holds, in the third step
we apply
a plug-in method
for estimating the optimal thresholds under fairness for each protected group.
We summarize our contributions as follows.
\begin{itemize}
\item We show that Bayes-optimal classifiers satisfying predictive parity may or may not be group-wise thresholding rules (GWTRs), depending on the data distribution.
\item We identify a sufficient condition under which all fair Bayes-optimal classifiers are GWTRs.
We show that, when the sufficient condition is not satisfied, imposing predictive parity while maximizing accuracy may lead to within-group unfairness for the minority group.
\item We propose the FairBayes-DPP algorithm for fair classification.
We demonstrate that FairBayes-DPP is computationally efficient, showing a solid performance in our experiments on a number of synthetic and standard empirical datasets.
\end{itemize}
\section{Related Literature}
\subsection{Fairness Measures}
\label{fm}
Various fairness metrics have been proposed to measure aspects of disparity in ML.
Group fairness \citep{CFM2009,DHPT2012,HPS2016} targets statistical parity across
protected groups, while individual fairness \citep{JKMR2016,PKG2019,RBMF2020} aims to provide nondiscriminatory predictions
for similar individuals.
In general, group fairness measures can be categorized into three categories.
The first group consists of independence-based measures, which require independence between predictions and protected attributes; this includes demographic parity \cite{KC2012,ZWKT2013} and conditional statistical parity \cite{CSFG2017,I2020}. In credit lending, independence means that the proportion of approved candidates is the same across different protected groups.
However, as discussed in \cite{HPS2016}, independence-based measures have limitations;
and applying them often leads to a substantial loss of accuracy.
The second group consists of separation-based measures,
which require conditional independence between predictions and protected attributes, given label information.
Typical examples in this group are equality of opportunity \cite{HPS2016,ZLM2018} and equalized odds \cite{HPS2016,ZVGGK2017}.
In credit lending, separation-based measures require that the individuals who will pay back (or default on) their loan
have an equal probability of
getting the loan, despite their race or gender.
Compared to independence-based measures, separation-based measures take label information into account, allowing for perfect predictions that equal the label.
However, these measures can be hard to validate, as the label information is often unknown for some groups. For example, the repayment status is missing for individuals whose loan application is declined.
As a result, measuring predictive bias is more widely applicable.
This leads to the third class, sufficiency-based measures \cite{PMK2017,C2017fair,LSMH2019}, where the label is required to be conditionally independent of the protected attributes, given the prediction. In credit lending, this requires that among the approved applications, the proportion of individuals who pay back the loan is equal across different groups.
Unlike independence- and separation-based measures that are well studied with solid theoretical benchmarks and efficient algorithms, sufficiency-based measures are less commonly investigated.
A possible reason is that conditioning on the prediction
leads to a complex constraint, which is thus challenging to study and enforce algorithmically.
\subsection{Algorithms Aimed at Fairness}
Literature on algorithms for fairness has grown explosively over the past decade.
Existing algorithms for fairness can be categorized broadly into three categories.
The first category is pre-processing algorithms aiming to remove biases from the training data.
Examples include transformations \citep{FFMS2015,KJ2016,CFDV2017,JL2019}, fair representation learning \citep{ZWKT2013,CKYMR2016,DETR2018,CMJW2019} and fair data generation \citep{XYZ2018,SHC2019,XYfairplus,RKSR2021}.
The second group is
in-processing algorithms, which handle fairness constraints during the training process.
Two common strategies are penalized optimization \citep{GCAM2016,NH2018,CJG2019,CHS2020} and adversarial training \citep{ZLM2018,CFC2018,XWYZW2019,LV2019}.
The former incorporates fairness measures as a regularization term into the optimization objective, while the latter tries to minimize
the predictive ability of the model with respect to
the protected attribute.
The third group is post-processing algorithms, aiming to remove disparities from the model output.
The most common post-processing algorithm is the thresholding method \citep{BJA2016,MW2018,I2020,SC2021,ZDC2022}, adjusting thresholds for every protected group to achieve fairness. In this paper, we propose a post-processing algorithm, FairBayes-DPP, to estimate the fair Bayes-optimal classifier under predictive parity.
\section{Problem Formulation and Notations}\label{Pre_and _Not}
We consider
classification problems where two types of feature are observed: the usual feature $X\in\mathcal{X}$, and the protected feature $A\in\mathcal{A}$.
For example, in loan applications, $X$ may refer to common features such as
education level and income, and $A$ may correspond to the race or gender of a candidate.
As multiclass protected attributes are often encountered in practice, we allow $\mathcal{A}$ to have any number $K\ge1$ of classes, and let $\mathcal{A}=\{1,2,...,K\}$.
We denote by $Y$ the ground truth label.
In credit lending, $Y$ may correspond to the status of repayment or defaulting on a loan.
The output $\widehat{Y}$ of the classifier aims to predict $Y$ based on observed features $X,A$.
We consider randomized classifiers defined as follows:
\begin{definition}[Randomized classifier]
A randomized classifier is a measurable function\footnote{We assume that, whenever needed, the sets considered are endowed with appropriate sigma-algebras, and all functions considered are measurable with respect to the appropriate sigma-algebras.}
$f:\mathcal{X}\times\mathcal{A} \to [0,1]$, indicating the probability of predicting $\widehat{Y}=1$ when observing $X=x$ and $A=a$. We denote by $\widehat{Y}_f=\widehat{Y}_f(x)$ the prediction induced by the classifier $f$.
\end{definition}
Group-wise thresholding rules (GWT rules/classifiers or GWTRs) over conditional probabilities are of special importance.
Consider an appropriate sigma-finite measure $\mu$ on $\mathcal{X}$ (such as the Lebesgue measure for measurable subsets of $\mathbb{R}^d$, $d\ge 1$, or the uniform measure for finite sets), and suppose that
for all $a\in\mathcal{A}$ and $y\in \mathcal{Y}$,
the features $X$ have a conditional distribution $P_{X|a,y}$
given $A=a,Y=y$
with a density $dP_{X|a,y}$
with respect to $\mu$.
For all\footnote{To be precise, this conditional density is defined for $\mu$-almost every $x\in\mathcal{X}$; however for simplicity we say for all $x\in\mathcal{X}$. We use this convention without further mentioning through the paper.}
$x\in\mathcal{X}$
and $a\in\mathcal{A}$, let $\eta_a(x)=P(Y=1|X=x,A=a)$.
\begin{definition}[GWT rule/classifier]
A classifier $f$ is a GWTR if there are constants $t_a$, $a\in\mathcal{A}$, and functions $\tau_a:\mathcal{X}\to[0,1]$, $a\in\mathcal{A}$, such that for all $x\in\mathcal{X}$ and $a\in\mathcal{A}$
\begin{equation}\label{gwtr}
f(x,a)=I(\eta_a(x)>t_a)+\tau_a(x) I(\eta_a(x)=t_a),
\end{equation}
where $I(\cdot)$ is the indicator function.
\end{definition}
Clearly, GWTRs choose individuals with the highest conditional probability in each group.
This property is a minimal requirement for within-group fairness.
For example, a GWT recruitment tool ensures that the most qualified candidates are approved in every protected group.
We consider predictive parity, which aims to ensure the same positive predictive value among protected groups:
\begin{definition}[Predictive Parity]
A classifier $f$ satisfies predictive parity if for all $a\in\mathcal{A}$,
$$P(Y=1 |\widehat{Y}_f = 1,A = a) =P(Y=1|\widehat{Y}_f = 1).$$
\end{definition}
We follow \cite{CHS2020, ZDC2022} to use the difference between positive predictive values to measure the degree of unfairness, defining the Difference in Predictive Parities (DPP) of a classifier $f$ as
$$\text{DPP}(f)=\sum_{a\in\mathcal{A}}|P(Y=1 |\widehat{Y}_f = 1,A = a) -P(Y=1|\widehat{Y}_f = 1)|.$$
\section{Fair Bayes-optimal Classifiers under Predictive Parity}
Since predictive parity is commonly considered in scenarios where false positives are particularly harmful \cite{LBRSP2021},
we study cost-sensitive classification.
For
a cost parameter $c\in[0,1]$, the cost-sensitive 0-1 risk\footnote{When $c=1/2$, cost-sensitive risk reduces to the usual zero-one risk.} of the classifier $f$ is defined as
$$R_c(f)=c\cdot P(\widehat{Y}_f=1,Y=0)+(1-c)\cdot P(\widehat{Y}_f=0, Y=1).$$
An unconstrained Bayes-optimal classifier for the cost-sensitive risk is any minimizer $f^\star\in \text{argmin}_f R_c(f).$
A classical result is that all Bayes-optimal classifiers have the form
$f^\star(x,a)=I(\eta_a(x)>c)+\tau I(\eta_a(x)=c)$, where $\tau\in[0,1]$ is arbitrary \cite{C2001CS, MW2018}.
Taking predictive parity into account, a fair Bayes-optimal classifier is any minimizer of the cost-sensitive risk among fair classifiers:
\begin{equation}\label{bao}
f_{PPV}^\star\in \underset{f: \text{DPP}(f)=0}{\text{argmin}} R_c(f).
\end{equation}
\subsection{GWT Fair Bayes-Optimal Classifiers under Predictive Parity}
We first identify a sufficient condition under which all fair Bayes-optimal classifier under predictive parity are GWTRs.
\begin{condition}[Sufficient condition for Bayes-optimal classifiers to be GWTRs]\label{suff-con}
\begin{equation*}
\underset{a\in\mathcal{A}}{\min}\,
P(Y=1|\,\eta_a(X)\geq c,A=a)
\geq\underset{a\in\mathcal{A}}{\max}\, P({Y=1}|A=a). \end{equation*}
\end{condition}
The sufficient condition \ref{suff-con} requires that the minimal group-wise positive predictive value
$P(Y=1|\,\eta_a(X)\geq c,A=a)$
of the unconstrained Bayes-optimal classifier is lower bounded by the maximal proportion of positive labels $P({Y=1|A=a})$ among groups.
In other words, the performances of different groups vary only moderately:
the average performance of the most qualified class of each group---the points with $x$ such that $\eta_a(x)\geq c$---should be better than the overall performance $P({Y=1|A=a})$ of any of the other groups.
Condition \ref{suff-con} holds if
$P({Y=1|A=a})\leq c$ for all $a\in\mathcal{A}$,
because $P(Y=1|\,\eta_a(X)\geq c,A=a)\geq c$.
This condition is particularly applicable in settings where $c$ is large, such as in credit lending, where false positives are more societally harmful than false negatives.
They are also relevant if $P({Y=1|A=a})$, $a\in\mathcal{A}$ are small,
such as in online advertising where the probability of clicks is small.
Under this condition, we present our main result.
\begin{theorem}[Main result]\label{thm-GWTR}
Consider the cost-sensitive $0$-$1$ risk with cost parameter $c$. If
Condition \ref{suff-con} holds,
then all fair Bayes-optimal classifiers under predictive parity are GWTRs.
Thus, for all $f_{PPV}^\star$ from \eqref{bao},
there are $(t_a)_{a=1}^{|\mathcal{A}|} \in [0,1]^{|\mathcal{A}|}$ and functions $\tau_a(x):\mathcal{X}\to[0,1]$, $a\in\mathcal{A}$,
such that \eqref{gwtr} holds.
\end{theorem}
Unlike for demographic parity or for equality of opportunity,
where the fairness constraint is linear with respect to the probability predictions of the classifier $f$ \cite{MW2018}, the DPP constraint is non-linear with respect to $f$.
As a consequence, previously used theoretical tools such as the Neyman-Pearson argument from hypothesis testing \cite{ZDC2022} are no longer valid in this case.
Instead, we prove the result using a novel constructive argument.
When Condition \ref{suff-con} is satisfied,
for any classifier satisfying predictive parity, which is not a GWTR, we construct a GWTR that satisfies predictive parity and achieves a smaller classification error.
As a result, under Condition \ref{suff-con}, all fair Bayes-optimal classifiers are GWTRs.
Overall, the proof of Theorem \ref{thm-GWTR} is quite involved, and requires a lot of careful casework and analysis.
The proof rests on three novel lemmas, Lemmas \ref{lem1}, \ref{lem2}, and \ref{lem3}, which establish some fundamental properties of the positive predictive value of group-wise thresholding rules; such as monotonicity properties with respect to the thresholding parameter.
These tools may be valuable in future work.
\subsection{Fair Bayes-optimal Classifiers under Predictive Parity do not Need to be Thresholding Rules}
Next, we consider the case when the sufficient condition \ref{suff-con} does not hold.
For simplicity, we consider a binary protected attribute $a\in\{0,1\}$ with
\begin{equation}\label{unsuff-con}
{P(Y=1|\,\eta_a(x)\geq c,A=1)}<P(Y=1|A=0). \end{equation}
Our result shows that, under condition \eqref{unsuff-con}, there exist class probabilities $p_a$, $a\in\mathcal{A}$, such that no Bayes-optimal classifier under predictive parity is a GWTR.
\begin{theorem}\label{thm-imGWT}
Suppose that condition \eqref{unsuff-con} holds.
Denote $t_1=\inf\{t: P(Y=1|\,\eta_1(X)\geq t,A=1)>P({Y=1|A=0})\}$. Suppose there exist $\delta_1,\delta_2>0$ such that $P(c+\delta_1<\eta_A(X)<t_1|A=1)=\delta_2>0$.
Then, for all $p_1>2/(2+\delta_1\delta_2)$, no fair Bayes-optimal classifier under predictive parity is a GWTR.
\end{theorem}
The condition involving the constants $\delta_1,\delta_2>0$ ensures that $\eta_1(X)$ has positive probability to be strictly larger than $c$, which is a technical condition needed in the proof.
Theorem \ref{thm-imGWT} shows that predictive parity may lead to within-group unfairness, whereby the most qualified individuals are predicted to be unqualified, for a better overall accuracy.
By definition, predictive parity requires that the qualifications of selected individuals are similar across the protected groups. Suppose there exists a highly qualified minority group in which most individuals are qualified.
Selecting the most qualified individuals in this group leads to a very high standard.
As a result, many qualified individuals in other majority groups may be predicted to be unqualified using this standard,
leading to accuracy loss.
Conversely, if we select less qualified individuals in the highly qualified group, the lower standard allows more qualified individuals from the other groups to be selected, and increases accuracy.
\section{FairBayes-DPP: Adaptive Thresholding for Fair Bayes-optimality}\label{sec-alg}
In this section, we propose the FairBayes-DPP algorithm (Algorithm \ref{alg:pre-par}) for fair Bayes-optimal classification under predictive parity.
As mentioned, the DPP constraint is non-linear with respect to the classifier $f$, and is also highly non-convex with respect to the model parameters, even if both the classifier $f$ and the risk function are convex with respect to these parameters.
In such cases,
incorporating fairness constraints as a penalty in the training objective may be hard due to potential harmful local minima.
Therefore, we consider a different approach, developing a new three-step plug-in method based on Theorem \ref{thm-GWTR}.
Suppose we observe data points $(x_i,a_i,y_i)_{i=1}^n$ drawn independently and identically from a distribution $\mathcal{D}$ over the domain
$\mathcal{X} \times \mathcal{A} \times \mathcal{Y}$.
\textbf{Step 1.}
In the first step, we apply standard machine learning algorithms to learn the feature- and group-conditional label probabilities $\eta$ based on the whole dataset.
Consider a loss function $L(\cdot, \cdot)$ and the function class $\mathcal{F}=\{f_\theta: \theta\in\Theta \}$ parametrized by $\theta$.
The estimator of $\eta$ is obtained by minimizing the empirical risk, $\widehat\eta_a(x):=f_{\widehat\theta}(x,a)$, where
\begin{align}\label{ht}
\widehat{\theta}&\in \underset{\theta\in\Theta}{\text{argmin}}\frac1{n}\sum_{i=1}^nL(y_{i},f_{\theta}(x_{i},a_i)).
\end{align}
Here we use the cross-entropy loss, as minimizing the empirical 0-1 risk is generally not tractable.
At the population level,
the minimizers of the risks induced by the 0-1 and cross-entropy losses are both the
true conditional probability function \citep{miller1993loss}.
\begin{algorithm}[tb]
\caption{FairBayes-DPP}
\label{alg:pre-par}
\begin{algorithmic}
\STATE {\bfseries Input:} Datasets $S=\cup_{a=1}^{|\mathcal{A}|}S_a$ with $S=\{x_{i},a_i,y_{i}\}_{i=1}^{n}$ and $S_a=\{x^{(a)}_{j},y^{(a)}_{j}\}_{j=1}^{n_a}$. Cost parameter $c\in[0,1]$.
\STATE {\textbf{Step 1}:} Estimate $\eta_a(x)$ by $\widehat\eta = f_{\widehat\theta}$, with $\widehat\theta$ from \eqref{ht}
\STATE {\textbf{Step 2}:} Check sufficient condition \ref{suff-con}.
\STATE Define, for all $t$,\qquad
$\widehat{\text{PPV}}_{a}(t)=\frac{{\sum}_{j=1}^{n_a} I(y^{(a)}_j=1,\widehat{\eta}_{a}(x^{(a)}_j)\geq t)}{{\sum}_{j=1}^{n_a}I(\widehat{\eta}_{a}(x^{(a)}_j)\geq t)}$, \qquad $\widehat{P}(Y=1|A=a)=\frac{1}{n_a}\sum\limits_{i=1}^{n_a}y^{(a)}_j$.
\IF{$\min_a\widehat{\text{PPV}}_{a}(c)< \max_a\widehat{P}(Y=1|A=a)$}
\STATE {We recommend considering other fairness measures.}
\ELSE
\STATE {\textbf{Step 3}:} Find the optimal thresholds.
Let {$t_{\min} = \min\{t:
\widehat{\text{PPV}}_{1}(t)\ge \max_a\widehat{P}(Y=1|A=a)
\}.$}
\FOR{$t \in \mathcal{T}=[t_{\min}, \max_j\widehat\eta_1(x^{(1)}_{j})]$}
\FOR{$a\in\mathcal{A} \setminus\{1\}$}
\STATE Find $\widehat{t}_a(t)$ such that {$\widehat{\text{PPV}}_{a}(\widehat{t}_a(t))\approx\widehat{\text{PPV}}_1(t).$}
\ENDFOR
\STATE Let $
\widehat{f}(x,a,t)
=\tilde{f}\left(x,a;\widehat t_1(t),\widehat t_2(t),...,\widehat t_{|\mathcal{A}|}(t)\right)
=I\left(\widehat\eta_{a}(x)\geq \widehat t_a(t)\right).$
\STATE Let $R_c(t)=\frac1n\sum_{i=1}^n c^{(1-y_i)}(1-c^{y_i})I(y_i\neq \widehat{f}(x_i,a_i,t)).
$
\ENDFOR
\STATE Find $\widehat{t}=\underset{t\in \mathcal{T}_n}{\text{argmin}}R_c(t).$
\STATE {\bfseries Output:}
$\widehat{f}_{PP}(x,a)=I(\widehat\eta_{a}(x)\geq\widehat{t}_a(\widehat{t}))$
\ENDIF
\end{algorithmic}
\end{algorithm}
\textbf{Step 2.}
In the second step, we check the empirical version of Condition \ref{suff-con} for the classifier derived in the first step.
To be more specific, we divide the data into $|\mathcal{A}|$ parts,
according to the value of $A$: for $a\in\mathcal{A}$,
$S_a=\{x^{(a)}_{j},y^{(a)}_{j}\}_{j=1}^{n_a}$, where $a^{(a)}_{j}=a$.
Let, for all $t$ for which it is defined, $$\widehat{\text{PPV}}_{a}(t)=\frac{{\sum}_{j=1}^{n_a} I(y^{(a)}_j=1,\widehat{\eta}_{a}(x^{(a)}_j)\geq t)}{{\sum}_{j=1}^{n_a}I(\widehat{\eta}_{a}(x^{(a)}_j)\geq t)} \qquad \text{ and } \qquad \widehat{P}(Y=1|A=a)=\frac{1}{n_a}\sum\limits_{i=1}^{n_a}y^{(a)}_j.$$
We only divide by nonzero quantities here and below.
To ensure that the quantities we divide by are nonzero, we restrict to $t_a\in[0,\max_j(\widehat\eta_a(x^{(a)}_{j}))]$ when evaluating $\widehat{\text{PPV}}_{a}(t_a)$.
We check whether $\min_{a}\, \widehat{\text{PPV}}_{a}(c)\geq\max_a\, \widehat{P}(Y=1|A=a) $.\footnote{One could modify this to allow some slack; and perform a formal statistical hypothesis test of our sufficient condition.}
If this is not satisfied, we recommend considering other fairness measures, as predictive parity may not be appropriate in this case, see the discussion after Theorem \ref{thm-imGWT}.
If it is satisfied,
we then move to step 3 to adjust the thresholds of the classifier aiming for predictive parity.
\textbf{Step 3.} In the last step, we find the optimal thresholds under fairness.
Based on Theorem~\ref{thm-GWTR}, we consider the following deterministic classifiers:
\begin{equation}\label{classifier1}
\tilde{f}(x,a;t_1,t_2,...,t_{|\mathcal{A}|})=I\left(\widehat\eta_{a}(x)\geq t_a\right),
\end{equation}
where $\widehat{\eta}$ is the estimate of $\eta$ from the first step, and $t_a$, $a\in\mathcal{A}$, are parameters to learn.
We use the following strategy to estimate $t_a$, $a\in\mathcal{A}$: First, we fix the threshold for the group with $a=1$, say $t$.
The positive predictive value for this group
can then be estimated by $\widehat{\text{PPV}}_{1}(t)$.
To achieve predictive parity, we need to find thresholds for the other groups such that the positive group-wise predictive values are the same\footnote{Since a sample mean $n^{-1}\sum_{i=1}^n Z_i$ of iid random variables $Z_i$ has a variability of order $O_P(n^{-1/2})$, even if the true predictive parities are equal, the empirical versions may differ by $O_P(n^{-1/2})$. However, in our case we simply find the values $t_a,t$ for which they are as close as possible.}, i.e., find $t_a$, $a=2,3,\ldots,|\mathcal{A}|$, such that
\begin{equation}\label{thresh-estimate}
\widehat{\text{PPV}}_{a}(t_a)
\approx
\widehat{\text{PPV}}_{1}(t)
, \ \ \ \text{ for } a = 2,3,...,|\mathcal{A}|.
\end{equation}
As proved in Lemma \ref{lem1}, the positive predictive value for each group in the population is non-decreasing with the respective threshold $t_a$.
As a consequence, we can search over
$t_a$, $a=2,3,\ldots,|\mathcal{A}|$,
efficiently via, for instance, the bisection method.\footnote{The empirical PPV is only approximately monotonic, but this does not cause problems.}
Correspondingly, we consider the following range of $t$: $\mathcal{T}=[t_{\min},\max_j \eta_1(x^{(1)}_{j})]$ with
$$t_{\min} = \min\left\{t:
\widehat{\text{PPV}}_{1}(t)\ge \max_a\widehat{P}(Y=1|A=a)
\right\}.$$
We denote by $\widehat t_a(t)$, $a=2,3,\ldots,|\mathcal{A}|$, the estimated thresholds given by \eqref{thresh-estimate}, writing $\widehat t_1(t)=t$ for convenience.
We consider the classifier \eqref{classifier1} with these thresholds:
\begin{equation*
\widehat{f}(x,a,t)
=\tilde{f}\left(x,a;\widehat t_1(t),\widehat t_2(t),...,\widehat t_{|\mathcal{A}|}(t)\right)
=I\left(\widehat\eta_{a}(x)\geq \widehat t_a(t)\right).
\end{equation*}
Finally, we find $t$ that minimizes the cost-sensitive risk on the training data by searching over a grid $\mathcal{T}_n$ within $\mathcal{T}$:
$
\widehat{t}=\underset{t\in \mathcal{T}_n}{\text{argmin}}\left\{\frac1n\sum_{i=1}^n c^{(1-y_i)}(1-c^{y_i})I(y_i\neq \widehat{f}(x_i,a_i,t))\right\}.
$$
Our final estimator of the fair Bayes-optimal classifier is $\widehat{f}_{PP} = \widehat f_{\widehat t}$.
The FairBayes-DPP algorithm is related to the algorithms proposed for other fairness measures in \cite{ZDC2022}, where a binary protected attribute is considered and closed-form optimal thresholds are derived.
In contrast, FairBayes-DPP can handle multi-class protected attributes and does not rely on closed-form thresholds.
Similar to the methods from \cite{ZDC2022}, our algorithm enforces fairness
only in the fast second step, where no gradient-based technique is applied.
Thus, it is computationally efficient and the non-convexity of fairness constraint is no longer problematic.
Our experimental results demonstrate that our method removes disparities and preserves accuracy.
\section{Experiments}
\begin{table
\caption{Classification accuracy and DPP of the true fair Bayes-optimal classifier and our estimator trained via logistic regression on a synthetic data example. See Section \ref{synth} for details.}
\label{table_synth}
\vspace{-0.2cm}
\begin{center}
\setlength{\tabcolsep}{5.7pt}
\renewcommand{\arraystretch}{0.95}
\begin{small}
\begin{sc}
\begin{tabular}{cc|cc|cc|cc}
\hline
\multicolumn{4}{c|}{Theoretical Value} & \multicolumn{4}{c}{Logistic regression} \\\hline
\multicolumn{2}{c|}{Fair} &\multicolumn{2}{c|}{Unconstrained} &\multicolumn{2}{c|}{FairBayes-DPP} &\multicolumn{2}{c}{Unconstrained} \\\hline
$p$ & ACC &DPP& ACC & DPP& $\widehat{\text{ACC}}$&$\widehat{\text{DPP}}$& $\widehat{\text{ACC}}$ \\\hline
0.2 & 0.814 & 0.000 & 0.814 & 0.049\,(0.036) & 0.813\,(0.005) & 0.046\,(0.037) & 0.813\,(0.005)\\
0.3 & 0.794 & 0.024 & 0.794 & 0.037\,(0.029) & 0.794\,(0.006) & 0.040\,(0.033) & 0.794\,(0.005)\\
0.4 & 0.781 & 0.050 & 0.781 & 0.035\,(0.029) & 0.781\,(0.006) & 0.054\,(0.029) & 0.782\,(0.005)\\
0.5 & 0.775 & 0.078 & 0.777 & 0.042\,(0.032) & 0.775\,(0.006) & 0.081\,(0.036) & 0.777\,(0.006)\\
0.6 & 0.778 & 0.113 & 0.781 & 0.038\,(0.031) & 0.778\,(0.006) & 0.113\,(0.037) & 0.781\,(0.006)\\\hline
\end{tabular}
\end{sc}
\end{small}
\end{center}
\vspace{-0.4cm}
\end{table}
\subsection{Synthetic Data}
\label{synth}
We first study a synthetic dataset to compare our method with the true Bayes-optimal fair classifier derived analytically using the true data distribution.
\textbf{Statistical model.}
Let $X=(X_1,X_2)\in \mathbb{R}^2$ be a generic feature, $A\in\{0,1\}$ be the protected attribute and $Y\in\{0,1\}$ be the label.
We generate $A$ and $Y$ according to the probabilities $P(A=1)$, $P(Y=1|A=1)$ and $P(Y=1|A=0)$, specified below.
Conditional on $A=a$ and $Y=y$, $X$ is generated from a bivariate
Gaussian distribution $N((2a-1,2y-1)^\top,2^2I_2)$, where $I_p$ is the $p$-dimensional identity covariance matrix.
In this model, $\eta_a(x)$ has a closed form, and we use it to find the true fair Bayes-optimal classifier numerically under the Condition \ref{suff-con}. More details about this synthetic model can be found in Section \ref{appendix-synth}.
\textbf{Experimental setting.}
We randomly sample $50,000$ training data points and $5,000$ test data points.
In the Gaussian case, the Bayes-optimal classifier is linear in $x$
and thus we employ logistic regression to learn $\eta_1(\cdot)$ and $\eta_0(\cdot)$.
We then
search over a grid with spacings equal to $0.001$ over the range we identified in Section \ref{sec-alg}
for the empirically optimal thresholds under fairness.
We denote $\widehat{f}$ and $\widehat{f}_{PPV}$ the estimators of the unconstrained and fair Bayes-optimal classifiers, respectively.
We first evaluate the FairBayes-DPP algorithm under the Condition \ref{suff-con}. We set the cost parameter $c=0.5$, while $P(A=1)=0.3$ and $P({Y=1|A=0})=0.2$.
It can be calculated that $P(Y=1|\,\eta_0(X)>0.5,A=0)\approx 0.613$, using \eqref{suff-gaussian} in the appendix.
To consider settings with varied levels of fairness in the population,
we vary $p=P({Y=1|A=1})$ from $0.2$ to $0.6$, with the DPP of unconstrained Bayes-optimal classifier grows from $0$ to $0.113$.
Table \ref{table_synth} presents the classification accuracy and DPP of the true fair Bayes-optimal classifier and FairBayes-DPP trained via logistic regression over 100 simulations\footnote{Here, the randomness of the experiment is due to the random generation of the synthetic data.}.
Our first observation is that, under predictive parity, the accuracy of the true unconstrained and fair Bayes-optimal classifiers is almost identical, indicating that predictive parity under Condition \ref{suff-con} requires a very small loss of accuracy.
This finding is consistent with the results in \cite{LSMH2019} that sufficiency-based measures are favored by unconstrained learning.
Second, our FairBayes-DPP method closely tracks the behavior of the fair Bayes-optimal classifier, controlling the accuracy metric ACC and unfairness metric DPP on the test data effectively.
When $|P({Y=1|A=1})-P({Y=1|A=0})|$ is small, FairBayes-DPP performs similarly to the unconstrained classifier.
However, when the data is biased against protected groups and $|P({Y=1|A=1})-P({Y=1|A=0})|$ is large,
FairBayes-DPP mitigates the disparity of the unconstrained classifier effectively, while preserving model accuracy. We further conduct extensive simulations to evaluate the FairBayes-DPP algorithm with different model and training setups, as shown in the appendix. In particular, we also consider the multi-class protected attribute case.
\subsection{Empirical Data Analysis}
{\bf Dataset.} We test FairBayes-DPP on two benchmark datasets for fair classification: ``Adult'' \cite{Dua2019} and ``COMPAS'' \cite{JJSL2016}.
For each dataset, we randomly sample (with replacement) 70\%, 50\% and 30\% as the training, validation and test set, respectively.
To further test the performance of our algorithm on a large-scale dataset, we conduct experiments on the CelebFaces Attributes (CelebA) Dataset \cite{CelebAdata}.
\begin{compactitem}
\item {\it Adult:} The target variable $Y$ is whether the income of an individual is more than \$50,000. Age, marriage status, education level and other related variables are included in $X$, and the protected attribute $A$ refers to gender.
\item {\it COMPAS:} In the COMPAS dataset, the target is to predict recidivism. Here $Y$ indicates whether or not a criminal will reoffend, while $X$ includes prior criminal records, age and an indicator of misdemeanor.
The protected attribute $A$ is the race of an individual,``white-vs-non-white''.
\item {\it CelebA:} CelebA dataset is a large-scale dataset with more than 200,000 face images,
each with 40 attributes (including protected attribute ``gender'' and other 39 different attributes for prediction tasks).
Our goal is to predict the face attributes $Y$ based on the images $X$ and remove bias with respect to gender $A$ from the output.
\end{compactitem}
\begin{figure}[t]
\centering
\includegraphics[scale=0.36]{both}
\vspace{-0.2cm}
\caption{Accuracy and difference in predictive parity on the Adult and COMPAS datasets.}
\vspace{-0.3cm}
\label{fig-ad_co}
\end{figure}
\textbf{Experimental setting.}
As algorithms for predictive parity are rarely considered in the literature, we use unconstrained learning as a baseline for our experiments.
For the ``Adult'' and ``COMPAS'' datasets, we adopt the same training setting as in \cite{CHS2020, ZDC2022}. The conditional probabilities are learned via a three-layer fully connected neural network architecture with 32 hidden neurons per layer.
For ``CelebA'', we apply the training setting from \cite{WQK2020}. We learn the conditional probabilities by training a ResNet50 mode; \citep{HZRS15Res}, pretrained on ImageNet \cite{DDSLKL2009}. For all the datasets,
Over the course of training the model on
the training set, we select the one with best performance on the validation set.
In addition, we learn the optimal thresholds over the validation set to avoid overfitting.
All experiments use PyTorch. We refer
readers to the Appendix for more training details, including optimizer, learning rates, batch sizes and training epochs.
We repeat the experiment 100 times for the Adult and COMPAS datasets and five times for the CelebA dataset.\footnote{For the Adult and COMPAS datasets, the randomness of the experiment comes from the random selection of the training, validation and test data, as well as the
stochasticity of the batch selection in the optimization algorithm.
For the CelebA dataset, the randomness is caused by the stochasticity of the optimization method.}
Figure \ref{fig-ad_co} presents the average performances of FairBayes-DPP and unconstrained learning on the Adult and COMPAS datasets.
Our method achieves almost the same accuracy as the unconstrained classifier, and has a smaller disparity.
To better compare our fair classifier with the unconstrained one, we use the paired $t$-test to compare the DPP of the proposed algorithm ($\text{DPP}_{Fair}$) and of unconstrained learning ($\text{DPP}_{Base}$).
We consider the following one sided test:
$$\mathcal{H}_0: \text{DPP}_{fair}=\text{DPP}_{Base} \ \ \ \text{ v.s. } \ \ \ \mathcal{H}_1:\text{DPP}_{fair}<\text{DPP}_{Base} .$$
The $p$-values of the tests are $3.90\times 10^{-4}$ for the Adult dataset and $3.09\times 10^{-8}$
for the COMPAS dataset.
In both cases, these results provide evidence
that our FairBayes-DPP method achieves a smaller disparity than unconstrained learning.
Finally, we test FairBayes-DPP on the CelebA dataset,
Here, we only consider 27 attributes\footnote{Among the 39 attributes, 12 are heavily skewed with $ \min(P(Y=1|M),P(Y=1|F))<0.01$ or $\max(P(Y=1|M),P(Y=1|F))>0.99$ (where $M$ represents Male and $F$ represents Female) in the training, validation or test set. They are:
``5 o'Clock Shadow'', ``Bald'', ``Double Chin'', ``Goatee'', ``Gray Hair'', ``Heavy Makeup'', ``Mustache'', ``No Beard'', ``Rosy Cheeks'', ``Sideburns'', ``Wearing Lipstick'' and ``Wearing Necktie''.}
with $0.01\leq P(Y=1|M),P(Y=1|F)\leq 0.99$ in the training, validation, and test sets; to ensure that the training, validation and test sample sizes are large enough for each subgroup.
We further identify one attribute, ``Young'', that violates Condition \ref{suff-con}. We calculate the per-attribute accuracies and DPPs on the test set. Table \ref{Table-celeba} presents the results of the first six attributes; the remaining results are in the appendix.
As we can see, even for the large-scale CelebA dataset with high dimensional image features, our algorithm mitigates the gender bias effectively, with almost no loss of accuracy.
\begin{table
\caption{Per-attribute accuracy and DPP of the FairBayes-DPP algorithm and unconstrained optimization on the CelebA dataset.}
\label{Table-celeba}
\vspace{-0.2cm}
\begin{center}
\setlength{\tabcolsep}{4.8pt}
\renewcommand{\arraystretch}{0.98}
\begin{small}
\begin{sc}
\begin{tabular}{l|cc|cc}\hline
\multirow{2}[1]{*}{Attributes} & \multicolumn{2}{c|}{Per-attribute Accuracy}& \multicolumn{2}{c}{Per-attribute DDP}\\\cline{2-5}
& FairBayes-DPP& Unconstrained & FairBayes-DPP& Unconstrained \\\hline
Arched Eyebrows & 0.838 & 0.838 & 0.028 & 0.105 \\
Attractive & 0.825 & 0.827 & 0.080 & 0.157 \\
Bags Under Eyes & 0.853 & 0.852 & 0.021 & 0.071 \\
Bangs & 0.959 & 0.958 & 0.012 & 0.081 \\
Big Lips & 0.706 & 0.718 & 0.026 & 0.123 \\
Big Nose & 0.845 & 0.845 & 0.076 & 0.130 \\\hline
\end{tabular}
\end{sc}
\end{small}
\end{center}
\vspace{-0.5cm}
\end{table}
\section{Summary and Discussion}
In this paper, we investigated fair Bayes-optimal classifiers under predictive parity. We proved that when the overall performances of different protected groups vary only moderately, all fair Bayes-optimal classifiers under predictive parity are GWTRs. We further proposed a post-processing algorithm to estimate the optimal GWTR. The derived post-processing algorithm removes the disparity in unconstrained classifiers effectively, while preserving a similar test accuracy.
However, when our sufficient condition is not satisfied, the fair Bayes-optimal classifier under predictive parity may lead to within-group unfairness for the minority group. In the current literature, many algorithms directly apply penalized/constrained optimization to impose fairness.
Our negative finding, however, is an important reminder that careful analysis is required
before employing a fairness measure.
The improper use of a measure may result in severe unintended consequences.
\bibliographystyle{plain}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,851 |
/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
/*
Copyright (C) 2012 Klaus Spanderen
This file is part of QuantLib, a free-software/open-source library
for financial quantitative analysts and developers - http://quantlib.org/
QuantLib is free software: you can redistribute it and/or modify it
under the terms of the QuantLib license. You should have received a
copy of the license along with this program; if not, please email
<quantlib-dev@lists.sf.net>. The license is also available online at
<http://quantlib.org/license.shtml>.
This program is distributed in the hope that it will be useful, but WITHOUT
ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
FOR A PARTICULAR PURPOSE. See the license for more details.
*/
/*! \file sparsematrix.hpp
\brief typedef for boost sparse matrix class
*/
#ifndef quantlib_sparse_matrix_hpp
#define quantlib_sparse_matrix_hpp
#include <ql/qldefines.hpp>
#if !defined(QL_NO_UBLAS_SUPPORT)
#include <ql/math/array.hpp>
#if defined(QL_PATCH_MSVC)
#pragma warning(push)
#pragma warning(disable:4180)
#pragma warning(disable:4127)
#endif
#if defined(__clang__) && BOOST_VERSION > 105300
#pragma clang diagnostic push
#pragma clang diagnostic ignored "-Wunused-function"
#endif
#include <boost/numeric/ublas/matrix_sparse.hpp>
#if defined(QL_PATCH_MSVC)
#pragma warning(pop)
#endif
#if defined(__clang__) && BOOST_VERSION > 105300
#pragma clang diagnostic pop
#endif
namespace QuantLib {
typedef boost::numeric::ublas::compressed_matrix<Real> SparseMatrix;
typedef boost::numeric::ublas::matrix_reference<SparseMatrix>
SparseMatrixReference;
inline Disposable<Array> prod(const SparseMatrix& A, const Array& x) {
Array b(x.size());
for (Size i=0; i < A.filled1()-1; ++i) {
const Size begin = A.index1_data()[i];
const Size end = A.index1_data()[i+1];
Real t=0;
for (Size j=begin; j < end; ++j) {
t += A.value_data()[j]*x[A.index2_data()[j]];
}
b[i]=t;
}
return b;
}
}
#endif
#endif
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,450 |
{"url":"https:\/\/lavelle.chem.ucla.edu\/forum\/viewtopic.php?f=149&t=44301","text":"## 15.37\n\n$\\frac{d[R]}{dt}=-k[R]^{2}; \\frac{1}{[R]}=kt + \\frac{1}{[R]_{0}}; t_{\\frac{1}{2}}=\\frac{1}{k[R]_{0}}$\n\nkanikubari\nPosts: 30\nJoined: Fri Dec 07, 2018 12:18 am\n\n### 15.37\n\nHow would you go about finding the equation to use for this problem?\n\nAshley Zhu 1A\nPosts: 69\nJoined: Fri Sep 28, 2018 12:16 am\n\n### Re: 15.37\n\nThis problem states that it is a first order decomposition of SO2Cl2, so you know that the only reactant is SO2Cl2 because that's what decomposition means. All parts of the question don't actually require the use of a complete chemical equation for the decomposition, so you don't actually need to figure out what it decomposes into. With the knowledge that SO2Cl2 is the only reactant, you can apply all of the first order reaction equations.","date":"2020-08-10 16:29:56","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 1, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.3923775553703308, \"perplexity\": 1092.1489622988142}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-34\/segments\/1596439736057.87\/warc\/CC-MAIN-20200810145103-20200810175103-00163.warc.gz\"}"} | null | null |
{"url":"http:\/\/www.computer.org\/csdl\/trans\/tc\/1978\/12\/01675017-abs.html","text":"Subscribe\nIssue No.12 - December (1978 vol.27)\npp: 1144-1152\nC. Halatsis , Digital Systems Laboratory, Computer Center, Nuclear Research Center\nABSTRACT\nThe paper presents systematic procedures for decomposing a sequential machine into submachines some or all of which are realized by polylinear sequential circuits. This polylinear decomposition is based upon classes of subsets of the set of states which possess the substitution property rather than upon partitions with substitution property. These classes are easily found using the backward state transition function of the machine and enables one to treat uniformly and simultaneously the problems of state minimization, machine decomposition in the classical or the polylinear way, finding the lattice of the partitions with substitution property, and periodic decomposition based upon classes that correspond to cyclic partitions. Procedures are developed for deriving optimal all polylinear decompositions and \"good\" partial polylinear decompositions of a machine.\nINDEX TERMS\nsynchronous sequential machines, Backward state transition function, connected class, lattice of partitions with substitution property, periodic decomposition, polylinear decomposition, state minimization, strongly connected class\nCITATION\nC. Halatsis, M. Sigala, G. Philokyprou, \"Polylinear Decomposition of Synchronous Sequential Machines\", IEEE Transactions on Computers, vol.27, no. 12, pp. 1144-1152, December 1978, doi:10.1109\/TC.1978.1675017","date":"2015-04-21 09:51:36","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8944206833839417, \"perplexity\": 3832.6462236706157}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2015-18\/segments\/1429246641266.56\/warc\/CC-MAIN-20150417045721-00268-ip-10-235-10-82.ec2.internal.warc.gz\"}"} | null | null |
ROBERTO MANCINI believes teams are so scared of Manchester United they are beaten before they start.
City boss Mancini cranked up the stakes ahead of tonight's Manchester derby by suggesting clubs have a "soft mentality" and roll over when they face United – but try harder against City.
Mancini said: "No one plays well against United because they only play with fear.
"Every team that plays against them plays very soft because they think this game is difficult – they can't beat them. Their mentality is poor.
"They think they can do nothing. But it is not true. If you play strong against United, you can beat them like they can beat us.
Mancini was stunned by Sunderland's limp display against United last Saturday that prompted Martin O'Neill's dismissal.
Sunderland have beaten City three years running at the Stadium of Light and were the only team to take a point at the Etihad Stadium last season.
But Mancini admits City do not carry the same fear factor for opponents yet because they have not been at the top for as long as their neighbours.
Mancini refuses to accept that United are a better team than City, even though they could stretch their lead to 18 points by completing the double over them tonight.
Mancini maintains that, had he been able to sign top targets such as Robin van Persie, Eden Hazard and Javi Martinez last summer, City would have retained the crown. | {
"redpajama_set_name": "RedPajamaC4"
} | 2,922 |
This is a sample README file for a Brackets extensions.
##Install from URL
1. Open the the Extension Manager from the File menu
2. Copy paste the URL of the github repo or zip file
##Install from file system
1. Download this extension using the ZIP button above and unzip it.
2. Copy it in Brackets' `/extensions/user` folder by selecting `Help > Show Extension Folder` in the menu.
3. Reload Brackets.
##Instructions
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,812 |
The Class B XI engines of the Royal Bavarian State Railways (Königlich Bayerische Staatsbahn) were built between 1895 and 1900 by the firm of Maffei for deployment in Bavaria. The first delivery comprised 39 vehicles with two-cylinder, saturated steam operation, a further 100 two-cylinder compound locomotives followed in the period up to 1900.
The Deutsche Reichsbahn took over 8 engines from the first batch as Class 36.7, with operating numbers 36 701 to 36 708, and 76 of the second batch, with numbers 36 751 to 36 826.
These steam locomotives were equipped with Bavarian 3 T 12, 3 T 14.5 and 2'2' T 18 tenders. These vehicles were the first locos that were give a four-axle tender.
See also
Royal Bavarian State Railways
List of Bavarian locomotives and railbuses
References
This page incorporates a translation of the equivalent page on the German language Wikipedia
4-4-0 locomotives
B 11
Standard gauge locomotives of Germany
Railway locomotives introduced in 1892
2′B n2 locomotives
2′B n2v locomotives
Maffei locomotives
Krauss locomotives
Passenger locomotives | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 23 |
// ------------------------------------------------------------------------------
// <auto-generated>
// This code was generated by SpecFlow (http://www.specflow.org/).
// SpecFlow Version:2.4.0.0
// SpecFlow Generator Version:2.4.0.0
//
// Changes to this file may cause incorrect behavior and will be lost if
// the code is regenerated.
// </auto-generated>
// ------------------------------------------------------------------------------
#region Designer generated code
#pragma warning disable
namespace Augurk.CSharpAnalyzer.Specifications.Features
{
using TechTalk.SpecFlow;
[System.CodeDom.Compiler.GeneratedCodeAttribute("TechTalk.SpecFlow", "2.4.0.0")]
[System.Runtime.CompilerServices.CompilerGeneratedAttribute()]
[Microsoft.VisualStudio.TestTools.UnitTesting.TestClassAttribute()]
public partial class AnalyzeACProjectWithASingleTestProjectFeature
{
private static TechTalk.SpecFlow.ITestRunner testRunner;
private Microsoft.VisualStudio.TestTools.UnitTesting.TestContext _testContext;
#line 1 "Analyze.feature"
#line hidden
public virtual Microsoft.VisualStudio.TestTools.UnitTesting.TestContext TestContext
{
get
{
return this._testContext;
}
set
{
this._testContext = value;
}
}
[Microsoft.VisualStudio.TestTools.UnitTesting.ClassInitializeAttribute()]
public static void FeatureSetup(Microsoft.VisualStudio.TestTools.UnitTesting.TestContext testContext)
{
testRunner = TechTalk.SpecFlow.TestRunnerManager.GetTestRunner(null, 0);
TechTalk.SpecFlow.FeatureInfo featureInfo = new TechTalk.SpecFlow.FeatureInfo(new System.Globalization.CultureInfo("en-US"), "Analyze a C# project with a single test project", null, ProgrammingLanguage.CSharp, ((string[])(null)));
testRunner.OnFeatureStart(featureInfo);
}
[Microsoft.VisualStudio.TestTools.UnitTesting.ClassCleanupAttribute()]
public static void FeatureTearDown()
{
testRunner.OnFeatureEnd();
testRunner = null;
}
[Microsoft.VisualStudio.TestTools.UnitTesting.TestInitializeAttribute()]
public virtual void TestInitialize()
{
if (((testRunner.FeatureContext != null)
&& (testRunner.FeatureContext.FeatureInfo.Title != "Analyze a C# project with a single test project")))
{
global::Augurk.CSharpAnalyzer.Specifications.Features.AnalyzeACProjectWithASingleTestProjectFeature.FeatureSetup(null);
}
}
[Microsoft.VisualStudio.TestTools.UnitTesting.TestCleanupAttribute()]
public virtual void ScenarioTearDown()
{
testRunner.OnScenarioEnd();
}
public virtual void ScenarioInitialize(TechTalk.SpecFlow.ScenarioInfo scenarioInfo)
{
testRunner.OnScenarioInitialize(scenarioInfo);
testRunner.ScenarioContext.ScenarioContainer.RegisterInstanceAs<Microsoft.VisualStudio.TestTools.UnitTesting.TestContext>(_testContext);
}
public virtual void ScenarioStart()
{
testRunner.OnScenarioStart();
}
public virtual void ScenarioCleanup()
{
testRunner.CollectScenarioErrors();
}
[Microsoft.VisualStudio.TestTools.UnitTesting.TestMethodAttribute()]
[Microsoft.VisualStudio.TestTools.UnitTesting.DescriptionAttribute("When calls directly into a single entrypoint")]
[Microsoft.VisualStudio.TestTools.UnitTesting.TestPropertyAttribute("FeatureTitle", "Analyze a C# project with a single test project")]
public virtual void WhenCallsDirectlyIntoASingleEntrypoint()
{
TechTalk.SpecFlow.ScenarioInfo scenarioInfo = new TechTalk.SpecFlow.ScenarioInfo("When calls directly into a single entrypoint", null, ((string[])(null)));
#line 3
this.ScenarioInitialize(scenarioInfo);
this.ScenarioStart();
#line 5
testRunner.Given("\'Cucumis.Specifications\' contains feature files", ((string)(null)), ((TechTalk.SpecFlow.Table)(null)), "Given ");
#line 6
testRunner.When("an analysis is run", ((string)(null)), ((TechTalk.SpecFlow.Table)(null)), "When ");
#line hidden
TechTalk.SpecFlow.Table table1 = new TechTalk.SpecFlow.Table(new string[] {
"Kind",
"Local",
"Expression/Signature"});
table1.AddRow(new string[] {
"When",
"",
"entrypoint is invoked directly"});
table1.AddRow(new string[] {
"Public",
"true",
"Cucumis.Gardener.PlantGherkin(), Cucumis"});
table1.AddRow(new string[] {
"Public",
"",
"System.Console.WriteLine(string), mscorlib"});
#line 7
testRunner.Then("the resulting report contains \'When entrypoint is invoked directly\'", ((string)(null)), table1, "Then ");
#line hidden
this.ScenarioCleanup();
}
[Microsoft.VisualStudio.TestTools.UnitTesting.TestMethodAttribute()]
[Microsoft.VisualStudio.TestTools.UnitTesting.DescriptionAttribute("Entrypoint is surrounded by other invocations")]
[Microsoft.VisualStudio.TestTools.UnitTesting.TestPropertyAttribute("FeatureTitle", "Analyze a C# project with a single test project")]
public virtual void EntrypointIsSurroundedByOtherInvocations()
{
TechTalk.SpecFlow.ScenarioInfo scenarioInfo = new TechTalk.SpecFlow.ScenarioInfo("Entrypoint is surrounded by other invocations", null, ((string[])(null)));
#line 13
this.ScenarioInitialize(scenarioInfo);
this.ScenarioStart();
#line 15
testRunner.Given("\'Cucumis.Specifications\' contains feature files", ((string)(null)), ((TechTalk.SpecFlow.Table)(null)), "Given ");
#line 16
testRunner.When("an analysis is run", ((string)(null)), ((TechTalk.SpecFlow.Table)(null)), "When ");
#line hidden
TechTalk.SpecFlow.Table table2 = new TechTalk.SpecFlow.Table(new string[] {
"Kind",
"Local",
"Expression/Signature"});
table2.AddRow(new string[] {
"When",
"",
"entrypoint is surrounded by other invocations"});
table2.AddRow(new string[] {
"Public",
"",
"System.Console.WriteLine(string), mscorlib"});
table2.AddRow(new string[] {
"Public",
"true",
"Cucumis.Gardener.PlantGherkin(), Cucumis"});
table2.AddRow(new string[] {
"Public",
"",
"System.Console.WriteLine(string), mscorlib"});
table2.AddRow(new string[] {
"Public",
"",
"Cucumis.Specifications.Support.ConsoleWriter.WriteDefaultMessage(), Cucumis.Speci" +
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table2.AddRow(new string[] {
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"System.Console.WriteLine(string), mscorlib"});
#line 17
testRunner.Then("the resulting report contains \'When entrypoint is surrounded by other invocations" +
"\'", ((string)(null)), table2, "Then ");
#line hidden
this.ScenarioCleanup();
}
[Microsoft.VisualStudio.TestTools.UnitTesting.TestMethodAttribute()]
[Microsoft.VisualStudio.TestTools.UnitTesting.DescriptionAttribute("When invokes two seperate entrypoints")]
[Microsoft.VisualStudio.TestTools.UnitTesting.TestPropertyAttribute("FeatureTitle", "Analyze a C# project with a single test project")]
public virtual void WhenInvokesTwoSeperateEntrypoints()
{
TechTalk.SpecFlow.ScenarioInfo scenarioInfo = new TechTalk.SpecFlow.ScenarioInfo("When invokes two seperate entrypoints", null, ((string[])(null)));
#line 26
this.ScenarioInitialize(scenarioInfo);
this.ScenarioStart();
#line 28
testRunner.Given("\'Cucumis.Specifications\' contains feature files", ((string)(null)), ((TechTalk.SpecFlow.Table)(null)), "Given ");
#line 29
testRunner.When("an analysis is run", ((string)(null)), ((TechTalk.SpecFlow.Table)(null)), "When ");
#line hidden
TechTalk.SpecFlow.Table table3 = new TechTalk.SpecFlow.Table(new string[] {
"Kind",
"Local",
"Expression/Signature"});
table3.AddRow(new string[] {
"When",
"",
"two separate entrypoints are invoked"});
table3.AddRow(new string[] {
"Public",
"true",
"Cucumis.Gardener.PlantGherkin(), Cucumis"});
table3.AddRow(new string[] {
"Public",
"",
"System.Console.WriteLine(string), mscorlib"});
table3.AddRow(new string[] {
"Public",
"true",
"Cucumis.Gardener.WaterPlants(), Cucumis"});
table3.AddRow(new string[] {
"Public",
"",
"System.Console.WriteLine(string), mscorlib"});
#line 30
testRunner.Then("the resulting report contains \'When two separate entrypoints are invoked\'", ((string)(null)), table3, "Then ");
#line hidden
this.ScenarioCleanup();
}
[Microsoft.VisualStudio.TestTools.UnitTesting.TestMethodAttribute()]
[Microsoft.VisualStudio.TestTools.UnitTesting.DescriptionAttribute("When an asynchronous entrypoint is invoked")]
[Microsoft.VisualStudio.TestTools.UnitTesting.TestPropertyAttribute("FeatureTitle", "Analyze a C# project with a single test project")]
public virtual void WhenAnAsynchronousEntrypointIsInvoked()
{
TechTalk.SpecFlow.ScenarioInfo scenarioInfo = new TechTalk.SpecFlow.ScenarioInfo("When an asynchronous entrypoint is invoked", null, ((string[])(null)));
#line 38
this.ScenarioInitialize(scenarioInfo);
this.ScenarioStart();
#line 40
testRunner.Given("\'Cucumis.Specifications\' contains feature files", ((string)(null)), ((TechTalk.SpecFlow.Table)(null)), "Given ");
#line 41
testRunner.When("an analysis is run", ((string)(null)), ((TechTalk.SpecFlow.Table)(null)), "When ");
#line hidden
TechTalk.SpecFlow.Table table4 = new TechTalk.SpecFlow.Table(new string[] {
"Kind",
"Local",
"Expression/Signature"});
table4.AddRow(new string[] {
"When",
"",
"an asynchronous entrypoint is invoked"});
table4.AddRow(new string[] {
"Public",
"true",
"Cucumis.Plant.Procreate(), Cucumis"});
table4.AddRow(new string[] {
"Public",
"",
"System.Threading.Thread.Sleep(int), mscorlib"});
table4.AddRow(new string[] {
"Public",
"",
"System.Threading.Tasks.Task.GetAwaiter(), mscorlib"});
table4.AddRow(new string[] {
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"System.Runtime.CompilerServices.TaskAwaiter.GetResult(), mscorlib"});
#line 42
testRunner.Then("the resulting report contains \'When an asynchronous entrypoint is invoked\'", ((string)(null)), table4, "Then ");
#line hidden
this.ScenarioCleanup();
}
}
}
#pragma warning restore
#endregion
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,484 |
ACCEPTED
#### According to
Index Fungorum
#### Published in
Saccardoa 2: 146 (1963)
#### Original name
Limacinia helianthemi Maire
### Remarks
null | {
"redpajama_set_name": "RedPajamaGithub"
} | 4,049 |
The Sri Lanka Railways M4 is a class of diesel-electric locomotive manufactured in Canada by Montreal Locomotive Works. Fourteen of the manufacturers' model MX-620 were imported in 1975.
This was the longest locomotive in the Sri Lanka Railways previously. All units are still operational. On regular operation they are not used beyond Polgahawela on the main line, and are not used on the hill country section. They used to travel on the full main line regularly until the previous decade.
They are painted in a livery of dark blue, light blue, silver with yellow striping; although No. 752 Point Pedro was painted in a special ICE (Intercity Express) livery.
Fleet
Gallery
References
M04
MLW locomotives
Co-Co locomotives
Railway locomotives introduced in 1975
5 ft 6 in gauge locomotives | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,750 |
Q: How to know the population distribution at the limit with a Leslie matrix So I have a Leslie Matrix:
\begin{bmatrix}0&3&1\\1/2&0&0\\0&7/10&0\end{bmatrix}
And I'm asked to find the population distribution at the limit. (how the poppulation stays after a lot of years).
I can't find it in any textbook nor the internet, I only find how to do it with Markov 2x2 matrices.
Since I've seen a few exercicies like this in past exams I really need to understand it.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,003 |
Q: Confusion over how Cross Origin Resource Sharing (CORS) works From what I understand about CORS, this is how it works: I have a site foo.com which serves a page X. X wants to post data to another domain bar.com. If bar.com is CORS enabled (its headers produce Access-Control-Allow-Origin foo.com) then page X can now send data to bar.com.
As I understand to get CORS to work it's all about settingit up on bar.com, and has nothing to do with foo.com. It's all about making sure bar.com doesn't accept requests from any old domain.
However this really doesn't make sense to me. I thought CORS was designed to enable foo.com to dictate who X is allowed to communicate with. If we go back to the previous example but this time X is compromised by dodgy script so that it sends data secretly to evil.com, how is CORS going to stop that? evil.com is CORS enabled, and set to *, so it will accept requests from anything. That way a user thinking they using a site foo.com, are unwittingly sending data to evil.com.
If it is really all about bar.com protecting itself, then why does it make the browser enforce the policy?. The only conceivable situation in which this makes sense if you have evil.com serving up page Y that impersonates foo.com, that tries to send data to bar.com. But CORS is enforced by the browser, all you'd have to do is make evil.com a proxy that sends faked origin requests to bar.com (data goes from Y to evil.com, evil.com sets its fake origin to foo.com then sends it to bar.com).
It only makes sense to me if it works the other way round. foo.com is CORS enabled, and its headers are set to Access-Control-Allow-Origin bar.com. That way rouge scripts would get denied access evil.com by the browser. It then makes sense for the browser to enforce the policy because its running the scripts that could go rouge. It won't stop rouge sites from trying to send rouge data to bar.com, but bar.com can protect itself with a username/password. If foo.com has endpoints that it's expecting data back from X, then you can embed tokens into X, to ensure evil.com doesn't send data to it instead.
I feel like I'm not understanding something fundamentally important here. Would really appreciate the help.
A:
However this really doesn't make sense to me. I thought CORS was designed to enable foo.com to dictate who X is allowed to communicate with.
No, it's about bar.com controlling use of its content.
But CORS is enforced by the browser, all you'd have to do is make evil.com a proxy that sends faked origin requests to bar.com...
Yup. And if you do, and the people at bar.com notice and care, they disallow requests from your server. You move it, they disallow the new one. Whack-a-mole time. But painful as that game of whack-a-mole is, it's a lot less painful than if the requests come directly from each individual user of foo.com, from their desktop.
Having foo.com enforce what foo.com can do doesn't make any sense. foo.com already enforces what foo.com can do, because it's foo.com that serves foo.com's content and scripts.
A: It isn't about Foo.com, nor about Bar.com. It is about user.
There are two things that CORS protects against. The first is access to resources behind the firewall. The second are resources that are normally protected, unless a request is sent from a browsers with authentication or other sensitive data cookies.
CORS is a Browser technology, with support from servers, that allows foo limited freedom to call outside of its domain. It is a restricted hole punched in the restriction against cross domain scripting.
Anyone can fake the ORIGIN header and create a CORS preflight or simple request -- Of course, anyone can directly connect to the Bar server directly and make the requests without using CORS at all. Any browser can directly connect to bar.com and get data. But a modern browser will not run a script from foo.com that access a bar.com resource. People visiting websites are protected against visiting a site designed to exploit cookies or the fact that the browser is behind the corporate firewall.
So the accepted answer is WRONG. It isn't about bar.com protecting its resources -- it does this through authentication and authorization. You don't have to create a proxy to send CORS requests -- you create a proxy to strip out the CORS requests (automatically responding to the preflight request, and returning the proper headers to the browser, but sending a normal request to bar.com). But you will still need authentication to get bar.com's resources, and foo.com would still need to somehow get you to install a proxy to exploit the cross domain scripting hole that CORS protects against.
But the concluding sentence is correct -- foo.com isn't in control of the resources -- it is the browser, with a quick check with bar.com to ask it if this is something that was intended.
From the OP:
If it is really all about bar.com protecting itself, then why does it
make the browser enforce the policy?. The only conceivable situation
in which this makes sense if you have evil.com serving up page Y that
impersonates foo.com, that tries to send data to bar.com. But CORS is
enforced by the browser, all you'd have to do is make evil.com a proxy
that sends faked origin requests to bar.com (data goes from Y to
evil.com, evil.com sets its fake origin to foo.com then sends it to
bar.com).
evil.com can already contact bar.com -- just like any human using a browser can (or curl or wget, etc). The issue is can evil.com force your browser to connect to bar.com, which may have IP filters, cookies, firewalls, etc protecting it, but javascript can connect to using your browser. So the Browser is the thing that protects the user. By disallowing cross domain scripting. But sometimes it is useful (ex: google apis, or a bank connecting to a bill paying service, etc) to cross domain script. CORS tells the browser that it is OK in this instance.
That isn't to say that there are no holes, or the the standard is the best, or that there aren't holes in implementation in the browser, or that sites are too permissive. But those are different questions...
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,067 |
package com.jgeppert.struts2.jquery.richtext.views.freemarker.tags;
import com.jgeppert.struts2.jquery.richtext.components.Ckeditor;
import com.opensymphony.xwork2.util.ValueStack;
import org.apache.struts2.components.Component;
import org.apache.struts2.views.freemarker.tags.TagModel;
import javax.servlet.http.HttpServletRequest;
import javax.servlet.http.HttpServletResponse;
/**
* @author <a href="http://www.jgeppert.com">Johannes Geppert</a>
* @see Ckeditor
*/
public class CkeditorModel extends TagModel {
public CkeditorModel(ValueStack stack, HttpServletRequest req, HttpServletResponse res) {
super(stack, req, res);
}
@Override
protected Component getBean() {
return new Ckeditor(stack, req, res);
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,884 |
Q: Why I can't query a subset of data from Datastax DSE 5.0.x Graph without getting allow_scan is disabled error? Hi I have disabled scans in my schema.
I know that queries like those wouldn't be allowed:
g.V()
g.V().hasLabel("User")
org.apache.tinkerpop.gremlin.driver.exception.ResponseException: Could
not find an index to answer query clause and graph.allow_scan is
disabled:
I wonder why even those:
g.V().limit(2)
g.V().hasLabel("User").limit(2)
cause the same exception to be thrown! This is frustrating since they are bounded queries and they certainly don't cause full cassandra table scans..
Thanks
A: There's an ongoing discussion about what kind of queries (if any at all) should be allowed with scans disabled. For now the rule is simple: If the initial step is not an index lookup, then it's considered to be a full scan.
It's easy to say that:
g.V().hasLabel("user").limit(2)
...for example should be allowed, but if that is not considered to be a full scan, then what about these:
g.V().hasLabel("user").limit(10)
g.V().hasLabel("user").limit(100)
g.V().hasLabel("user").limit(1000)
g.V().hasLabel("user").limit(10000)
g.V().hasLabel("user").limit(100000)
Where do we draw the line? I'm not expecting you to answer this question, just want to show that it's not as easy as it may seem at first.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,662 |
Race riot lawsuit to be subject of bill, hearing
A published report says a 2003 lawsuit filed by survivors of the 1921 Tulsa race riot will be the subject of legislation and a hearing in Congress next week. Michigan Congressman John Conyers Junior plans to introduce on Monday a bill that would extend the statute of limitations on the lawsuit filed against the city of Tulsa and the state of Oklahoma. Conyers chairs the House Judiciary Committee. A federal judge dismissed the survivors' lawsuit in 2004, stating the statute of limitations had expired 80 years earlier. According to the Tulsa World's Washington bureau, a House subcommittee will hold a hearing Tuesday on the proposed Tulsa-Greenwood Riot and Accountability Act of 2007. Conyers and subcommittee chairman Jerrold Nadler of New York say the riot is worthy of congressional attention because evidence suggests governmental officials deputized and armed the mob and that the National Guard joined in the destruction. Tulsa Mayor Kathy Taylor's couldn't be reached for comment. A spokesman for Governor Brad Henry declined comment, saying the office hadn't been notified about the legislation. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,336 |
package com.chrisrm.idea.ui;
import javax.swing.*;
import javax.swing.plaf.ComponentUI;
/**
* @author Konstantin Bulenkov
*/
public final class MTCheckBoxMenuItemUI extends MTMenuItemUIBase {
public static ComponentUI createUI(final JComponent c) {
return new MTCheckBoxMenuItemUI();
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,034 |
Among the compromises, the 8150 Sunset Blvd. project will see its tallest residential tower reduced by 56 feet.
A planned mixed-use development from Frank Gehry and located on Sunset Boulevard will be moving forward after the Planning and Land Use Management Committee voted unanimously to approve the plans.
As the Los Angeles Times reports, however, a few compromises were required before the project could move forward. As it stands now, the project consists of two residential towers comprising 229 units (38 of which are for low-income residents), 65,000 sf of commercial space, and a pedestrian plaza. The project arrived at these numbers after the area's councilman, David Ryu, said he would not support the development unless the 15-story residential tower was downsized, included more affordable housing units, and had additional parking spots added.
All of these conditions were met causing the tower to shrink from 234 feet to 178 feet, the number of parking spots to increase to 494, and the sidewalks to widened to 15 feet.
Even with the compromises, including the smaller size that makes the new tower more in line with the height of surrounding buildings, some neighboring residents are expressing a bout of NIMBYism. They believe the development will still be out of scale with the surrounding area, will ruin their views, and may produce heavy traffic in the area due to all of the new residents. Despite their concerns, the scaled-back project is moving forward. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,665 |
import React from 'react'
import { Link } from 'react-router'
const HomePage = () => {
return (
<ul className="main-menu">
<li><Link to="play">New Game</Link></li>
<li><Link to="options">Options</Link></li>
<li><Link to="credits">Credits</Link></li>
</ul>
)
}
export default HomePage
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,142 |
Woking Homes would like to wish you all a very Merry Christmas and a Happy New Year.
Our residents Christmas Party will take place on the 14th December. Watch this space for some lovely photos. Unfortunately due to the dining room development we are limited for space this year. Therefore the party will be for residents only. We apologise for this as we know that many residents families enjoy the occasion.
Remembrance windows The ladies group have produced some beautiful stained glass poppy windows with the assistance of Marjorie our Activities organiser. Click on Ladies Group above to see photos. | {
"redpajama_set_name": "RedPajamaC4"
} | 6,727 |
She had every reason to stop singing – but she refused to give up.
Cathrine Sadolin is among the leading voice researchers in the world.
Her writing and master classes have now developed into a vocal Institute with a team of trained associates who work all over the globe.
We thought it would be interesting to ask Cathrine to tell us about her personal journey with vocal technique.
You've taught singing technique to thousands of singers, but when did you first 'bump into' the need for vocal technique in your own singing?
When I opened my mouth I had every issue imaginable! You see, I was not your typical singer. I originally began singing to help my asthma. Of course I fell in love with music and performance, but was plagued by difficulties: I was constantly hoarse, had lots of air in my voice, a restricted range and no volume.
How did you deal with these issues?
At first I didn't. My voice teachers didn't have much experience dealing with these problems with their students. They were great with their suggestions of how to fine tune technique and harness the power of imagination, but none of this helped me. I needed more basic help and this didn't seem available.
At this time you were beginning to perform – how did you cope?
I relied on my acting and interpretive abilities to pull me through; however, I was struggling very seriously. I tried different vocal teachers, but I was getting nodules and knew I just couldn't keep going on.
Tell us about your turning point.
I decided to lock myself in a room with the latest books on anatomy and physiological research on the voice. I was determined not to emerge until I solved these problems for myself!
Was there a central insight on technique that unlocked the door to your vocal issues?
Yes. The vocal folds provide the vibrations necessary to make sound, but the tone of your sound is created in the vocal tract above the vocal cords. This means that, with proper support, it is possible to make any sound you want – in any genre you want – without hurting the vocal cords.
Click here for more on how Cathrine applied these insights.
At what point did you decide that you wanted to share what you were learning with other singers?
This happened naturally. I had many friends who were rock singers; they were coming to me complaining about hoarseness. Even though I hadn't yet explored their unique sounds in my own singing, I was able to share with them the basic physiology about how sound is produced and we began to apply this to making the kinds of sounds they wanted to make.
Are you saying that classical and contemporary singing technique are the same?
I was actually told that if I wanted to sing classical I should never dream about singing another style – for example, rock. I knew that couldn't be true. Just think about your hand: it can do different things though it is the same hand. So, too, is this true with the voice.
Does this mean that there is one vocal technique?
Certainly there are different techniques that one would employ above the vocal folds to make different sounds you wish to make – I discuss this in great detail in my book. But there is one set of methods to properly support these techniques so that your vocal cords are free to do their work without any negative impact.
Your book on technique has been widely used by contemporary singers – is this because you do not feel that classical training covers all of the bases?
It's not written for mostly contemporary singers in mind; it is written for all singers. It's funny that there has been a traditional divide between two types of singing: classical and non-classical. I disagree with this – classical music is just one style out of thousands of styles. Because of its distinguished history it is regarded as a "basic" style – which actually isn't true because folk singing is earlier.
This brings us back to your basic insight about the physiology of singing.
Yes. Basic healthy technique is neither classical nor contemporary. What you achieve after you have developed healthy support for your voice is an individual choice – a darker "color" may be classical and a lighter "color" may be involved in certain jazz interpretations. Thus my book doesn't aim at just one genre, but all genres of singing.
Cathrine Sadolin is a voice researcher, vocal instructor, vocal coach, producer, author, singer, composer and instrumentalist. She is the author of Complete Vocal Technique and the founder of the Complete Vocal Institute (CVI) with branches across the world. Her research across all vocal styles, combined with her own experiences as a professional singer, has inspired innovative thinking across the field.
Interviewed by Greg Barker. You can see more of Greg's work here.
This is very interesting and I couldn't agree more. I can't wait to read some of the material she has written!
I can't wait to buy her book.
I like this "Basic healthy technique is neither classical nor contemporary." I took lessons once from an opera-trained instructor and it was helpful to learn that stuff, but I've had to learn how to apply some of the technique to pop/rock music. I'm interested in checking out the book!
My first voice teacher was a nun with a sharpened pointer and her first words to me were that I could not breathe. Over the years I learned much about breathing, support and posture and all that constitutes production of consistent sound. We can never have enough training and reminders of correct use of our instrument. Because it is a voice and it came with our human package, we can dismiss that importance so much more quickly. Thanks to Cathrine and others who spend time and effort making that information accessible.
If this is all necessary……..How come Susan Boyle did not need any of it?
Hi Anthony, It's a good question. Having been reading VoiceCouncil for some time (I am a layperson!) here's what I think: I expect that all of this is necessary and that Susan Boyle was actually doing this unconsciously – or had some very good choir training along the way. Some singers don't have these problems because they rather naturally don't put strain on their larynx but let other parts of their vocal tract, face and body play a part in creating their sound.
She probably grew up listening to singing that was technically very good, for example classic musical theatre stuff (think Julie Andrews). So to re-create those sounds she simply had to develop good technique whether she was conscious of it or not.
We are all very different in our way of functioning, relating to our bodies and we all have a different degree of physical intelligence.
So some people go natural, others don't (for I-don't-know-how-many reasons). For example, my husband has never had a drum lesson in his entire live and ended up being a professional (he's had percussion training, but that's a different instrument). Same story for me (before I got involved in CVT, back in 2006).
As a singing teacher working across many styles I really can't wait to get hold of this book. My personal philosophy is that there is good basic functional singing technique and then there are stylistic and interpretive choices we make in how to apply that technique. Great to see such a comprehensive book produced on the topic.
Thanks Voice Council you're a great read!
I have always found that there are certain times when singing that a particularly connected level of emotion kicks in and you suddenly hear your voice doing things you either could not have "just made happen normally" or even imagined that you even could do. It seems to me that this "vocal empowerment" it is linked inextricably to do with the depth of your mental involvement and how the "emotion of what you are singing" affects you.
Maybe there are ways of physically controlling your vocal folds and throat and chest muscles and diaphragm one can learn so as to try and achieve the same thing. To me that is a bit like having to keep on going to daily Zumba or fitness classes, something that interferes with time management for other things……sadly.
I think everyone should investigate the link between "mental involvement and voice production" far more.
Maybe I should publish a manual on how to get into the mental groove for best voice production?.
What with having to keep drinking the right concoctions to obviate hoarseness or throat problems, voice exercises, physical exercise, music studies, instrument practice, endless hours on the DAW etc etc. I'm beginning to feel like a young Arnie Schwarzenegger in training for the Mr Atlas body building competition!
When really…..it's all in the mind!
Hmmmmm… might make me more money than writing music !
depending on the mode you want to use, the effect you want to add, with the right support. I extended my vocal reach, have the courage to apply effects while following the rules. I'm still growing more and more in applying the CVT technique. i think it is great. It is an eye opener. | {
"redpajama_set_name": "RedPajamaC4"
} | 568 |
Q: passing selected datarow into modal bootstrap I have a table which has a button in each row to show details of each record. what I need is to pass selected data row to modal.
here is my PHP code for display data
<?php
include 'koneksi.php';
$sql = "SELECT UPPER(id_site) AS id_site, UCASE(name_site) AS name_site, witel, UPPER(uplink) AS uplink, port_uplink, ip_uplink, UPPER(olt) AS olt, ip_olt, port_olt, ip_ont FROM data_site ORDER BY witel ASC";
$result = $conn->query($sql);
if($result->num_rows > 0){
//output data
while ($row = $result->fetch_assoc()) {
echo "<tr>
<td>$row[id_site]</td>
<td>$row[name_site]</td>
<td>$row[witel]</td>
<td>$row[olt]</td>
<td>$row[ip_olt]</td>
<td>$row[port_olt]</td>
<td>$row[ip_ont]</td>
<td><button type='button' data-id='".$row['id_site']."' class='btn btn-xs btn-success' data-toggle='modal' data-target='#detail'>DETAIL</button></td>
</tr>";
}
}
?>
this one for display data in modal
<?php
include 'koneksi.php';
$sql = "SELECT * FROM data_site WHERE id_site = '".$row['id_site']."'";
$result = $conn->query($sql);
if($result->num_rows > 0){
while ($row = $result->fetch_assoc()) {
echo "<table>
<tr>
<td>ID Site </td>
<td>: $row[id_site]</td>
</tr>
<tr>
<td>Nama Site </td>
<td>: $row[name_site]</td>
</tr>
<tr>
<td>Witel</td>
<td>: $row[witel]</td>
</tr>
<tr>
<td>OLT Hostname </td>
<td>: $row[olt]</td>
</tr>
<tr>
<td>IP OLT </td>
<td>: $row[ip_olt]</td>
</tr>
<tr>
<td>Port OLT</td>
<td>: $row[port_olt]</td>
</tr>
<tr>
<td>IP ONT</td>
<td>: $row[ip_ont]</td>
</tr>
</table>
";
}
}
?>
and this is my modal
<div id="detail" class="modal fade" role="dialog">
<div class="modal-dialog">
<!-- Modal content-->
<div class="modal-content">
<div class="modal-header">
<h4 class="modal-title"></h4>
</div>
<div class="modal-body">
</div>
<div class="modal-footer">
<button type="button" class="btn btn-default" data-dismiss="modal">Close</button>
</div>
</div>
</div>
</div>
and my javascript
<script>
$('#detail').on('show.bs.modal', function (e) {
var uniqueId = $(e.relatedTarget).data('id');
$('.modal-title').html('Detail site ' + uniqueId);
$('.modal-body').html(uniqueId);
});
</script>
A: What you need is an Ajax call inside $('#detail').on('show.bs.modal', function (e) { } to fetch data against $row['id_site']
<script>
$(document).ready(function() {
$('#detail').on('show.bs.modal', function (e) {
var uniqueId = $(e.relatedTarget).data('id');
//Ajax Method
$.ajax({
type : 'post',
url : 'file.php', //Here you will fetch records
data : 'uniqueId='+ uniqueId, //Pass uniqueId
success : function(response){
$('.modal-title').html('Detail site ' + uniqueId);
$('.modal-body').html(response);
}
});
});
});
</script>
file.php
<?php
include 'koneksi.php';
if($_POST['uniqueId']) {
$uniqueId = $_POST['uniqueId']; //escape the string
$sql = "SELECT * FROM data_site WHERE id_site = '".$uniqueId."'";
$result = $conn->query($sql);
if($result->num_rows > 0){
while ($row = $result->fetch_assoc()) {
echo "<table>
<tr>
<td>ID Site </td>
<td>: $row[id_site]</td>
</tr>
<tr>
<td>Nama Site </td>
<td>: $row[name_site]</td>
</tr>
<tr>
<td>Witel</td>
<td>: $row[witel]</td>
</tr>
<tr>
<td>OLT Hostname </td>
<td>: $row[olt]</td>
</tr>
<tr>
<td>IP OLT </td>
<td>: $row[ip_olt]</td>
</tr>
<tr>
<td>Port OLT</td>
<td>: $row[port_olt]</td>
</tr>
<tr>
<td>IP ONT</td>
<td>: $row[ip_ont]</td>
</tr>
</table>
";
}
}
}
?>
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,275 |
{"url":"http:\/\/math.stackexchange.com\/questions\/788\/why-is-it-so-hard-to-find-the-roots-of-polynomial-equations","text":"# Why is it so hard to find the roots of polynomial equations?\n\nThe question that follows was inspired by this question:\n\nWhen trying to solve for the roots of a polynomial equation, the quadratic formula is much more simple than the cubic formula and the cubic formula is much more simple than the quartic formula.\n\n1. That the general solutions to various polynomial equations are so complex and difficult to derive seems to suggest a fundamental limitation in the problem solving capabilities of the mathematical machinery. Does this intuition of mine make any sense? What should I make of it?\n2. Why is it that with each successive degree in a polynomial equation, the solution becomes so much more complex? Can I gain some intuition about what makes finding the roots so hard?\n3. According to the Abel-Ruffini theorem: \"there is no general algebraic solution\u2014that is, solution in radicals\u2014 to polynomial equations of degree five or higher.\" What is so special about the quintic that makes it the cut-off for finding a general algebraic solution?\n-\nI think this should be closed - it's mentioned twice in the question you link to that there is no formula for polynomials of degree > 4, so asking how fast the complexity of the formulas grows is meaningless \u2013\u00a0BlueRaja - Danny Pflughoeft Jul 27 '10 at 19:33\nit's not so hard to find roots numerically... \u2013\u00a0Jason S Jul 27 '10 at 19:33\npossible duplicate of Is there a general formula for solving 4th degree equations? \u2013\u00a0Akhil Mathew Jul 27 '10 at 19:37\nThe article you linked to in 3 answers your question 3, and possibly your question 1 if I am understanding it correctly. If there is something about the proof you don't understand, that should be part of your question. Right now I don't really know what you're asking. \u2013\u00a0Larry Wang Jul 27 '10 at 19:54\n@Kaestur: the question is about gaining an intuition. Intuition about why this problem is so hard is very different than a proof that this problem is hard. I recognize that one possible answer to my question might be: it is impossible to gain any intuition about why quintic is unique, all we know how to do is prove it. I'm just curious to see if anyone can surprise me with a particularly elegant way of thinking about this problem. \u2013\u00a0Ami Jul 27 '10 at 19:59\n\nThe idea is basically:\n\nAny monic polynomial can be factored as $f(x) = \\prod (x - a_i)$, where $a_{1,\\dots,n}$ are the roots of the polynomial.\n\nNow if we expand such a product:\n\n$(x - a_1)(x - a_2) = x^2 - (a_1 + a_2)x + a_1a_2$ $(x - a_1)(x - a_2)(x - a_3) = x^3 - (a_1 + a_2 + a_3)x^2 + (a_1a_2 + a_1a_3 + a_2a_3)x - a_1a_2a_3$\n\nAnd so on. The pattern should be clear.\n\nThis means that finding the roots of a polynomial is in fact equivalent to solving systems like the following:\n\nFor a quadratic polynomial $x^2 - px + q$, find $a_1,a_2$, such that\n\n$p = a_1 + a_2$\n\n$q = a_1a_2$\n\nFor a cubic polynomial $x^3 - px^2 + qx - r$, find $a_1,a_2,a_3$, such that\n\n$p = a_1 + a_2 + a_3$\n\n$q = a_1a_2 + a_1a_3 + a_2a_3$\n\n$r = a_1 a_2 a_3$\n\nAnd similarly for higher degree polynomials.\n\nNot surprisingly, the amount of \"unfolding\" that needs to be done to solve the quadratic system is much less than the amount of \"unfolding\" needed for the cubic system.\n\nThe reason why polynomials of degree 5 or higher are not solvable by radicals, can be thought of as: The structure (symmetries) of the system for such a polynomial just doesn't match any of the structures that can be obtained by combining the structures of the elementary operations (adding subtracting, multiplication, division, and taking roots).\n\n-\n\nWhen you try to solve a degree $n$ equation, there are $n$ roots you have to find (in principle) and none of them is favoured over any of the others, which (in some metaphorical sense) means that you have to break an $n$-fold symmetry in order to write down the roots.\n\nNow the symmetry group of the n roots becomes more and more complicated the larger $n$ is. For $n = 2$, it is abelian (and very small!); for $n = 3$ and $4$ it is still solvable (in the technical sense of group theory), which explains the existence of an explicit formula involving radicals (this is due to Galois, and is a part of so-called Galois theory); for $n = 5$ or more this group is non-solvable (in the technical sense of group theory), and this corresponds to the fact that there is no explicit formula involving radicals.\n\nSummary: The complexity of the symmetry group of the $n$ roots leads to a corresponding complexity in explicitly solving the equation.\n\n-\n\nFor a different take, some practical problems are discussed in Wilkinson's classic article The Perfidious Polynomial. If you can't access it, check what Wikipedia has to say on the subject.\n\n-\nJim Wilkinson's prize-winning article can be read here; briefly, one frequent source of trouble in numerical polynomial root finding is our habit of expressing polynomials in the monomial basis, and it happens that there are polynomials like $\\prod_i (x-i)$ that numerically behave very poorly in rootfinding when expressed in the monomial basis. \u2013\u00a0J. M. Apr 20 '11 at 8:51\n@\uff2a.\uff2d. This link is broken. \u2013\u00a0I. J. Kennedy Oct 25 '14 at 23:48\n@I.J.Kennedy, Neil has given you a working link; thank him. \u2013\u00a0J. M. May 1 at 13:42\n\nAs an introduction to this important issue in mathematics, I shall first consider the whole issue as simplified polynomial or (general trinomial equation) by providing only and absolutely one correct root (absolutely convergent to algebraic real irrational number always) to the following trinomial equation (without using any approximation methods), say for $n$-th degree polynomial equation $$x^n + x^m = 1,$$ where $m < n$ are two distinct positive integers, the real root is given by the following power series:\n\n\\begin{align} x &= 1-\\frac{1}{n}+ \\sum_{i=2}^\\infty \\frac{(-1)^i \\prod_{j=1}^{i-1}(im-jn+1)}{i!n^i} \\\\ &= 1-\\frac{1}{n}+\\frac{2m-n+1}{2!n^2}- \\frac{(3m-2n+1)(3m-n+1)}{3!n^3} +\\frac{(4m-3n+1)(4m-2n+1)(4m-n+1)}{4!n^4} -\\frac{(5m-n+1)(5m-2n+1)(5m-3n+1)(5m-4n+1)}{5!n^5} +\\dotsb \\end{align}\n\nIn so many cases the root can be radical form, but the big question is that always can be converted to radical form, for a more useful reference see the link below:\n\nReference: (1994), For a reference to more general solution of this Trinomial equation $$ax^n +bx^m +c = 0$$ click, OK, then click National then type in the search button \"solution of equations by power series\" at this link http:\/\/opac.nl.gov.jo\/uhtbin\/cgisirsi\n\n-\nI don't think this answers the question. I also think it's odd that this is identical to another recent answer of yours. \u2013\u00a0pjs36 Nov 26 at 16:12\nI thought that I could provide a little insight to the problem even repeatedly, I would also appreciate the downvoters if they prove me wrong openly with their real identities, I know also many have deleted my same answer for several times without any obvious reason, it is quite simple, just assume any arbitrary example even with radical known roots and see then if my answer is wrong, mentioning that what is given in the answer is very little from the original reference that I strictly believe is very useful subject, I can provide infinitely many radical solution from the provided series \u2013\u00a0bassam karzeddin Nov 26 at 16:30","date":"2015-12-01 17:17:22","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 1, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8965930342674255, \"perplexity\": 298.9313477387225}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2015-48\/segments\/1448398468396.75\/warc\/CC-MAIN-20151124205428-00239-ip-10-71-132-137.ec2.internal.warc.gz\"}"} | null | null |
Q: Adding 'Link to Item' in SPView Programmatically I am trying to add Link to Item to existing SPView in SharePoint 2010.
The same view is further used in ListViewWebpart.
I am using below code to add LinkToItem attribute for field 'SpecificGoal'. Code working fine as I could see that attribute being added to xmlDoc.OuterXml.
However, ListViewWebpart is not displaying Link to Item.
Is this a valid way to update the view? Any other way to programmatically adding 'LinkToItem'?
Thank you.
string fieldXml = _view.GetViewXml();
fieldXml = CreateLinkToItem(fieldXml);
_view.SetViewXml(fieldXml);
_view.Update();
private string CreateLinkToItem(string fieldXml)
{
XmlDocument xmlDoc = new XmlDocument();
xmlDoc.LoadXml(fieldXml);
const string xpathSourceNodes = @"//FieldRef[contains(@Name,'SpecificGoal')]";
XmlNode targetNode = xmlDoc.SelectNodes(xpathSourceNodes)[0];
if (null != targetNode)
{
XmlAttribute newAttribute = xmlDoc.CreateAttribute("LinkToItem");
newAttribute.Value = "TRUE";
targetNode.Attributes.Append(newAttribute);
break;
}
return xmlDoc.OuterXml;
}
A: According to the documentation,
Attempts to set this property are ignored if LinkToItemAllowed is not
set to Allowed.
So, I would try adding both:
LinkToItem="TRUE"
LinkToItemAllowed="Required"
| {
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Q: Compactness implies subsequence converges to a point? Suppose we have a compact set $K$ in a region $\Omega.$ We have that for every $l \geq 0,$ there exists points $z_l \in K, w_l \in \Omega^C$ such that $|z_l - w_l| < 1/l.$ So apparently by the compactness of $K$ there exists a subsequence of $\{z_l\}$ that converges to a point $z \in K.$ Why is this? I get that since $K$ is compact it is closed meaning that it has all its limit points. But the $\{z_l\}$ could be all over the region of $K.$ Why should they be limiting to any one point?
| {
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{"url":"https:\/\/zbmath.org\/?q=an:0696.34005","text":"## The Cauchy problem for fuzzy differential equations.(English)Zbl\u00a00696.34005\n\nSummary: The classical Peano theorem states that in finite dimensional spaces the Cauchy problem $$x'(t)=f(t,x(t)),$$ $$x(t_ 0)=x_ 0$$, has a solution provided f is continuous. In addition, Godunov has shown that each Banach space in which the Peano theorem holds true is finite dimensional. For differential inclusions, the existence of a solution to the Cauchy problem is also guaranteed under various assumptions on the right-hand side. In this paper, we study the Cauchy problem for fuzzy differential equations. To be more specific, let U be a subspace of normal, convex, upper semicontinuous, compactly supported fuzzy sets defined in $${\\mathbb{R}}^ n$$ and assume that $$f: [t_ 0,t_ 0+a]\\times U\\to U$$ is continuous. We show that the Cauchy problem has a solution if and only if U is locally compact.\n\n### MSC:\n\n 34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations\n\n### Keywords:\n\nCauchy problem; Banach space; fuzzy differential equations\nFull Text:\n\n### References:\n\n [1] Aubin, J.P.; Cellina, A., Differential inclusions, (1984), Springer-Verlag Berlin [2] De Blasi, F.R.; Lasota, A., Daniell\u2019s method in the theory of the Aumann-Hukuhara integral of set-valued functions, Atti acad. naz. lincei rendiconti ser. 8, 45, 252-256, (1968) \u00b7 Zbl\u00a00174.09202 [3] De Blasi, F.R.; Pianigiani, G., Differential inclusions in Banach spaces, J. differential equations, 66, 208-229, (1987) \u00b7 Zbl\u00a00609.34013 [4] Debreu, G., Integration of correspondences, (), 351-372, Part 1 \u00b7 Zbl\u00a00211.52803 [5] Diamond, P.; Kloeden, P., Characterization of compact subsets of fuzzy sets, Fuzzy sets and systems, 29, 341-348, (1989) \u00b7 Zbl\u00a00661.54011 [6] Dubois, D.; Prade, H., Fuzzy sets and systems: theory and applications, (1980), Academic Press New York \u00b7 Zbl\u00a00444.94049 [7] Feron, R., Ensembles aleatoires dont la fonction d\u2019appartenance prend ses valeurs dans un treillis distributif ferme, Publ. econometriques, 12, 81-118, (1979) \u00b7 Zbl\u00a00406.46010 [8] Godunov, A.N., Peano\u2019s theorem in Banach spaces, Funct. anal. appl., 9, 53-55, (1975) \u00b7 Zbl\u00a00314.34059 [9] Goetschel, R.; Voxman, W., Topological properties of fuzzy numbers, Fuzzy sets and systems, 10, 87-99, (1983) \u00b7 Zbl\u00a00521.54001 [10] Himmelberg, C.J.; Van Vleck, F.S., Existence of solutions for generalized differential equations with unbounded right-hand side, J. differential equations, 61, 295-320, (1986) \u00b7 Zbl\u00a00582.34002 [11] Hukuhara, M., Integration des applications measurables dont la valeur est un compact convexe, Funkcialaj. ekvacioj., 10, 205-223, (1967) \u00b7 Zbl\u00a00161.24701 [12] Kaleva, O., Fuzzy differential equations, Fuzzy sets and systems, 24, 301-317, (1987) \u00b7 Zbl\u00a00646.34019 [13] Klement, E.P.; Puri, M.L.; Ralescu, D.A., Limit theorems for fuzzy random variables, (), 171-182 \u00b7 Zbl\u00a00605.60038 [14] Lang, S., Analysis II, (1969), Addison-Wesley Reading, MA [15] Nguyen, H.T., A note on the extension principle for fuzzy sets, J. math. anal. appl., 64, 369-380, (1978) \u00b7 Zbl\u00a00377.04004 [16] Puri, M.L.; Ralescu, D.A., Differentials for fuzzy functions, J. math. anal. appl., 91, 552-558, (1983) \u00b7 Zbl\u00a00528.54009 [17] Puri, M.L.; Ralescu, D.A., The concept of normality for fuzzy random variables, Ann. probab., 13, 1373-1379, (1985) \u00b7 Zbl\u00a00583.60011 [18] Puri, M.L.; Ralescu, D.A., Fuzzy random variables, J. math. anal. appl., 114, 409-422, (1986) \u00b7 Zbl\u00a00592.60004 [19] Tolstonogov, A.A., Differential inclusions in Banach space with nonconvex right-hand side, Siberian J. math., 22, 625-637, (1981) \u00b7 Zbl\u00a00529.34058 [20] Vitale, R.A., Lp metrics for compact, convex sets, J. approx. theory, 45, 280-287, (1985) \u00b7 Zbl\u00a00595.52005\nThis reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.","date":"2022-11-30 06:43:37","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.864791989326477, \"perplexity\": 4288.689735142968}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-49\/segments\/1669446710733.87\/warc\/CC-MAIN-20221130060525-20221130090525-00596.warc.gz\"}"} | null | null |
Give your brain a workout with aPuZzLr.
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aPuZzLr is a challenging and brain teasing puzzle game and is a universal app for the iPhone, iPad, and iPod Touch. This game is fun for both adults and kids.
aPuZzLr has been lovingly crafted by an independent developer - if you enjoy playing, your positive reviews on the App Store will help me build more levels for your entertainment!
But enough talk! Download now FREE and see if you can earn all of the achievements. Enjoy!
This game hooked me. After the first 50 puzzles (some only take a second or two to complete) you have to pay to download. And then there are only 120 total! Completed them all in less than an hour. I'd love more and disappointed that I paid for the app thinking there would be more after the 120. | {
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Широколи́ственные леса́ — разновидность лиственных лесов, образованных листопадными деревьями с широкими листовыми пластинками.
Широколиственные леса приурочены к влажным и умеренно-влажным районам умеренного климата с ослабленной континентальностью, равномерным распределением осадков в течение года и относительно высокими температурами.
Почвы широколиственных лесов — обычно серые, тёмно-серые и бурые лесные; реже — чернозёмы и дерново-подзолистые.
Распространение
Распространены в Европе, Северной Америке, Восточной Азии, на юге Чили и островах Новой Зеландии.
Климат
Умеренно тёплый климат характеризуется тёплым, продолжительным летом и мягкой зимой. Годовая сумма осадков, равномерно распределённых в течение года, несколько выше испарения, что существенно снижает уровень заболоченности грунтов. Средняя годовая температура летом +19…24°С
Растительный мир
Древесные породы таких лесов в Европе представлены, главным образом, буком и дубом; реже — грабом, липой. Обычны также ясень, вяз, клён. В Северной Америке наиболее широко распространены кленово-буковые, дубово-гикоровые, дубовые, а в прошлом — дубово-каштановые леса. Для подлеска Восточной Европы обычны лещина, полевой и татарский клёны, черёмуха, бересклеты. В травяном покрове господствуют мезофильные, эвтрофные или мезотрофные виды, в европейских лесах — пролесник, сныть, зеленчук, копытень, медуница, ясменник, осока волосистая и другие.
Широколиственным лесам свойственны весенние эфемероиды-геофиты: хохлатка, ветреница, зубянка, подснежник, пролеска, гусиный лук, которые успевают пройти цикл развития от распускания листьев до созревания семян весной, до развёртывания листьев деревьев.
Животные
Часть Голарктической зоны.
Млекопитающие
Млекопитающие широколиственных лесов Западной и Восточной Европы: дикий кабан, олени (благородный олень; в лесной зоне вне Европы встречаются такие его подвиды, как, например, марал, вапити, изюбрь), лось, косуля, зубр, лесная белка, бурундук, бобр, рыжая полёвка, желтогорлая мышь, сони (полчок, лесная соня, садовая и орешниковая), заяц-беляк, заяц-русак, обыкновенный ёж, лесная кошка, рысь, лисица, волк, бурый медведь, барсук, горностай, лесная куница, ласка, норка европейская. Крупные животные сохранились лишь в отдельных массивах, не уничтоженных хозяйственной деятельностью человека (например, в Карпатских горах).
Фауна муссонных широколиственных лесов на Дальнем Востоке (амурско-уссурийских, китайских, корейских) особенно богата, так как туда проникают виды с юга. Из млекопитающих характерны уссурийский крот, маньчжурский заяц, пятнистый олень, горал, гималайский, или белогрудый медведь, харза, енотовидная собака, амурский тигр, дальневосточный леопард, дальневосточный лесной кот.
Необычное животное широколиственных лесов Америки — скунс. По чёрной шубке скунса от головы к большому пушистому хвосту проходят две широкие белые полосы. Другой лесной обитатель, свойственный Америке, — енот, или енот-полоскун.
Птицы
Встречаются ястреб-тетеревятник, иволга, поползень, чиж, рябчик, тетерев-косач, глухарь, дрозд-рябинник, соловей, синица, снегирь, жаворонки, серый журавль, зяблик, кукушка, дубонос, большой пёстрый, зелёный и средний дятлы, осоед, канюк, совы, филины, чёрные стрижи, голуби, аисты, вороны, галки, сороки, грачи, гуси, утки и др.; в Азии: личинкоед, белоглазка, голубая сорока, широкорот, фазан, утка-мандаринка.
Рептилии и амфибии
Рептилии: уж, гадюка, эскулапов полоз, змея медянка (неправильно считающаяся ядовитой), безногая ящерица веретеница, зелёная и живородящая ящерицы, болотная черепаха. Если рассматривать территорию России, то нельзя не упомянуть дальневосточную черепаху, или китайского трионикса (Pelodiscus sinensis).
Земноводные: квакша, прудовая, травяная и остромордая лягушки, зелёная жаба, жерлянки, гребенчатый тритон, пятнистая саламандра.
Рыбы
Сом, щука, пескарь, карповые, лососёвые и другие.
Литература
Экология растений
Типы леса
Зоны растительности | {
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{"url":"https:\/\/pendulumedu.com\/qotd\/quantitative-aptitude-time-speed-and-distance-26-august-2021","text":"Question of The Day26-08-2021\n\nTrain A started from Kanpur station and at the same time train B started from Agra station toward each other. After crossing each other they take 4 hours and 36 hours respectively to reach the destination (i.e. Start point of other train). If the speed of train A is 180 km\/h, then find the speed of train B.\n\nCorrect Answer : b ) 60 km\/h\n\nExplanation :\n\nAccording to the question\n\nWe know that,\n\n$${s1 \\over s2}=\\sqrt{T_2 \\over T_1}$$\n\nWhere, s1 = speed of train A and s2 = Speed of train B and T1 be the time taken by train A after crossing and T2 be the time taken by the train B after crossing.\n\nT1 = 4hrs and T2 = 36hrs\n\n$$s1 \\over s2$$=$$\\sqrt{36 \\over 4}$$$$\\sqrt 9$$\n\n$$s_1\\over s_2$$=$$3 \\over 1$$\n\nSpeed of train A = 180km\/h\n\n3unit = 180 km\/h\n\n1 unit = 60 km\/h\n\nSpeed of train B = 60km\/h\n\nHence, (b) is the correct answer.\n\n0","date":"2023-01-29 08:42:37","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.26221928000450134, \"perplexity\": 1030.0422256600114}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-06\/segments\/1674764499710.49\/warc\/CC-MAIN-20230129080341-20230129110341-00140.warc.gz\"}"} | null | null |
Is the price per tour or per person?
This is the total price for a private guided tour only for you.
Is it compulsory a minimum number of participants on the tour ?
No. It is sufficient one person.
At what time can we start this tour ?
We can start this tour in any moment.
Which languages are spoken in Your office if we wish to talk to you directly by phone ?
In our office we speak a fluent Italian, English and French.
Can we start our excursion from any spot ?
Yes. We can start any excursion from any spot (hotel, port, airport) You tell us.
Is it compulsory to reserve this tour in certain specific days ?
No, we can arrange this tour any day of the week in any season.
Is the admission for the Bolognini Chapel in the Basilica of Saint Petronio included in the total price of the tour ?
No, the admission for the Bolognini Chapel in the Basilica of Saint Petronio is additional and it has to be paid apart. | {
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\section{Introduction}\label{sec:intro}
We are entering the post-PC era where an ever-larger variety of smart Internet-of-things (IoT) devices including wearables and smart sensors are increasing the demands for low power communication technologies. Wi-Fi has recently become a contender for this regime due to its compatibility with IP networks and wide deployments of hotspots. Growing numbers of wearables in the market have enabled Wi-Fi connection to the Internet without relying on gateways. Many commercial smart watches including Apple Watch have already equipped with built-in Wi-Fi chipsets. Google's latest Android Wear 2.0 release~\cite{androidwear} allows Android smartwatches to directly connect to Wi-Fi networks.
With the proliferation of sensor-equipped smart objects in streets, homes, and offices, IoT connectivity is envisioned to be massive~\cite{cisco,wang2016less}. Thus, Wi-Fi for IoT devices should be not only low power but also spectrum efficient. As a mature and spectrum efficient multiplexing access technology widely adopted in the latest Wi-Fi standards, OFDM continues its adoption in the new Wi-Fi standard for IoT, i.e., IEEE 802.11ah~\cite{wifihalow}, which reuses OFDM frame format that conforms to IEEE 802.11ac to ensure spectrum efficiency.
\begin{figure}[t]
\center
\includegraphics[width=0.49\textwidth]{rationale}\vspace{0.3cm}
\caption{The concept of Wi-Fi Teeter-Tooter. A IoT device lowers its TX power and pushes all decoding and energy burdens to the AP side. As the compensation for decreased TX power, the AP takes full advantage of its high clock rate.}\label{fig:rationale}
\end{figure}
A number of recent research efforts have been devoted to reducing the power consumption of OFDM-based Wi-Fi communications. Downclocking receivers' radios is proposed to reduce the power consumption while receiving packets~\cite{wang2017sampleless,mobicom14enfold} or idle listening~\cite{mobicom11emili,wang2017wideband}. Efficient sleep modes~\cite{psm,khorov2015survey} are proposed to reduce the power consumption of conventional IEEE 802.11 nodes by allowing them to enter extremely low power state during idle listening. Despite these growing attempts and extensive efforts, most of them have focused on energy-efficient data reception. A variety of IoT applications require data-rich sensors such as cameras and microphones to frequently upload sampled data to servers, which incurs a large amount of energy consumption for uplink transmission.
In this paper, we argue that the fundamental hurdle for energy efficient uplink transmission lies in the transceiver design. Existing Wi-Fi transceivers are originally designed for symmetric nodes with equal hardware capabilities and power constraints. This assumption no longer stands for IoT applications, where a Wi-Fi AP is much more powerful and almost energy-unconstrained compared to IoT devices. This evokes a similar picture in cellular networks, where base stations are equipped with 100$\times$ sensitive RF front-end and have 20~dB more transmission (TX) power than mobile clients. Analogous to the hardware asymmetry in cellular networks, APs are equipped with Wi-FI chipsets that can support up to 160~MHz bandwidth in IEEE 11ac/11ax, while IoT devices normally support only 1-2MHz bandwidth~\cite{wifihalow}. Thus, there is significant potential to exploit asymmetric PHY configurations to enable low power uplink transmission without undermining decoding performance.
Our line of attack starts from the rationale that we trade the computing and energy resources of APs for the TX power of IoT devices in a \textit{teeter-totter} fashion, that is, we allow IoT devices to transmit uplink packets using the lowest power while pushing all the decoding burdens to the AP side, as illustrated in Figure~\ref{fig:rationale}. In particular, the AP takes full advantage of its clock rate that are tens of times higher than those of IoT's to decode packets with low signal-to-noise ratio (SNR). IoT devices reap benefits from such an asymmetric design by configuring TX power lower than the minimum power required for decoding. On the other hand, the AP with enough computing capabilities and constant power source can afford the extra cost induced by overclocking. Such an asymmetric design trades resources that are cheap to AP for power consumptions that are expensive to IoTs. Such a design completely conforms to Wi-Fi protocols, and thus can be readily integrated with existing standards. The only changes are standalone update of computational logics and RF settings at APs.
A key challenge in realizing the teeter-totter design is how to effectively leverage overclocking in OFDM to decode conventionally undecodable packets under low SNR conditions. Our fundamental insight is that APs yield correlated signals by exploiting the \textit{time shift} effect between overclocked samples. Specifically, when the AP sets its analog-to-digital converter (ADC) clock at a much higher rate than IoT devices, it yields multiple interpolated samples from one transmitted sample. These interpolated samples can be considered as time-shifted versions of the transmitted sample. When transformed into the frequency domain, time-shifted samples result in different phase rotations at different subcarriers. Thus, we can leverage this phase rotation effect to combine these samples for packet decoding. Another challenge stems from the lack of knowledge in modeling the noise distributions between these redundant samples. Instead of blindly reusing conventional decoders, we turn to a data-driven approach. In particular, we build a noise map from preambles without making distribution assumptions. Then, we combine redundant samples by employing a maximum likelihood (ML) decoder based on the joint probability of these samples.
We implement the above teeter-totter design, referred to as \texttt{T-Fi}, on the GNURadio/USRP platform. Evaluation results validate \texttt{T-Fi} in reliably receiving and decoding Wi-Fi packets at low SNR across a wide range of scenarios. Furthermore, \texttt{T-Fi} can reduce the IoT's TX power when the AP is overclocked by a factor of up to 8, which is still lower than the rate used for IEEE 802.11a/g/n.
The contributions of this paper are summarized below.
\begin{itemize}
\item We provide a thoughtful study towards enabling low-power OFDM transmissions for IoT Wi-Fi standards. Our solution can be seamlessly integrated into existing Wi-Fi standards without modifying the legacy frames or protocols.
\item We explore the fundamental structure of overclocked reception in OFDM, and propose a reception pipeline to decode legacy packets at lower SNRs than the conventional transceivers. The key underlying technique is a new decoding algorithm that exploits the time shift effects in oversampled signals.
\item We build a full prototype of \texttt{T-Fi} and quantify the merits of our design in a wide range of scenarios.
\end{itemize}
The remainder of the paper is structured as follows. We begin in Section~\ref{sec:motivation} with the design motivation. Section~\ref{sec:overclock} analyzes the effect of overclocking in OFDM, which is the underpinnings of our design. Section~\ref{sec:design} elaborates the detailed reception pipeline of our design. System implementation and experimental evaluation are introduced in Section~\ref{sec:implementation} and Section~\ref{sec:evaluation}, respectively. Section~\ref{sec:relatedwork} gives a brief survey of related work, followed by conclusion in Section~\ref{sec:conclusion}.
\vspace{0.3cm}
\section{Motivation of Transceiver Asymmetry}\label{sec:motivation
In this section, we ask and answer questions to explore practical and suitable transceiver architectures. Through this exploration, we hope to convince readers that the teeter-totter design is a practical design in that it elegantly fits the architecture of IoT Wi-Fi while imposing affordable costs.
\subsection{IoT Scenarios and Requirements}
It is envisioned that the number of smart devices that need to wirelessly share data or access the Internet is growing exponentially. To embrace the coming wave of \textit{massive numbers} of wearables, driverless cars, smart sensors, IEEE 802.11ah was announced in 2016 to tailor Wi-Fi protocols specially for these IoT devices. The target envisioned in this IoT Wi-Fi standard is to challenge Bluetooth and cellular networks by achieving the best of both worlds: to enable \textit{long-distance} communications using relatively \textit{low amounts of power}. Specifically, IEEE 802.11ah defines the following requirements: (i) more than a fourfold number of devices supported by one AP compared to legacy Wi-Fi, (ii) doubled transmission range compared to legacy Wi-Fi links, and (iii) low energy consumption with guaranteed data rates of at least 100~Kb/s.
To meet the above requirements, IEEE 802.11ah introduces a bundle of new features in PHY/MAC. Particularly, the new standard reuses the formerly-adopted OFDM waveform consisting of 32/64 subcarriers while tailoring it to a narrowband channel of 1~MHz or 2~MHz in 900~MHz unlicensed spectrum. OFDM is used to maintain spectrum efficiency for massive connectivity, and narrowband transmission at lower frequencies are used to facilitate low-power, long-range transmissions. In addition, a set of MAC mechanisms is designed to reduce the power consumption in channel access.
\subsection{Hardware Asymmetry}
Although the new standard specifies new PHY and MAC techniques and features to meet the requirements of IoTs, Wi-Fi transceivers still work in a conventional fashion: today's Wi-Fi radios work in a symmetric manner in that the transmitter and receiver are configured with the same clock rate. This symmetric transceiver design has worked well over the past decade as both sides are expected to be (more or less) equally powerful in conventional scenarios. While this assumption no longer stands in IoT scenarios where the disparity between the hardware capabilities of APs and IoTs makes such a design very inefficient. It is either an overwhelming burden for IoTs to match the radio configurations of APs, or makes APs largely under-utilize their hardware capabilities to shoehorn the proper settings of IoTs' radios.
The above situation evokes a similar picture in cellular networks, where base stations are much more powerful than mobile stations and are designed to support thousands of hardware-constrained mobile devices. Cellular networks equip base stations with higher TX power, RF sensitivity, and large form-factor antennas to compensate the hardware gap. The wisdom of cellular networks motivates us to design an asymmetric PHY that fully reap the benefits of the hardware gap between APs and IoTs.
Our target is to explore an asymmetric transceiver design that can better fit the IoT scenarios and be seamlessly integrated into existing and future OFDM-based Wi-Fi protocols. The fundamental insight is that APs have already equipped with high clock-rate radios to support high-speed Wi-Fi such as IEEE 802.11ac/ax, and the clock rates are tens of times to those of IoTs'. When APs use such overclocked rates to sample IoT's transmissions, a large amount of redundancy between these samples can be used to boost the performance of packet reception, which in turn relaxes TX power requirements imposed on IoTs.
\vspace{0.3cm}
\section{Exploiting Overclocking Opportunities}\label{sec:overclock
\begin{figure*}[t]
\center
\includegraphics[width=7in]{ofdm_schematic}\vspace{0.3cm}
\caption{OFDM schematic under overclocking.} \label{fig:ofdm_schematic}
\end{figure*}
At the core of the asymmetric transceiver design, we exploit the redundancy in OFDM signals sampled by overclocked radios. To this end, we discuss the opportunities in retaining the OFDM samples at an overclocked rate and utilizing them to improve symbol decoding. We start by analyzing the correlations among these samples.
In a standard OFDM system, as illustrated in Figure~\ref{fig:ofdm_schematic}, one OFDM symbol contains a sequence of bits that are modulated into a set of lattice points $X[f]$ on orthogonal subcarriers in the frequency domain. The information-bearing $X[f]$. For OFDM transmissions, inverse fast Fourier transform (IFFT) are used to transform $X[f]$ into a time-domain sequence
\begin{equation}
x[n] = {1 \over F} \sum_{f=0}^{F-1} X[f] e^{j 2\pi f n /F}, n=0,...,F-1,\notag
\end{equation}
where $F$ is the number of subcarriers. Note that we omit cyclic prefix for simple illustration. The discrete-time signal $x[n]$ passes an digital-to-analog converter (DAC) and a pulse shaping filter to transform itself into a continuous-time signal in baseband form, which is written as
\begin{equation}
x(t) = \sum_{f=0}^{F-1} x[n] p(t-f T_s),\notag
\end{equation}
where $p(t)$ is the impulse response of the pulse-shaping filter and $1/T_s$ is the baud rate. $x(t)$ is transmitted and propagates a wireless channel with impulse response $h(t)$. A receiver receives the signals with a matched filter $p(-t)$, and obtain
\begin{equation}
y(t) = \sum_{n=0}^{F-1} x[n] p(t-f T_s) \ast h(t-f T_s) \ast p(f T_s-t) + w(t),\notag
\end{equation}
where $\ast$ denotes convolution, and $h(t)$, $w(t)$ are the impulse response of the channel and complex Gaussian noise, respectively. In the presence of a time-dispersive channel and additive noise, the received continuous time-domain baseband signal $y(t)$ can be expressed as
\begin{equation}
y(t) = {1 \over F} \sum_{f=0}^{F-1} X[f] H(f) e^{j {2\pi f t \over T} } + w(t),\notag
\end{equation}
where $H(f)$ is the channel frequency response at subcarrier $f$, and $T$ is the symbol duration.
Now we consider a concrete example in which $F=64$ subcarriers are used to convey information. When using the same clock rate to sample received signal $y(t)$, the sampling instances are at $t={n \over 64} T$. Thus, the receiver yields
\begin{equation}
y_1[n] = {1 \over 64} \sum_{f=0}^{63} X[f] H(f) e^{j {2\pi f n \over 64} } + w_1(n).\notag
\end{equation}
When the receiver doubles the clock rate, the number of samples in each FFT segment is 128. As such, the sampling instances are at $t={n \over 128} T$, and the time domain sample sequence is expressed as
\begin{align}
y[n] =& {1 \over 64} \sum_{f=0}^{63} X[f] H(f) e^{j {2\pi f n \over 128} } + w(n), n = 0,...,127 \notag\\
= & {1 \over 64} \sum_{f=0}^{63} \left( X[f] H(f) e^{j {2\pi f \over 128} 2n } + X[f] H(f) e^{j {2\pi f \over 128} (2n+1) } \right) \notag\\
&+ w_1(2n) + w_1(2n+1), n = 0,...,63 \notag\\
= & y_1[n] + y_2[n] + w_2(n), n = 0,...,63, \notag\\
\end{align}
where $y_2[n] = {1 \over 64} \sum_{f=0}^{63} X[f] H(f) e^{j {2\pi f \over 64} (n+1/2)}$ and $w_2(n)$ is the new noise sequence in oversampled signals. We see that the oversampled signal $y[n]$ consists of two polyphase components $y_1[n]$ and $y_2[n]$. Notice that $y_2[n]$ is a time-shifted version (delayed by a half sample) of $y_1[n]$. The frequency response of $y_1[n]$ can be derived by performing FFT.
\begin{align}
Y_1[l] &= \sum_{n=0}^{63} y_1[n] e^{-{j 2\pi n l \over 64}}, l=0,...,63 \notag \\
&= {1 \over 64} \sum_{n=0}^{63} \sum_{f=0}^{63} X[f] H(f) e^{j {2\pi f n \over 64} } e^{-{j 2\pi n l \over 64}} + W_1(l) \notag \\
&= X[l] H(l) + W_1(l),\notag
\end{align}
where $ W_1(l)$ is the noise expressed in the frequency domain. Similarly, the frequency response of $y_2[n]$ is expressed as
\begin{align}
Y_2[l] &= \sum_{n=0}^{63} y_2[n] e^{-{j 2\pi n l \over 64}}, l=0,...,63 \notag \\
&= {1 \over 64} \sum_{n=0}^{63} \sum_{f=0}^{63} X[f] H(f) e^{j {2\pi f \over 64} (n+1/2)} e^{-{j 2\pi n l \over 64}} + W_2(l) \notag \\
&= X[l] H(l) e^{-{j \pi l \over 64}} + W_2(l).\notag
\end{align}
Ignoring the noises, $Y_2[l]$ is a phase-shifted version of $Y_1[l]$, and the amount of phase rotation is linear to the subcarrier frequency $l$.
Hence, doubling the receiver's clock rate yields two receptions, which transforms the system into a single-input-multiple-output (SIMO) system. When we employ $G$-fold clock rate at the receiver, we can obtain $G$ phase-shifted versions of $Y_1[l]$ described as
\begin{equation}\label{e:yg}
Y_g[l] = X[l] H(l) e^{-{j 2 \pi gl \over 64G}} + W_g(l), g=0,...,G-1.
\end{equation}
Figure~\ref{fig:phaseshift} shows the phase shifts across all subcarriers in a real Wi-Fi packet when received at eight-fold clock rate using USRP testbeds. The predictable phase shifts in $Y_g[l]$ can be easily compensated to obtain $G$ copies of the transmitted signal. We can combine these copies to improve the received signal strength while amortizing noise, and thereby enhance the decoding capability under low SNR conditions.
\begin{figure}[t]
\center
\includegraphics[width=3.5in]{phaseshift}\vspace{0.3cm}
\caption{Phase shift induced by oversampling a real Wi-Fi packet. Phases are normalized according to the phases of $Y_1$. Subcarriers 0-25, 38-63 carrying payload data are compared.} \label{fig:phaseshift}
\end{figure}
\vspace{0.3cm}
\section{Asymmetric Transceiver Design}\label{sec:design
In this section, we describe our design of \texttt{T-Fi}, an asymmetric transceiver architecture for IoT Wi-Fi. \texttt{T-Fi} can fully interoperate with standard IEEE 802.11 devices, with no modifications to existing protocols. \texttt{T-Fi} leverages the high clock rate of APs to enable low power transmissions for IoT devices. This section elaborates the detailed reception pipeline design that embraces this design rationale.
\subsection{Frame structure and reception pipeline
A Wi-Fi frame starts with a preamble prepending payload to perform time synchronization, carrier frequency offset (CFO) compensation, and channel estimation. In particular, a preamble consists of a Short Training field (STF) and a Long Training Field (LTF). The legacy STF contains two OFMD symbols that are comprised of ten repetitions of a 16-sample sequence, and the LTF contains two identical 64-sample (80-sample including cyclic prefix) OFDM symbols. A PHY header, known as signaling field (SIG), containing modulation and coding scheme (MCS) information, duration, and service field, sits between the preamble and payload. The preamble and the first 24 bits of the PHY header are encoded using the lowest MCS mode to ensure their correct reception, while the rest can be encoded using different MCS modes based on a certain rate adaptation algorithm.
Under overclocking settings, a receiver increases the sampling rate of the ADC by $G$ folds. The incoming analog signals are oversampled and converted into $G$ baseband streams, which are fed into the synchronization block to identify the start of a Wi-Fi frame. Meanwhile, the CFO between the sender and the receiver is estimated and compensated using preamble sequences to eliminate the effect of the disparity between clock oscillator frequencies. Since there may still be residual frequency offset, phase compensation is employed to correct the phase rotation throughout the entire payload. In Wi-Fi, four subcarriers are used as pilots to estimate phase rotation in each data symbol. Then, the receiver performs FFT on the samples to obtain their frequency domain values. After performing FFT, the receiver estimates channel responses by comparing the received values in LTF and the actual values transmitted. Finally, the receiver decodes the bits in the payload symbols.
In what follows, we systematically exploit the merits as well as issues brought by overclocking, and propose corresponding techniques to optimize the packet reception performance.
\subsection{Timing Synchronization
Timing synchronization aims to detect the presence of a Wi-Fi packet and then identify the start of Wi-Fi symbols. Wi-Fi receivers take advantage of the periodicity in STF to detect the Wi-Fi preamble. It measures the energy of the sampled signal $\mathbf{y}$: $P = \sum_{k=1}^L \left|y[k]\right|^2$, where $L$ is the measuring window. If $P_s$ is higher than a given threshold, it then performs auto correlation to check whether the incoming signal is a Wi-Fi frame. Specifically, it tracks the auto correlation of the signal with $d$ delayed samples:
\begin{equation}
c_{\text{auto}}=\sum_{k=1}^{L} y[n+k] r^{*}[n+k-d].\notag
\end{equation}
$d$ is normally set to be 16, which is the period of one short sequence in STF. For a Wi-Fi packet, $c_{\text{auto}}$ spikes and forms a plateau. Our experiments show that even at very low SNR, the plateau is still clear to be separated from noise. Therefore, we configure normal clock rate for packet detection to avoid unnecessary energy consumption in idle listening and carrier sensing.
\begin{figure}[t]
\center
\includegraphics[width=3.5in]{sync_eg}
\caption{Cross correlation responses at various clock rates. We set up a USRP link to capture raw data samples pf a real Wi-Fi packet at 8$\times$ and 1$\times$ Nyquist rates, respectively.}\label{fig:sync_eg}
\end{figure}
After successfully detecting a Wi-Fi packet, the receiver precisely locates the boundary between OFDM symbols. While the IEEE 802.11 specifications do not mandate any specific algorithm, a typical synchronization algorithm is to use the cross-correlation property of the LTF. In particular, the receiver correlates the received signal with the transmitted time-domain LTF sequence. However, when the SNR is poor, cross correlation cannot cancel out noise, thereby resulting in multiple false peaks. We exploit correlations in oversampled LTF signals to overcome this predicament. The standard LTF consists of 52 subcarriers with the identical magnitude and $0$ or $\pi$ phase. When the LTF is sampled by an overclocked ADC, the correlation between oversampled data can be described by Eq.~\ref{e:yg}. Hence, the receiver can treat all the samples as a new LTF, which has stronger correlation properties. Figure~\ref{fig:sync_eg} illustrates the merits of using the oversampled LTF for synchronization. Under low SNR conditions, the cross correlation result of a standard LTF produces multiple comparable peaks, which makes the receiver easily aligns to a wrong peak. The cross correlation result of an oversampled LTF produces a single highest peak corresponding to the full alignment of OFDM symbols.
We set normal clock rate for idle listening to avoid unnecessary energy consumption. After packet detection, the receiver switches to the overclocking mode, which incurs latency for clock switch. As the receiver keeps the same frequency synthesizer and the center frequency of its analog circuit, the latency comes from the digital phase-locked loop (PLL) stabilization. Wi-Fi radios take merely several microseconds (e.g., 8~$\mu s$ in MAXIM2831~\cite{maxim}) to stabilize PLL. As the state-of-the-art IoT Wi-Fi, i.e., IEEE 802.11ah, shrinks the conventional 20~MHz bandwidth to 2~MHz while retaining the same number of subcarriers, the duration of one OFDM symbol is extended to 40~$\mu s$. Both STF and LTF last 80~$\mu s$, which leaves enough time for clock switch. As illustrated in Figure~\ref{fig:autocorr}, which is tested with SNR=10 dB, we observe that the first nine repetitions are enough to produce a plateau for packet detection. Thus, we take the first nine repetitions in STF to perform auto correlation while leaving the last repetition of STF for clock switch.In our experiment results, we show that it does not need to extra packet loss and the whole timing synchronization perform is boosted compared to the standard algorithm under low SNR conditions.
\begin{figure}[t]
\center
\includegraphics[width=3.5in]{autocorrelation_snr10}\vspace{0.3cm}
\caption{Auto correlation response.} \label{fig:autocorr}
\end{figure}
\subsection{Frequency Offset Compensation}
CFO varies over time and must be estimated and compensated for each frame. In practice, the received baseband signal, instead of being centered at DC (0 Hz), is centered at a frequency offset $\Delta f$.
\begin{align}
y_{\text{CFO}}(t) &= y(t)e^{\frac{j2\pi\Delta ft}{F_{s}}},\notag
\end{align}
where $F_s$ is the sampling frequency. CFO induces phase rotation over time that not only undermines the payload decoding but also affects the phase correlation among $Y_g[l]$. As our overclocking design relies on the phase correlation, CFO must be precisely estimated and calibrated.
Standard Wi-Fi receiver estimates CFO by comparing the phase rotation between the two identical OFDM symbols in LTF. Such an estimation is accurate enough for normal packet decoding, while the residual offset cannot be neglected when using the phase correlation among oversampled data. We make a fine-grained calibration in the following data symbols by exploiting the CFO effect among oversampled sequences $y_g[n]$. Let $\mathbf{y}_g = [y_g[0],...,y_g[N-1]]^\top$ denotes $g$th copy of the oversample signals, and the received data values in frequency domain $\mathbf{d} = [X[0]H(0),...,X[N-1]H(N-1)]^\top$. FFT is expressed in the form of a $N \times N$ matrix $\mathbf{F}$, where each entry $f_{ij}=e^{j (i-1)(j-1)}$. The effect of CFO on the first copy can be expressed as $\mathbf{P}=\text{diag}\left(1 \; e^{j {\Delta f\over F_{s}}} \cdot \cdot \cdot e^{j {\Delta f (N-1)\over F_{s}}}\right)$. The phase shifts due to overclocking can be described by $\mathbf{O}_g=\text{diag}\left(1 \; e^{-{j 2 \pi g \over NG}} \cdot \cdot \cdot e^{-{j 2 \pi g(N-1) \over NG}} \right)$. Then, the $g$th copy of the oversample signals $\mathbf{y}_g$ can be expressed as
\begin{equation}
\mathbf{y}_g = e^{\frac{j2\pi\Delta fg}{F_{s}G}} \mathbf{P} \mathbf{F} \mathbf{O}_g \mathbf{d} + \mathbf{W_g}.
\end{equation}
In the absence of noise, we obtain the following the relationship
\begin{align}\label{e:cfo}
\mathbf{F}^{\text{H}} \mathbf{P}^{\text{H}} \mathbf{y}_g &= e^{-\frac{j2\pi\Delta fg}{F_{s}G}} \mathbf{O}_g^{\text{H}} \mathbf{F}^{\text{H}} \mathbf{P}^{\text{H}} \mathbf{y_0},
\end{align}
where $(\cdot)^{\text{H}}$ is the conjugate transpose of a matrix. It is intuitive to find the unique $\Delta f$ that satisfying the above equation. In the presence of noises, we minimize the sum of distances between the left and the right hands of Eq.~\ref{e:cfo} for all $g>0$. We employ ML estimator to derive $\Delta f$.
\subsection{Decoding
After timing synchronization and frequency offset compensation, channel response needs to be understood before packet decoding. Conventional Wi-Fi receivers utilize the two training symbols in LTF to estimate the channel response at each subcarrier. As the data bits encoded on each subcarrier in LTF are pre-known by the receiver, the channel response is obtained by comparing the received data values and the transmitted ones. Based on the estimated channel response, the receiver removes the effect of wireless channel on the following OFDM symbols to extract the transmitted data values. Noises are assumed to be independent and Gaussian distributed, and thus the receiver demodulates each data value by mapping it to the nearest lattice point in the constellation map. We extend the above rationality to the overclocking case by averaging multiple copies of compensated samples before feeding them into the standard ML decoder.
\subsection{Putting Everything Together}
An overview of \texttt{T-Fi} system architecture is shown in Figure~\ref{fig:architecture}. \texttt{T-Fi} leverages the AP's high clock rate and processing capability to grant lower TX power for IoT devices. The reception pipeline conforms to the logic of a standard Wi-Fi receiver but is tailored to the asymmetric design to make the most of the redundancy induced by overclocking. An AP uses the first nine repetitions of the STF for packet detection and then switches to the high clock rate mode within the STF reception within the last one repetition. As the symbol length in IoT Wi-Fi packets is 10$\times$ of that in conventional Wi-Fi packets, it leaves enough time for the state-of-the-art chips to switch and stabilize the clock. Under the overclocking setting, the AP yields redundant LTF and payload samples, and exploits the correlations among these samples to boost the performance of synchronization and CFO estimation under low SNR conditions. Then synchronization algorithm is performed on the LTF, while CFO estimation is performed on the LTF and then calibrated in the following data symbols in the payload. Multiple copies of oversampled samples are averaged and fed to the standard ML decoder.
\begin{figure}[t]
\center
\includegraphics[width=3.5in]{architecture}\vspace{0.3cm}
\caption{System architecture of \texttt{T-Fi}.} \label{fig:architecture}\vspace{0.3cm}
\end{figure}
\section{Implementation}\label{sec:implementation
\texttt{T-Fi} can be realized in the existing OFDM PHY with no hardware changes. We prototype \texttt{T-Fi} on top of the OFDM implementation on the GNURadio/USRP platform. We implement the entire PHY design specified in Section~\ref{sec:design} directly in the USRP Hardware Drive (UHD). We use USRP B210 nodes connecting to PCs with Intel i7 quad-core processor and 8~GB memory for the testbed setup. Nodes in our experiments are configured to operate in the 2.4-2.5~GHz or 900~MHz range.
The Transmitter is configured to continuously send Wi-Fi packets conforming to IEEE 802.11ah PHY format. It operates on a 2~MHz or 1~MHz channel, of which 52 subcarriers are configured to carry data values while 4 subcarriers are pilot tones. The symbol duration is 40~$\mu s$. We adopt the legacy PHY layer convergence procedure (PLCP) format of IEEE 802.11ah, where the PLCP preamble consists of 2 STF OFDM symbols and 2 LTF OFDM symbols. There are 10 MCSs starting from 1/2 BPSK to 5/6 256QAM. Due to hardware limitations, we only implement BPSK, QPSK, 16QAM, and 64QAM modulations.
\begin{figure}[t]
\center
\includegraphics[width=3.5in]{floorplan}\vspace{0.3cm}
\caption{Experimental floorplan.} \label{fig:floorplan}
\end{figure}
\section{Evaluation}\label{sec:evaluation
In this section, we present a detailed experimental evaluation of \texttt{T-Fi}. Our experiments center around two questions: (i) How much decoding performance improvement can \texttt{T-Fi} provide in real wireless environments under proper overclocking settings? (ii) How much TX power can \texttt{T-Fi} save without compromising the decoding performance? To answer these questions, we conduct a set of experiments to evaluate the performance of synchronization, decoding, the overall packet reception, as well as TX power.
\subsection{Experimental Setup}
We conduct our experiments using USRP B210 nodes deployed in an indoor environment with metal/wood shelves and stone walls. Figure~\ref{fig:floorplan} illustrates the floorplan of the environment, where nodes are placed in a lab, a meeting room, a hallway, and a corridor. We control the TX power by adjusting the programmable-gain amplifier (PGA) in USRP. As such, we sweep the SNR range from 9~dB to 35~dB. For each setting, a USRP node sends approximate 3000 packets. For each setup, we vary the clock rates of the receiver from the normal clock rate ($1\times$) to overclocked rates ($2\times$, $4\times$, $8\times$). Note that the clock rate of IEEE 802.11ac/ax receiver is $10$-$80\times$. We use the standard IEEE 802.11 packet reception at the normal clock rate as the baseline for comparison.
\subsection{Synchronization}
\begin{figure}[t]
\center
\includegraphics[width=3.5in]{1_1_sync_errorbar}\vspace{0.3cm}
\caption{Synchronization error.} \label{fig:sync}
\end{figure}
First, we show that \texttt{T-Fi} addresses the synchronization issue under low SNR conditions. Recall that under low SNR conditions, conventional cross correlation algorithm cannot cancel out noise and yields multiple false peaks. We instead exploit the inherent correlation in the oversampled LFT signals to enhance the correlation property. Since we cannot directly obtain the ground truth of synchronization, we capture raw USRP samples and conduct offline analysis. In particular, we log raw samples received by a USRP node and feed them into an emulator to rehearse the synchronization process. We run the synchronization \texttt{T-Fi} at multiple clock rates (2$\times$, 4$\times$, and 8$\times$ clocking rates) and the standard synchronization algorithm at the normal clock rate (1$\times$ clock rate).
Figure~\ref{fig:sync} compares the timing estimation error, which is measured in number of samples. Error bars show the standard deviation of time synchronization offset. Under all SNR conditions, synchronization error diminishes with the increment of clock rates. The standard synchronization algorithm works well for SNR$>15$, while the synchronization error grows substantially when SNR is low (9~dB - 12~dB). Empowered by overclocking, \texttt{T-Fi} largely reduces the synchronization error. When SNR=9~dB, the average synchronization error at $8\times$ clock rate is 0.91, which is merely 13\% of the average error at the standard clock rate. These results confirm the merit of using over-clocking to do time synchronization.
\subsection{Decoding}
\textbf{Decoding performance under various SNR conditions.} Our second set of experiments evaluates the decoding performance at different MCS modes under various environments and SNR conditions. To focus on the decoding performance, we use the same synchronization algorithm at the same clock rate for all decoders. In particular, we set $8\times$ clock rate to receive packets, and perform synchronization and CFO compensation. Unless otherwise stated, USRP nodes operate at 900~MHz, which is the frequency band specified in IEEE 802.11ah.
In Figure~\ref{fig:modulation}, we compare the decoding performance under different modulation schemes. The clock rate of \texttt{T-Fi} receiver is $8\times$ of the standard 802.11 receiver. As expected, for all modulations demonstrated, \texttt{T-Fi} outperforms the standard 802.11 receiver. We also observe that \texttt{T-Fi} substantially reduces BER compared to the standard receiver for high-order modulations. When SNR=9~dB, the BER of \texttt{T-Fi} is merely 25\% of the standard receiver's BER. The reason is that high-order modulations are more error-prone under low SNR conditions, which leaves more room for overclocking to correct these errors.
Another observation is that the decoding performance of \texttt{T-Fi} for 64QAM (16QAM) is comparable to that of the standard receiver for 16QAM (QPSK). This result implies that \texttt{T-Fi} can use higher-order modulations for packet transmission compared to standard Wi-Fi, and thus delivers roughly $4\times$ data rate. Hence, we can envision a new dimension of the \texttt{T-Fi} is to improve the throughput in IoT Wi-Fi.
\textbf{Impact of wireless environment.} As analyzed in Section~\ref{sec:overclock}, the decoding performance depends on channel response and noise. In our previous experiments, all nodes are in the same lab, and we focus on the decoding performance under different SNR conditions. In this experiment, we evaluate the impact of wireless environments, which lead to different multi-path fading and shadowing. We repeat the previous set of experiments at three additional locations in the same building but with significantly different propagation environments. The locations of IoT nodes and the AP include a hallway in the lab, a corridor outside the lab, and a meeting room next to the lab, as illustrated in Figure~\ref{fig:floorplan}. We use a fixed PGA to set the same TX power for all locations.
Figure~\ref{fig:loc_rate} compares the decoding performance under different clock rates at different locations. For all locations, \texttt{T-Fi} outperforms the standard 802.11 decoding, and higher clock rate \texttt{T-Fi} receivers achieve lower BER than the lower clock rate ones. Compared to the standard receiver, \texttt{T-Fi} at $8\times$ clock rate reduces BER to 56\%-23\% across all locations. \texttt{T-Fi} yields higher performance gain in the hallway and the corridor. The reason behind this is that the BER in these two locations falls in a sweet range where there is room for BER improvement while noise can be amortized by exploiting overclocking. The results confirm that \texttt{T-Fi} achieves stable performance gain over a wide range of wireless environment. Figure~\ref{fig:loc_mod} further shows the BER performance for different modulation schemes at the four locations. Though the BER of different modulations varies significantly across different locations, \texttt{T-Fi} achieves steady BER gain over the standard receiver.
\begin{figure}[t]
\center
\includegraphics[width=3.5in]{2_2_Modulation_250k_900M}
\caption{BER for different modulations.} \label{fig:modulation}\vspace{0.1cm}
\end{figure}
\begin{figure*}
\centering
\begin{minipage}[b]{0.66\textwidth}\centering
\subfigure[\scriptsize Different sample rates]
{\label{fig:loc_rate}\includegraphics[width=0.5\textwidth]{2_3_Location_250k_900M_16QAM_16x_70dB}}\hspace{-0.2cm}
\subfigure[\scriptsize Different modulations]
{\label{fig:loc_mod}\includegraphics[width=0.5\textwidth]{2_4_Location_QPSK64QAM_1x8x_70dB}}\vspace{0.3cm}
\caption{BER in different locations.}
\label{fig:loc}
\end{minipage} \vspace{0.2cm}
\hspace{-0.2cm}
\begin{minipage}[b]{0.33\textwidth}\centering
\includegraphics[width=1\textwidth]{2_5_carrier_freq_250k_16QAM}\vspace{0.45cm}
\caption{BER under different carrier frequencies.} \label{fig:freq}
\end{minipage} \vspace{0.2cm}
\end{figure*}
\textbf{Impact of carrier frequency.} Previous experiments use 900~MHz as the carrier frequency, which is the band specified by IEEE 802.11ah. Recently, several leading companies have been exploring the possibility of using 2.4~GHz for IoT transmissions. To meet this potential demands, we now evaluate the BER performance in both carrier frequencies under various SNR conditions.
Figure~\ref{fig:freq} shows the BER for 16QAM decoding. We have the following observations: (i) The BER performance at both frequencies has the same trend and is comparable in most cases; (ii) \texttt{T-Fi} achieves lower BER compared to the standard receiver at both frequencies. The results validate the feasibility of \texttt{T-Fi} at both carrier frequencies.
\vspace{0.3cm}
\section{Related Work}\label{sec:relatedwork
\textbf{IoT Wi-Fi standard.} IEEE 802.11ah, known as Wi-Fi HaLow~\cite{wifihalow}, is the first and only Wi-Fi protocol dedicatedly designed for IoT. The target of this IoT Wi-Fi protocol is to utilize the Wi-Fi technology to provide low power and long range transmission for the emerging low-end IoT devices. It reuses the OFDM PHY as in IEEE 802.11ac, but provides a bundle of low power features, particularly at the MAC layer. Research efforts have focused on enhancing the MAC layer~\cite{park2014enhancement,liu2013power}. Our goal is to design a transceiver architecture for IoT Wi-Fi networks that (i) completely conforms to the IoT Wi-Fi standard without any protocol modifications, and at the same time (ii) take full advantage of AP to further reduce the TX power of IoT devices. Our compliant design can be seamlessly integrated into the existing IoT Wi-Fi protocol. Existing Wi-Fi power models~\cite{qiao2003miser,jung2002energy,baiamonte2006saving,bruno2002optimization,carvalho2004modeling,chen2008edca,chen2008edca,ergen2007decomposition,garcia2011energy} show that power amplifier operations dominate the total power consumption of Wi-Fi radios. Thus, the saved power budget by our design can facilitate lower power or longer range transmission, which fully aligns with the targets of IoT Wi-Fi protocols.
\textbf{Downclocking.} Recently, researchers have demonstrated that downclocking Wi-Fi radios can effectively reduce the energy consumption during packet reception and idle listening. E-Mili~\cite{mobicom11emili} pioneers this kind of mechanisms to downclock receiver's clock rate during idle listening, and switches to full clock rate for packet reception. SloMo~\cite{nsdi13slomo} enables packet decoding in IEEE 802.11b while downclocking by exploiting the sparsity in direct sequence spread spectrum (DSSS) PHY. SEER~\cite{wang2017wideband} designs a special preamble to allow narrowband receiver sensing a much wider band signal without boosting the sampling rate. Downclocking for packet reception has been extended to OFDM-based Wi-Fi by leveraging the gap between modulation and SNR~\cite{mobicom14enfold}. Instead of exploiting PHY redundancy, Sampleless Wi-Fi~\cite{wang2017sampleless,wang2016rateless} utilizes the retransmission opportunities to decode packets at downclocked rates without relying on PHY redundancy, and thus makes downclocking OFDM packet reception feasible under low SNR conditions. \texttt{T-Fi} takes an opposite approach by overclocking the receiver's radio to reduce the power consumption of the transmitter, which is complementary to downclocking approaches by bringing low power to the transmitter side.
\vspace{0.3cm}
\section{Conclusion}\label{sec:conclusion}
This paper introduces \texttt{T-Fi}, an asymmetric transceiver paradigm for IoT that pushes power burden to the AP side, and enables IoT devices to transmit packets at power levels that are even lower than the minimal power required by conventional receivers. We think this is an important design point in IoT communications, and the teeter-totter fashion in \texttt{T-Fi} has significant ramifications in the new transceiver design for IoT Wi-Fi protocols. Our experimental evaluation confirms the benefits of \texttt{T-Fi} in real environments. We hope the design can contribute the wireless community by providing some insights for future transceiver design that takes advantage of the hardware asymmetry between APs and IoT devices.
\vspace{0.3cm}
\section*{Acknowledgment}
The research was supported in part by the National Science Foundation of China under Grant 61502114, 91738202, 61729101, and 61531011, Major Program of National Natural Science Foundation of Hubei in China with Grant 2016CFA009, and the Fundamental Research Funds for the Central Universities with Grant number 2015ZDTD012.
\balance
\bibliographystyle{IEEEtran}
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} | 3,520 |
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Home - News - Pangs 2022 Half-Year Work Report It's Time to Chase Dreams in Your Life
Pangs 2022 Half-Year Work Report It's Time to Chase Dreams in Your Life
In July 2022, Pangs Chem held a meeting to summarize the achievements of each department in the first half of the year. The meeting was divided into three parts: the team leaders' reports, the department managers' reports, and the summary by Mr. Pang.
Team Leader Work Report
Jessie, the team leader from second business department, has continued to grow in performance in the first half of the year. Her team members can also develop customers independently, own ideas for their team development and have growth. In her opinion, it's necessary to continue to pay attention to the shortage of talents reserve and to inject "fresh blood" into the department in a timely manner. The market is changing rapidly, we need timely respond to the situation that the low conversion rate of potential customers and the shortage of new customers , and pay attention to shift the sales strategies. In second half of this year, she said attention should be paid to developing new markets, risk management and control, and talents development.
Marcelo, one team leader from third business department, said that in the first half of the year personal growth requires self-discipline and a change in thoughts. Also team growth requires mutual trust. In his team, it deeply reflect on problems such as loose management, fixed thought, and poor collaboration. It is necessary to identify the "pain points" and apply "medicine" according to indications. In second half, he plans to expand customer development channels, develop new markets and new products. And to increase repurchase rate of customers after closing. Meantime, to focus on team training, to improve team tacit understanding and to work efficiency.
Gabriel, another team leader from third business department, thought that it was necessary to build up a team, to optimize the organization and to have plan to carry out the business; to do good job in the analysis of successful and failed cases; to strengthen the understanding of the changes of foreign trade and the diversification of products. In second half, their team are going to regularly organize analysis and discussion on significant customers difficult to close and to find solutions. For such customers, it is necessary to teamwork and tackle difficulties together.
Olivia, one team leader from fourth business department, helped newcomers integrate into new environment and adapt to new job, and gave them development directions. She said: " Due to my limited experience, I have not been able to set proper goals for newcomers. I still need to keep learning, increase my business knowledge, and strive for more correct guidance and help for newcomers." Her plan is to shift from stocking to strict management, to set "big" goals for newcomers, and to strictly implement customers development tasks. To improve the level, to benefit others and oneself.
Sebastian, another team leader from third business department, completed the goals at the beginning of this year on schedule in the first half of the year, and made steady progress. The join of new staff brings new energy and powers to the small team. Due to that the formation of his team is not long, it takes a certain amount of time to hone. Through with systematic study and sharing the thought on leadership of our boss, his management knowledge has been improved to a certain extent. In second half, he expressed that his team would strive to achieve an increase in sales and settled profits compared with the second half of last year. To help newcomers grow quickly, to continue to enrich their professional quality and ability to transform their identity.
Aida, the team leader from fifth business department, did not set small goals and plans for newcomers due to the lack of innovation in the team management model. The problem exposed: their total number of customers is small and the major customers are less. Her team had an idea to explore new markets and new products. To research the market and products, to find competitive supply chain and to targeted develop new customers. In second half, the work plan is to adheres to zero risk and newcomers growth. To find major customers and middlemen and to develop new markets and new products.
Alex, the team leader from sixth business department, has got orders regularly owing to the stable maintenance of his customers. There was continuous demand for new products of his customers, so he got orders and the samples were sent several years ago. There are still some deficiencies in his work, such as a lack of a thorough understanding of the products, a lack of grasping the opportunities and time to sign orders and the lack of new customers. In second half, he and his team will focus on developing new customers and changing the situation between the customers, pay attention to new products and new opportunities according to the new strategy of our boss-Mr.Pang. To let the newcomers of the department become proficient as soon as possible, and to keep up with the footsteps of the whole department and the whole company.
Department Manager Work Report
Wang Dongjuan, the manager of the finance department, in the first half year learned the courses arranged by company personnel administration department. She is a person that thought in her work. She found that her business ability and management ability still need to be improved. It's useful to learn more from our boss and try her best to improve the management level. To communicate more with colleagues in the finance department to improve problems' feedback and resolution speed. In second half, she plans to complete the daily work on time, arrange the work reasonably in advance and to organize everyone's work and to require staff being more meticulous and standardized.
Yuan Jiang, manager of the purchasing department, said that the department of the supply chain was generally stable in the first half of the year and the number of competitive products of Pangs increased significantly. The cooperation and coordination between with major suppliers had been optimized; Doing a good job in market research, estimated procurement and price negotiation. To detail management of planned procurement contracts and processes in second half; meantime to improve procurement team and business team coordination; to do market research and strategic procurement and to expand new products.
Eduardo, manager of first business department, expressed that the team and the department business performance is generally excellent. Adhering to the principle-to be formal staff based on receiving order, which the newcomers have received orders in less than two months under the leadership of the team and it is very good. He has rich experience in the aspect of newcomers growth and progress, however, there is still room for improvement in terms of business scale.
Daniel, manager of second business department, said his department formation was basically completed, the business structure was reasonable and the risk control was mature. But the lack of continuity of business and "hematopoietic" capability of his department is a hidden danger. In the second half, it is hoped that the number of monthly matching sales will reach the expected target. To adjust the business direction appropriately and to improve the "hematopoietic" function quickly .
Justin, manager of third business department, expressed that the sales in the first half reached the standard of the whole one of last year. The number of personnel in the department increased in total, and each teams formed an atmosphere of competition and chased after each other. As a department manager, he feels that he lacks management experience and fails to completely abandon the old work pattern. It was found that the department has single product sold and had not developed new products and new markets. In second half, he want to strictly implement the wore process required by our boss, to strictly control risks and to require zero error and zero risk.
Harvey, manager of fourth business department, said that due to the shortage of department funds, orders lost and shipment delayed, for which sales fell off a cliff. He found that there was a gap of each salesmen performance, this problem was serious and so as other problems. He figured out a way to slightly loosen credit sales restrictions and encourage team members to take more credit sales, in order to close or to achieve the company' average level. In second half, it is necessary to set up customer development targets, to hold weekly summary and other methods. He hoped to ensure sales performance, adjust sales strategies, pay more attention to the training and incentives of new coming and company experienced salesmen and promote all department progress.
Arron, manager of fifth business department, indicated that the overall operation of his department was stable and had a certain ability to resist risks. On this basis, there were also some problems, such as the lack of new customers and the lack of new markets and new products. The pace of new employees development lagged behind other departments over the same period. He figures out focusing on researching our company competitive products and taking full advantage of the team. New markets and new products should be carefully studied and other team developing solutions. In second half, it is hoped that the team will build better and develop major customers that can promote development of the department, also decrease the risk to zero.
Leonard, manager of sixth business department, concluded that training newcomers must form a working atmosphere of helping each other. To effectively and purposefully improve rate of closing orders by new customers, it is necessary to actively guide bestsellers or popular products and provide valuable customers. So as teamwork, resource sharing and other ideas. In second half, to make a difference, he plans to keep the department stable, the spirit positive, the performance promoting and the risks controlling.
Mr. Pang summed up: The world's great things must be done in detail. The only one we need to conquer is ourselves without being influenced by others and to become a better version of ourselves.
Previous : 【Transaction Strategy and Price Game】--Business Sharing Session
Next : July pangs Travel Notes
Hundreds of barges strive for strong waves--- Happy 20th birthday to PANGS [ 2022-12-14 ]
The Third Quarter Summary meeting "IT'S GONNA BE A LONG HARD ROAD, BUT WE'LL MAKE IT" [ 2022-10-27 ]
[Risk Control and Risk Analysis] Business Sharing Meeting [ 2022-09-29 ]
Congratulations on the relocation of Pangs Chemical Chongqing Division of Labor, and the future can be expected [ 2022-08-09 ]
【Transaction Strategy and Price Game】--Business Sharing Session [ 2022-08-02 ]
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\chapter*{ }
\vskip -5truecm
\centerline {\Large\bf Quantization and eigenvalue distribution}
\centerline {\Large\bf of noncommutative scalar field theory}
\vskip 10truemm
\centerline {\bf Harold Steinacker}
\vskip 5truemm
\centerline{Institute for Theoretical Physics, University of Vienna}
\centerline{Boltzmanngasse 5, A-1090 Vienna, Austria}
{\renewcommand{\thefootnote}{}
}
\setcounter{page}{1}
\markboth{ }
{ }
\vskip 5truemm
\begin{quote}
\small{\bf Abstract.} {\it
The quantization of noncommutative scalar field theory
is studied from the matrix model point of view, exhibiting
the significance of the eigenvalue distribution.
This provides a new framework to study
renormalization, and predicts a
phase transition in the noncommutative $\phi^4$ model.
In 4-dimensions, the corresponding
critical line is found to terminate at a non-trivial point.
}
\end{quote}
\begin{quote}
\small{\bf Key words:} {\it Noncommutative field theory, phase transition,
matrix model, renormalization }
\end{quote}
\vskip 5truemm
\centerline {\sc 1. INTRODUCTION: NC FIELD THEORY AND UV/IR MIXING}
\vskip 3truemm
Noncommutative (NC) field theory has been studied intensively in recent years,
see e.g. \cite{douglas, Szabo} and references therein.
Part of the motivation has been the hope that the UV divergences of Quantum Field
Theory should be controlled and perhaps regularized by noncommutativity,
which in turn should be related to quantum fluctuations of geometry
and quantum gravity.
In the simplest case of the quantum plane
${\Bbb R}^{d}_\theta$, the coordinate functions $x_i, i=1,...,d$ satisfiy
the canonical commutation relations
\begin{equation}
[x_i,x_j] = i \theta_{ij}.
\label {CC}
\end{equation}
We assume for simplicity that $\theta_{ij}$ is
nondegenerate and maximally-symmetric, characterized by a single
parameter $\theta$ with dimension $({\rm length})^2$. This
implies uncertainty relations among the coordinates,
\begin{equation}
\Delta x_i \Delta x_j \geq \frac 12 \theta_{ij} \approx \theta.
\label{uncertainty}
\end{equation}
This leads to an energy scale
\begin{equation}
\Lambda_{NC} = \sqrt{\frac 1{\theta}},
\end{equation}
and noncommutativity is expected to be important for energies
beyond $\Lambda_{NC}$.
The most naive guess would be that $\Lambda_{NC}$ plays the role of a UV cutoff
in Quantum Field Theory.
However, this turns out not to be the case.
The quantization of NC field theory
not only suffers from divergences in the UV, but also from new divergencies
in the IR, which are in general not under control up to now.
This type of behavior can to some extent be expected from \eq{CC}.
To understand this, expand a scalar field on ${\Bbb R}^{d}_\theta$
in a basis of plane waves,
$\phi(x) = \int d^d p \,\phi(p) \exp(i p x)$, and
consider the range of momenta $p$. Formally, $p \in {\Bbb R}^d$
has neither an IR nor a UV cutoff.
As usual in field theory, it is necessary however
to impose cutoffs $\Lambda_{IR} \leq p \leq \Lambda_{UV}$, with the hope that the
dependence on the cutoffs can be removed in the end;
this is essentially the defining property of a renormalizable field theory.
In view of \eq{uncertainty}, if $\Delta x_i$ is minimized according to
$\Delta x_i = \Lambda_{UV}^{-1}$,
it implies that some $\Delta x_j \geq \theta \Lambda_{UV}$.
This means that momenta near
$\Lambda_{UV}$ are inseparably linked with momenta in the IR,
and suggests $\Lambda_{UV} \Lambda_{IR} = \Lambda_{NC}^2$.
A consistent way to impose IR and UV cutoffs in NC field theory is to
replace ${\Bbb R}^n_\theta$ by some compact NC spaces such as fuzzy tori, fuzzy
spheres etc. The algebra of functions is then given by a
finite matrix algebra $Mat({\cal N}\times{\cal N},{\Bbb C})$. One then finds
quite generally
\begin{equation}
\Lambda_{IR} = \sqrt{\frac 1{N\theta}} \quad\ll\quad \Lambda_{NC} =
\sqrt{\frac 1{\theta}} \quad\ll\quad \Lambda_{UV} = \sqrt{\frac N{\theta}}
\end{equation}
where ${\cal N} \approx N^{d/2}$
(depending on the details of the compact NC space) is the
dimension of the representation space. This shows that
the scales in the UV and IR are intimately related, which is
the origin of $UV/IR$ mixing in NC field theory. It is then
not too surprising that a straightforward Wilsonian renormalization
scheme involving only $\Lambda_{UV}$
will generically fail \cite{SMV}.
The remainder of this paper is a qualitative discussion of a
new approach to quantization of euclidean NC scalar field theory,
which was proposed in \cite{NCscalarFT} with the goal to
take into account more properly the specific features of NC field theory.
\vskip 5truemm
\centerline {\sc 2. NONCOMMUTATIVE SCALAR FIELD THEORY}
\vskip 3truemm
Consider the scalar $\phi^4$ model on a suitable $d$-dimensional NC space,
\begin{equation}
S = \int d^d x\, \left(\frac 12 \phi \Delta \phi
+ \frac 12 m^2 \phi^2 + \frac{\lambda}{4} \phi^4\right)
= S_{kin} + \int d^d x\, V(\phi).
\label{Rd-action}
\end{equation}
We assume that the algebra of functions on the NC space is represented
as operator algebra on some {\em finite-dimensional} Hilbert space
(this is the defining property of the so-called fuzzy spaces), so that
$\phi \in Mat({\cal N}\times {\cal N},{\Bbb C})$ is a Hermitian matrix.
The integral is then replaced by the
appropriately normalized trace, $\int d^d x = (2\pi\theta)^{n/2} Tr()$.
In this finite setup, the quantization is defined by an integral over all
Hermitian matrices, e.g.
\begin{eqnarray}
Z &=& \int [{\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F} \phi] e^{-S}, \label{Z-full}\\
\langle \phi_{i_1 j_1} \cdots \phi_{i_n j_n} \rangle &=&
\frac 1Z \int [{\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F} \phi] e^{-S}\; \phi_{i_1 j_1} \cdots \phi_{i_n j_n}
\end{eqnarray}
As a warm-up, we recall the analogous but simpler case of
\paragraph{Pure matrix models.}
These are defined by an action of the form
\begin{equation}
S^{M.M.} = {\cal N} Tr(V(\phi))
\end{equation}
for some polynomial $V(\phi)$,
where again $\phi \in Mat({\cal N}\times {\cal N},{\Bbb C})$ is a Hermitian matrix.
These pure matrix models can be solved exactly by the following change
of coordinates in matrix space:
\begin{equation}
\phi = U^{-1} D U
\end{equation}
where $D$ is a diagonal matrix with real eigenvalues $\phi_i$, and $U \in U({\cal N})$.
Using $[{\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F} \phi] = \Delta^2(\phi_i) dU d\phi_1 ... d \phi_N$, where $dU$ denotes the
Haar measure for $U({\cal N})$ and $\Delta(\phi_i) = \prod_{i<j}(\phi_i-\phi_j)$
is the Vandermonde-determinant.
The integral over $dU$ can be carried out trivially,
reducing the path integral to an integral over eigenvalues,
\begin{equation}
Z = \int d\phi_i \Delta^2(\phi_i-\phi_j)\exp(-{\cal N} \sum_i V(\phi_i))
= \int d\phi_i \exp(-S_{eff}^{M.M}(\vec\phi))
\label{Z-MM}
\end{equation}
with an ``effective action'' of the form
\begin{equation}
\exp(-S_{eff}^{M.M.}(\phi_i)) = \exp(\sum_{i\neq j}
\log|\phi_i-\phi_j| -{\cal N} \sum_i V(\phi_i)).
\label{Seff-MM}
\end{equation}
We can assume that the eigenvalues are ordered,
$\phi_1 \leq \phi_2 \leq ... \leq \phi_{\cal N}$, and
denote with $\cE_{\cal N}$
the space of ordered ${\cal N}$-tupels $\vec \phi = (\phi_1, ..., \phi_{\cal N})$.
They will be interpreted as
coordinates of a point in the space of eigenvalues $\cE_{\cal N}$.
The remaining integral over $\phi_i$ can be evaluated with a variety of techniques
such as orthogonal polynomials, Dyson- Schwinger resp. loop equations,
or the saddle-point method. It turns out that the large ${\cal N}$ limit
is correctly reproduced by the saddle-point method,
because the Vandermone-determinant
$\Delta^2(\phi_i) = \exp(\sum_{i\neq j} \ln|\phi_i-\phi_j|)$ corresponds to a
strongly repulsive potential between the eigenvalues, which
leads to a strong localization in $\cE_{\cal N}$.
This means that the effective action $\exp(-S_{eff}^{M.M.}(\vec\phi))$
is essentially a delta-function in $\cE_{\cal N}$,
\begin{equation}
\exp(-S_{eff}^{M.M.}(\vec \phi)) \approx \exp(-{\cal N} Tr(V(\vec \phi_0))) \,
\delta(\vec \phi - \vec \phi_0).
\label{localiz-delta}
\end{equation}
One now proceeds to determine the
localization of the maximum $\vec \phi_0 \in \cE_{\cal N}$ by solving
$\frac{\partial}{\partial \phi_i} S_{eff} =0$, which can be written as an
integral equation in the large ${\cal N}$ limit.
Expectation values can then be computed as
\begin{equation}
\langle f(\vec\phi)\rangle = f(\vec\phi_0)
= f(\langle\vec\phi \rangle)
\end{equation}
which is indeed correct in the large ${\cal N}$ limit of pure matrix models,
consistent with \eq{localiz-delta}.
This "factorization'' of
expectation values is characteristic for a delta-function
integral density, and can be used to test the localization hypothesis
\eq{localiz-delta} resp.
the sharpness of the (approximate) delta function.
\paragraph{Strategy for the full model.}
This analogy suggests to apply similar techniques
also for the field theory \eq{Z-full}.
At first sight, this may appear impossible,
because the action is no longer invariant under
$U({\cal N})$. However, let us assume that we could somehow evaluate the
``angular integrals''
\begin{equation}
\int dU \exp(-S_{kin}(U^{-1}(\phi) U)) = : e^{-\cF(\vec\phi)},
\end{equation}
Note that the resulting $\cF(\phi_i)$ is a totally symmetric, analytic function
of the eigenvalues $\phi_i$ of the
field $\phi$, due to the integration over the unitary matrices $U$.
Using this definition, the partition function
can indeed be cast into the same form as \eq{Z-MM},
\begin{equation}
Z = \int d \phi_i \Delta^2(\phi_i)\exp(-\cF(\vec\phi) -(2\pi\theta)^{d/2}\;Tr V(\vec\phi))
= \int d\phi_i \exp(-S_{eff}(\vec\phi))
\label{Z-F}
\end{equation}
(with the obvious interpretation of $Tr V(\vec\phi)$), defining
\begin{equation}
\exp(-S_{eff}(\vec\phi)) =\Delta^2(\phi_i)\exp(- \cF(\vec\phi) - (2\pi\theta)^{d/2}\;Tr V(\vec\phi)).
\label{S-eff}
\end{equation}
Of course we are only allowed
to determine observables depending on the eigenvalues with this effective action
(the extension to other observables is discussed in \cite{NCscalarFT}).
These are determined by
$S_{eff}(\vec\phi)$ in the same way as for pure matrix models,
since the degrees of freedom
related to $U$ are integrated out.
This demonstrates that all thermodynamic properties
of the quantum field theory, in particular
the phase transitions, are determined
by $S_{eff}(\vec\phi)$ and the resulting eigenvalue distribution
in the large ${\cal N}$ limit.
The advantage of this
formulation is that it is very well suited to include interactions,
and naturally extends to the non-perturbative domain.
So far, all this is exact but difficult to apply, since we will not be
able to evaluate $\cF(\vec\phi)$ exactly.
However, the crucial point in \eq{S-eff} is the presence of the
Vandermonde-determinant
$\Delta^2(\phi_i) = \exp(\sum_{i\neq j}\ln|\phi_i-\phi_j|)$,
which prevents the eigenvalues from
coinciding and strongly localized them. This effect cannot be
canceled by $\cF(\vec\phi)$, because it ts analytic.
This and the analogous exact analysis of the pure matrix models
suggests the following {\em central hypothesis} of our approach:
\eq{S-eff} is strongly localized, and
the essential features of the full QFT
are reproduced by the following approximation
\begin{equation}
\exp(-S_{eff}(\vec\phi)) \approx \exp(-\cF(\vec\phi_0)
- (2\pi\theta)^{d/2}\;Tr V(\vec\phi_0)) \,
\delta(\vec \phi - \vec \phi_0),
\label{localiz-delta-full}
\end{equation}
where $\vec \phi_0$ denotes the maximum (saddle-point) of $S_{eff}(\vec\phi)$.
The
remaining integral over the eigenvalues $\phi_i$ can
hence be evaluated by the saddle-point method.
This will be justified to some extent and made more precise
below, by studying
$\cF(\vec\phi)$ using the free case.
If $\cF(\vec\phi)$ is
known, the free energy can be found by determining the minimum
$\phi_0$ of $S_{eff}(\vec\phi)$.
However, \eq{localiz-delta-full} strongly suggests that also
correlation functions can be computed by restricting
the full matrix integral to the
dominant eigenvalue distribution $\vec\phi_0$. This
would provide nonperturbative control over the full NC QFT.
It is important to keep in mind that
a saddle-point approximation {\em before} integrating out $U({\cal N})$ would be
complete nonsense. It is only appropriate for the effective action
\eq{localiz-delta-full}.
The possibility of separating the path integral into
these 2 steps is only available for NC field theory.
\vskip 5truemm
\centerline {\sc 3. THE CASE OF FREE FIELDS}
\vskip 3truemm
For free fields, we can verify the validity of the basic hypothesis
\eq{localiz-delta-full}, and compute the unknown function
$\cF(\vec\phi)$ explicitly at least in some domain. This
will then be applied in the interacting case.
To check localization in the sense of \eq{localiz-delta-full},
we need to show that
the expectation values of all observables which depend only on the eigenvalues
$\vec\phi$ factorize, and can be computed by evaluating the observable at a
specific point $\vec \phi_0$.
A complete set of such observables is given by the product of traces of
various powers of $\phi$, which in field theory language is
$\langle(\int d^d x \phi(x)^{2n_1})...(\int d^d x \phi(x)^{2n_k}) \rangle$.
It is shown in \cite{NCscalarFT} using Wicks theorem
that in the large $N$ resp. $\Lambda_{UV}$ limit,
\begin{eqnarray}
\frac{\langle\big( \frac 1V\int d^d x \phi(x)^{2n_1}\big)
\big( \frac 1V\int d^d x \phi(x)^{2n_k}\big) \rangle}
{\langle \frac 1V\int d^d x \phi(x)^{2} \rangle^{n_1}
\langle \frac 1V\int d^d x \phi(x)^{2} \rangle^{n_k}}
&=& \frac{\langle \frac 1V\int d^d x \phi(x)^{2n_1}\rangle}
{\langle \frac 1V\int d^d x \phi(x)^{2} \rangle^{n_1}}\;
\;\frac{\langle \frac 1V\int d^d x \phi(x)^{2n_k} \rangle}
{\langle \frac 1V\int d^d x \phi(x)^{2} \rangle^{n_k}} \nonumber\\
&=& N_{Planar}(2n_1)\; ... \;N_{Planar}(2n_k).
\label{cluster}
\end{eqnarray}
Here $N_{Planar}(2n)$ is the number of planar contractions of a
vertex with $2n$ legs.
The non-planar contributions always involve oscillatory
integrals, and do not contribute to the above ratio in the large $N$ limit.
This is exactly the desired factorization property, which implies
that the effective action of the eigenvalue sector localizes
as in \eq{localiz-delta-full}, after suitable rescaling.
This is only true for
noncommutative field theories, where nonplanar (completely contracted) diagrams
are suppressed by their oscillating behavior, so that only
the planar diagrams contribute\footnote{this is only true for observables of
the above type, not e.g. for propagators}.
To complete the analysis of the free case, we need to find the dominant
eigenvalue distribution $\vec\phi_0$. That is an easy task using the known
techniques from pure matrix models described above.
Writing
$\varphi(s) = (\phi_0)_j, \quad s = \frac j{{\cal N}}$ for $s \in [0,1]$
in the continuum limit,
the saddle-point $\vec\phi_0$ then corresponds to a density of eigenvalues
$\rho(\varphi) = \frac{ds}{d\varphi}$
with $\int_{-\infty}^{\infty} \rho(p) dp =1$. This turns out to be
the famous Wigner semi-circle law
\begin{equation}
\rho(p) = \left\{\begin{array}{ll}\frac 2{\pi} \sqrt{1-p^2} & \;\; p^2<1\\
0, & \mbox{otherwise}.
\end{array}\right.
\label{wigner-law}
\end{equation}
This is the same as for the pure Gaussian matrix model, and it follows that
the effective action for the eigenvalue sector is given by
\begin{equation}
S_{eff}^{free}(\vec\phi)
= f_0(m) -\sum_{i\neq j}\ln|\phi_i-\phi_j| + \frac{2{\cal N}}{\alpha_0^2(m)} (\sum\phi_i^2).
\label{eff-act-eig}
\end{equation}
Here
$f_0(m)$ is some numerical function of $m$ which is not important here,
and
\begin{equation}
\alpha_0^2(m) = \left\{\begin{array}{ll} \frac{1}{4\pi^{2}}\;
\Lambda_{UV}^2 \left(1 - \frac{m^2}{\Lambda_{UV}^2}\ln(1+\frac{\Lambda_{UV}^2}{m^2})\right), & d=4\\
\frac{1}{\pi}\ln(1+\frac{\Lambda_{UV}^2}{m^2}) , & d=2
\end{array}\right.
\label{alpha}
\end{equation}
depending on the dimension of the field theory.
\vskip 5truemm
\centerline {\sc 3. APPLICATION: THE $\phi^4$ MODEL}
\vskip 3truemm
We can now apply the result \eq{eff-act-eig} to the interacting case.
Consider for example the $\phi^4$ model \eq{Rd-action},
which is obtained from the free case by adding
a potential of the form
$$
S_{int}(\phi) = \frac{\lambda}{4}\int d^d x \phi^4(x)
= \frac{\lambda}{4} (2\pi\theta)^{d/2}
Tr \phi^4,
$$
The goal is to derive some properties (in particular
thermodynamical, but also others) of the interacting model.
The basic hypothesis is again \eq{localiz-delta-full}, where the
eigenvalue distribution $\vec\phi_0$ in the interacting case
is to be determined.
It is quite easy to understand the main effect of the interaction term
using the above results \eq{eff-act-eig}:
since $S_{int}$ only depends on the eigenvalues, it seems
very natural to simply add $S_{int}$ to \eq{eff-act-eig}, suggesting
\begin{equation}
Z_{int} = \int d\phi_i e^{-S_{eff}^{free}(\vec\phi) - S_{int}(\vec\phi)}.
\label{Z-perturb}
\end{equation}
To justify this, we can expand the
interaction term into a power series in $\lambda$.
Using the factorization property \eq{cluster},
we can immediately compute any expectation value of
observables which depend only on the eigenvalues, and sum up the
expansion in $\lambda$. As in the case of pure matrix models \cite{BIPZ},
the effect of this perturbative computation is reproduced
for ${\cal N} \to \infty$
by the second (nonperturbative) point of view, which is that
\eq{Z-perturb} is localized at a new eigenvalue distribution
$\vec\phi_0^{\lambda}$, which now depends on the interaction.
It can again be found by the saddle-point approximation.
We conclude with some remarks and results of this analysis \cite{NCscalarFT}:
\begin{enumerate}
\item
A caveat: the validity of \eq{eff-act-eig} has been established only
``locally'', i.e. the function $\cF(\vec\phi)$ is really known only
for $\vec \phi \approx\vec \phi_0$. The only free parameter available
is the mass\footnote{this can be enhanced by adding other
terms such as the one in \cite{wulki}, which will be pursued
elsewhere},
which enters in \eq{eff-act-eig} through $\alpha_0$ and allows
to test a one-dimensional submanifold of $\cE_{\cal N}$. Therefore we
can trust \eq{eff-act-eig} only if $\vec\phi_0^{\lambda} \approx \vec\phi_0$.
This leads to the second observation:
\item
The physical properties of the interacting model will be close to those of
the free case only if $\vec\phi_0^{\lambda} \approx \vec\phi_0$.
In particular, the correlation length (i.e. physical mass) of the interacting model
will be reproduced best by the free action whose EV distribution
$\vec\phi_0$ is closest to $\vec\phi_0^{\lambda}$.
Working this out \cite{NCscalarFT} leads to a very simple understanding of mass
renormalization. Namely, it turns out that the bare mass must be
adjusted to
\begin{equation}
m^2 = m_{phys}^2 - \frac{3}{16\pi^{2}}\;
\Lambda_{UV}^2 \left(1 -\; \frac{m_{phys}^2}{\Lambda_{UV}^2}\;
\ln(\frac{\Lambda_{UV}^2}{m_{phys}^2})\right)\; \lambda.
\label{mass-ren-4d}
\end{equation}
in the 4-dimensional case. This reproduces precisely the result of
a conventional one-loop computation, however it is now based on
a nonperturbative analysis and not just a formal expansion.
\item
Stretching somewhat the range where \eq{eff-act-eig} has been tested,
one can study the
thermodynamical properties and phase transitions of the interacting model.
Indeed a phase transition is found at the point where the eigenvalue distribution
$\vec\phi_0^\lambda$ breaks up into 2 disjoint pieces ("2 cuts"), due to the
interaction term.
This is expected to be
the transition between the ``striped'' and disordered phase
found in NC scalar field theory \cite{gubser,bieten}.
In the most interesting case of 4 dimension, a critical line
is found which ends at a nontrivial critical point $\lambda_c >0$
\cite{NCscalarFT}.
This is strongly
suggestive for a {\em nontrivial} NC $\phi^4$ model in 4 dimensions,
since the endpoint of the critical line $\lambda_c >0$ should correspond to
a fixed-point under a suitable RG flow.
Since the mechanism of the phase transition is very generic and does
not depend on the details of the potential, the
qualitative features of this result are expected to be correct.
\end{enumerate}
Full details can be found in \cite{NCscalarFT}.
\paragraph{Acknowledgments} It is a pleasure to thank the organizers
of the II. Southeastern European Workshop BW 2005, Vrnjacka Banja,
Serbia
and the XIV. Workshop of Geometric Methods in Physics,
Bialowieza, Poland
for providing an enjoyable and stimulating
atmosphere. This work was supported by the FWF project
P16779-N02.
\nopagebreak
\renewcommand {\bibname} {\normalsize \sc References}
\nopagebreak
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,925 |
package org.apache.tinkerpop.gremlin.util;
import org.apache.tinkerpop.gremlin.TestHelper;
import org.junit.Test;
import static org.junit.Assert.assertEquals;
/**
* @author Stephen Mallette (http://stephen.genoprime.com)
*/
public class GremlinTest {
@Test
public void shouldBeUtilityClass() throws Exception {
TestHelper.assertIsUtilityClass(Gremlin.class);
}
@Test
public void shouldGetVersion() {
// the manifests lib should be solid at doing this job - can get by with a
// light testing here - this will be good for at least version 3
assertEquals('3', Gremlin.version().charAt(0));
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,601 |
Medicaid typically does not pay for housing costs for individuals with mental illness or developmental disabilities. However, Medicaid does provide for Personal Care in Specialized Residential Settings. This is care provided on a 24 hr. basis, in a community residence, for individuals who require such care. Personal Care in Specialized Residential Settings assists an adult with mental illness or developmental disabilities with activities of daily living, self-care and basic needs, while they are living in a specialized residential setting in the community.
MCCMH also offers other supports and services to help adults with mental illness or developmental disabilities live successfully in their own homes. Call the Access Center or talk to your Case Manager or Supports Coordinator to learn more. | {
"redpajama_set_name": "RedPajamaC4"
} | 6,727 |
{"url":"http:\/\/zqjr.jf-huenstetten.de\/matlab-code-for-maximum-lyapunov-exponent.html","text":"# Matlab Code For Maximum Lyapunov Exponent\n\n34 The exponent is positive for a chaotic. The algorithm has just been converted to Matlab. Local Bifurcations of the Chen System 2259 When t= t 0, system (2) is a two-dimensional linear system with constant coe cients. , the so-called Perron effect) (Leonov and Kuznetsov 2006) A strictly positive maximal Lyapunov exponent is often considered as a definition of deterministic chaos. The idea behind the Lyapunov exponent is knowing if a (dynamical) system is \"chaotic\" in the sense of. 105-181 19179 Blanco Rd #181 San Antonio, TX 78258 USA. You may have to register before you can post: click the register link above to proceed. zip - When processing a signal frequency analysis, Calculate the maximum eigenvalue judgment matrix of AHP, By matlab code. For most flows of practical importance, the FTLE varies as a function of space and time. Consider two iterations of the logistic map starting from two values of x which are close together. Collins, C. The maximum Lyapunov exponent (MLE) is usually used to estimate the level of chaos. dimension and for each (integer) scale specified by the. Lyapunov exponents found with our method agreed well with those obtained using the Matlab code by V. A strictly positive maximal Lyapunov exponent is synonymous of exponential instability, but one should be warned that in some special cases, this may not be true (see, e. The purpose of this paper is to present some results on the effects of parametric perturbations on the Lyapunov exponents of discrete time-varying linear systems. That way, this paper shows the experimental realization of an optimized multiscroll chaotic oscillator based on saturated function series. Figure 5 shows the maximum Lyapunov exponent under two different \u03bc values. Indications of chaos in terms of Lyapunov exponents would be to have at least one positive exponent and the larger the more sensitive dependence exists. Chapter 14 Bifurcations in the Quadratic Map We will approach the study of the universal period doubling route to chaos by \ufb01rst investigating the details of the quadratic map. IGEEKS TECHNOLOGIES Software Training Division Academic Projects for BE,ME,MCA,BCA and PHD Students IGeekS Technologies (Make Final Year Project) No: 19, MN Complex, 2nd Cross, Sampige Main Road, Malleswaram, Bangalore- 560003. nargout Report the number of output arguments from a. Four representative examples are considered. Protasov and Rapha el M. If the Lyapunov exponent is negative, then these points converge exponentially quickly, such as toward a fixed point or limit cycle. This test performs the test for chaotic dynamics of a noisy time series based on the Lyapunov exponent. My problem is that I don't know first Lyapunov exponent is average of all the lyapunovs when k = 1 or average of all the lyapunovs for the first time that data[i] - data[j] < eps? Is this right implementation for Lyapunov exponent? And this is the Numerical Calculation of Lyapunov Exponent. matfile_listvar \u2014 Lists variables of a Matlab binary MAT-file. Largest Lyapunov exponent. To compute the largest Lyapunov exponent, you first need to determine the expansion range needed for accurate estimation. It allow to user select embedding lag( tau) and embedding dimension(m), but if a user cannot give any value to this parameters the code will select automatically this values. Govorukhin, which is given in the folder lyapounov2. \"global\" shares by variable name. I found a MatLab program lyaprosen. matlab\u7f16\u5199\u7684Lyapunov\u6307\u6570\u8ba1\u7b97\u7a0b\u5e8f % Lyapunov exponent calcullation for ODE-system. , 1985) to calculate the maximum Lyapunov exponent. dim and max. Numerical Calculation of Largest Lyapunov Exponent J. The dynamics of the system are investigated extensively by using this model. Also, the programs to obtain Lyapunov exponents as function of the bifur-cation parameter and as function of the fractional order are described. Question: Lyapunov exponent Tags are words are used to describe and categorize your content. I mean, by using the variational equations or by monitoring the deviation between two initially nearby orbits? If it is the latter, then I could provide such a Mathematica code. Govorukhin, which is given in the folder lyapounov2. The experimental data were used to compute the Poincar\u00e9 maps and the largest Lyapunov exponent. Since the Lyapunov exponent increases with bifurcation parameter, the worst case scenario is D1 domain; the sequence generation was done in this domain and D7 for comparison. Here I will demonstrate how to use the function lyap_k. The Lyapunov exponent measures the divergence rate between two points which are initially close in the state space. Swinney, and J. out file into Agilent's ADS (0). All these systems also show a strange attractor for certain parameter values. m calculate maximal lyapunov exponent from a single point, using two different (but close) initial conditions. Free Online Library: Sources of irreversibilities in a perfectly axisymmetric geometry at low Reynolds numbers. In this quick tutorial, I'll show you a cleaner way to get the Lyapunov exponent for the specific case of the logistic map, and then using a really short script in Mathematica, plot it against r. Let t+T t de ne a function which maps a particle at time x(t) to time x(t+ T). , see the methods implemented in TISEAN (Hegger et al. You may have to register before you can post: click the register link above to proceed. , its numerical stability or Lyapunov exponent). An essential ingredient is some estimate of the local Jacobians, i. De Luca, A practical method for calculating largest Lyapunov exponents from small data sets, Physica D 65, 117 (1993) See Also mutual, false. nearest for the choice of optimal embedding parameters. find the stationary points, ii. The time series was. NaN Return a scalar, matrix, or N-dimensional array whose elements are all equal to the IEEE symbol NaN (Not a Number). Rossler was inspired by the geometry of flows in dimension three and, in particular, by the re-injection principle, which is based on the feature of relaxation-type. A wikipedia page gives a general idea about the lyapunov stability. Refer to the GNU Octave web site for information about how to install GNU Octave itself on your specific OS. Join GitHub today. I ask code for this paper. trajectories deviate from each other with time is characterized by a quantity called the Lyapunov exponent. Contact Author. We compute the viscosity \u03b7 of a fluid consisting of a large number of particles, N=108 and 864, as a function of shear rate \u03b3 from its maximum and minimum Lyapunov exponents. The Logistic chaotic mapping and Tent chaotic mapping are. Steve McIntyre discusses this here. Lyapunov exponent of ion motion in microplasmas Pierre Gaspard Center for Nonlinear Phenomena and Complex Systems Universit\u00b4e Libre de Bruxelles, Code Postal 231, Campus Plaine, B-1050 Brussels, Belgium Dynamical chaos is studied in the Hamiltonian motion of ions con\ufb01ned in a Penning trap and forming so-called microplasmas. I\u2019ve since had a lot of requests for the code which one of my coauthors has been helpfully emailing to anyone who asked. I am currently doing research in non-linear dynamical systems, and I require to calculate Lyapunov exponents from time series data frequently. Negative Lyapunov exponents are associated with dissipative systems; Lyapunov exponents equal to zero are associated with conservative systems; and positive Lyapunov exponents are associated with chaotic systems (provided the system has an attractor). The calculation is based on an extension of Smale's pairing rule of Lyapunov exponents for Hamiltonian systems to non-Hamiltonian systems in contact with a heat bath. In this study, we propose the chaotic cipher combined with Mersenne Twister that is an extremely good pseudo-random number generator for the secure communications. Nunn The numerical simulation of VLF nonlinear wave-particle interactions in collision-free plasmas using the Vlasov hybrid simulation technique. If you've ever wondered how logistic population growth (the Verhulst model), S curves, the logistic map, bifurcation diagrams, sensitive dependence on initial conditions, \"orbits\", deterministic chaos, and Lyapunov exponents are related to [\u2026]. To estimate a Lyapunov exponent, The Chaos textbook reports the largest exponent of a chaotic Chua orbit to be 0. 5 for more details):. All these systems also show a strange attractor for certain parameter values. The well-known technique of phase space reconstruction with delay coordinates [2, 33, 34] makes it possible to obtain from such a time series an attractor whose Lyapunov spectrum is identical to that of the original attractor. dimension and for each (integer) scale specified by the. 2- And has the maximum or the minimum or the negative values a meaning knowing tht I found that the Lyapunov exponent unit is the bits\/s. (Report) by \"Annals of DAAAM & Proceedings\"; Engineering and manufacturing Fluid-film bearings Mechanical properties Models Hydrodynamics Hydrofoil boats Nonlinear theories Analysis. How to use Mathematica to compute Lyapunov spectrum of a smooth dynamical system Numerical bifurcation methods and their application to fluid dynamics ( PDF ) Computing Lyapunov exponents from time series ( PDF ) ( Matlab code ). Divergence of nearest trajectories can be seen on the graph. The dynamics of the system are investigated extensively by using this model. calculate lyapunov of the logistic map lyap=zeros(1,1000); j=0; for (r=3:0. Swinney, and J. Through the analysis of the Lyapunov exponent, the new hybrid chaotic map has one positive Lyapunov exponent, which can be theoretically convinced that the hybrid mapping system is in a chaotic state. Numerical calculation of Lyapunov exponents for three-dimensional systems of ordinary di erential equations Clyde-Emmanuel Estorninho Meador We consider two algorithms for the computation of Lyapunov exponents for systems of ordinary di erential equations: orbit separation and continuous Gram-Schmidt orthonormal-ization. From its documentation: The software also supports the calculation of Conditional Lyapunov Exponents or CLEs (see Pecora et al, Chaos Vol. SeHyoun Ahn & Greg Kaplan & Benjamin Moll & Thomas Winberry & Christian Wolf, 2017. For most flows of practical importance, the FTLE varies as a function of space and time. Refer to the GNU Octave web site for information about how to install GNU Octave itself on your specific OS. dim and max. From previous lectures we know that there are ef\ufb01cient algorithms that. Lyapunov exponents describe how a system expands and contracts in phase space. In time-series analysis, the existence of at least one positive Lyapunov exponent is interpreted as a strong indi-cator for chaos. If you've ever wondered how logistic population growth (the Verhulst model), S curves, the logistic map, bifurcation diagrams, sensitive dependence on initial conditions, \"orbits\", deterministic chaos, and Lyapunov exponents are related to one another\u2026 this post explains it in just 10 steps, each with some code in R so you can explore it all yourself. Mehdizadeh, Sina. series (see buildTakens). It can be seen, say, that when a=2 , the LE is positive and chaos is in action [Check the bifurcation diagram to get the same intuition for that value of a ]. If the Lyapunov exponent is negative, then these points converge exponentially quickly, such as toward a fixed point or limit cycle. However, their prediction becomes a challenge when they have different values of their maximum Lyapunov exponent (MLE), which is associated to the degree of unpredictability of a chaotic system. The calculation is based on an extension of Smale's pairing rule of Lyapunov exponents for Hamiltonian systems to non-Hamiltonian systems in contact with a heat bath. Vladimir Y. m - this MATLAB file simulates the logistic difference equation u(n+1)=a u(n) (1-u(n)) and carries out a bifurcation analysis by varying a. It allow to user select embedding lag( tau) and embedding dimension(m), but if a user cannot give any value to this parameters the code will select automatically this values. A method to estimate the (positive) largest Lyapunov exponent (LLE) from data using interval extensions is proposed. Using the code is very simple, it needs only an scalar time series, number of lags and number of hidden unites. When a Lyapunov exponents is positive, we will say that the system is chaotic. Help to find source of Largest Lyapunov Exponent Matlab Code I have been trying to find the source\/theoretical basis for the following code. This Demonstration implements a number of methods used in the analysis of such systems: bifurcation plots, Poincar\u00e9 maps, phase portraits, time series, and power spectra. Even in cases for which the log-likelihood is well. then the exponent is called the Lyapunov exponent. lyapunov spectrum (all Lyapunov exponents). The paper by Wolf et. IM3SHAPE is a modular C code with a significant amount of Python glue code to enable setting up new models and their options automatically. The experimental data were used to compute the Poincar\u00e9 maps and the largest Lyapunov exponent. Following posts gives a very basic example to hel user use the lyapunov function in Matlab. Nowadays, different kinds of experimental realizations of chaotic oscillators have been already presented in the literature. Lyapunov exponents estimate the rate of divergence of nearby trajectories, a key component of chaotic dynamics. A n-dimensional system will have n Lyapunov exponents. For an in-depth exploration of \u201cirregular non-cyclic motion\u201d by the three different methods, we used the maximum Lyapunov exponent and bifurcation diagrams to determine and compare non-cyclic motion. I have another question. Any data used for programming examples should be embedded in the question or code to generate the (fake) data must be included. The method you describe about how to find the MLE of a 1D map can be expanded into the method described in the link. Now, all we have to do is to write a Matlab program that will compute the Lyapunov exponent using equation (4). Also, the samples after 120 sec of plasma treatment show a definite improvement of bacterial adhesion. Lyapunov equations arise in several areas of control, including stability theory and the study of the RMS behavior of systems. After estimation of network weights and finding network with minimum BIC, derivatives are calculated. Lyapunov spectrum of the new hybrid chaotic map (the first 500 iterations are ignored to avoid the influence of initial state). 1 Naive numerical evaluation of 1. Issuu is a digital publishing platform that makes it simple to publish magazines, catalogs, newspapers, books, and more online. Related Calculus and Beyond Homework Help News on Phys. A version of MATLAB1 m-code. Eykholt, and R. If the answer is the time data field, then this is a time series data set candidate. As the result shows, all maximum Lyapunov exponents are above zero. 34 The exponent is positive for a chaotic. Lyapunov exponent calcullation for ODE-system. This may be done through the eigenvalues of the Jacobian matrix J 0 (x 0). series (see buildTakens). Determination of Chaos in Different Dynamical Systems A Thesis Presented to the Graduate School of Clemson University In Partial Ful llment of the Requirements for. In the figure we plot the graph of the maximum lyapunov exponent against parameter values ( ) varies from 0. 2- And has the maximum or the minimum or the negative values a meaning knowing tht I found that the Lyapunov exponent unit is the bits\/s. Any help would be greatly appreciated. Lagrangian coherent structures of flow in a ramp-up motion (Finite-Time Lyapunov Exponents) - Duration: 2:08. The finite-time Lyapunov exponent, FTLE, which we will denote by , is a scalar value which characterizes the amount of stretching about the trajectory of point over the time interval [t, t + T]. All these systems also show a strange attractor for certain parameter values. the Matlab code was my own. 3 The Finite-Time Lyapunov Exponent. 66, Number 25. The de nition of Lyapunov exponents and a method to nd such exponents [6] is also provided. Wolf's paper Determining Lyapunov Exponents from a Time Series states that:. The largest Lyapunov exponent (LyE) is an accepted method to quantify gait stability in young and old adults. Lyapunov exponents estimate the rate of divergence of nearby trajectories, a key component of chaotic dynamics. The purpose of this paper is to present some results on the effects of parametric perturbations on the Lyapunov exponents of discrete time-varying linear systems. Join GitHub today. The algorithm has just been converted to Matlab. Lyapunov exponents were carried ou t using codes written in Matlab environment. Downloadable! This code uses Rosenstein et al. rar > Lyapunov. Related Calculus and Beyond Homework Help News on Phys. is Abstract: Lyapunov functions are a mathematical generalization of the dissipative energy concept of physics. LYAPUNOV EXPONENTS 3 For almost any initial point U. It allow to user select embedding lag( tau) and embedding dimension(m), but if a user cannot give any value to this parameters the code will select automatically this values. , then there is a quadratic Lyapunov function that proves it (we'll prove this later) Basic Lyapunov theory 12-20. It has just been converted to Matlab. Indications of chaos in terms of Lyapunov exponents would be to have at least one positive exponent and the larger the more sensitive dependence exists. Lin will talk about how to automatically bridge the semantic gap with a number of program analysis techniques from the hypervisor layer, and demonstrate a set of new applications, such as using the native command for guest-OS introspection, and automated guest-OS management. Analysis of the spectrum of Lyapunov exponents is widely used to study the complex dynamics in systems of ordinary differential equations and models that can be reduced to maps. The algorithm was distributed for many years by the authors in Fortran and C. All your les and all work which you do will be in this folder. matfile_listvar \u2014 Lists variables of a Matlab binary MAT-file. function [Texp,Lexp]=Lyapunov(n,rhs_ext_fcn,fcn_integrator,tstart,stept,tend,ystart,ioutp. Sprott Department of Physics, University of Wisconsin, Madison, WI 53706, USA October 15, 1997 (Revised January 8, 2015) The usual test for chaos is calculation of the largest Lyapunov exponent. To estimate the Lyapunov exponent of time series, several approaches were suggested. \"global\" shares by variable name. on the local \ufb01nite time Lyapunov exponent in each dimension. Q&A for Work. Let the two starting values be x0 and x0 +dx0. Vladimir Y. 3 stated in Eckhardt (1993). dim and max. MATLAB 1 Matlab basics 1. To this end, Q2. Combine multiple words with dashes(-), and seperate tags with spaces. Divergence of nearest trajectories can be seen on the graph. 1 Convex search for storage functions The set of all real-valued functions of system state which do not increase along system. The goal of our work is to calculate Lyapunov exponent to types of local bifurcation by Mathlab program. Time-optimal nonlinear feedback control technique was used in order to take full advantage of the maximum torques that the controller can deliver. A detailed post on the Lyapunov Stability Criteria will be uploaded soon. Consider two iterations of the logistic map starting from two values of x which are close together. It allow to user select embedding lag( tau) and embedding dimension(m), but if a user cannot give any value to this parameters the code will select automatically this values. For example, an attractive limit cycle has only negative Lyapunov exponents (except possibly one at zero corresponding to the freedom of. It is approximately 2. com > Lyapunov. Re: matlab source code for lyapunov exponent of eeg signals What parameter of EEG have u decided to take for studying its chaotic nature ? You might have to look for an ERP and use wavelet methods which is best for feature extraction and then formulate a non-linear dynamical mathematical structure. In the figure we plot the graph of the maximum lyapunov exponent against parameter values ( ) varies from 0. m - this MATLAB file simulates the logistic difference equation u(n+1)=a u(n) (1-u(n)) and carries out a bifurcation analysis by varying a. is, [email\u00a0protected] As the result shows, all maximum Lyapunov exponents are above zero. The code chaostest can detect the presence of chaotic dynamics. If it were not, there would be a step change in the Lyapunov Exponent as we looked at longer and longer windows on the weather. By analyzing different parts of the code separately, we demonstrate that corrective maintenance does not necessarily lead to code deterioration, that adaptive maintenance may improve some quality metrics, and that growth is largely the result of continued development as part of perfective maintenance. Computer Physics Communications Volume 60, Number 1, August, 1990 D. Rossler was inspired by the geometry of flows in dimension three and, in particular, by the re-injection principle, which is based on the feature of relaxation-type. (Lyapunov exponents) Consider the 1D map x n+1 = rx n(1 2x n)(1 2x n) , x n 2[0;1]. To estimate the Lyapunov exponent of time series, several approaches were suggested. In the intermediate range, $$\\ell$$ tells us how the growth of a perturbation is affected by nonlinearities. Let's estimate the maximal Lyapunov exponent of the Lorenz system, which is known to be chaotic. How to get the maximal Lyapunov exponent from Alan Wolf's new Matlab code? I'm using Prof. \u2022 if A is stable, Lyapunov operator is nonsingular \u2022 if A has imaginary (nonzero, i\u03c9-axis) eigenvalue, then Lyapunov operator is singular thus if A is stable, for any Q there is exactly one solution P of Lyapunov equation ATP +PA+Q = 0 Linear quadratic Lyapunov theory 13-7. This alternate definition will provide the basis of our spectral technique for experimental data. The Matlab program prints and plots the Lyapunov exponents as function of time. The sum of Lyapunov exponents is negative for dissipative systems. Then, I would like to store that R and Lyapunov exponent and plot them together. Created Date: 10\/29\/2001 8:46:42 AM. Lyapunov exponent calcullation for ODE-system. zip - To achieve the recognition of 10 digital sound, matlab development toolbox support vector machine, Least-squares regression analysis algorithm. There are some chaos related libraries in Rats, for example R\/S Statistics, Hurst exponent. The source code and files included in this project are listed in the project files section, please make sure whether the listed source code meet your needs there. ) I found this method during my Masters while recreating the results of an interesting paper on how some standard tests for chaos fail to distinguish chaos from stochasticity (Stochastic neural network\u2026. Fluctuation of mean Lyapunov exponent for turbulence; The number of packets generated Poisson distribution with the exponential distribution of packet length generator. If the sum of all Lyapunov exponents is negative than the system has an attractor. method with some modifications based on below references. First it is proved that the considered class of fractional-order systems admits the necessary variational system necessary to \ufb01nd the Lyapunov. In Physica 16D (1985) we presented an algorithm that estimates the dominant Lyapunov exponent of a 1-D time series by monitoring orbital divergence. A n-dimensional system will have n Lyapunov exponents. For example consider a simple dynamical system. A new technique involving the products of the ratios of successive differentials is used to find maximal Lyapunov exponents in systems of differential equations. Abstract: This M-file calculates Lyapunov exponents with minimum RMSE neural network. lag) that shall be used to construct the Takens' vectors. The Lyapunov exponents capture the average exponential growth or decay rate of the principal axes of the ellipsoid and the maximal Lyapunov exponent captures the long-term behavior of the dominating direction. Whereas positive Lyapunov means nearby trajectories are exponentially diverging. matlab source code for lyapunov exponent of eeg signals (1) Reading MATLAB data into a VHDL file (3) Reading data from Matlab. Chaos and Lyapunov Exponents Exercise Chaotic dynamical systems have sensitive dependence on initial conditions. In this paper the Benettin-Wolf algorithm to determine all Lyapunov exponents for a class of fractional-order systems modeled by Caputo's derivative and the corresponding Matlab code are presented. The dynamics of the system are investigated extensively by using this model. Lyapunov exponent calcullation for ODE-system. (summa cum laude), Cornell University, Ithaca, NY, USA, 2010. One of the features of chaos is exponential divergence (sensitivity to initial conditions). This is commonly described as the \"butterfly effect\": the flap of a butterfly's wing in Brazil can build up to change a tornado later in Texas. Again, because there would be a large number of graphs, we only. Numerical calculation of Lyapunov exponents for three-dimensional systems of ordinary di erential equations Clyde-Emmanuel Estorninho Meador We consider two algorithms for the computation of Lyapunov exponents for systems of ordinary di erential equations: orbit separation and continuous Gram-Schmidt orthonormal-ization. The Lyapunov exponents capture the average exponential growth or decay rate of the principal axes of the ellipsoid and the maximal Lyapunov exponent captures the long-term behavior of the dominating direction. A detailed post on the Lyapunov Stability Criteria will be uploaded soon. nolds module\u00b6. It tests the positivity of the dominant (or largest) Lyapunov exponent \u03bb at a specified confidence level. Related Calculus and Beyond Homework Help News on Phys. If the sum of all Lyapunov exponents is negative than the system has an attractor. For a bound phase space if we get a positive LE that means we have chaos. However, there are some issue should be discussed on the. First it is proved that the considered class of fractional-order systems admits the necessary variational system necessary to \ufb01nd the Lyapunov. Lyapunov exponents for f(x,a)=a cos x, when a is the range [0. Refer to the GNU Octave web site for information about how to install GNU Octave itself on your specific OS. Recurrence plots [8] can well solve the. A MATLAB code was used that calculated. The finite-time Lyapunov exponent, FTLE, which we will denote by , is a scalar value which characterizes the amount of stretching about the trajectory of point over the time interval [t, t + T]. dim) and the time lag (time. The chaotic attractor with only one stable equilibrium can be generated via a period-doubling bifurcation. This is commonly described as the \"butterfly effect\": the flap of a butterfly's wing in Brazil can build up to change a tornado later in Texas. If the motion is chaotic, the orbits will, by definition, separate at an exponential rate. I am using the code which was previously posted on Mathworks. Any data used for programming examples should be embedded in the question or code to generate the (fake) data must be included. This may be done through the eigenvalues of the Jacobian matrix J 0 (x 0). The source code and files included in this project are listed in the project files section, please make sure whether the listed source code meet your needs there. Join GitHub today. Wang, Smith-Miles and Hyndman (2009) \u201cRule induction for forecasting method selection: meta- learning the characteristics of univariate time series\u201d, Neurocomputing, 72, 2581-2594. It means that chaotic phenomenon appears in the structure system. To compute the largest Lyapunov exponent, you first need to determine the expansion range needed for accurate estimation. Consider two iterations of the logistic map starting from two values of x which are close together. Rosenstein, James J. The Lyapunov exponent is a measure of how in\ufb01nitesimally close trajectories of dynamical system vary or diverge in a certain dimension of phase space (Fig. For most flows of practical importance, the FTLE varies as a function of space and time. Lyapunov Function Veri cation: MATLAB Implementation Skuli Gudmundsson Sigurdur F. lyap solves the special and general forms of the Lyapunov equation. Daca aveti cont Ad Astra si de Facebook, intrati pe pagina de profil pentru a da dreptul sa va logati pe site doar cu acest buton. nearest for the choice of optimal embedding parameters. Impact Factor 2019: 1. Rosenstein, James J. My problem is that I don't know first Lyapunov exponent is average of all the lyapunovs when k = 1 or average of all the lyapunovs for the first time that data[i] - data[j] < eps? Is this right implementation for Lyapunov exponent? And this is the Numerical Calculation of Lyapunov Exponent. Combine multiple words with dashes(-), and seperate tags with spaces. series (see buildTakens). Any data used for programming examples should be embedded in the question or code to generate the (fake) data must be included. We use it as a measure of performance for the tendency of the system to synchronize. China India US. 2 Lyapunov Analysis 9 2. The Logistic chaotic mapping and Tent chaotic mapping are. Population_Fit. The Lyapunov exponent measures the divergence rate between two points which are initially close in the state space. zip - When processing a signal frequency analysis, Calculate the maximum eigenvalue judgment matrix of AHP, By matlab code. Some of the algorithms are available as Matlab functions or can be simply programmed for Matlab. These are points whose third return is associated with a strong El Ni\u00f1o event. In this file, you will notice Python code that has already been written, but it mostly consists of hints to help you flesh out the code, i. Finally, the program to find the Lyapunov exponent is run, and the stride frequency and Lyapunov exponent are returned to Excel. , kansas state university, 2007. LYAPROSEN: MATLAB function to calculate Lyapunov exponent. Lyapunov exponent calcullation for ODE-system. The largest Lyapunov exponent (LyE) is an accepted method to quantify gait stability in young and old adults. Thus, early warnings about the undesired synchronous behavior can be given to the control system for continuous optimization of the performance. The algorithm was also compared with K-means clustering based on Kolmogorov complexity (KC), the highest value of Kolmogorov complexity spectrum (KCM), and the largest Lyapunov exponent (LLE). method - iank\/lyapunov_estimation. Standard methods are then applied to compute Lyapunov exponents. This is one of the 100+ free recipes of the IPython Cookbook, Second Edition, by Cyrille Rossant, a guide to numerical computing and data science in the Jupyter Notebook. Govorukhin, which is given in the folder lyapounov2. If the Lyapunov exponent\u2019s value is positive, then the two points move apart over time at an exponential rate. We simulate it by using Matlab\/Simulink software. The algorithm has just been converted to Matlab. Any help would be greatly appreciated. For example, an attractive limit cycle has only negative Lyapunov exponents (except possibly one at zero corresponding to the freedom of. The chaotic attractor with only one stable equilibrium can be generated via a period-doubling bifurcation. Lyapunov Exponents. Now, all we have to do is to write a Matlab program that will compute the Lyapunov exponent using equation (4). It means that chaotic phenomenon appears in the structure system. Chapter 14 Bifurcations in the Quadratic Map We will approach the study of the universal period doubling route to chaos by \ufb01rst investigating the details of the quadratic map. The numerical integration is done with an explicit Euler method. For a bound phase space if we get a positive LE that means we have chaos. leading Lyapunov exponent. , kansas state university, 2007. of Mathematics Macalester College 1600 Grand Ave. If the Lyapunov exponent\u2019s value is positive, then the two points move apart over time at an exponential rate. The reason for taking the logarithm and dividing by the magnitude of the integration time T is the parallel that we then establish with autonomous systems. Even in cases for which the log-likelihood is well. Matlab 2013 14 papers astract 1. Height at max: a 2 Monday, January 27, 14 7. species3driver. Sandri in order to determine the maximum Lyapunov exponent as well as all the Lyapunov exponents. This alternate definition will provide the basis of our spectral technique for experimental data. A Novel Design of Maximum Power Point\/Droop Controllers for Photovoltaic Sources in DC Microgrids Voltage Stabilization in a DC MicroGrid by an ISS-Like Lyapunov. Free Online Library: Sources of irreversibilities in a perfectly axisymmetric geometry at low Reynolds numbers. matlab source code for lyapunov exponent of eeg signals (1) Reading MATLAB data into a VHDL file (3) Reading data from Matlab. dim) and the time lag (time. The surrogated largest Lyapunov exponent values in the PAD group were significantly higher than the original data only for the ankle (Table II). The following Matlab project contains the source code and Matlab examples used for calculation lyapunov exponents for ode. narginchk Check for correct number of input arguments. method with some modifications based on below references. It tests the positivity of the dominant (or largest) Lyapunov exponent \u03bb at a specified confidence level. We fix our attention on the greatest and smallest exponents. Through the analysis of the Lyapunov exponent, the new hybrid chaotic map has one positive Lyapunov exponent, which can be theoretically convinced that the hybrid mapping system is in a chaotic state. 1 Gait Study Data Collection 17 2. The dynamics of the system are investigated extensively by using this model. Calculating the Lyapunov Exponent of a Time Series (with python code) Posted on July 22, 2014 by Neel ( In a later post I discuss a cleaner way to calculate the Lyapunov exponent for maps and particularly the logistic map, along with Mathematica code. We compute the viscosity \u03b7 of a fluid consisting of a large number of particles, N=108 and 864, as a function of shear rate \u03b3 from its maximum and minimum Lyapunov exponents. (Note that the exponent of is simply a quadratic function. Then, I would like to store that R and Lyapunov exponent and plot them together. Here I will demonstrate how to use the function lyap_k. 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\subsection{Angular momentum of the Cooper pair}
Next, we treat the superconducting phase where the Cooper pairs consist of electrons on the
conduction band Fermi surface which encloses a Berry flux of $\phi_{B}(k_{\mathrm{F}})$.
In the weak coupling regime, we can describe the superconducting ground state using the BCS ansatz
\begin{align}
|\mathrm{BCS}\rangle =\prod_{k} \left(u_\mathbf{k} + v_\mathbf{k} c^{\dagger}_{\mathbf{k}}c^{\dagger}_{-\mathbf{k}}\right) |0\rangle,
\label{eq:BCSWF}
\end{align}
where $|0\rangle$ is the electronic vacuum of the conduction band. The BCS ansatz parametrizes the expectation value of the
Cooper pair $\langle \mathrm{BCS} |c_{\mathbf{k}} c_{-\mathbf{k}}|\mathrm{BCS}\rangle = u_{\mathbf{k}}^*v_{\mathbf{k}} \equiv F_{\mathbf{k}}$.
Given the BCS gap equation $\Delta_{\mathbf{k}} = \sum_{\mathbf{k}'} U_{\mathbf{k}\mathbf{k}'} F_{\mathbf{k}'} $, the expectation value $F_{\mathbf{k}}$ can be
written as $|F_{\mathbf{k}}| e^{i l_0 \phi_{\mathbf{k}_F}}$, where $l_0$ maximizes $|U_{\mathbf{k}_{F},\mathbf{k}_{F}'}^{l_0}|$ under the
constraint that the Cooper pair wavefunction in a given (pseudo-) spin representation is anti-symmetric.
Naively, one could conclude from the above discussion that the expectation value of the many-body angular momentum operator
is $\hbar l_0 n_c$, where we defined the number of condensed pairs as $n_c~\equiv~\sum_{\mathbf{k}} F_{\mathbf{k}}$.
However, a more careful calculation shows that this conclusion is incorrect in the presence of the non-trivial Berry curvature.
The expectation value of the relative angular momentum operator is (see Supplementary Material)
\begin{align}
\nonumber & \langle L_{\mathrm{rel}}\rangle \equiv - \frac{1}{2} \langle \mathrm{BCS}|\sum_{i \neq j} (\hat{r}_i - \hat{r}_j) \times (\hat{p}_i - \hat{p}_j)|\mathrm{BCS}\rangle \\
\nonumber &= \hbar \sum_{\mathbf{k}} F^*(\mathbf{k}) \left(- i \partial_{\phi}\right) F(\mathbf{k}) + \sum_{\mathbf{k}}F^{*}(\mathbf{k}) \hbar \frac{\phi_{\mathrm{B}}(k_{\mathrm{F}})}{\pi} F(\mathbf{k})\\
&= n_c \hbar \left(l_0 + \frac{\phi_{B}(k_{\mathrm{F}})}{\pi} \right).
\label{eq:AngMomExp}
\end{align}
We observe that the expectation value of the angular momentum operator is shifted with respect
to the naive guess above by twice the angular momentum associated with $\phi_{\mathrm{B}}(k_{\mathrm{F}})$ in
Eq.~($\ref{eq:BerryPhase}$). Importantly, the Berry flux contribution can be traced back to the
fact that the second quantized position operator projected onto the conduction band is \cite{Blount1962,Girvin1999}
\begin{align}
\nonumber \hat{r} &= \sum_{\mathbf{k},\mathbf{k}'}\langle c(\mathbf{k}') |\hat{r}| c(\mathbf{k}) \rangle c_{\mathbf{k}'}^{\dagger}c_{\mathbf{k}}\\
& = \sum_{\mathbf{k},\mathbf{k}'}\left[ -i \partial_{\mathbf{k}} + A_c(\mathbf{k}) \right] \delta_{\mathbf{k},\mathbf{k}'} c_{\mathbf{k}'}^{\dagger}c_{\mathbf{k}}.
\end{align}
Equivalently, the angular momentum of the Cooper pair is given by the eigenvalue of the covariant derivative
in momentum space, $D_{\phi} = -i\partial_{\phi} + \frac{2\phi_{\mathrm{B}}(k_{\mathrm{F}})}{2\pi}$.
This result can be intuitively understood by considering the analogy between the single-particle
wavefunctions on the Fermi surface enclosing the Berry flux $\phi_{\mathrm{B}}$, and the wavefunctions
of a particle on a ring which encloses a magnetic flux $\Phi_{\mathrm{Mag}}$ (see Fig. $\ref{fig:Analogy}$). As discussed
in Ref. \cite{Wilczek1982a,Wilczek1982b}, in both cases, the eigenfunctions have angular momenta
whose values are shifted with respect to their conventional integer values by a fraction associated with $\phi_{\mathrm{B}}$ or
$\Phi_{\mathrm{Mag}}$. Thus, the unconventional expectation value of the Cooper pair angular momentum
can be understood as the sum of the shifted angular momenta of their constituent particles.
What are the physical effects of the fractional angular momentum expectation value of the Cooper
pairs? We show that one interesting consequence of the fractional angular momentum
is its effect on the orbital currents.
We calculate the orbital current contribution in the Ginzburg Landau (GL) free energy from the microscopic model
using the formalism of functional integrals (Supplementary Material) \cite{Altland2010,Mudry2014}.
To this end, we consider a configuration of the order parameter where the relative orientation
of the Cooper pairs is adiabatically rotated as a function of their center of mass position $R$ [see Fig.~$\ref{fig:SQUID}$ (a)].
As a result, the gradient term in the GL free energy density can be shown to be
\begin{align}
\nonumber &F_{\mathrm{grad}} = \frac{\hbar^2 \rho_s}{2m^*} \bigg( -2\frac{e}{\hbar} \mathbf{A}_{\mathrm{e.m.}}(R)\\
&+ \nabla \theta(R)+\left( l_0 +\frac{\phi_{\mathrm{B}}}{\pi} \right)\nabla \phi_k(R)\bigg)^2,
\label{eq:GradEnergy}
\end{align}
where $\rho_s \equiv |\Delta|^2$ is the superfluid density for an isotropic superconductor, $m^*$ is the mass of the Cooper pair,
$\mathbf{A}_{\mathrm{e.m.}}(R)$ is the electromagnetic vector potential, and $\theta(R)$ is the
$U(1)$ phase of the order parameter. We omit the $k_{\mathrm{F}}$ dependence of the Berry flux for simplicity.
Taking the variation of the free energy with respect to $\mathbf{A}_{\mathrm{e.m.}}$, we find that the supercurrent
including the effect of a variation in the relative orientation of the Cooper pair is
\begin{align}
\nonumber \mathbf{J}_s &= \frac{\delta F}{\delta \mathbf{A}_{\mathrm{e.m.}}} = \frac{\hbar^2 \rho_s}{m^*} \bigg[ -2\frac{e}{\hbar} \mathbf{A}_{\mathrm{e.m.}}(R)\\
&+ \nabla \theta(R)+(l_0+ \phi_{\mathrm{B}}/\pi) \nabla \phi_k(R) \bigg].
\label{eq:SuperCurr}
\end{align}
While the first and the second term in the above expression can be attributed to the charge and
mass currents in a conventional superconductor, respectively, the third term contains the effect of the Berry flux
enclosed in the Fermi surface. We note that because the normal state has a rotationally symmetric (isotropic) Fermi surface,
the energy associated with the orbital currents goes to zero for configurations where the
wavelength of the variation is taken to infinity.
The form of the orbital current contribution in Eq.~($\ref{eq:SuperCurr}$), modifies the possible vortex states of the superconductor.
As is well known, the magnetic field inside a superconductor is screened due to the Meissner effect~\cite{Orlando1991, Altland2010},
resulting in the condition that away from the vortex core, the supercurrents should vanish.
This statement is captured by the following equation:
\begin{align}
\oint_{C} d\mathbf{l} \cdot \mathbf{J}_s = 0,
\label{eq:Quant}
\end{align}
where $C$ denotes a path around the vortex core, and we assumed that
the distance between any point on $C$ and the vortex core is much larger than the
effective London penetration length \cite{Pearl1964,Fetter1980} of the 2D superconductor. Inserting $
(\ref{eq:SuperCurr})$ into ($\ref{eq:Quant}$), we find the minimum value of the
magnetic flux $\Phi_v \equiv \oint_{C} dl \cdot \mathbf{A}_{\mathrm{e.m.}} $ the vortex can accommodate to be
\begin{align}
\Phi_v = \frac{2\pi \hbar}{2e} \left[p +(l_0+\phi_{\mathrm{B}}/\pi) q \right],
\label{eq:flux}
\end{align}
where $p,q \in \mathds{Z}$ denote the winding numbers of the $U(1)$ phase and the orientation angle $\phi_{\mathbf{k}}$
of the Cooper pairs along the contour, respectively.
The minimal magnetic flux $\Phi_{\mathrm{min}}$ enclosed in the vortex core in a superconductor on a band with
non-trivial Berry curvature is given by setting $q=1$ and $p = -l_0$:
\begin{align}
\Phi_{\mathrm{min}} = \frac{h}{2e} \phi_{\mathrm{B}}/\pi.
\end{align}
Given that $2 \phi_{\mathrm{B}}$ can be interpreted as the Aharanov-Bohm phase acquired by the Cooper pair wavefunction as the orientation
of the pair is adiabatically rotated by $2\pi$, the configuration with $q=1$ corresponds to a vortex with fractional flux quantum $\Phi_{min}$
that is stable due the screening of the electromagnetic field by the Aharanov-Bohm phase [see Fig. $\ref{fig:SQUID}$ (a)].
The modification of the flux quantization in the presence of non-trivial Berry curvature and an isotropic Fermi surface also bears interesting
consequences for a superconducting quantum interference device (SQUID) [see Fig.~$\ref{fig:SQUID}$ (b)].
The total current $i_T$ through the SQUID consisting of two Josephson junctions is
\begin{align}
i_T = i_1+i_2 = 2 I_c \cos{\left(\frac{\phi_1-\phi_2}{2}\right)}\sin{\left(\frac{\phi_1+\phi_2}{2}\right)},
\label{eq:SQUID}
\end{align}
where $I_c$ is the critical current of each junction and $\phi_j$ denotes the phase difference
between the two sides of the $j^{th}$ Josephson junction. When the thickness of the superconducting regions
connecting the two Josephson junctions is much larger then the penetration length, we can define a contour
$C$ for which Eq.~($\ref{eq:Quant}$) is satisfied. Then the relation between the
phase difference $\phi_1-\phi_2$ and the total flux
$\Phi$ is
\begin{align}
\frac{\Phi}{\Phi_0} = \frac{\phi_1-\phi_2}{2\pi} + p + (l_0 + \phi_{\mathrm{B}}/\pi) q,
\label{eq:SQUIDrel}
\end{align}
where the winding numbers $p,q \in \mathds{Z}$ have the same meaning as in
Eq. ($\ref{eq:flux}$).
Inserting ($\ref{eq:SQUIDrel}$) into ($\ref{eq:SQUID}$), the total current simplifies to
\begin{align}
i_T = 2 I_c \cos{\left( \phi_{\mathrm{B}} q + \pi \frac{\Phi}{\Phi_0}\right)}\sin{\left(\phi_1+ \phi_{\mathrm{B}} q+ \pi \frac{\Phi}{\Phi_0}\right)}.
\end{align}
Maximizing the driving current, we obtain
\begin{align}
i_{\mathrm{max}} \approx 2I_c \left| \cos{\left( \phi_{\mathrm{B}}q+ \pi \frac{\Phi_{\mathrm{ext}}}{\Phi_0}\right)}\right|.
\label{eq:SQUIDmax}
\end{align}
where we neglected the self-inductance of the superconducting loop
(i.e., $\Phi = \Phi_{\mathrm{ext}}$). The value of the winding number $q$ in Eq.~($\ref{eq:SQUIDmax}$)
is given by the configuration which minimizes the kinetic energy of the system for a given $\Phi_{\mathrm{ext}}$. As a result, the
Josephson current through the SQUID has a periodicity which is a fraction of the conventional SQUID.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.5\textwidth]{VortexDevice.png}
\caption{The illustration of the configuration of the superconducting order parameter which allows
stabilization of a vortex with fractional magnetic flux quantum $\Phi_0 \left(\frac{\phi_{\mathrm{B}}}{\pi}\right)$.
The single-valuedness of the order parameter is satisfied only is the relative coordinate of the Cooper
pairs rotate by a multiple $q$ of $2\pi$ around the vortex. Such a configuration realizes an adiabatic
evolution of the order parameter as a function of the center of mass position. The Berry flux acquired
by the order parameter contributes to the overall phase of the superconductor, modifying the conventional fluxoid quantization.
(b) The modification of the fluxoid quantization also allows for building SQUID's with tunable periodicity.}
\label{fig:SQUID}
\end{center}
\end{figure}
Before we conclude, we discuss possible experimental realizations of the physics presented above.
The experimental setups where the single-particle Hamiltonian
in Eq.~($\ref{eq:Dirac}$) can be realized include (i) surface modes of a 3D TI in the \textcolor{black}{proximity of a ferromagnetic insulator} \cite{Fu2008,Chen2010}
(ii) an isotropic anomalous Hall system \cite{Nagaosa2010} (iii) a single $K$ - valley of a TMD monolayer
\cite{Xu2014, Srivastava2015a} and (iv) a single $K$ - valley of a bilayer graphene biased by an in plane
electric field \cite{Oostinga2008}.
The effects discussed above will also be present if each electron comprising the Cooper pair
can be described by a separate massive Dirac Hamiltonian with the same Berry flux enclosed by each Fermi surface.
Such systems include (v) a graphene monolayer on a hexagonal anti-ferromagnetic substrate \cite{Qiao2014}
and (vi) a cold atom realization of a weakly-doped Chern insulator \cite{Aidelsburger2015,Jotzu2014}.
For systems (i), (ii), and (v) above, the effects discussed above can be in principle observed by inducing
a superconducting gap via proximity effect to a conventional $s$-wave superconductor.
However, this strategy is not viable for systems which do not break time-reversal (TR) symmetry [(iii) and (iv)]
since proximity effect only allows pairing between zero total momentum Cooper pairs, and the $K$ and $-K$
valleys in (iii) and (iv) carry opposite Berry curvature.
Consequently, the superconducting gap in (iii) and (iv) should be opened via intrinsic attractive
\cite{ Rahimi2017} or repulsive \cite{Hsu2017} interactions which
favor intravalley pairing. Lastly, in cold atom settings (vi), reliable Feshbach resonances allow greater freedom to
realize pairing with desired properties.
In conclusion, we have demonstrated a direct relation between a non-trivial single-particle Berry phase and
the modification of the fluxoid quantization in a BCS superconductor. A natural extension of our treatment is
to include the effects of anisotropy and disorder.
We also emphasize that our approach to combine
the projected many-body position operators and with a mean field ansatz can be applied to other weakly correlated systems
consisting of excitons \cite{Yao2008} or polarons \cite{Sidler2016}.
We acknowledge useful discussions with Manfred Sigrist, Dima Geshkenbein, Aline Ramires, Ovidiu Cotlet, Mark Fisher, and Tomas Bzdusek , Jose Luis Lado. This work is supported by a European Research Council (ERC) Advanced investigator grant (POLTDES) and Consolidator grant (TopMechMat).
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,771 |
Q: Using Boolean Statements to Address a Pandas Series I have some data in a Pandas DF and would like to isolate specific portions of it based on some boolean conditions. The following two lines work as I want them to:
df['test'] = df[df.N == 30].my_variable
df['test2'] = df[df.Y >0.4].my_variable
Not being that familiar with Pandas yet I then tried to write:
df['test'] = df[df.N == 30 & df.Y >0.4].my_variable
But this throws the following error:
TypeError: ufunc 'bitwise_and' not supported for the input types, and
the inputs could not be safely coerced to any supported types
according to the casting rule ''safe''
Can anyone suggest the correct (or a better) way of achieving what I'm trying to do?
A: As mentioned, you can wrap this with parenthesis to force the correct precedence:
df[(df.N == 30) & (df.Y > 0.4)].my_variable
It's worth mentioning that you can use loc (which I think is slightly cleaner):
df.loc[(df['N'] == 30) & (df['Y'] > 0.4), 'my_variable']
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,556 |
/*jslint browser:true, regexp:false */
/*global window, $*/
$(document).ready(function () {
//58% (2512/4354) vs. 36% (28055/78704) {20a82645-c095-46ed-80e3-08825760534b} (Microsoft .NET Framework Assistant, http://www.windowsclient.net/)
var corrRegExp = /[^\d]*(\d*)%\s*\(.*vs[^\d]*(\d*)%\s*\([^/]*\/[^)]*\)\s*(.*)/,
lineCache = {},
percentageDifference,
correlationItem,
sortByCorrelation,
firstLineOrEmptyString;
percentageDifference = function (correlationLine) {
var parts;
if (lineCache[correlationLine]) {
return lineCache[correlationLine];
} else {
parts = corrRegExp.exec(correlationLine);
if (parts && 4 === parts.length) {
lineCache[correlationLine] = parseInt(parts[1], 10) - parseInt(parts[2], 10);
return parseInt(parts[1], 10) - parseInt(parts[2], 10);
} else {
return 0;
}
}
};
correlationItem = function (correlationLine) {
var parts = corrRegExp.exec(correlationLine);
if (parts && 4 === parts.length) {
return parts[3];
} else {
return '';
}
};
sortByCorrelation = function (a, b) {
var aDiff = percentageDifference(a),
bDiff = percentageDifference(b);
if (aDiff < bDiff) {
return 1;
} else if (aDiff > bDiff) {
return -1;
} else {
return 0;
}
};
/**
* Sorts the correlation reports and updates
* the HTML.
* @param string jQuery compatible selector for overall element
* @return void Updates the dom as a side effect
*/
window.socSortCorrelation = function (jQueryId) {
// jQueryId = '#cpu_correlation'
$(jQueryId + ' .correlation pre').each(function () {
var lines = $(this).text().split('\n');
lines.sort(sortByCorrelation);
$(this).text(lines.join('\n'));
});
};
firstLineOrEmptyString = function (lines) {
if (typeof lines !== 'string') {
return '';
}
var parts = lines.split('\n');
if (parts.length > 0) {
return parts[0];
}
};
/**
* Determines which of the correlation reports in the DOM
* has the highest correlation. Items must already be sorted
* via socSortCorrelation
* @param string jQuery compatible selector for overall element
* @return string Formatted line for the highest correlation. Will
* start with UNKNOWN if none is available.
*/
window.socDetermineHighestCorrelation = function (jQueryId) {
var cpuLine = firstLineOrEmptyString($('.cpus .correlation pre', jQueryId).text()),
addonLine = firstLineOrEmptyString($('.addons .correlation pre', jQueryId).text()),
moduleLine = firstLineOrEmptyString($('.modules .correlation pre', jQueryId).text()),
cpu = percentageDifference(cpuLine),
addon = percentageDifference(addonLine),
module = percentageDifference(moduleLine);
if (cpu === 0 && addon === 0 && module === 0) {
return 'UNKNOWN: No Data';
} else if (cpu >= addon &&
cpu >= module) {
return 'CPU: ' + correlationItem(cpuLine);
} else if (addon >= cpu &&
addon >= module) {
return 'ADD-ON: ' + correlationItem(addonLine);
} else {
return 'MODULE: ' + correlationItem(moduleLine);
}
};
}); | {
"redpajama_set_name": "RedPajamaGithub"
} | 8,664 |
<?php
use yii\db\Schema;
use yii\db\Migration;
class m150118_104303_following extends Migration
{
public function up()
{
$this->createTable('following', [
'id' => Schema::TYPE_PK,
'user_id' => Schema::TYPE_INTEGER . ' NOT NULL',
'follow_to' => Schema::TYPE_INTEGER . ' NOT NULL',
'created_at' => Schema::TYPE_INTEGER. ' NOT NULL',
]);
}
public function down()
{
$this->dropTable('following');
return true;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,601 |
package vrpsim.simulationmodel.dynamicbehaviour.impl.usutil;
import static org.junit.Assert.*;
import java.util.Arrays;
import java.util.HashMap;
import java.util.Map;
import org.apache.commons.math3.ml.distance.EuclideanDistance;
import org.junit.Test;
import org.slf4j.Logger;
import org.slf4j.LoggerFactory;
public class UnstringingT1Test {
private static Logger logger = LoggerFactory.getLogger(UnstringingT1Test.class);
public static final void main(String[] args) {
new UnstringingT1Test().testPerformOperator();
}
private final Map<Integer, Integer[]> buildIsCorrect() {
HashMap<Integer, Integer[]> correct = new HashMap<>();
correct.put(0, new Integer[] { 1, 5, 6, 4, 3, 2 });
correct.put(1, new Integer[] { 0, 3, 2, 5, 4, 6 });
correct.put(2, new Integer[] { 0, 1, 3, 5, 4, 6 });
correct.put(3, new Integer[] { 0, 1, 2, 5, 4, 6 });
correct.put(4, new Integer[] { 0, 6, 5, 2, 3, 1 });
correct.put(5, new Integer[] { 2, 1, 0, 3, 4, 6 });
correct.put(6, new Integer[] { 2, 1, 0, 3, 4, 5 });
return correct;
}
/**
* Returns true if the order of the result is like in the given correct list
* (the correct list is determined with the index within the given map).
*
* @param index
* @param result
* @param correct
* @return
*/
private final boolean isOrderCorrect(int index, Integer[] result, Map<Integer, Integer[]> correct) {
boolean isCorrect = true;
Integer[] correctOrder = correct.get(index);
int startIndex = 0;
boolean start = true;
for (Integer r : result) {
if (start) {
start = false;
startIndex = determineStartIndex(correctOrder, r);
}
int indexInCorrect = startIndex % correctOrder.length;
indexInCorrect = indexInCorrect < 0 ? indexInCorrect + correctOrder.length : indexInCorrect;
if (r != correctOrder[indexInCorrect]) {
isCorrect = false;
break;
}
startIndex++;
}
return isCorrect;
}
/**
* Returns the index from the given value in the list.
*
* @param values
* @param value
* @return
*/
private final int determineStartIndex(Integer[] values, int value) {
int result = -1;
for (int i = 0; i < values.length; i++) {
if (values[i] == value) {
result = i;
break;
}
}
return result;
}
@Test
public final void testPerformOperator() {
Map<Integer, Integer[]> correct = buildIsCorrect();
UnstringingT1 unstringT1 = new UnstringingT1();
double[][] tour = getSimpleTour();
for (int i = 0; i < tour.length; i++) {
// int i = 0;
Integer[] result = unstringT1.performOperator(transform(tour), i, 4, getDistancecalculator(tour)).getTour();
logger.debug("Result tour for index {} is {}", i, Arrays.toString(result));
if (!isOrderCorrect(i, result, correct)) {
fail("False result for index " + i + ".");
}
}
}
private IDistanceCalculator getDistancecalculator(double[][] tour) {
return new IDistanceCalculator() {
@Override
public double getDistance(Integer i, Integer j) {
EuclideanDistance dc = new EuclideanDistance();
return dc.compute(tour[i], tour[j]);
}
};
}
private final Integer[] transform(double[][] tour) {
Integer[] result = new Integer[tour.length];
for (int i = 0; i < tour.length; i++) {
result[i] = i;
}
return result;
}
private final double[][] getSimpleTour() {
// 0
// 1
// 3 2
// 4 5
// 6
//
// Tour is: 0-1-2-3-4-5-6
double[][] tour = new double[7][2];
// 0
tour[0][0] = 0;
tour[0][1] = 3.5;
// 1
tour[1][0] = 1.6;
tour[1][1] = 3.3;
// 2
tour[2][0] = 2;
tour[2][1] = 2.2;
// 3
tour[3][0] = 1;
tour[3][1] = 2.2;
// 4
tour[4][0] = 1;
tour[4][1] = 1.1;
// 5
tour[5][0] = 2;
tour[5][1] = 1.1;
// 6
tour[6][0] = 0;
tour[6][1] = 0;
return tour;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,284 |
The Holiday Season is a great time of the year made more enticing when all the festive decorations go up. Our bold & colorful Christmas Bunting should be part of your decorations to make your home stand out for all to see and enjoy. You can drape these bunting from your porch, fence or even from your roof. Add a Christmas ribbon bow at each top swag and in an instant your home becomes a Christmas Wonderland.
We offer our Christmas Bunting in 4 different color variations, all using the traditional colors of Christmas, which are pine green(evergreen) and heart red.
Christmas Bunting is made with either a heavy weight cotton blend, which is much sturdier than the industry standard; or for outdoor use we suggest using the 100% Solar-Max nylon.
Christmas Buntings are available in 2 widths, 18" & 36" and are hemmed & double stitched, not printed, with a clean finish hem on top and bottom and a reinforced header on each cut end.
Then on the checkout page, enter the specific lengths you want in the "Order Comments" section. There is a $1.50 per piece charge for any cuts under 5 yards.
Any of our buntings can have grommets put on the top and/or bottom. Please call or email us for pricing.
Along with our Christmas Flag Bunting, Pull Down Banners and hand-tied Christmas Bows you will have no problem selecting just the right Christmas Decorations for your home to help it look beautiful for the Holidays. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,383 |
Q: How do I convert a trackball mouse into a 360° scroller? Is it possible to use the ball of a trackball mouse as a 360° scroller (or a 2D mouse wheel)?
A: I found a very nice solution for my question and thought others might be interested. It works on any mouse not just trackballs.
open or create the file /etc/X11/xorg.conf.d/10-evdev.conf and add the following section:
Section "InputClass"
Identifier "Mouse0"
Driver "evdev"
MatchIsPointer "on"
MatchDevicePath "/dev/input/event*"
Option "EmulateWheel" "true"
Option "EmulateWheelButton" "9"
Option "ZAxisMapping" "4 5"
Option "XAxisMapping" "6 7"
EndSection
Logout and relogin or restart. Now by pressing button #9 you can use the trackball (or mouse) as a 360° scroller. Very handy for photo editing for example.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 0 |
\section{Introduction}
Throughout this paper we consider a possible blow up for a model from surface
growth. Our main motivation is to carry over the program developed for
3D-Navier stokes to this equation, in order to study the possible blow up
of solutions. This paper is the starting point focusing mainly on Hilbert space
theory.
Details on the model can be found in Raible et al.~\cite{Ra-Ma-Li-Mo-Ha-Sa:00},
\cite{Ra-Li-Ha:00} or Siegert \& Plischke \cite{Si-Pl:94:2nd}.
In its simplest version, it is given by
\begin{equation}\label{e:SG}
\partial_t h=-\partial_x^4 h -\partial_x^2 (\partial_x h)^2
\end{equation}
subject to periodic boundary conditions on $[0,L]$ and $\int_0^L h dx=0$.
Although the surface is not periodic,
these boundary conditions together with the assumption of a moving frame
are the standard conditions in models of this type.
Sometimes the model has been considered also
on the whole real line without decay condition at infinity,
even though we do not examine this case here.
From a mathematical point of view Neumann
or Dirichlet boundary conditions are quite similar for the
problem studied here. The key point ensured by any of these
boundary conditions is that there is a suitable cancellation
in the non-linearity, namely
\begin{equation}\label{e:key}
\int_0^L h\,(h_x^2)_{xx}\,dx =0\;,
\end{equation}
which is the main (and probably only) ingredient to
derive useful a-priori estimates.
The main terms in the equation are the dominant linear operator, and the
quadratic non-linearity. Sometimes the equation is considered with
a linear instability $-h_{xx}$,
which leads to the formation of hills,
and the Kuramoto-Shivashinky-type nonlinearity $(h_x)^2$
leading to a saturation in the coarsening of hills.
Both terms are neglected here.
They are lower order terms not important for questions regarding regularity and blow up.
Moreover, the presence of these terms complicates calculations significantly
(cf. \cite{BlGuRa02}).
Furthermore, the equation is usually perturbed
by space-time white noise (see for instance~\cite{BlFlRo08}), which we also neglect here, although many results do hold for the stochastic PDE also.
For general surveys on surface growth processes and molecular beam
epitaxy see Barab\'asi \& Stanley \cite{BaSt95} or Halpin-Healy \& Zhang
\cite{HaZh95}.
\subsection{Existence of solutions}
There are two standard ways of treating the existence of solutions.
The first one relies on the spectral Galerkin method and shows energy
type estimates for the approximation, which
by some compactness arguments ensure the convergence of a subsequence.
See \cite{StWiP}, or for the stochastically
perturbed equation \cite{BlGu02,BlHa04,BlFlRo08}.
In all cases initial conditions in $L^2$ ensure the existence, but not uniqueness, of
global solutions.
The second way uses fixed point arguments
to show local uniqueness and regularity using the mild formulation.
See \cite{BlGu04}, which could not treat the optimal case.
In Section \ref{s:critical} we give a local existence,
which is optimal in the sense that initial conditions are
in a critical space.
We also establish uniqueness among mild solutions and, less
trivially, among weak solutions.
For these smooth local solutions we can easily
show energy estimates,
and discuss possible singularities and blow-up.
Standard arguments assure uniqueness of global solutions
using a fixed point argument in $C^0([0,T],H^1)$
for sufficiently small regular data in $H^1$.
We can even go below that for uniqueness of solutions
in $H^\alpha$ for any $\alpha\geq\tfrac12$. This improves
results of~\cite{BlGu04}. But we are still not able to
prove uniqueness of global solutions without smallness
condition on the initial data. Nevertheless, we can give
easily several conditions that imply uniqueness of global
solutions. All of them assume regularity in critical spaces
or more regularity (cf.\ Section \ref{sec:reg}).
In Section \ref{sec:blowup} we study possible
singularities and blow up. Based on energy-type estimates,
we establish Leray-type estimates for lower bound on blow-up
in terms of $H^\alpha$-norms.
Moreover, we study an upper bound on the Hausdorff-dimension
set of singularities in time,
and show that a blow-up to $-\infty$ is more likely.
\begin{remark}
All results for regularity and Leray-type estimates
are based on energy estimates.
These are optimal in the sense that they hold also hold for complex
valued solutions. Furthermore, using the ideas of \cite{LiSin08},
\cite{LiSin08a}, one should be able to construct a complex valued
solution with strictly positive Fourier coefficients that actually
blows up in finite time. This is the subject of a work in progress.
This would show that results based on energy-estimates are useful
to describe a possible blow-up, but they alone will never be able
to rule it out.
\end{remark}
\subsection{Energy inequality}
\label{ss:ee}
We outline the standard idea for energy estimates,
which is to our knowledge the only useful idea for this equation.
If we formally multiply the equation by $h$
and integrate with respect to $x$, then we obtain using (\ref{e:key}),
\begin{equation}\label{e:energy}
|h(t)|_{L^2}^2 + 2\int_0^t|\partial_x^2 h(s)|^2_{L^2}\,ds
\leq |h(0)|_{L^2}^2.
\end{equation}
Thus, using Poincare inequality,
$$
|h(t)|_{L^2} \le e^{-ct} |h(0)|^2_{L^2}
\quad\text{and} \quad
\int_0^\infty |h(t)|^2_{H^2}\,dt\leq |h(0)|^2_{L^2}.
$$
As explained before this estimate is only valid for smooth local solutions,
or one could use spectral Galerkin approximation to verify it for global solutions.
Note that this regularity is lower than critical regularity.
It is enough for existence of solutions,
but not sufficient for uniqueness.
\subsection{A Lyapunov-type functional}\label{sec:LF}
We can prove another a-priori estimate either for smooth local solutions or via
spectral Galerkin approximations,
\begin{align*}
\tfrac1{\alpha^2}\partial_t \int_0^L \e^{\alpha h}\,dx
& = \int_0^L \e^{\alpha h}h_x h_{xxx}\,dx + 2\int_0^L \e^{\alpha h}h_x^2 h_{xx}\,dx\\
& = - \int_0^L \e^{\alpha h} h_{xx}^2 \,dx + (2 - \alpha) \int_0^L \e^{\alpha h}h_x^2 h_{xx}\,dx\\
& = - \int_0^L \e^{\alpha h} h_{xx}^2 \,dx - \tfrac13(2 - \alpha)\alpha \int_0^L \e^{\alpha h}h_x^4\,dx.
\end{align*}
Thus, for $\alpha\in(0,2)$,
$$
\int_0^L \e^{\alpha h(t)}\,dx \leq \int_0^L \e^{\alpha h(0)}\,dx \quad\text{for all }t>0
$$
and
$$
\tfrac{(2 - \alpha)}{3}\alpha^3\int_0^\infty\int_0^L \e^{\alpha h}h_x^4\,dx\,dt
+ \alpha^2\int_0^\infty\int_0^L\e^{\alpha h} h_{xx}^2 \,dx\,dt
\leq \int_0^L \e^{\alpha h(0)}\,dx.
$$
With some more effort (cf. Stein-Winkler \cite{StWiP}), one knows that these terms are
bounded independently of $h(0)$ for large $t$.
The positive part $h^+=\max\{0,h\}$ now has much more regularity
than the negative part $h^-=\max\{0,h\}$, so a possible blow up
seems to be more likely to $-\infty$ than to $+\infty$.
We will illustrate this in Subsection \ref{ss:negblow}. But unfortunately,
this is still not sufficient regularity for uniqueness of solutions.
\begin{figure}
\includegraphics[scale=.5]{pic.png}
\caption{A snapshot of a numerical solution
to the surface growth equation
with additional linear instability $-30\partial_x^2h$.
The hills look like parabola with
sharp valleys in between.}
\end{figure}
\section{Existence and uniqueness in a critical space}
\label{s:critical}
Prior to the details on some regularity criteria for equation \eqref{e:SG},
we introduce the \emph{scaling heuristic} which explains the formulae that
relate the different exponents in the results of the paper. An account on the
scaling heuristic for the Navier-Stokes equations can be found for
example in Cannone \cite{Can04}, such argument are on the ground
of the celebrated result on partial regularity for Navier-Stokes
of Caffarelli, Kohn \& Nirenberg \cite{CafKohNir82}. A recent paper
by Tao \cite{Tao08} discusses the scaling heuristic in the
framework of dispersive PDE.
The rationale behind the method is the following. First, notice
that the equations are invariant for the scaling transformation
\begin{equation}\label{e:scaling}
h(t,x) \longrightarrow h_\lambda(t,x)=h(\lambda^4 t, \lambda x).
\end{equation}
If $X$ is a functional space for $h$ (for example $L^\infty(0,T;L^2(0,L))$),
we can consider how the norm of $X$ scales with respect to the transformation
\eqref{e:scaling} above. Say the following relation holds,
$$
\|h_\lambda\|_X = \lambda^{-\alpha}\|h\|_X.
$$
We have the three cases
\begin{enumerate}
\item\emph{sub-critical} case for $\alpha<0$,
\item\emph{critical} case for $\alpha=0$,
\item\emph{super-critical} case for $\alpha>0$.
\end{enumerate}
The super-critical case corresponds to small-scales behaviour and is related
to low regularity, typically to topologies where possibly existence can be
proved, but no regularity or uniqueness. For example, one gets $\alpha=\frac12$
(hence, super-critical) for $X=L^\infty(0,\infty;L^2)$ or $X=L^2(0,T;\dot H^2)$,
which are the spaces where existence of global weak solutions can be proved.
The general scheme is the following. Consider spaces $X$ (depending on
the space variable) and $Y_T$ (depending on both variables, with $t$
up to $T>0$), then in order to have a regularity criterion based
on $Y_T$, the following statements must hold,
\begin{enumerate}
\item there is a unique local solution for every initial condition in $X$,
\item the unique local solution provided by (1) is regular,
\item the solution from (1) can be continued up to time $T$, as long
as its norm in $Y_T$ stays bounded.
\end{enumerate}
The above analysis has been extensively carried on by a large number
of authors for the three dimensional Navier-Stokes equations (see for
examples references in Cannone~\cite{Can04}). The first paper dealing
with such aims were Prodi~\cite{Pro59} and Serrin~\cite{Ser62}, see also
Beale, Kato \& Majda~\cite{BeaKatMaj84}.
\subsubsection{Function spaces}
We shall mainly work in the hierarchy of Sobolev spaces of Hilbert type.
Since the equations are considered on $[0,L]$ with periodic boundary
conditions and zero space average, we shall use the following homogeneous
fractional Sobolev spaces. For $\alpha>0$,
$$
\dot H^{\alpha}
=\Bigl\{u\in L^2(0,L): u(\cdot+L)=u(\cdot),\quad u_0=0,\quad\sum_{k\not=0} k^{2\alpha}|u_k|^2<\infty\Bigr\},
$$
where $u_k$ is the $k^\text{th}$ Fourier coefficient, and $\dot H^{-\alpha}=(\dot H^\alpha)'$.
We shall consider the norm on $\dot H^\alpha$ defined by
\begin{equation}\label{e:normdotH}
|u|_{\alpha}^2 = \sum_{k\not=0} k^{2\alpha}|u_k|^2,
\end{equation}
which is equivalent to the norm of the Sobolev space $H^\alpha(0,L)$ on $\dot H^\alpha$.
We also use the space $L^p$ with norm $|\cdot|_{L^p}$
for the Lebesgue space of functions with integrable $p$-th power,
the space $W^{k,p}$ with norm $|\cdot|_{W^{k,p}}$ for the
Sobolev space, where the $k$-th derivative is in $L^p$, and the
space $C^k$ of $k$-time continuously differentiable functions
with the supremum-norm.
\subsection{Existence and uniqueness in \texorpdfstring{$\dot H^{\frac12}$}{H1/2}}
This section is devoted to the proof of existence and uniqueness in the
critical space $\dot H^{\frac12}$,
which improves significantly some results of Bl\"omker \& Gugg \cite{BlGu04}.
Here we shall follow the results of Fujita \& Kato~\cite{FujKat64}
on the Navier-Stokes equations with initial conditions in the
critical Sobolev Hilbert space. This is optimal in the sense
that local existence and uniqueness with lower regularity should imply uniqueness by rescaling.
\begin{definition}\label{d:Sa}
Given $T>0$, $\delta$ and $\alpha\in(0,\frac12)$, define the complete
metric space $\mathcal{S}_\alpha=\mathcal{S}_\alpha(T)$ as
$$
\mathcal{S}_\alpha(T)=\Big\{u\in C((0,T]; \dot H^{1+\alpha}):
\sup_{s\in(0,T]}\{s^{\frac{2\alpha+1}{8}}|u(s)|_{1+\alpha}\}<\infty\Big\},
$$
with norm
$$
\|u\|_{\alpha,T} = \sup_{s\in(0,T]}\bigl\{s^{\frac{2\alpha+1}{8}}|u(s)|_{1+\alpha}\bigr\}
$$
and the $\delta$-ball
$$
\mathcal{S}_\alpha^\delta(T) = \Big\{u\in\mathcal{S}_\alpha(T)\ :\ \|u\|_{\alpha,T}\leq \delta\Big\}.
$$
\end{definition}
Let us remark that for any $h\in \mathcal{S}_\alpha(T)$, $\widetilde\alpha\in(0,\alpha)$ and $\delta>0$
we find $\widetilde{T}\in(0,T)$ such that $h\in\mathcal{S}_{\widetilde\alpha}^\delta(\widetilde{T})$.
\begin{theorem}\label{t:criticalexuniq}
Given an arbitrary initial condition $h_0\in\dot H^{\frac12}$, there
exists a time $T_\bullet>0$, depending only on $h_0$, such
that there is a solution $h\in C([0,T_\bullet);\dot H^{\frac12})$ to problem~\eqref{e:SG}.
Moreover,
\begin{enumerate}
\item $h\in C^\infty((0,T_\bullet)\times[0,L])$,
\item the solution satisfies the energy equality
$$
|h(t)|_{L^2}^2 + 2\int_0^t|h_{xx}|_{L^2}^2 = |h(0)|_{L^2}^2,
$$
for all $t<T_\bullet$,
\item there exists $a_\bullet>0$ such that $T_\bullet=+\infty$ if $|h_0|_{\frac12}\leq a_\bullet$.
\item Either the solution blows up in $\dot H^\beta$ for all $\beta>\frac12$ at $T=T_\bullet$ or $T_\bullet=\infty$.
\end{enumerate}
\end{theorem}
\begin{remark}\label{r:not_in_Hahalf}
If the maximal time $T_\bullet$ of a solution $h$ is finite, while we
know that $\|h(t)\|_\beta\to\infty$ as $t\uparrow T_\bullet$ for $\beta>\frac12$,
we cannot conclude that the same is true for $\|h(t)\|_{\frac12}$. Indeed,
$h$ can be discontinuous in the maximal time $T_\bullet$, so either
$\|h(t)\|_{\frac12}$ is unbounded, or is bounded and discontinuous in $T_\bullet$.
The reason behind this is that a solution in $\dot H^{\frac12}$ can be continued
as long as there is a control on the quantity $K_0$ of the type~\eqref{e:K0to0},
and this quantity is not uniformly convergent to $0$ in bounded subsets of $\dot H^{\frac12}$.
In different words, $K_0$ can be controlled as long as one can control the way
the mass of $h(0)$ is partitioned among Fourier modes.
\end{remark}
The proof of this theorem is developed in several steps, which we will prove in
the remainder of this section.
First, we prove existence and uniqueness (together with the global existence
statement). Then we prove an analogous result in $\dot H^\beta$, for all $\beta>\frac12$.
By a standard bootstrap technique, this implies the smoothness of solutions.
Let $A$ be the operator $\partial_x^4$ with domain $\dot H^4$. It is a standard
result that $A$ generates an analytic semigroup. Using
for example the Fourier series expansion,
it is easy to verify that
\begin{equation}\label{e:sgestimate}
|A^\gamma\e^{-tA}|_{\mathcal{L}(\dot{H}^\beta)}
\leq c_\gamma t^{-\gamma},
\end{equation}
for every $t>0$, where $\gamma\geq0$ and $\beta\in\mathbb{R}$.
Moreover, it is easy to verify that the norm $|A^\frac{\beta}{4}\cdot|_{L^2} $,
which we will use several times in the paper, coincides with the standard
norm~\eqref{e:normdotH} on $\dot H^\beta$.
Proposition~\eqref{p:inequality} implies that for $\alpha\in(0,\frac12)$,
\begin{equation}\label{e:nonlin}
|A^{\frac18(4\alpha-5)}(h_x^2)_{xx}|_{L^2}
\leq c_\alpha |h|_{1+\alpha}^2
\end{equation}
(just apply the proposition with $\alpha=\beta$, $\gamma=\tfrac12-2\alpha$
and use the dual formulation of $L^2$ norm).
Consider now the right hand side of the mild formulation,
\begin{equation}\label{e:mild}
\mathcal{F}(h)(t)=\e^{-tA}h_0 + \int_0^t\e^{-(t-s)A}(h_x^2)_{xx}(s)\,ds,
\end{equation}
and define
\begin{gather*}
K_0(t)=\sup_{s\in(0,t]}\bigl(s^{\frac18(2\alpha+1)}|\e^{-sA}h_0|_{1+\alpha}\bigr),
\quad\text{for }h_0\in \dot{H}^{\tfrac12}, \\
K(t,h)=\sup_{s\in(0,t]}\bigl(s^{\frac18(2\alpha+1)}|h(s)|_{1+\alpha}\bigr),
\quad\text{for }h\in\mathcal{S}_\alpha(T),\ t\in[0,T].
\end{gather*}
Obviously, $K(t,h+k) \le K(t,h) + K(t,k) $
and
\begin{lemma}
For $h_0\in \dot{H}^{\tfrac12}$ we have
\begin{equation}\label{e:K0to0}
K_0(t)\to0\quad\text{as } t\to0.
\end{equation}
Furthermore, for each $\beta\in[\tfrac12,1+\alpha]$ there is a constant $c_\beta>0$
such that
\begin{equation}\label{e:K0bound}
K_0(t)\leq c_\beta t^{\frac18(2\beta-1)} |h_0|_\beta.
\end{equation}
\end{lemma}
\begin{proof}
By assumption $A^{\frac18}h_0\in L^2$, hence by Lemma~\ref{lem:SG} for $s\to 0$,
$$
s^{\frac18(2\alpha+1)}|\e^{-sA}h_0|_{1+\alpha}
= |s^{\frac18(2\alpha+1)}A^{\frac18(1+2\alpha)}\e^{-sA} A^{\frac18}h_0|_{L^2}
\to 0.
$$
For the second claim use (\ref{e:sgestimate}) to show
$$
K_0(t)
= \sup_{s\in(0,t]} s^{\frac18(2\alpha+1)}|A^{\frac14(1+\alpha-\beta)} e^{-sA}A^{\frac{\beta}{4}}h_0|_{L^2}
\leq c_\beta t^{\frac18(2\beta-1)}|h_0|_\beta.
$$
\end{proof}
Now we proceed to find a solution of $h=\mathcal{F}(h)$.
\begin{lemma}
There is a small constant $\delta>0$ depending on $\alpha$ such that for all
$h_0\in \dot H^{\frac12}$ there exists a time $T$ sufficiently small,
such that the map $\mathcal{F}$ is a contraction on $\mathcal{S}_\alpha^\delta(T)$.
\end{lemma}
\begin{proof}
First we show that $\mathcal{F}$ maps $\mathcal{S}_\alpha^\delta$ into itself for $T$ and $\delta$
sufficiently small.
To be more precise, there is a number $c_\alpha>0$ such that for all $t\in[0,T]$
and all $h\in\mathcal{S}_\alpha^\delta$
\begin{equation}\label{e:claiminto}
K(t,\mathcal{F}(h))\leq K_0(t) + c_\alpha K(t,h)^2 \leq K_0(T)+ c_\alpha \delta^2.
\end{equation}
Thus for $\delta\leq c_\alpha/2$ and $T$ sufficiently small $\mathcal{F}$
maps $\mathcal{S}_\alpha^\delta$ into itself.
In order to prove \eqref{e:claiminto} we consider
$$
|\mathcal{F}(h)(t)|_{1+\alpha}
\leq |\e^{-tA}h_0|_{1+\alpha} + \int_0^t|\e^{-(t-s)A}(h_x^2)_{xx}|_{1+\alpha}\,ds.
=I_0+I_1.
$$
For the first term,
$$
t^{\frac18(2\alpha+1)}I_0
=t^{\frac18(2\alpha+1)}|A^{\frac18(2\alpha+1)}\e^{-tA}h_0|_{\frac12}
\leq K_0(T)\to 0
$$
for $T\to0$.
For the second term we use \eqref{e:nonlin}, as well as \eqref{e:sgestimate}, to obtain
\begin{align*}
I_1
&= \int_0^t|A^{\frac{5-4\alpha}8+\frac{1+\alpha}{4}}\e^{-(t-s)A}A^{\frac18(4\alpha-5)}(h_x^2)_{xx}|_{L^2}\,ds\\
&\leq c K(t, h)^2\int_0^t s^{-\frac14(2\alpha+1)}(t-s)^{-\frac18(7-2\alpha)}\,ds\\
&=C_\alpha t^{-\frac18(1+2\alpha)}K(t, h)^2,
\end{align*}
where $C_\alpha=cB(\frac14(3-2\alpha),\frac18(1+2\alpha))$ and
$B(x,y)=\int_0^1 t^{x-1}(1-t)^{y-1}\,dt$ is the \emph{Beta} function.
Now let us show that $\mathcal{F}$ is a contraction on $\mathcal{S}_\alpha$.
If $h,k\in \mathcal{S}_\alpha$, then by following essentially the
above estimate of $I_1$, one can derive the following estimate
\begin{equation}\label{e:contraction}
K(T, \mathcal{F}(h)-\mathcal{F}(k))
\leq C_\alpha K(T, h-k)K(T, h+k) \leq 2\delta C_\alpha K(T, h-k)
\end{equation}
Thus $\mathcal{F}$ is a contraction, if $\delta\leq 1/(4C_\alpha)$.
\end{proof}
The following corollary is obvious, if we use \eqref{e:K0bound}
for $\beta>0$. The same conclusion cannot be drawn in the case
$\beta=\frac12$ (see Remark~\ref{r:not_in_Hahalf}).
\begin{corollary}
If $h_0\in \dot H^\beta$ for $\beta>\frac12$, then the time $T$
in the previous lemma depends only on a bound on $|h_0|_\beta$ and
not directly on $h_0$.
\end{corollary}
Thus, as long as a solution is bounded in any $\dot H^\beta$
with $\beta>\frac12$, the interval of existence can by extended
by a fixed length $T$, which depends only on the bounding constant.
The next lemma shows that the solution to the fixed point
$h=\mathcal{F}(h)$ in $\mathcal{S}_\alpha$ is continuous with values in $H^{1/2}$.
\begin{lemma}
If $h\in\mathcal{S}_\alpha(T)$, then $\mathcal{F}(h)\in C^0((0,T],\dot{H}^{1/2})$.
\end{lemma}
\begin{proof}
Obviously, it is enough to show that $\mathcal{F}(h)$ is continuous in $t=0$.
First, $e^{-tA}h_0\to 0$ in $\dot{H}^{\frac12}$ by continuity of the semigroup.
It remains to show that
$$
\int_0^t e^{-(t-s)A}(h_x(s)^2)_{xx}\,ds \to 0
$$
in $\dot H^{\frac12}$ for $t\to0$. We know already by \eqref{e:nonlin} that
$f(s) = s^{\frac14(2\alpha+1)}A^{\frac18(4\alpha-5)}(h_x^2)_{xx}$
is bounded in $L^2$ for $s\in(0,T]$ with $|f(s)|_{L^2}\leq cK(s,h)^2$.
Thus from Lemma~\ref{lem:SG}),
$$
\int_0^ts^{-\frac14(2\alpha+1)} A^{-\frac18(4\alpha-5)+\frac18 } e^{-(t-s)A}f(s) \to 0
$$
in $L^2$, for $t\to0$.
\end{proof}
\begin{proposition}\label{p:critical}
Given $h_0\in\dot H^{\frac12}$ and $\alpha\in(0,\frac12)$, there exists
$T_0>0$ and $\delta_0$, depending only on $\alpha$ and $h_0$, such that
there is a unique solution in $\mathcal{S}_\alpha^{\delta_0}(T_0)$ to problem~\eqref{e:SG}
starting at $h_0$.
Moreover, the solution is in $C^0([0,T_0),\dot{H}^{1/2})$ and
there exists $a_0>0$ small enough such that, if $|h_0|_{1/2}\leq a_0$,
then $T_0=\infty$.
\end{proposition}
\begin{proof}
Most of the proof is already done. We need to prove the last
statement of the proposition. By \eqref{e:sgestimate},
$K_0(t)\leq c_0 |h_0|_{\frac12}$, so that, if we choose
$a_0\leq (c_0c_\alpha)^{-1}$ (where $c_\alpha$ is the
constant in formula \eqref{e:claiminto}) and
$K = (2c_\alpha)^{-1}(1-\sqrt{1-c_0c_\alpha a_0})$,
by \eqref{e:claiminto} it follows that, for $K(t,h)\leq K$,
$$
K(t,\mathcal{F}(h))
\leq K_0(t) + c_\alpha K(t,h)^2
\leq c_0 a_0 + c_\alpha K^2
\leq K,
$$
independently of $t$. Hence, $T_0=\infty$.
\end{proof}
\begin{remark}[Criticality of $\mathcal{S}_\alpha(T)$]
Following the same notation used in Section \ref{s:critical},
we have that if $h\in\mathcal{S}_\alpha(T)$, then
$h_\lambda\in\mathcal{S}_\alpha(T_\lambda)$ and $K(T_\lambda,h_\lambda)$
scales as $\lambda^{\frac18(1-6\alpha)}K(T,h)$.
So, apparently, the $\|\cdot\|_{\alpha,T}$ does not obey
the scaling heuristic. On the other hand, this information
is of no use. Indeed, the scaling behaviour is hidden,
as it is shown by Lemma~\ref{l:addreg}, where the
boundedness in a space which is almost $\mathcal{S}_\alpha$
implies boundedness in the critical space
$L^q(0,T;\dot H^{1+\alpha})$, with $q=\frac8{1+2\alpha}$.
\end{remark}
Next, the case of more regular initial condition is considered.
The result is stated for integer exponents only, since for showing
regularity the present version is sufficient (we already know
that solutions with initial value in $\dot H^{\frac12}$ are
continuous in $\dot H^1$). It is easy to adapt the proposition
to noninteger exponents, with some slight changes.
\begin{proposition}\label{p:critimorereg}
Let $n\in\mathbf{N}$, $n\geq1$. Given an arbitrary $h_0\in \dot H^n$,
there exist $T>0$ and a solution
$h\in C([0,T);\dot H^n)\cap L^2_\text{loc}([0,T);\dot H^{n+2})$
to problem \eqref{e:SG}, with initial condition $h_0$.
\end{proposition}
\begin{proof}
We only prove the core \emph{a-priori} estimate for the
Theorem. Existence of a solution can be proven by means
of Proposition \ref{p:critical} or by an approximation
procedure (such as finite dimensional approximations).
Start by $n=1$,
$$
\frac{d}{dt}|h|_1^2
= 2\scal{h,\partial_t h}_1
= - 2|h|_3^2 - 2\scal{h_{xx},(h_x^2)_{xx}}.
$$
By integration by parts and Sobolev,
interpolation and Young's inequalities, we get
$$
\begin{aligned}
2\scal{h_{xx},(h_x^2)_{xx}}
&= - 2\scal{h_{xxx}, 2h_x h_{xx}}\\
\text{\Tiny (by H\"older's inequality)}\quad
&\leq 2|h_{xxx}|_{L^2}|h_x|_{L^6}|h_{xx}|_{L^3}\\
\text{\Tiny (by Sobolev embedding)}\quad
&\leq c|h|_3 |h|_{\frac43} |h|_{\frac{13}6}\\
\text{\Tiny (by interpolation)}\quad
&\leq c|h|_3^{\frac74}|h|_1^{\frac54}\\
\text{\Tiny (by Young's inequality)}\quad
&\leq |h|_3^2 + c|h|_1^{10}.
\end{aligned}
$$
In conclusion, if we denote by $\varphi(t)=|h(t)|_1^2+\int_0^t|h|_3^2$,
the above inequality reads
$$
\dot\varphi
= \frac{d}{dt}|h|_1^2 + |h|_3^2
\leq c|h|_1^{10}
\leq c\varphi^5
$$
and by solving the differential inequality, we have a time $T$
such that $h$ is bounded in $C([0,T);\dot H^1)$ and in
$L^2_\text{loc}([0,T);\dot H^3)$.
The method is similar for $n\geq 2$. By computing the derivative
of $|h(t)|_n^2$, it turns out that it is necessary to estimate
the term originating from the nonlinear part. By integration
by parts and Leibnitz formula,
$$
\begin{aligned}
2\scal{D^{2n}h,(h_x^2)_{xx}}
&= 2\scal{D^{n+2}h,D^n(h_x^2)}\\
&= 2\sum_{k=0}^n\binom{n}{k}\scal{D^{n+2}h,(D^{k+1}h)(D^{n+1-k}h)}.
\end{aligned}
$$
By applying H\"older's inequality and Sobolev embedding, the above
sum can be estimated as above. All terms $|h|_a$ with $a\leq n$
can be controlled by $|h|_n$, while all terms with $a\in(n,n+2)$
can be controlled by $|h|_n$ and $|h|_{n+2}$ by interpolation. We
finally get the estimate
$$
\frac{d}{dt}|h|_n^2 + 2|h|_{n+2}^2
\leq |h|_{n+2}^2 + c_n |h|_n^{a_n},
$$
with suitable $c_n$ and $a_n$, depending only on $n$. By solving,
as above, the implied differential inequality, the solution $h$
turns out to be bounded in $C([0,T);\dot H^n)$ and in
$L^2_\text{loc}([0,T);\dot H^{n+2})$.
\end{proof}
Everything is now ready to carry on the proof of the main theorem
of this section.
\begin{proof}[Proof of Theorem \ref{t:criticalexuniq}]
The existence of solutions with initial condition in $\dot H^{1/2}$,
as well as the $T_\bullet=\infty$ statement, follow
from Proposition \ref{p:critical}.
The regularity statement (\textbf{1}) follows from Proposition
\ref{p:critimorereg}. Indeed, by Proposition \ref{p:critical},
a solution starting in $\dot H^{1/2}$ is continuous with
values in $\dot H^1$. By applying Proposition \ref{p:critimorereg}
on each $h(t)\in\dot H^1$, for $t\leq T_\bullet$, it follows that the solution
is $C((0,T_\bullet);\dot H^1)$ and $L^2_\text{loc}((0,T_\bullet);\dot H^3)$.
The last statement implies that $h(t)\in\dot H^3$, for almost every
$t\in(0,T_\bullet)$ and so Proposition \ref{p:critimorereg} can be
used with $n=3$, and so on. By iterating the procedure, it follows
that $h\in C((0,T_\bullet);\dot H^\beta)$ for all $\beta\geq1$.
Time regularity now follows from this space regularity and the
mild form~\eqref{e:mild}.
The energy equality in (\textbf{2}) is now easy using the space-time
regularity in $(0,T_\bullet)$ and the continuity at $t=0$ in the $L^2$ norm.
\end{proof}
\subsection{Uniqueness among weak solutions}
A weak solution to equation~\eqref{e:SG} is a function
$h\in L^\infty_{\text{\Tiny loc}}([0,\infty);L^2)\cap L^2_{\text{\Tiny loc}}([0,\infty);\dot H^2)$
which satisfies the equation in distributions. Existence of such
solutions for all initial data in $L^2$ has been established in
\cite{StWiP} (or \cite{BlGuRa02,BlFlRo08}). The following theorem shows that the solutions provided
by Theorem~\ref{t:criticalexuniq} are unique in the class of all weak
solutions $h$ that satisfy the energy inequality~\eqref{e:energy}.
\begin{theorem}\label{t:weakstrong}
Let $h_0\in\dot H^{\frac12}$ and let $h\in C([0,T_\bullet);\dot H^{\frac12})$ be the solution
to~\eqref{e:SG} provided by Theorem~\ref{t:criticalexuniq} and defined up to its maximal
time $T_\bullet$. Then every weak solution to~\eqref{e:SG} starting at $h(0)$ coincides with
$h$ on $[0,T_\bullet)$.
\end{theorem}
In order to prove the theorem, we shall proceed in several steps.
We will essentially prove that any solution in $\mathcal{S}_\alpha(T)$
with an additional integrability condition is unique in the class
of weak solutions (Proposition~\ref{p:weakstrong} below). Then we
prove that solutions in $\mathcal{S}_\alpha(T)$ satisfy the additional
condition (Lemma \ref{l:addreg} and \ref{l:last_step}). It is worth
remarking that the additional integrability condition~\eqref{e:ass_integ}
turns out to correspond to the critical space $L^{\frac{8}{1+2\alpha}}(H^{1+\alpha})$
(see Section~\ref{ss:criticality}).
\begin{proposition}\label{p:weakstrong}
Let $h\in \mathcal{S}_\alpha(T)$ be a solution to~\eqref{e:SG} and assume moreover that
\begin{equation}\label{e:ass_integ}
\int_0^T|h(t)|_{1+\alpha}^{\frac8{1+2\alpha}}\,dt
=\|h\|^{8/(2\alpha+1)}_{L^{8/(2\alpha+1)} ([0,T],H^{1+\alpha})} <\infty.
\end{equation}
Then $h$ is the unique weak solution starting at $h(0)$.
\end{proposition}
\begin{proof}
Let $k$ be any weak solution starting at $h(0)$. Since $h\in C^\infty((0,T]\times[0,L])$
and $h$ is continuous in $\dot H^{\frac12}$, it follows that
$$
\scal{h(t),k(t)} + 2\int_0^t\scal{h_{xx},k_{xx}}\,ds
= -\int_0^t\int(h_{xx}k_x^2+k_{xx}h_x^2)\,ds
$$
which, together with the energy inequality for $k$ and the energy equality
(see Theorem~\ref{t:criticalexuniq}) for $h$ implies that the difference $w=h-k$
satisfies the following energy inequality,
$$
\begin{aligned}
|w(t)|_{L^2}^2 +2\int_0^t|w_{xx}|_{L^2}^2\,ds
&\leq 2\int_0^t\int (h_{xx}k_x^2+k_{xx}h_x^2)\,dx\,ds\\
&= 4\int_0^t\int k_xw_xw_{xx}\,dx\,dt
\end{aligned}
$$
where we have used~\eqref{e:key} since
$$
h_{xx}k_x^2+k_{xx}h_x^2=2k_xw_xw_{xx}+w_{xx}w_x^2+h_{xx}h_x^2+k_{xx}k_x^2.
$$
The conclusion now follows from the assumption~\eqref{e:ass_integ} and
Gronwall's lemma, since
$$
4\int_0^t\int k_xw_xw_{xx}\,dx\,dt
\leq c|w|_2 |k|_{1+\alpha} |w|_{\frac{3-2\alpha}2}
\leq |w|_2^2 + c|k|_{1+\alpha}^{\frac{8}{1+2\alpha}}|w|_{L^2}^2,
$$
where we have used H\"older inequality (with exponents $2$, $\frac1{\alpha}$,
and $\frac{2}{1-2\alpha}$), the Sobolev embeddings $L^{\frac2{1-2\alpha}}\subset H^{\frac{3-2\alpha}2}$
and $L^{\frac1\alpha}\subset H^{1+\alpha}$, interpolation of $H^{\frac{3-2\alpha}2}$
between $L^2$ and $H^2$, and finally Young's inequality.
\end{proof}
Assumption~\eqref{e:ass_integ} cannot be obviously satisfied
by any arbitrary element of $\mathcal{S}_\alpha(T)$, hence we are led to
prove additional regularity for the solutions of~\eqref{e:mild}.
To this end, define for $T>0$ and $\alpha\in(0,\frac12)$,
$$
(\|u\|_{\alpha,T}^\star)^2
=\sum_{k\not=0} k^{2(1+\alpha)}\Big(\sup_{s\leq T} \{ s^{\frac18(1+2\alpha)}|u_k(s)|\}\Big)^2
$$
and
$$
\mathcal{S}_\alpha^\star(T) = \{u\in\mathcal{S}_\alpha(T): \|u\|_{\alpha,T}^\star<\infty\}.
$$
Assuming that $\mathcal{S}_\alpha^\star(T)\subset \mathcal{S}_\alpha(T)$ is not restrictive,
since it is easy to verify that $\|\cdot\|_{\alpha,T}\leq\|\cdot\|_{\alpha,T}^\star$.
\begin{lemma}\label{l:addreg}
If $h\in\mathcal{S}_\alpha^\star(T)$, then $\mathcal{F}(h)$ satisfies~\eqref{e:ass_integ} on $[0,T]$.
\end{lemma}
\begin{proof}
We write $\mathcal{F}(h)(t)=H_0(t)+H_1(t)$ where $H_0(t)=\e^{-tA}h(0)$
and $H_1$ contains the nonlinearity. Now,
$$
|H_0(t)|_{1+\alpha}^2 = \sum_{k\not=0} k^{2(1+\alpha)}\e^{-2ctk^4}|h_k(0)|^2,
$$
and so, if $\varphi\in L^q(0,T)$ with $p=\frac4{1+2\alpha}$ and $\frac1p+\frac1q=1$,
$$
\begin{aligned}
\int_0^T\varphi(t)|H_0(t)|_{1+\alpha}^2\,dt
&= \sum_{k\not=0}^\infty k^{2(1+\alpha)}|h_k(0)|^2\int_0^T\varphi(t)\e^{-2ctk^4}\,dt\\
&\leq \|\varphi\|_{L^q}\sum_{k\not=0}
k^{2(1+\alpha)}|h_k(0)|^2\Bigl(\int_0^T\varphi(t)\e^{-2ctpk^4}\,dt\Bigr)^{\frac1p}\\
&\leq c_p\|\varphi\|_{L^q} |h(0)|_{\frac12}^2.
\end{aligned}
$$
By duality, the $L^{\frac8{1+2\alpha}}$ norm of $|H_0|_{1+\alpha}$ is finite. The second term is
more delicate, we shall proceed as in the proof of Proposition~\ref{p:inequality},
$$
\begin{aligned}
|H_1(t)|_{1+\alpha}^2
&= \sum_{k\not=0} k^{2(1+\alpha)}\Bigl(\int_0^t\e^{-c(t-s)k^4}[(h_x^2)_{xx}]_k\Bigr)^2\\
&= \sum_{k\not=0} k^{2(3+\alpha)}\Bigl(\sum_{l+m=k}|lm|\int_0^t\e^{-c(t-s)k^4}|h_l(s) h_m(s)|\,ds\Bigr)^2\\
&\leq \sum_{k\not=0} k^{2(3+\alpha)}\Bigl(\sum_{l+m=k}|lm|h_l^\star h_m^\star\Bigr)^2\Bigl(\int_0^t\e^{-c(t-s)k^4}s^{-\frac{1+2\alpha}4}\,ds\Bigr)^2,
\end{aligned}
$$
where $h_k^\star=\sup_{s\leq T}s^{\frac{1+2\alpha}8}|h_k(s)|$. Hence, for every
$\varphi\in L^q(0,T)$,
\begin{multline*}
\int_0^T\varphi(t)|H_1(t)|_{1+\alpha}\,dt\leq\\
\leq \sum_{k\not=0} k^{2(3+\alpha)}\Bigl(\sum_{l+m=k}|lm|h_l^\star h_m^\star\Bigr)^2
\int_0^T\varphi(t)\Bigl(\int_0^t\e^{-c(t-s)k^4}s^{-\frac{1+2\alpha}4}\Bigr)^2\,dt.
\end{multline*}
If we prove that
\begin{equation}\label{e:elem_claim}
\int_0^T\varphi(t)\Bigl(\int_0^t\e^{-c(t-s)k^4}s^{-\frac{1+2\alpha}4}\,ds\Bigr)^2\,dt
\leq c\|\varphi\|_{L^q}k^{2\alpha-7},
\end{equation}
then we can proceed as in the proof of Proposition~\ref{p:inequality} (where the $h_k^\star$
replace the Fourier components and $\gamma=\frac12-2\alpha$) to obtain that
$$
\int_0^T\varphi(t)|H_1(t)|_{1+\alpha}\,dt
\leq c\|\varphi\|_{L^q}(\|h\|_{\alpha,T}^\star)^2,
$$
and, again by duality, boundedness of $\mathcal{F}(h)$.
So, everything boils down to proving~\eqref{e:elem_claim}. Using H\"older
inequality and (twice) a change of variables,
$$
\begin{aligned}
\lefteqn{\int_0^T\varphi(t)\Bigl(\int_0^t\e^{-c(t-s)k^4}s^{-\frac{1+2\alpha}4}\Bigr)^2\,dt\leq}\qquad\\
&\leq \|\varphi\|_{L^q}\Bigl[\int_0^T\varphi(t)\Bigl(\int_0^t\e^{-c(t-s)k^4}s^{-\frac{1+2\alpha}4}\,ds\Bigr)^{2p}\,dt\Bigr]^{\frac1p}\\
&\leq \|\varphi\|_{L^q} k^{2\alpha-7}\Bigl[\int_0^\infty\Bigl(\int_0^t\e^{-(t-s)}s^{-\frac{1+2\alpha}4}\,ds\Bigr)^{2p}\,dt\Bigr]^{\frac1p},
\end{aligned}
$$
and it is elementary to verify that the integral on the right-hand side is convergent.
Indeed,
$$
\int_0^{\frac{t}2}\e^{-(t-s)}s^{-\frac{1+2\alpha}4}\,ds
\leq c t^{\frac{3-2\alpha}4}\e^{-\frac{t}2},
$$
which is in $L^p(0,\infty)$, as well as
$$
\int_{\frac{t}2}^t\e^{-(t-s)}s^{-\frac{1+2\alpha}4}\,ds
\leq c t^{-\frac{2\alpha+1}4}(1-\e^{-\frac{t}2}),
$$
since $t^{-2p\frac{2\alpha+1}4}=t^{-2}$.
\end{proof}
The final step is to prove that solutions exist in the smaller space
$\mathcal{S}_\alpha^\star$. This is then the unique weak solution and the
solution given by Theorem \ref{t:criticalexuniq}.
\begin{lemma}\label{l:last_step}
Let $h_0\in\dot H^{\frac12}$ and $\alpha\in(0,\frac12)$. Then there
is $T_\star>0$ such that there exists a solution $h$ in $\mathcal{S}_\alpha^\star(T_\star)$.
\end{lemma}
\begin{proof}
The proof is essentially a fixed point argument, as in Proposition~\ref{p:critical}.
So, it is sufficient to show the following facts:
\begin{enumerate}
\item $\|H_0\|_{\alpha,T}^\star\leq\|h(0)\|_{\frac12}$,
\item $\|H_0\|_{\alpha,T}^\star\longrightarrow0$ as $T\to0$,
\item there is $c>0$ (independent of $T$) such that for all $h\in\mathcal{S}_\alpha^\star(T)$,
$\|\mathcal{F}(h)\|_{\alpha,T}^\star\leq \|H_0\|_{\alpha,T}^\star + c(\|h\|_{\alpha,T}^\star)^2$,
\item there is $c>0$ (independent of $T$) such that $\|\mathcal{F}(g)-\mathcal{F}(h)\|_{\alpha,T}^\star\leq c\|g-h\|_{\alpha,T}^\star \|g+h\|_{\alpha,T}^\star$
for all $g$, $h\in\mathcal{S}_\alpha^\star(T)$,
\end{enumerate}
where $H_0(t)=\e^{-tA}h(0)$ and $H_1=\mathcal{F}(h)-H_0$. Notice that
$$
\sup_{s\leq T}s^{\frac{1+2\alpha}8}|[H_0(t)]_k|
= |h_k(0)|\sup_{s\leq T}s^{\frac{1+2\alpha}8}\e^{-csk^4}
\leq c k^{-\frac{1+2\alpha}2}|h_k(0)|
$$
and so
$$
(\|H_0\|_{\alpha,T}^\star)^2
\leq \sum_{k\not=0} k^{2(1+\alpha)}|h_k(0)|^2 ck^{-(1+2\alpha)}
\leq c\|h(0)\|_{\frac12}^2.
$$
In order to prove the second property, we have to refine the previous
computation. Fix $\varepsilon>0$ such that $\varepsilon\leq c_\alpha$
(where $c_\alpha^4$ is the point where the function $s^{\frac{1+2\alpha}8}\e^{-s}$
attains its maximum), then
$$
\begin{aligned}
(\|H_0\|_{\alpha,T}^\star)^2
&= \Bigl(\sum_{|k|\leq\varepsilon T^{-\frac14}} + \sum_{|k|>\varepsilon T^{-\frac14}}\Bigr)
k^{2(1+\alpha)}|h_k(0)|^2(\sup_{s\leq T} s^{\frac{1+2\alpha}4}\e^{-2csk^4})\\
&\leq \sum_{|k|\leq\varepsilon T^{-\frac14}}(k^4T)^{\frac{1+2\alpha}4} |k||h_k(0)|^2
+ \sum_{|k|>\varepsilon T^{-\frac14}} k|h_k(0)|^2\\
&\leq \varepsilon^{1+2\alpha}\|h(0)\|_{\frac12}^2 + c\sum_{|k|>\varepsilon T^{-\frac14}} |k||h_k(0)|^2.
\end{aligned}
$$
Now, $\limsup_{T\to0}\|H_0\|_{\alpha,T}^\star\leq\varepsilon^{1+2\alpha}\|h(0)\|_{\frac12}^2$
and, as $\varepsilon\downarrow0$, the conclusion follows.
In order to prove the last fact, we follow the proof
of Lemma~\ref{l:addreg},
$$
|(H_1)_k(t)|
\leq k^2\sum_{l+m=k}|l m| h_l^\star h_m^\star\Bigl(\int_0^t\e^{-c(t-s)k^4}s^{-\frac{1+2\alpha}{4}}\Bigr)
$$
and so
$$
\sup_{t\leq T}t^{\frac{1+2\alpha}8}|(H_1)_k(t)|
\leq k^2\sum_{l+m=k}|l m| h_l^\star h_m^\star\Bigl(\sup_{t\leq T}t^{\frac{1+2\alpha}8}\int_0^t\e^{-c(t-s)k^4}s^{-\frac{1+2\alpha}{4}}\Bigr).
$$
Assume that the term in round brackets in the above formula is bounded by $c_\alpha k^{-\frac{7-2\alpha}2}$
(we shall prove this later), then, as in the proof of Proposition~\ref{p:inequality},
$$
(\|H_1\|_{\alpha,T}^\star)^2
\leq c_\alpha\sum_{k\not=0} k^{6+2\alpha}\Bigl(k^{-\frac{7-2\alpha}2}\sum_{l+m=k}|l m| h_l^\star h_m^\star\Bigr)^2
\leq c(\|h\|_{\alpha,T}^\star)^4.
$$
As it regards the rounded brackets term, we use the inequality $\e^{-c(t-s)k^4}\leq c_\alpha [k^4(t-s)]^{-\frac{7-2\alpha}{8}}$
to get
$$
\Bigl(\sup_{t\leq T}t^{\frac{1+2\alpha}8}\int_0^t\e^{-c(t-s)k^4}s^{-\frac{1+2\alpha}{4}}\,ds \Bigr)
\leq c_\alpha B(\tfrac{3-2\alpha}4,\tfrac{1+2\alpha}8)k^{-\frac{7-2\alpha}2},
$$
and $B$ is the \emph{Beta} function.
The proof of the last fact is similar. Indeed, if $g$, $h\in\mathcal{S}_\alpha^\star(T)$, then
$$
\begin{aligned}
|[\mathcal{F}(g)(t)-\mathcal{F}(h)(t)]_k|
&\leq k^2\int_0^t\e^{-c(t-s)k^4}|[(g-h)_x(g+h)_x]_k|\,ds\\
&\leq k^2\sum_{l+m=k}|l m|\int_0^t\e^{-c(t-s)k^4}|(g_l-h_l)(g_m+h_m)|\,ds
\end{aligned}
$$
and so, by proceeding as above, the last fact follows.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{t:weakstrong}]
Given $h(0)\in\dot H^{\frac12}$, let $h\in C([0,T_\bullet);\dot H^{\frac12})$
be the solution provided by Theorem~\ref{t:criticalexuniq} and fix $T<T_\bullet$.
By Lemma~\ref{l:last_step} we know that $h\in\mathcal{S}_\alpha^\star(T_\star)$,
so Lemma~\ref{l:addreg} implies that $h$ satisfies the integrability
condition~\eqref{e:ass_integ} on $[0,T_\star]$. By property (\textbf{1})
of Theorem~\ref{t:criticalexuniq}, $h$ satisfies trivially~\eqref{e:ass_integ}
on $[T_\star,T]$. So Proposition~\ref{p:weakstrong} applies and the
conclusion follows.
\end{proof}
\section{Regularity}
\label{sec:reg}
\subsection{Criticality}\label{ss:criticality}
In this section, we carry out the program described in the beginning of the previous section.
We will find spaces $Y_T$ such that boundedness in these spaces imply
uniqueness for solutions starting in $H^{1/2}$.
Let us first discuss regularity criteria in Lebesgue spaces.
Set $T_\lambda=\lambda^{-4}T$ and $L_\lambda=\lambda^{-1}L$ and consider the space
$X(\lambda)=L^q(0,T_\lambda;L^p(0,L_\lambda))$, for some values of $p$ and $q$.
Under the scaling \eqref{e:scaling} we have that
$$
\|h_\lambda\|_{X(\lambda)} = \lambda^{-\frac{4}{q}-\frac{1}{p}}\|h\|_{X(1)}.
$$
so that the space $L^\infty((0,T)\times(0,L))$ turns out to be the only critical
space in this class. All other Lebesgue spaces are super-critical.
The conjecture now is that solutions in $L^\infty(0,T;L^\infty(0,L))$
or $C((0,T)\times(0,L))$ are unique and regular.
We believe that with similar methods, as in the existence for
initial conditions in $H^{1/2}$, one should be able to
prove existence of unique local solutions.
But this is much more involved.
In order to consider Sobolev spaces, we set
$X(\lambda)=L^q(0,T_\lambda;\dot W^{k,p}(0,L_\lambda))$ and
$$
\|h_\lambda\|_{X(\lambda)} = \lambda^{k-\frac{4}{q}-\frac{1}{p}}\|h\|_{X(1)}.
$$
(this is easy for integer $k$ and tricky for non-integer values, but
it can be done). Hence, the space is critical for
$$
\frac{4}{q} + \frac{1}{p} = k.
$$
In the following subsection, we will give
the corresponding criteria for $p=2$, $k$ arbitrary
and $p=4$, $k=1$.
The extension to $k=1$ and $p$ arbitrary is straightforward
and not presented here.
Let us finally remark, that in the following,
we also give regularity criteria for $L^4(0,T,C^1(0,L))$,
which is also a critical space.
\subsection{Regularity Criteria}
In principle the following {\em Meta-theorem} should hold:
If a solution is bounded in a critical space,
then it is unique, and does not have a blow up.
This means that the unique local solution exists as long
as at least one (hence all, as the solution is then proved to be regular)
of the critical norms is finite over the time horizon.
For simplicity, in the rest of the section we focus only
on some examples and we consider solutions with
sufficiently smooth initial condition, in order to have energy type
estimates for the $H^1$-norm without any trouble at $t=0$.
We just remark that energy estimates in any
other $H^s$-space with $s>\tfrac12$ yield exactly the same result.
\begin{theorem}\label{t:regul}
Let $h_0\in\dot H^1$, let $h=h(\cdot,h_0)$ be the unique local
solution started at $h_0$ and let $\tau(h_0)$ be the maximal
time of $h$. Then $h$ is $C^\infty$ in
space and time on $(0,\tau(h_0))$ and for every $\alpha\in(\frac12,\frac92)$,
$$
\int_0^{\tau(h_0)}\|h(s)\|_{H^\alpha}^{\frac{8}{2\alpha-1}}\,ds
=\infty.
$$
Moreover,
$$
\int_0^{\tau(h_0)}\|h(s)\|_{W^{1,4}}^{16/3}\,ds
=\infty
\quad\text{and}\quad
\int_0^{\tau(h_0)}\|h(s)\|_{C^1}^4\,ds
=\infty\;.
$$
\end{theorem}
\begin{proof}
We already know by Theorem \ref{t:criticalexuniq}
that there is a unique local solution in $C((0,\tau); \dot H^1)$
for initial conditions in $\dot H^1$, which is actually smooth. Indeed
$h\in C^\infty((0,\tau)\times(0,L))$. Furthermore, the $H^1$-norm blows up at $t\to \tau$.
Now fix $\alpha\in(\frac12,\frac92)$, then by integration by parts and the Sobolev embedding
$H^\frac16 \subset L^3$,
\begin{align} \label{e:crit_H}
\frac{d}{dt}|h|_{1}^2 + 2|h|_{3}^2
& = -2\langle h, (h_x^2)_{xx}\rangle_{1}
= -2\langle h_x, (h_x^2)_{xxx}\rangle_{L^2}\nonumber\\
& = -4\int_0^L h_x h_{xx} h_{xxx}\,dx
= 2\int_0^L h_{xx}^3\nonumber\\
&\leq c|h|_{{\frac{13}6}}^3.
\end{align}
By interpolation, it is easy to see that
$$
|h|_{\frac{13}6}^3\leq |h|_{1}^{\frac{2\alpha-1}4}|h|_{\alpha}|h|_{3}^{\frac{9-2\alpha}4},
$$
and so using Young's inequality,
$$
\frac{d}{dt}|h|_{1}^2 + 2|h|_{3}^2
\leq |h|_{3}^2 + c|h|_{\alpha}^{\frac8{2\alpha-1}}|h|_{1}^2.
$$
Finally, by Gronwall's lemma, the proof of the first statement is complete.
Let us turn again to (\ref{e:crit_H}).
Using Sobolev embedding $H^\frac14\subset L^4$ yields
\begin{equation}\label{e:crit_W}
\frac{d}{dt}|h|_{1}^2 + 2|h|_{3}^2 \le C|h|_{W^{1,4}}|h|_{2+\frac14}|h|_3\;.
\end{equation}
Again by interpolation and Young inequality
$$
\frac{d}{dt}|h|_{1}^2 + 2|h|_{3}^2
\leq |h|_{3}^2 + C|h|_{W^{1,4}}^{16/3}|h|_{1}^2\;,
$$
which yields the result using Gronwall.
The last claim follows similarly, using
\begin{equation}\label{e:crit_C}
\frac{d}{dt}|h|_{1}^2 + 2|h|_{3}^2 \le C|h|_{C^1}|h|_2|h|_3\;.
\end{equation}
\end{proof}
\subsection{\texorpdfstring{$H^3$}{H3}-regularity}
In this section we show $L^p(0,T, H^3)$
for some small $p$ which is possibly less than $1$.
We gain spatial regularity by paying time regularity.
The main result is:
\begin{theorem}
If for a solution $h\in L^r(0,T,H^1)$ for some $r\in(0,10)$, then $h\in L^{r/5}(0,T,H^3)$.
Moreover,
$$
\int_0^T |h_{xxx}|^{r/5} \,dt
\le C \Big(\int_0^T |h_x|^r \, dt \Big)^{(10-r)/5}\;.
$$
\end{theorem}
\begin{remark}
It is easy to check that the space $L^{r/5}(0,T,H^3)$ is critical
if and only if $L^r(0,T,H^1)$ is critical. Thus this result respects
the criticality heuristic.
\end{remark}
\begin{remark}
If $h\in L^\infty(0,T,H^{1/2})$ (critical)
then by interpolation of $H^{1/2}$ and $H^2$ we obtain from energy estimates $h\in L^6(0,T,H^1)$,
and now $h\in L^{6/5}(0,T,H^3)$.
Then by interpolation of $H^{1/2}$ and $H^3$ we recover $h\in L^2(0,T,H^2)$.
Thus this regularity result gives no improvement of the regularity given by the
energy estimate in Section \ref{ss:ee}. It respects the level
of criticality of the spaces.
\end{remark}
\begin{proof}
For some $p>0$ where $|\cdot|$ denotes the norm in $L^2$
$$
\partial_t \frac{|h_x|^2}{1+|h_x|^p}
= 2 \frac{\langle h_{xt},h_x\rangle}{1+|h_x|^p} + p \frac{|h_x|^2 \langle h_{xt},h_x \rangle}{(1+|h_x|^p)^2}\\
=: \varphi_p(|h_x|) \langle h_{xt},h_x\rangle
$$
where $ \varphi_p(z)=(2+(2+p)z^2)/(1+z^p)^2$.
Thus using the PDE and integration by parts
$$
\partial_t \frac{|h_x|^2}{1+|h_x|^p}
= \varphi_p(|h_x|) (\tfrac12 \int h_{xx}^3\,dx - |h_{xxx}|^2)
$$
Using the embedding of $H^{1/6}$ into $L^3$, interpolation, and Young yields
$$
| \int h_{xx}^3\,dx|
\le C |h|_{{13/6}}^3
\le C |h|_{1}^{5/4} |h|_{3}^{7/4}
\le C |h|_{1}^{10} + |h|_{3}^2
$$
Combining both results yields
$$
\partial_t \frac{|h_x|^2}{1+|h_x|^p} = -\tfrac12\varphi_p(|h_x|) |h_{xxx}|^2 + C \varphi_p(|h_x|) |h_x|^{10}
$$
Now, $\varphi_p(|h_x|) |h_x|^{10}\in L^1(0,T)$ if $12-r-2p\le0$ and thus $p\ge 6-\tfrac{r}2$.
We derive
$$ \int_0^T \tfrac12\varphi_p(|h_x|) |h_{xxx}|^2 \,dt <\infty \quad\text{for}\quad p= 6-\tfrac{r}2>1.
$$
Using H\"older inequality for some $\alpha\in(0,2)$ yields
\begin{eqnarray*}
\int_0^T |h_{xxx}|^\alpha \,dt
&\le&\Big(\int_0^T |h_{xxx}|^2\varphi_p(|h_x|) \,dt \Big)^{\alpha/2}
\Big(\int_0^T\varphi_p(|h_x|)^{-\alpha/(2-\alpha)} \,dt \Big)^{(2-\alpha)/2}\\
&\le& C \Big(\int_0^T |h_x|^{(2p-2)\alpha / (2-\alpha)} \,dt \Big)^{(2-\alpha)/2}
\end{eqnarray*}
Fixing $\alpha=r/5$ yields the claim.
\end{proof}
\subsection{Blow up below criticality}
In this section we will study the blow up in a space below criticality,
i.e.\ in some $H^s$ with $s<\frac12$.
This is a slight generalisation of Theorem \ref{t:regul}
and prepares the results of Leray-type shown later.
For $ \frac14\le \delta \le 1$ we obtain:
\begin{equation} \label{e:leray}
\begin{split}
\tfrac12\partial_t |h|^2_{\delta}
&\leq -c |h|^2_{{2+\delta}} + 2 \int_0^L (-\partial_x^2)^{\delta} h_x \cdot h_x h_{xx}\,dx\\
&\leq -c |h|^2_{{2+\delta}} + C |h|_{{1+2\delta}}|h|_{{\frac94}}|h|_{{\frac54}}\\
&\leq -c |h|^2_{{2+\delta}} + C |h|_{{2+\delta}}^{(9-2\delta)/4} | h|_{\delta}^{(3+2\delta)/4}
\end{split}
\end{equation}
where we have used the Sobolev embedding $H^{\frac14}\subset L^4$.
\begin{remark}\label{r:interpol}
As it is used several times in the proofs,
we state the following elementary interpolation inequality.
For $\gamma>\alpha$ and $\beta\in[\alpha,\gamma]$,
$$
|h|_{\beta}
\le C |h|_{\alpha}^{\frac{\gamma-\beta}{\gamma-\alpha}}|h|_{\gamma}^{\frac{\beta-\alpha}{\gamma-\alpha}}.
$$
\end{remark}
Using interpolation between $H^\gamma$, $\gamma\le\tfrac54$ and $H^{2+\delta}$ implies
$$
\tfrac12\partial_t |h|^2_{\delta}
\le -c |h|^2_{{2+\delta}} +
C | h|_{\gamma}^{(3+2\delta)/(4+2\delta-2\gamma)} |h|_{{2+\delta}}^{(9+4\delta-6\gamma)/(4+2\delta-2\gamma)}
$$
If we suppose $\gamma>\tfrac12$, then using Young inequality with $p=(8+4\delta-4\gamma)/(9+4\delta-6\gamma)$ and $q= (8+4\delta-4\gamma)/(2\gamma-1)$
we derive
$$
\tfrac12\partial_t |h|^2_{\delta}
\le
C | h|_{\gamma}^{2(3+2\delta)/(2\gamma-1)}
$$
We proved the following Theorem:
\begin{theorem}
Let $h\in C^\infty([0,t_0)\times[0,L])$ be
a solution and fix $\gamma\in(\tfrac12,\tfrac54]$ and $\delta\in[\tfrac14,1]$.
Then
$$ |h(t)|_{\delta} \to \infty\ \text{for}\ t\nearrow t_0
\quad \Rightarrow \quad
\int_0^{t_0} | h(t)|_{\gamma}^{2(3+2\delta)/(2\gamma-1)}\,dt =\infty\;.
$$
\end{theorem}
Note that for a blow up below criticality with $\delta<\frac12$ the
$L^p([0,T],H^\gamma)$-norm in this theorem
has a smaller $p$ than assured by Theorem \ref{t:regul}.
The spaces in the above theorem should always have the
same level of criticality.
\section{Blow-up}
\label{sec:blowup}
In this section we discuss some properties of the blow up.
First, at a possible blow up time,
one expects that all norms with higher regularity
than the critical norms will blow up,
in particular all $H^s$-norm with $s>1/2$ should blow up.
In Subsection \ref{ss:leray}, we give a lower bound on the blow-up
in $H^s$-spaces, while in Subsection \ref{ss:singu}
we show a bound on the size of the set of singular times.
We illustrate that a blow up to $-\infty$ is
more likely, but first we give some remarks on
possible shapes of a blow-up.
\subsection{Some remarks}
Let us first give examples on which blow up profiles $v=h(\tau)$ are possible
at the blow up time $\tau$.
\begin{itemize}
\item If $v$ exhibits a {\em jump} like $\text{sign}(x)$,
then the Fourier-coefficients decay like $1/k$,
and thus $v$ is in $H^s$ if and only if $s<\frac12$.
\item If $v$ exhibits a {\em logarithmic pole} like $\log(|x|)$,
then the Fourier-coefficients decay like $1/k$,
and thus $v$ is in $H^s$ if and only if $s<\frac12$.
\item If $v$ exhibits a {\em cusp} like $|x|^\alpha$ for $\alpha\in(0,1)$,
then the Fourier-coefficients decay like $|k|^{-(1+\alpha)}$,
and thus $v$ is in $H^{1/2}$, and not a possible blow up.
\end{itemize}
\subsubsection{Stationary solutions}\label{sss:stationary}
The $L^2$ estimates~\eqref{e:energy} show that the only stationary solution
is $h \equiv 0$, as $|h(t)|_{L^2}\to0$ for $t\uparrow\infty$. On the other
hand the problem is one-dimensional, so it is worth trying to look for
solutions directly. The equation for stationary solutions is
$$
h_{xxxx}+(h_x^2)_{xx} = 0,
$$
so there are constants $A$, $B$ such that $h_{xx}+h_x^2=A x+B$. By the
periodic boundary conditions, $A=0$.
\emph{Case 1: $B=0$}. By direct computations, we get
$$
h(x) = c_1 + \log|1+c_2 x|,
$$
and the only periodic solution corresponds to $c_2=0$, a constant function.
Notice that, anyway, the solutions are singular with a log-like profile.
\emph{Case 2: $B=b^2$}. Again by direct computations,
$$
h(x) = c_1 + \log|\cosh b x + \frac{c_2}{b}\sinh b x|,
$$
and there are no periodic solutions. We remark that again the singularity
has a log-like profile.
\emph{Case 3: $B=-b^2$}. By elementary computations,
$$
h(x) = c_1 + \log|b\cos bx + c_2\sin bx|,
$$
all solutions are periodic on $[0,L]$ as long as $b=\frac{2\pi}{L}k$,
for some $k\in\mathbf{N}$. If $x_0$ is any zero of $b\cos bx+c_2\sin bx$,
we can write the solution as $h(x)=c_1 + \log|\sin b(x-x_0)|)$ (with a
different value of $c_1$). Again, the stationary profile is log-like.
\subsubsection{Self-similar solutions}
By exploiting the scaling \eqref{e:scaling}, we may look for solutions
of the following kind,
$$
h(t,x)=\varphi(\frac{x}{\root 4\of {T-t}}),
$$
where $\varphi$ is a suitable function. The equation for $h$ reads in terms
of $\varphi$ as
\begin{equation}\label{e:selfsimilar}
\varphi_{yyyy} + (\varphi_y^2)_{yy} + y\varphi_y = 0,
\qquad y\in\mathbf{R},
\end{equation}
and, by the regularity of weak solutions
one shows easily $\varphi$, $\varphi_{xx}\in L^2$
and hence $\varphi\in H^2(\mathbf{R})$.
Here for simplicity we have neglected boundary conditions and formulated the problem
on the whole real line. The problem above can be recast in weak form as
$$
\int\varphi\eta_{yyyy}\,dy + \int\varphi_y^2\eta_{yy}\,dy - \int\varphi\eta\,dy - \int y\varphi\eta_y\,dy = 0,
\qquad\eta\in C_c^\infty,
$$
where the solution $\varphi\in H^1_\text{loc}(\mathbf{R})$.
There is quite a strong numerical evidence that there are no solutions
to \eqref{e:selfsimilar} defined on the whole $\mathbf{R}$. This fact would rule
out self-similar solutions%
\footnote{Existence of self-similar solutions has been a long standing problem
for the Navier-Stokes equations. The problem was firstly posed by J. Leray
\cite{Ler34} in 1934 and finally solved by Ne\v{c}as,~R{\accent"17 u}\v{z}i\v{c}ka
\& \v{S}ver\'ak \cite{NecRuzSve96} in $1996$. Lately, Cannone \& Planchon~\cite{CanPla96}
proved existence of self-similar solution in Besov spaces.
Ne\v{c}as et al.~exploited a non-trivial maximum principle for $|u|^2+p$
(where $u$ is the velocity field and $p$ is the pressure). We remark
that no such fact seems to be true in this case.}.
\subsection{Leray-type results}
\label{ss:leray}
We will prove the following theorem, which is based on one of the several
celebrated results of Leray~\cite{Ler34} on the Navier-Stokes equations. This relies mainly
on a comparison result for ODEs (see Lemma~\ref{l:techLR})
and energy estimates. It improves the results of Theorem \ref{t:regul},
which states that at blow-up for $s>\tfrac12$ the function
$t\to|h(t)|_s^{8/(2s-1)}$ is not integrable. The result now says
that it behaves like $\tfrac1t$.
\begin{theorem}\label{t:leray}
Let $h\in C^\infty([0,t_0)\times [0,L])$ be a smooth local solution.
Then for $s>\frac12$
there is a universal constant $C>0$ such that
$|h(t)|_s \to \infty$ for $t\nearrow t_0$ (or for any subsequence)
implies
$$|h(t)|_s \ge C(t_0-t)^{-(2s-1)/8}
\quad\text{for all } t\in[0,t_0)\;.
$$
\end{theorem}
\begin{proof}
We proceed by using energy estimates.
Again use the notation $D=A^{1/4}=|\partial_x|$ and $B(u,v)=(u_x v_x)_{xx}$.
From (\ref{e:SG}) we obtain for $s=1+\delta$ with $\delta\in(-\tfrac12,\tfrac32)$
\begin{eqnarray*}
\partial_t|h|^2_{1+\delta}+ 2 |h|^2_{3+\delta}
&=& -2 \int D^{2\delta}h_{xx} B(h,h)\,dx\\
&=& 4 \int D^{2\delta}h_{x} B(h,h_x)\,dx\\
&\le & C |h|_{1+\delta+\epsilon}|h|_{\tfrac52-\epsilon}|h|_{3+\delta}\;,
\end{eqnarray*}
where we used Proposition \ref{p:inequality} with
$\alpha=2+\delta$, $\beta=\tfrac12-\epsilon$, and $\gamma=-\alpha+\epsilon$
for some small $\epsilon\in(0,\tfrac12)$
such that $\epsilon+\delta \in (-\tfrac12,\tfrac32)$.
Now using interpolation (cf. Remark \ref{r:interpol}) yields
$$
\partial_t|h|^2_{1+\delta}+ 2 |h|^2_{3+\delta}
\le C |h|_{1+\delta}^{\tfrac14(7-2\delta)} |h|_{3+\delta}^{\tfrac14(5+2\delta)}\;.
$$
As $(5+2\delta)<8$, we can apply Young's inequality
with $p=8/(7-2\delta)$ and $q=8/(1+2\delta)$
to derive
$$
\partial_t |h|^2_{1+\delta}
\le C | h|_{1+\delta}^{2(5+2\delta)/(1+2\delta)}=C | h|_s^{2(3+2s)/(2s-1)}\;.
$$
Thus Lemma \ref{l:techLR} implies the theorem for $s\in(\tfrac12,\tfrac52)$.
Consider now $s=2+\delta$ with $\delta\in(-\tfrac12,\tfrac32)$.
\begin{eqnarray*}
\partial_t|h|^2_{2+\delta}+ 2 |h|^2_{4+\delta}
&=& 2 \int D^{2\delta}h_{xxxx} B(h,h)\,dx\\
&=& -4 \int D^{2\delta}h_{xxx} B(h,h_x)\,dx\\
&=& -4 \int D^{2\delta}h_{xx} \left[B(h_x,h_x)+B(h,h_{xx})\right]\,dx\\
&\le & C |h|_{2+\delta+\epsilon}|h|_{\tfrac72-\epsilon}|h|_{3+\delta}+
C |h|_{3+\delta+\epsilon}|h|_{\tfrac72-\epsilon}|h|_{2+\delta}\;,
\end{eqnarray*}
where we again used Proposition \ref{p:inequality}
with the same choice of $\alpha$, $\beta$, $\gamma$ and $\epsilon$.
Now using interpolation
$$
\partial_t|h|^2_{2+\delta}+ 2 |h|^2_{4+\delta}
\le C |h|_{2+\delta}^{\tfrac14(7+2\delta)} |h|_{4+\delta}^{\tfrac14(5-2\delta)}\;
$$
and Young with $p=8/(5-2\delta)$ and $q=8/(3+2\delta)$
\begin{equation}\label{e:forsingular}
\partial_t |h|^2_{2+\delta}
\le C | h|_{2+\delta}^{2(7+2\delta)/(3+2\delta)}=C | h|_s^{2(3+2s)/(2s-1)}\;.
\end{equation}
Now Lemma \ref{l:techLR} finishes the proof for $s\in(\tfrac32,\tfrac72)$.
The general case is proven similarly, by distributing the derivatives
as evenly as possible on the trilinear terms, as in the proof of
Proposition~\ref{p:critimorereg}, and then applying Proposition~\ref{p:inequality},
possibly with different $\alpha$'s for different terms.
\end{proof}
\begin{remark}
\label{rem:leray}
We can also give a lower bound on the blow-up time $t_0$
depending on $| h(0)|_{\delta}$ for $\delta>\frac12$.
To be more precise, using the upper bound in Lemma \ref{l:techLR}
the following is straightforward to verify.
For all $s>\tfrac12$ there is a constant $c_s>0$
such that the solution is regular and smooth on $(t,t_\ast)$
if
$ c_s |h(t)|_s^{8/(2s-1)}(t_\ast-t)<1.$
On the other hand, Theorem \ref{t:leray} immediately
implies that near a blow up at $t_\ast$ we obtain
for all $r\in(t,t_\ast)$, that
$ c_s |h(r)|_s^{8/(2s-1)}(t_\ast-r) \ge 1.$
\end{remark}
\subsection{Criterion for point-wise blow up to \texorpdfstring{$-\infty$}{-infinity}}
\label{ss:negblow}
We show that for a blow up in $L^\infty$ the blow up to $-\infty$
is much more likely than the blow up to $\infty$.
This is mainly based on the a-priori estimate from Section \ref{sec:LF},
but first we use the following estimate:
$$
\begin{aligned}
\tfrac13 \partial_t \int_0^Lh^3\,dx
& = - \int_0^L h^2 h_{xxxx}\,dx - \int_0^L h^2 ((h_{x})^2)_{xx}\,dx\\
& = 2\int_0^L h h_x h_{xxx}\,dx + 4\int_0^L h (h_{x})^2 h_{xx}\,dx\\
& = - \int_0^L h (h_{xx})^2\,dx - \tfrac43 \int_0^L (h_{x})^4 \,dx,
\end{aligned}
$$
where we used the cancellation property~\eqref{e:key}.
Thus
$$ \int_0^T \int_0^L (h_{x})^4\,dx\,dt
\leq \int_0^L h^3(0)\,dx + \int_0^T \int_0^L h^- (h_{xx})^2\,dx\,dt + \int_0^L h^- h^2\, dx.
$$
This implies:
\begin{theorem}
Let $h\in C^\infty([0,\tau)\times[0,L])$ be a smooth local solution.
If $\int_0^L h^3(0)\,dx$ is finite and $\| h\|_{L^4(0,\tau,W^{1,4})}=\infty$
then the negative part $h^-$ has to blow up. In other terms,
there are $t_n\nearrow\tau$ and $x_n\in[0,L]$ such that $ h(t_n,x_n) \to -\infty$.
\end{theorem}
\begin{corollary}
If $\int_0^L h^3(0)\,dx<\infty$ and $h^-$ uniformly bounded, then
$\| h\|_{L^4(0,T,W^{1,4})}<\infty$ and $\int_0^T \int_0^L h^+ (h_{xx})^2\,dx\,dt<\infty$.
\end{corollary}
Let us now show that not only we have a point-wise
blow up, but also a blow up for some $\int_0^L e^{-\gamma h(t)}\,dx$,
while we know already by Section \ref{sec:LF}
that $\int_0^L e^{-\gamma h(t)}\,dx$ stays finite for $\gamma\in(0,2)$.
\begin{lemma}
Let $h\in C^\infty([0,\tau)\times[0,L])$ be a smooth local solution.
If
\begin{equation}
\label{e:exp_bl_ass}
\int_0^T\int|h_x|^\alpha|h|^k\,dx\,dt \to\infty
\quad \text{for} \quad T\nearrow\tau
\end{equation}
for some $\alpha\in(0,4)$ and $k\ge0$,
then
$$ \int e^{-\gamma h(t)}\,dx \to\infty
\quad \text{for} \quad t\nearrow \tau
$$
for all $\gamma\in(0,2\alpha/(4-\alpha))$.
\end{lemma}
Note that the corresponding metric is always not critical.
It has less regularity.
Furthermore, note that for $\alpha\le 2$, by H\"older and interpolation,
the quantity in (\ref{e:exp_bl_ass}) will never blow up.
\begin{proof}
Using H\"older and results of Section \ref{sec:LF}
yields for any $\epsilon\in(0,\alpha/2)$
(i.e.\ $4\epsilon/\alpha\in(0,2)$),
$$
\begin{aligned}
\lefteqn{\Big(\int_0^T\int|h_x|^\alpha|h|^{k}\,dx\,dt \Big)^{\alpha/4}\leq}\qquad\\
&\leq C \int_0^T \Big(\int|h_x|^\alpha|h|^{k}\,dx \Big)^{\alpha/4}\,dt\\
&\leq C \int_0^T \int|h_x|^4 e^{4\epsilon h /\alpha}\,dx \cdot
\Big(\int e^{-4\epsilon h /(4-\alpha)} |h|^{4k/(4-\alpha)}\,dx \Big)^{(4-\alpha)/\alpha}\,dt\\
&\leq C\sup_{[0,T]} \Big( \int e^{-4\epsilon h /(4-\alpha)} |h|^{4k/(4-\alpha)}\,dx \Big)^{(4-\alpha)/\alpha}\\
&\leq C\sup_{[0,T]} \Big(\int e^{-\gamma h}\,dx \Big)^{(4-\alpha)/\alpha}
\end{aligned}
$$
for $\gamma\in (0, 4\epsilon /(4-\alpha))$.
\end{proof}
\subsection{The set of singular times}
\label{ss:singu}
Let $h$ be a weak solution to~\eqref{e:SG} and consider the set of regular times of $h$,
$$
\mathscr{R} = \{t\in(0,\infty): u\text{ is continuous with values in $H^1$ in a neighbourhood of }t\}.
$$
By Proposition~\ref{p:critimorereg}, $\mathscr{R}$ is equal to the set of all times
$t$ such that $h$ is $C^\infty$ in space and time in a neighbourhood of $t$.
Define the set of singular times $\mathscr{S} = [0,\infty)\setminus\mathscr{R}$.
The next theorem proves (in the spirit of results of Leray~\cite{Ler34},
Scheffer~\cite{Sch76} for Navier-Stokes), that the set of singular times
is ``small''.
\begin{theorem}\label{t:singularset}
Given a weak solution $h$ to~\eqref{e:SG}, the set $\mathscr{S}$ of singular times
of $h$ is a compact subset of $[0,\infty)$ and
$$
\mathcal{H}^\frac14(\mathscr{S}) = 0,
$$
where $\mathcal{H}^\frac14$ is the $\tfrac14$-dimensional Hausdorff measure.
\end{theorem}
\begin{proof}
Fix a weak solution $h$ and define $\mathscr{R}$ and $\mathscr{S}$ as above.
The proof is divided in four steps.
\emph{1. $\mathscr{S}$ is compact}.
The set $\mathscr{R}$ is clearly open, hence $\mathscr{S}$ is closed.
We prove that $\mathscr{S}$ is bounded. Let $a_\bullet$ be the constant
given in part~\textbf{3} of Theorem~\ref{t:criticalexuniq}. Assume by
contradiction that $a_\bullet<|h(t)|_{\frac12}$ for all $t\geq0$.
By interpolation and using the energy inequality~\eqref{e:energy},
$$
a_\bullet^{\frac83} t
< \int_0^t |h(s)|_{\frac12}^{\frac83}\,ds
\leq \int_0^t |h(s)|_{L^2}^{\frac23} |h(s)|_{2}^2\,ds
\leq |h(0)|_{L^2}^{\frac23}\int_0^t |h(s)|_2^2\,ds
\leq 2 |h(0)|_{L^2}^{\frac83}.
$$
Hence for some $t_0>0$, $|h(t_0)|_{\frac12}\leq a_\bullet$ and Theorems
\ref{t:criticalexuniq} and \ref{t:weakstrong} imply that the solution
$h$ is regular in $[t_0,\infty)$.
\emph{2. $\mathscr{S}$ has Lebesgue measure $0$}.
As any open set of $\mathbf{R}$ is the countable union of
disjoint open intervals we have $\mathscr{R}=\bigcup_j I_j$,
where the open intervals $I_j$ are the connected
components of $\mathscr{R}$.
Define $\mathscr{S}_2=\{t:u(t)\not\in\dot H^2\}$.
Trivially, $\mathscr{R}\subset\mathscr{S}_2^c$, hence $\mathscr{S}_2\subset\mathscr{S}$.
If $t_0\in\mathscr{S}\setminus\mathscr{S}_2$, by Proposition~\ref{p:critimorereg}
$t_0$ is the endpoint of some $I_j$, hence $\mathscr{S}\setminus\mathscr{S}_2$ is
at most countable. Finally, the energy estimate \eqref{e:energy}
implies that $\mathscr{S}_2$ has measure $0$.
\emph{3. Estimate on the length of bounded $I_j$}.
Indeed, let $I_j$ be a bounded component of $\mathscr{R}$ and let $t_1,t_2\in I_j$.
From Remark \ref{rem:leray} we know $c(t_2-s)|h(s)|_2^{8/3}\geq1$,
and hence $c(t_2-s)^{-3/4}\leq |h(s)|_2^2$, for $s\in(t_1,t_2)$.
Integrating for $s\in(t_1,t_2)$ and using the energy inequality~\eqref{e:energy},
yields
\begin{equation}\label{e:intervalbound}
c(t_2-t_1)^{\frac14}
\leq \int_{t_1}^{t_2} |h(s)|_2^2\,ds
\leq \frac12|h(0)|_{L^2}^2.
\end{equation}
\emph{4. $\mathcal{H}^{\frac14}(\mathscr{S})=0$}.
Write $I_j=(a_j,b_j)$ for bounded intervals. From~\eqref{e:intervalbound}
it follows that
$$
\sum_j (b_j - a_j)^{\frac14}<\infty,
$$
while $\sum_j (b_j - a_j)<\infty$, by the first step of the proof. Now we
can proceed as in the proof of Theorem 2 of~\cite{Sch76} to get the conclusion.
\end{proof}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,502 |
Added on May 19, 2015 by analog robot.
The INNOCENTS DVD is available at Objectifs. You can purchase it at their store, or online here (ships internationally). You can also pickup a copy from local bookstores Kinokuniya and Books Actually. | {
"redpajama_set_name": "RedPajamaC4"
} | 349 |
{"url":"https:\/\/nbviewer.jupyter.org\/github\/ZikangXiong\/blogs\/blob\/main\/notebooks\/Neural%20Symbolic\/greg_neurosymbolic_nips20.ipynb","text":"# Neuro-symbolic Reinforcement Learning with Formally Verified Exploration\u00b6\n\n## Contribution\u00b6\n\nGuarantee the safe exploration in continuous control tasks.\n\n## Imitate Safely\u00b6\n\nGet a shield.\nfor t = $1, \\dots, T$ do,\n\n1. Initialize the system with a manually constructed shield.\n2. The neural network policy is imitated from the shield.\n3. The neural network is updated with gradient-based RL algorithms.\n4. Imitate the shielded neural network to get a new shield.\n\n## $Project_\\mathcal{G}$\u00b6\n\nRefine the piecewise linear shield.\nfor t = $1, \\dots, T$ do,\n\n1. Cutting space to 2 parts.\n2. Call imitate safely for both of the subspace, and get 2 verified policies $g_i^1$ and $g_i^2$ for the 2 spitted spaces.\n3. Compose $g_i^1$ and $g_i^2$ to $g'$.\n4. If $D(g', h) < D(g*, h)$, $g* \\leftarrow g'$. $D$ is the $L_2$ norm, h is the shielded neural policy.\n\n## Detailed Questions\u00b6\n\n1. In step 2 of $Project_\\mathcal{G}$, what if the learned the new shield cannot be verified? Even if redo this step 2 may find verified policy, but there is no guarantee. I guess if they cannot find it, they will keep the previous shield.\n2. If we already have a shielded policy at the beginning, why not keep using this shield to ensure the safety for exploration, while focusing on optimizing the neural network policy? The time calling the shield is much less than the time of calling the neural network. Thus, even the performance of the shield is not so good, since it is less likely to be involved, it should not affect the reward too much. Their results also show that neural network based DDPG can have best rewards in almost every case.\n\n## Other\u00b6\n\nThe proof is useful for an AI conference submission.","date":"2021-06-24 07:03:36","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6125076413154602, \"perplexity\": 1391.736585804454}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-25\/segments\/1623488551052.94\/warc\/CC-MAIN-20210624045834-20210624075834-00584.warc.gz\"}"} | null | null |
## 24™ Declassified
Head Shot
## David Jacobs
Based on the hit FOX series created by Joel Surnow & Robert Cochran
## Contents
1
The Following Takes Place Between the Hours of 3 A.M and 4 A.M. Mountain Daylight Time
2
The Following Takes Place Between the Hours of 4 A.M and 5 A.M. Mountain Daylight Time
3
The Following Takes Place Between the Hours of 5 A.M and 6 A.M. Mountain Daylight Time
4
The Following Takes Place Between the Hours of 6 A.M and 7 A.M. Mountain Daylight Time
5
The Following Takes Place Between the Hours of 7 A.M and 8 A.M. Mountain Daylight Time
6
The Following Takes Place Between the Hours of 8 A.M and 9 A.M. Mountain Daylight Time
7
The Following Takes Place Between the Hours of 9 A.M and 10 A.M. Mountain Daylight Time
8
The Following Takes Place Between the Hours of 10 A.M and 11 A.M. Mountain Daylight Time
9
The Following Takes Place Between the Hours of 11 A.M and 12 P.M. Mountain Daylight Time
10
The Following Takes Place Between the Hours of 12 P.M and 1 P.M. Mountain Daylight Time
11
The Following Takes Place Between the Hours of 1 P.M and 2 P.M. Mountain Daylight Time
12
The Following Takes Place Between the Hours of 2 P.M and 3 P.M. Mountain Daylight Time
13
The Following Takes Place Between the Hours of 3 P.M and 4 P.M. Mountain Daylight Time
14
The Following Takes Place Between the Hours of 4 P.M and 5 P.M. Mountain Daylight Time
15
The Following Takes Place Between the Hours of 5 P.M and 6 P.M. Mountain Daylight Time
16
The Following Takes Place Between the Hours of 6 P.M and 7 P.M. Mountain Daylight Time
17
The Following Takes Place Between the Hours of 7 P.M and 8 P.M. Mountain Daylight Time
18
The Following Takes Place Between the Hours of 8 P.M and 9 P.M. Mountain Daylight Time
19
The Following Takes Place Between the Hours of 9 P.M and 10 P.M. Mountain Daylight Time
20
The Following Takes Place Between the Hours of 10 P.M and 11 P.M. Mountain Daylight Time
21
The Following Takes Place Between the Hours of 11 P.M and 12 A.M. Mountain Daylight Time
22
The Following Takes Place Between the Hours of 12 A.M and 1 A.M. Mountain Daylight Time
23
The Following Takes Place Between the Hours of 1 A.M and 2 A.M. Mountain Daylight Time
24
The Following Takes Place Between the Hours of 2 A.M and 3 A.M. Mountain Daylight Time
About the Author
Other Books by David Jacobs
Copyright
About the Publisher
##
After the 1993 World Trade Center attack, a division of the Central Intelligence Agency established a domestic unit tasked with protecting America from the threat of terrorism. Headquartered in Washington, D.C., the Counter Terrorist Unit established field offices in several American cities. From its inception, CTU faced hostility and skepticism from other Federal law enforcement agencies. Despite bureaucratic resistance, within a few years CTU had become a major force in the war against terror. After the events of 9/11, a number of early CTU missions were declassified. The following is one of them.
##
## THE FOLLOWING TAKES PLACE BETWEEN THE HOURS OF 3 A.M AND 4 A.M. MOUNTAIN DAYLIGHT TIME
Red Notch, Colorado
The Zealot compound's front gate was chained, padlocked, and wrapped with the police's yellow-and-black POLICE: DO NOT ENTER tape.
The Toyota pickup truck that had driven Jack Bauer here from Denver stood idling about twenty feet from the gate, facing it with its headlights on. Jack said, "The police didn't leave any guards behind?"
Frank Neal said, "Why guard the henhouse when the chickens have already flown the coop?"
Jack countered one old saw with another. "Why not? They locked the barn door after the horses were stolen."
Neal made a sound that was half snort, half chuckle. Then he got serious. "The forensics teams have already come and gone. Local law enforcement's already shorthanded on account of Sky Mount and don't have any men to spare to station out here in hopes that Prewitt and his strayed lambs will come straggling home. That goes for us, too." By "us" he meant CTU/DENV, the outfit to which he was assigned.
Jack pointed out, "You're here."
Neal said, "Thanks to you." It was not an expression of gratitude. "You're the one who wanted to see the compound as soon as possible. Besides, I'm the in-house expert on Prewitt and his cult, if only by default. Lucky me. Who needs sleep?"
"Sorry to pull you out of bed at this hour. Thanks."
Neal made a dismissive gesture with his hand. "The boss says cooperate, so I'm cooperating." His boss was Orlando Garcia, head of CTU's divisional headquarters in Colorado covering the state's Rocky Mountain corridor. "I was up anyway. This whole business has got everybody working overtime. Sure stirred up a hornet's nest."
He reached into a side pocket of the red-and-black-checked jacket he was wearing, fishing out a key ring. "I've got the keys from Taggart, one of the state cops working the case." Neal cut a sidelong glance at Jack. "He thought it was a waste of time to go poking around here in the dead of night, but he's cooperating, too."
Jack said, "What do you think?"
Neal said, "I think I better go unlock the gate." He got out of the cab on the driver's side.
Jack had his own opinion on the subject. He was only recently arrived in the state and wanted to see the compound for himself at the first possible opportunity and this was it. He wanted to get a feel for the site and the terrain, and there was no substitute for firsthand knowledge. Call it the hunter's instinct, the need to physically experience the stalking ground.
Prewitt and his Zealot leadership cadre, an inner circle numbering more than two dozen men and women, had pulled their disappearing act at approximately this time last night. They hadn't gone alone. Two agents from the Bureau of Alcohol, Tobacco, and Firearms who'd been keeping the site under surveillance had vanished at the same time.
Either set of disappearances, that of the cultists or of the ATF agents, would have set off alarms among officialdom, especially in light of the heightened security attendant on the Sky Mount Round Table conference set to convene later on Friday, this day. Not far from here, more than two hundred of the richest and most powerful individuals in America, if not the world, were already gathering to meet in conclave from Friday through Sunday.
The vanishment of both the Zealots and the agents had thrown a shock wave into the array of law enforcement and national security personnel assigned to protect the three-day conference.
Jack climbed out of the passenger side, glad for the chance to stretch his legs after the ninety-minute-plus drive out from Denver to this site in the foothills of the eastern slope of the Rocky Mountains.
Here, at early Friday morning in the last week of July, the air was cool and crisp, a far cry from the thick, choking heat of Los Angeles that Jack had left behind no more than twelve hours ago. He was glad he'd thought to wear a light jacket for this late-night trip into the hills of Red Notch.
He was in his mid-thirties, sandy-haired, clean-shaven, athletic. He didn't feel athletic, though; he felt off his feed, with an oncoming headache knotting behind his eyes.
The compound was dark, except where the pickup's headlights shone through the gate, an eight-foot-tall construction of chicken wire stretched across a wooden beam framework. The secluded retreat was ringed by a chain-link, barbed-wire-topped fence. Prewitt was a man who jealously guarded his privacy and that of his followers.
Jack looked around. The area was dark in all directions, as far as he could see. The compound was the only sign of human habitation in these parts, save for a two-lane blacktop road at the foot of the rise. A winding dirt road linked the hilltop site to the paved roadway. The compound was presently uninhabited and the paved road was empty of all other vehicular traffic.
The Zealots' mysterious disappearance was part of the reason that Jack had been detached from his post as Special Agent in Charge of CTU/L.A. and assigned to temporary duty—TDY—here in the Rockies. Part of the reason, a big part, but not the only one.
Keys jingled as Frank Neal went to the gate, moving in front of the headlights, casting a long, weird shadow deep into the compound. Neal, a heavyset, fortyish African American, was an investigative agent assigned to CTU/DENV, the organization's regional field office in this sector.
CTU, the Counter Terrorist Unit, was a component of the Central Intelligence Agency. The Agency's founding charter back in 1947 had restricted its operations to foreign soil, prohibiting it from engaging in domestic activities in the United States. Cold War realities had ensured that that prohibition was honored more in the breach than the observance. The nightmare of 9/11 had finished off what remained of those antiquated laws, brushing them aside like cobwebs.
One result was the creation of CTU, an elite branch of CIA whose proactive mission was to detect, deter, and prevent foreign and domestic terrorists from striking within the borders of the United States.
The rugged natural grandeur of this remote highlands locale seemed an unlikely setting for a plot against the national security, but a combination of events had conspired to flag it with a Priority One alert that currently engaged the interests of local, state, and Federal law enforcement agencies, not least of which was CTU.
CTU/DENV had jurisdiction over the incident, and the Sky Mount conference, too, but an extra added element, a special spin on an already loopy curved ball, had sent Jack Bauer from his L.A. headquarters to liaise with the Mile High City team on their home turf.
Frank Neal stood sideways against the front gate, holding up the chain so that he could see the lock better in the headlights. Jack stood to one side of the beams by habit, to avoid being pinned by their glare and to minimize the loss of his night vision.
Neal fitted a key into the lock, opening it. He separated the prongs from the chain links they were holding together and cleared the chain from where it wrapped the upright gateposts, freeing them.
He opened the gate, exerting some muscle to tear open the POLICE: DO NOT ENTER tape. The thin plastic strands were tougher than they looked, and he had to dig in his heels and put a broad shoulder to the wooden gate frame, throwing his considerable weight into it to snap the tapes. He looped the chain around a crossbar where it would be out of the way and hung the padlock by its hook on one of the chicken wire loops. The yellow-and-black streamers had a forlorn look, like tattered ribbons left over in the aftermath of a party that had already been held and forgotten.
Neal pocketed the key ring and swung the gate open and out, clearing the way. It made a nerve-rattling groan evocative of the opening of a door to a haunted house in a horror movie.
He trudged back toward the truck and got behind the wheel. Jack went to the passenger side, pausing to look back the way they came. A dirt road switchbacked down a long, gentle slope, meeting at its base, at right angles, the blacktop road. No streetlamps on that road; no lamps anywhere, and no streets, either, except for that paved road.
The dirt road was lighter in color than its surroundings and stood out in the moonlight. The paved road was all hard and shiny, looking like a strip of black water in a long, thin canal. All else on the lower slope and beyond was boulders and rock formations, shot through with stands of scraggly pine and clumps of brush.
A big waxing moon hung halfway between the zenith and the midpoint of the western sky. It would have been hidden behind the mountains in most places along the eastern slope, but not here, not in Red Notch. The notch was just that, a gap between the mountains, though its upper levels were too steep and hazardous for it to serve as a pass for anything less nimble than a mountain goat.
The lowlands were less forbidding down near the base, where the compound was located. The dark hills bracketing it were mostly granite and basalt, but the notch was a different strata, an outcropping of softer, reddish-brown sandstone.
Here wind, water, and, above all, time, had done their work, eroding the sandstone into a jumble of fantastic shapes, forming needles, spires, arches, pinnacles, domes, and buttresses. It resembled a vest-pocket edition of the similar but infinitely more stupendous Garden of the Gods near Colorado Springs.
Beautiful in its way, but alien. Lonesome.
Jack hopped into the cab, where Neal was using the handheld microphone of the vehicle's scanner/communicator to report in to a temporary command post that CTU/DENV had established in nearby Pike's Ford to be closer to Sky Mount for the duration of the three-day conclave. The post's call sign was designated Central. He reported that his mobile unit was at Red Notch and about to enter the site. The dispatcher at Pike's Ford acknowledged the message.
Neal signed off, then drove the pickup through the gate and into the compound. It was set on a flat-topped rise, an oval whose long axis ran north-south. It was open on the east and rimmed on the other three sides by a gnarly, jagged border of weirdly angular rock formations that resembled a mouth of broken teeth. Barbed-wire fencing enclosed the entire area.
A handful of wooden frame and concrete block buildings, crude but functional, utilitarian, were grouped in an arc along the western edge of the oval. Their peaked roofs were steeply slanted, to resist heavy winter snows. A two-story structure with a veranda and second-floor balcony stood at the center. Long, shedlike buildings alternated with single-room cabins on either side of it, with a few shacks and blockhouses straggling off along the periphery.
The grounds consisted of mostly sandy soil speckled with patches of thin, dry, colorless grass. A maze of tire tracks and footprints crisscrossed the terrain. Neal drove diagonally toward a cubed blockhouse at the north end of the oval. He said, "The lab boys have already done their thing, going over the grounds, photographing tire tracks, taking moulage impressions and all the rest, so there's no worry about spoiling evidence. It's all been documented."
Jack said, "What evidence have they got?"
Neal said, "Too much—but of what? That's the question. Maybe you'll be able to supply the answer." The tone of his voice suggested that he believed the contrary. He went on, "All that's known is that sometime last night over two dozen human beings vanished from Red Notch, departing for points unknown. Since then, no one's seen hide nor hair of them."
"Or of the two ATF agents monitoring them."
"Or their patrol vehicle, either."
"They didn't get off any emergency calls or distress signals?"
"Nope. They radioed in at two A.M. according to schedule, reporting that everything was all quiet. That's the last time anything was heard from them. So whatever happened, happened after then."
After a pause, Neal added, "I knew both men, Dean and O'Hara. They were both pros, not the type to be caught napping."
Jack picked up on Neal's use of the past tense in referring to the missing pair. "Think they're dead?"
Neal shrugged massive shoulders. "If they could report in, they would. I can't see the Zealots taking them without a fight. To be honest, though, I can't see the Zealots taking them at all. They're a bunch of wimps, mostly. Armchair revolutionaries. The only thing violent about them is their rhetoric. Or at least that was the case till last night."
Jack said, "They have guns."
"Everybody in this part of the country has guns, it's part of the lifestyle. There were some weapons stockpiled at the compound, true. That's why the ATF was monitoring them. Ever since Waco, they like to keep an eye on cults with guns. But the weapons were ordinary shotguns, rifles, and handguns, legally bought and registered where registration was required. Prewitt's a stickler for that kind of detail; he wasn't going to leave himself open to an illegal firearms rap."
"What do you think happened last night?"
"Damned if I know. But something happened—something violent. There's some bullet holes and bloodstains around."
Jack said, "The background I got on Prewitt was that he and his group were nonviolent."
Neal said, "So it seemed. He took pains to put up a legitimate front. He took a fall on a tax evasion rap about ten years back and served eighteen months in a Federal pen. Since then he's been careful to be seen obeying the letter of the law."
"Until now."
"Cults like the Zealots are always basically unstable. That's because they're personality cults and the dominant personality is usually cracked."
"Was Prewitt cracked? He seemed more tightly wrapped than most cult leaders, at least publicly."
"Maybe he was wrapped so tightly that he just plain burst. Or maybe having two hundred of the richest folks in America gathered less than thirty miles away finally pushed him over the edge."
Jack said, "It never has before. The compound's been here during the last four or five annual Round Table meetings without incident."
Neal barked a laugh. "Hell, the reason Prewitt set up here was just so he could irritate the Sky Mount crowd. If he could have moved any closer to them than this, he would have, but the Round Table Trust has all the land locked up for miles around.
"Anyway, there's a first time for everything. Especially with the economy in the toilet the way it is now. Maybe Prewitt saw that as a sign his time has come."
Neal halted the pickup outside a concrete blockhouse with a steep-sided roof. He got out, taking a flashlight with him. Jack followed. The concrete cube had a solid, brown-painted metal door and narrow, horizontal slitted windows set high in the walls.
The door was locked. Neal said, "Can you hold this flash for a second?"
Jack said, "Sure," taking the flashlight and pointing the beam at the doorknob.
Neal held the key ring in the light, flipping through it before finding a likely looking candidate. He tried it on the keyhole in the doorknob but it wouldn't fit. After a couple more tries, he found a key that did, unlocked the door, and opened it.
A heavy gasoline smell came wafting out. Jack and Neal stepped away from the open doorway, letting the reeking fumes dissipate. The dark interior was dominated by heavy, hulking forms. Jack shone the flashlight beam inside, revealing a generator in the foreground whose base was bolted to the cement floor. Gasoline drums were stacked against a rear wall.
Neal said, "Gas-powered generator. We had to leave it locked up so nobody came back later to, er, liberate the fuel. Gas prices being what they are nowadays, even some of our local lawmen might be led into temptation."
Jack had nothing to say to that. He held the flashlight while Neal worked over the generator, trying to start it up. The machine sputtered, coughed, choked, spasmed, raced, shuddered like it was going to shake itself to pieces, and finally caught, turning over steadily. The racket was tremendous, sounding like the biggest leaf blower in all creation. The blockhouse filled with fumes, noxious blue-gray clouds that caused Jack and Neal to beat a hasty retreat outside.
Lights started coming on all over the compound. They winked on inside and outside the buildings, beginning as dim, fuzzy glowing patches and brightening as the generator continued supplying a steady source of power.
Some of the buildings had exterior-mounted floodlights that came on. Some had interior lights that came on, too. Other structures remained dark inside and out.
The lights did little to dispel the darkness that hung over the site. They were islands of brightness on a lake of black shadow.
The inside of the generator room remained dark. Neal reached inside the doorframe, groping around for the light switch. He flicked it on, a ceiling lamp filling the space with burnished brightness. The generator continued yammering noisily.
This was Neal's turf, and protocol dictated that he be the one to communicate directly with Central. He went to the pickup and spoke into the hand mic. "Central, this is Unit Three. Over."
Each mobile unit operating in the area had its own designation; Neal's was Unit Three. The comm system worked on a secure tight-beam band whose frequency was in constant automatic change to thwart electronic eavesdroppers. The reply came back: "Unit Three, this is Central. Over."
Neal said, "We have arrived at Red Notch and will be going temporarily out of service. I'll be switching to handset mode. Over."
"Roger that, Three. Over."
Neal signed off, going over and out. He switched off the dashboard-mounted communicator, its green "on" light fading to darkness. He switched on his portable handset, running a comm check with Central to make sure it was working properly. It checked out okay. Neal fitted the handset into a holster fastened to his belt.
Jack was equipped with a similar handset, which was tuned to Central's frequency. He ran a comm check on it, too, as a routine safeguard. It was functioning properly.
Neal killed the pickup's headlights, a zone of darkness springing into being where the twin beams had been. He turned off the engine, dropping the keys into his right front pants pocket.
Jack gave the scene a quick visual scan. Lighting the compound didn't help much. It added to the air of unreality, making it look like a stage set. Shadows were weird, elongated.
He found himself reaching under his jacket, adjusting the way his gun sat in the speed-rig shoulder harness under his left arm so that it settled the way he liked it.
Neal caught what he was doing and grinned. "Kind of gets to you, doesn't it?"
Jack said feelingly, "It looks like a prison camp on Mars."
Neal said, "Come on, I'll give you the grand tour."
Jack had been in his office in CTU's Los Angeles Domestic Unit headquarters twelve hours earlier on Thursday afternoon, meeting with Ryan Chappelle. Jack was the SAC, the Special Agent in Charge of the site. Chappelle was the Regional Division Director. That made him Jack's boss.
Jack regarded a meeting with Chappelle as something akin to having root canal work done. It was sure to be not only unpleasant but costly. The fact that Chappelle had come there to see Jack rather than summoning Jack to see him was a portent that Chappelle meant to hand him the dirty end of a stick. What remained to be seen was the size and shape of the problem he was about to dump in Jack's lap.
Chappelle began by saying, "What do you know about put options?"
Jack said, "Not much, except that it's the kind of tricky financial manipulation that casual investors like me would do well to steer clear of."
It was typical of Chappelle to come at him sideways, rather than just coming out and saying what it was he wanted. Jack sat back and decided to let Chappelle carry the conversational ball. It was an old interrogator's trick. You find out more when you let the subject tell the story in his own way, while at the same time committing yourself to nothing.
Chappelle looked mildly irked. "I didn't come down here to rope you into some stock deal. This is official CTU business and it could be an important lead."
Jack was all open-faced earnestness. "I know. You're a busy man who doesn't waste time on nonessentials. You wouldn't be here if it weren't important."
Chappelle nodded, accepting the other's remark at face value. He went back into his pitch. "I'll keep it simple. A put option is a financial instrument for short selling. Essentially, the investor is betting that certain stocks are about to experience a sudden drop in price. An extreme drop in price. By selling those stocks in advance of the drop, selling short as the term goes, the investor stands to make a steep profit—a killing.
"That's as long as the investor has guessed right, of course. In most cases, no guesswork is involved. The short seller is acting on the basis of inside information. Which is illegal, but easily gotten around by anybody who knows their stuff."
Jack nodded to show Chappelle he was with him so far. Chappelle continued, "The unit of fiscal analysts which I set up has detected a disturbing pattern of recent shorting on the market."
There was no denying that Chappelle was a wizard with numbers. Today's intelligence professionals tend to specialize in one of two areas: HUMINT and ELINT. HUMINT stands for human intelligence; that is, data collected by human sources. This is the side of the trade that concentrates on cultivating informants with access to data desired in a targeted area, including but not restricted to military personnel, government officials, scientists, technicians, diplomats and consular attachés, and members of other intelligence services—usually, but not always, citizens of other countries. Such informants may be motivated by altruism, greed, or extortionary pressures, depending on the individual. It is, in the words of former CIA counter-intelligence expert James J. Angleton, a "wilderness of mirrors," a shadowy world of spies and counterspies, defectors and double agents.
ELINT—electronic intelligence—is the other side of the coin. Here murky human ambiguities are replaced by hard data acquired by hardware. This is the arena of spy satellites, sonar and radar networks, signal traffic, telephone intercepts, and the myriad communications of cyber sphere and Internet. A vast array of electronic eavesdropping devices are deployed globally to monitor transactions in private and public sectors, vacuuming up mountains of data daily in all areas where information is a commodity. It is a realm of technicians, collectors, analysts, and data miners.
HUMINT and ELINT, the twin-chambered heart of the modern espionage apparatus. A built-in tension exists between practitioners of the two disciplines. The HUMINT crowd tends to view the other half as board operators and number crunchers, overly reliant on technology and tone-deaf to the human element. The ELINT crowd too often regard their counterparts as outdated relics of a bygone cloak and dagger age, trapped in a confusing labyrinth of deceitful and unreliable informants. Yet neither branch can operate successfully without the other.
Jack Bauer was an adept of HUMINT, a superb field operative who was equally skilled at the command level.
Ryan Chappelle was a disciple of ELINT, a technocrat supreme with a gift for selecting out significant data from signal noise, separating the wheat from the chaff. One human element he had not neglected, though, was the art of office politics and bureaucratic infighting. He'd risen far fast, and it was no secret that his goal was to win a berth on the seventh floor of CIA headquarters at Langley, the coveted precinct of the agency's intelligence mandarins.
The Fiduciary Special Investigative Unit was a pet project of Chappelle's. The SIU was a team of specialists who monitored the financial sphere, tracking the fluctuations of the global marketplace to detect patterns, profile motives, and forecast actions of private institutions and foreign governments. It had proved to be particularly useful in charting and deciphering the clandestine money movements and funding of both independent and state-sponsored terrorist groups.
Chappelle went on, "In the last few weeks, several million dollars' worth of put options have been bought in the marketplace. The stocks selected were all those of leading American companies and corporations. We're not talking about any failing, fly-by-night market dogs; these are all solid blue-chippers. Media conglomerates, software titans, genetic engineering, pharmaceuticals, even energy-related combines. Representative of the healthiest sector of the national economy—such as it is nowadays. These stocks have been bucking global recessionary trends by continuing to turn a profit."
Jack said, "But somebody is betting that they'll take a fall. Betting big."
Chappelle said, "Exactly."
"Who?"
"I don't know; not yet, that is. Our mystery shorter has taken great pains to disguise himself. He's covered his tracks by using a variety of shell companies, dummy corporations, and similar cutouts. It's like an onion. Peel back one layer and you find another, peel away that and there's another underneath. He's also been careful to spread out his operations in a variety of exchanges, foreign and domestic."
Chappelle's expression was like a clenched fist as he added, "But we'll get him. We'll peel back that onion to get to the heart of it, no matter how clever he thinks he is. It's just a matter of time."
Jack said, "Where do I come in?"
Chappelle's features relaxed, a crafty look coming into his eyes. "The catch is we may be running out of time. One pattern stands out: all of the stocks our mystery man is betting against are those of companies whose owners and CEOs are attending the Sky Mount Round Table."
The Round Table was a prestigious annual conclave of the movers and shakers of the U.S. economy. It was held each July in the luxurious and scenic splendor of the Sky Mount estate in the Colorado Rockies. Its invited guests were the elite of American business, members not of the Fortune 500 but of the Fortune 50. They occupied the apex of the national socioeconomic pyramid. It was a domestic counterpart to the periodic Bilderberger meetings of Europe's corporate masters.
Jack's bailiwick was the Los Angeles area, but he and his outfit had been much concerned lately with the imminent Sky Mount gathering. CTU/L.A. had increased its surveillance of local hate groups, militant foreign and domestic anti-American organizations, and the far wider pool of their sympathizers, fellow travelers, and enablers, monitoring them for any credible evidence of a plot aimed at this year's Round Table. Two hundred leading lights of big business gathered in one spot at the same time presented an attractive target to the nation's enemies. Or to any crackpot or group of crackpots who happened to hate rich people and wanted to strike a blow at the corporate empire.
No such plot had been detected by CTU/L.A. Which didn't mean that none existed. Such a conspiracy might have evaded their notice, or could be hatching somewhere in a different jurisdiction. Chappelle's information certainly put a new and sinister slant to the possibility.
Jack said, "The person or persons betting those stocks will take a sudden and dramatic fall may not be gambling at all. It could be a sure thing. A terror strike or other catastrophic event at Sky Mount could send those stocks tumbling. It could wreck the national economy. What's left of it, that is."
Chappelle said, "A logical conclusion. There was a lot of short selling of airline and insurance stocks in the days before 9/11. Somebody knew in advance that those stocks were going to take a big hit and reaped several billions of dollars due to that inside information. We've never been able to pinpoint the profiteer but there's no doubt about the pattern."
He'd been pacing back and forth in front of Jack's desk. He stopped abruptly, turning to face Jack and point a finger at him. "Now couple that with those Colorado cultists suddenly dropping off the board."
Reports about the disappearance of Abelson Prewitt and two dozen of his most fanatical followers from the Red Notch compound had already come to Jack's attention as part of his daily intelligence summary.
Jack said, "I don't doubt that Prewitt would like to bring a mountain crashing down on the Round Table's guests, but I can't see him playing the market to make a profit off it. That goes against his whole crackbrained ideology, what little I understand of it. His theories are a bit too opaque for me."
Chappelle made a hand gesture like he was shooing away gnats. "Maybe he decided that if you can't beat them, join them. He may be cracked but he knows the financial system inside out. Remember, he used to be an economics professor before he went all political."
"Where would he get a couple of million dollars to invest?"
"Good question. Maybe he's found a sponsor; a hostile foreign power, say. Interests inimical to the United States could be backing him. Using him for a cat's-paw to do their dirty work while they turn a big fat profit at the same time."
"Possibly."
"And I know just the man to find out the answer, too, Jack."
"Who?"
"You."
Jack had seen that one coming but there wasn't much he could do about it. He'd try, though. He said, "That's CTU/DENV's turf."
Chappelle said, "The SIU's findings about the short selling gives us an in."
"I don't think Lando Garcia's going to want us horning in." Garcia headed CTU/DENV.
Chappelle said, "You let me worry about Garcia."
Jack said, "I can't just drop everything here and take off for Colorado—"
"Sure you can. Nina Myers can hold down the fort while you're gone." Nina Myers was the assistant SAC of CTU/L.A. and Jack's chief of staff, fully qualified to take over in Jack's absence.
Chappelle said, "Things are relatively quiet here. Sky Mount is where the action is. You're a top field man, Jack. Garcia hasn't got anybody in your class."
"He's got some good people out there—"
"Not like you. You could make a difference. This is important, Jack. Big. I'm surprised you're not jumping at the chance."
"I've got a full plate here, Ryan. I don't like to leave in the middle of things."
"Nothing that won't keep for a few days. And that's all it'll take, a few days. The Round Table ends on Sunday, and by Monday the guests will all have gone their separate ways."
There was no way out. Jack had to say yes. Chappelle was his superior officer in the chain of command; he could order Jack to take the assignment. It didn't matter that Jack was in charge of CTU/L.A. with its mountainous workload, awesome responsibilities, and important ongoing projects and investigations. It didn't matter that Jack had recently ended a long and painful separation from his wife, Teri, and had moved back home with her and his teenage daughter, Kimberly, a delicate situation that was an emotional minefield of raw sensitivities, resentments, and bruised feelings.
It didn't matter that Garcia and his whole CTU/DENV outfit would see Jack as Chappelle's creature, giving the notoriously ambitious Regional Division Director an opportunity to extend his authority by injecting himself into their operations.
It was a classic Chappelle ploy. The SIU's discovery of the money manipulations gave him the opening wedge he needed to put Jack on temporary duty and strap him on Garcia's CTU/DENV as a consultant. If Jack turned up something at Sky Mount, Chappelle could claim a share of the credit. If things went sour, he could wash his hands of all responsibility and hang it on Jack. And if nothing happened and it all worked out a draw, Chappelle would still have the pleasure of having intruded on the turf of his longtime rival Orlando Garcia.
The hell of it was that Chappelle might just be on to something with the discovery of the suspicious stock manipulations being a warning sign of an anti–Round Table plot. But he couldn't just pass the information along to CTU/DENV for them to handle in their own way. No, he had to use it as a way to get his foot into Garcia's door, like a pushy salesman who won't take no for an answer. He certainly wouldn't take Jack's no for an answer.
Jack accepted the inevitable, stifling the sigh that sought to escape him and keeping a poker face. "When do I leave?"
Chappelle said, "Immediately." He rubbed his palms together, a gesture somehow suggestive of a fly anticipating a choice morsel. "I'm counting on you, Jack. They need you out there. I know you'll make Garcia and his crew look sick."
Jack smiled wanly. Chappelle said, "And stay in close contact with me here. Keep me posted on all developments at all times."
Chappelle had gotten what he wanted. The interview was over.
That was the prelude. Jack was now in the center of things, probing the Red Notch compound. The buildings were grouped close together, within walking distance. Neal indicated the central structure, a white, wooden frame two-story building. He said, "That's the admin building, Prewitt's headquarters."
They crossed toward it. It fronted east, its long axis running north-south. A television satellite receiving dish was mounted on the roof, pointed at a forty-five-degree angle at the sky. Floodlights were mounted at the tops of the building's southeast and northeast corner posts where they met the front ends of the second-floor balcony. The lights on the northeast post were dark. Lights were on inside the first and second floors, shining through the windows.
The northeast corner post floodlight hung at a twisted angle. Broken glass littered the ground below it. Jack stood under it, looking up. Neal said, "It was shot out. The lab crew recovered one of the bullets. It's from a handgun but they haven't typed it yet."
Jack nodded.
Four wide wooden stairs led to a porch, a veranda that fronted the building on three sides, all but the west side. The second-floor balcony was similarly constructed. It was as if the builders had shunned the rear of the building, its west face. Behind the back of the building, behind the entire cluster, rose a jumble of sandstone formations, pillars and needles and boulders, all eroded into angular, distorted shapes.
Jack and Neal climbed the stairs to the porch. The front door hung at a tilted angle, half torn off its hinges. The second-floor balcony roofed over the veranda.
A row of tall windows were set in the walls on either side of the doorway. A window to the right of the doorway was broken, leaving a mostly empty frame. Shards of broken glass were strewn on the porch below it. Dark reddish-brown stains, long-dried, mottled the outside of the windowsill and the wall beneath it. The porch planks under the window were stained, too. The stains were pretty big, the largest being bathmat-sized. White chalk markings had been drawn around the stains by the crime lab team. Each marking was tagged with an identifying letter-number combination written in chalk.
Jack said, "Looks like it was broken from the inside out. Like somebody jumped or was thrown through the window. Whoever it was must've been cut up pretty badly. Bad enough to have bled to death, if all that blood came from one person." He looked at Neal. "No bodies were found?"
Neal said, "None."
Jack took a closer look, while avoiding stepping on the chalked-off bloodstains. The stains went to the front edge of the porch. What looked like bloody handprints showed on the top rail of the waist-high balustrade bordering the edge of the porch.
He switched on his flashlight, shining it down on the ground below the porch. Bloodstains extended out in dribs and drabs for a dozen yards or so before coming to an abrupt halt. He said, "The injured party managed to get that far before dropping. Then what happened?"
Neal said, "Your guess."
Jack edged north along the porch, carefully picking his way through, trying to avoid treading on the broken glass and bloodstains. Neal followed, saying, "The lab crew's already photographed and diagrammed everything, so you don't have to worry about messing up anything or altering the scene." Jack noticed that Neal, too, despite his words made an effort, conscious or not, to avoid stepping on glass or bloodstains.
Jack turned left at the corner, following the north branch of the veranda to its end at the building's northwest corner. He shone the flashlight around, noticing a cluster of propane gas tanks connected by pipes to the building's rear.
An open space about twenty yards wide stretched from the backs of all the buildings in an arc to the foot of the jumbled sandstone formations. It was bordered by an eight-foot-tall chain-link fence topped with three strands of barbed wire. The fence extended in both directions, north and south, enclosing the western edge of the oval before curving eastward on both sides to complete the encirclement of the rest of the space.
Jack said, "The compound is completely fenced in?"
Neal, at his shoulder, said, "Yes."
Jack gestured with the flashlight so its beam played across the jagged rock rim beyond the fence. "Any roads back there?"
"A couple of game trails, maybe. Nothing you could get a vehicle through, not even a dirt bike."
"So the front gate's the only way in or out?"
"I suppose there's places along the fence line that could be hopped, if you were determined and athletic enough. But whoever did it would be walking, not riding. And they'd be in for a hell of a hike. Why?"
"No particular reason, just trying to get the lay of the land."
Jack turned, starting back the way he came, Neal falling into step behind him. Neal said, "The Zealots didn't troop out of here on foot, if that's what you're thinking. We know how they left."
Jack said, "How?"
"They've got an old school bus that they use to get around in. They're always driving in a group to the county seat or down into Denver or wherever to hold protest demonstrations or stage media events. Prewitt's big on that. A natural-born pest. The bus is painted blue, kind of a trademark so people'll know they're coming. They keep it in a garage up here and it's not there now, so we figure that's how they left the scene."
"A blue bus, eh? Sounds like it'd be hard to hide."
Neal said grimly, "You'd think so."
They went to the front of the building and went inside into a long, narrow hall. It ran straight through the building from front to back. There were four rooms on the first floor, two on either side of the hall. A staircase led up to the second floor.
The front room on the right was a kind of communications center. That was the room where the window was broken from the inside out. Not much seemed disturbed, apart from that. A floor lamp was knocked over and lay on its side. A mass of moths flew in circles under an overhead light. A couple of workstations were placed around the space, complete with computers, phone banks, printers, fax machines, and the like. A Styrofoam cup of coffee stood on one of the desks. It gave the impression that the desk's occupant had just stepped away for a minute, except that the cream in the coffee had curdled and the cup's contents were a gray-brown sludge.
Each workstation featured a hardcover book in a prominent position. They were all the same book. Jack picked one up and examined it. The title was: Whip Them with Scorpions. It was subtitled, Driving the Money-Changers from the Temple.
It was a very thick book, a real doorstopper, with lots of fine print and charts and graphs but no pictures.
Its author was Abelson Prewitt. The back cover displayed a black-and-white photo of Prewitt. A big, double-domed cranium topped a long, bony face. A few thin strands of black hair were plastered across his oversized skull. Dark, intent eyes glared behind thick-lensed black glasses. Thin lips were tightly compressed.
Neal said dryly, "His magnum opus. Ever read it?"
Jack said, "I'm waiting for the movie. You?"
"Part of it. I got farther through it than anybody else in my outfit. That's what makes me the expert."
"What's it like?"
"Let me put it this way: if you think people have some cracked ideas about sex—which they do—that's nothing compared to some of the crazy notions out there about economics."
"That bad, huh?"
"Prewitt's like most philosophers cracked or sane, if there are any of the latter. He comes up with a theory that he claims explains why everything works the way it does better than the theories of all the other thinkers who've done the same thing. They're all just chasing their tails around, as far as I'm concerned."
Jack said, "That makes you a philosopher, too." The book was heavy. Jack set it down. He said, "I take it there's no love lost between Prewitt and the Round Table?"
Neal said, "You can take that to the bank. No, better not. Prewitt's not too happy with the banking system, either."
The left front room was a kind of day room. There was a fireplace, a sofa, and a couple of armchairs. The mantle over the fireplace was lined with books, every one of them a copy of Whip Them with Scorpions. A card table with three opened folding chairs grouped around it stood in a corner. The fourth chair stuck half in and half out of the frame of a big-screen TV it had been pitched through.
The left back room was a storeroom for the cult's publications. One wall had a floor-to-ceiling bookshelf devoted solely to copies of Prewitt's masterwork. Tables were stacked with copies of Zealot newsletters and pamphlets. The subjects reflected such topics as the iniquity of the Federal Reserve banking system, the necessity of returning to the gold standard, the Wall Street/Washington, D.C. plot to repeal the Constitution and turn the United States into a slave state, and various world-historical conspiracies by the Illuminati and Freemasonry to rule the world.
The right back room was a combination kitchen and dining room area.
The second floor was reserved for a double set of private living quarters. Neal said, "It's split in two. One half is Prewitt's, the other belongs to Ingrid Thaler, his second-in-command."
Prewitt's rooms were stark, spare, ascetic, almost monklike in their spartan simplicity. There was a copy of his book on the bedside night table, another on his writing desk and a third on a small bookshelf that also contained copies of Plato's Republic, the Bible, a complete edition of Shakespeare's works, a dictionary and thesaurus.
Ingrid Thaler's suite was tastefully decorated, handsomely appointed, and expensively furnished. She certainly didn't lack for any of the creature comforts. One room was filled with nothing but her clothes, shoes, and accessories. The suite was bare of any copies of Prewitt's book. Her boudoir—you couldn't call it a bedroom, it was too luxurious for that—featured a photograph in a silver frame prominently displayed on top of a dresser cabinet. It depicted a glamorous woman in her forties with an upswept blond hairstyle, cool level eyes, a sensuous mouth, and a lot of strong jaw and determined chin.
Jack said, "Is that her?"
Neal nodded. "That's Madame Thaler."
"She doesn't like herself too much, keeping a framed photo of herself on the bureau. Is she Prewitt's mistress?"
Neal shrugged. "Nobody knows. That's one of the cult's best-kept secrets. But it's no secret that she's his lieutenant, his enforcer, chief executive officer, number two in the hierarchy. She sees that the great man's word becomes law among his followers. The compound is for the Zealots' leadership cadre, their inner circle. The outer circle, the rank and file, live in their own private homes. There's hundreds of them, a large part of whom live in this state."
Jack said, "So there's a good potential depth of backfield on short notice if Prewitt needs to call them up."
"Yes."
"Great."
Nothing in the upstairs living quarters showed any signs of violence, chaos, or disorder. It was as if their owners had just stepped out for a minute.
Jack and Neal went downstairs and outside, standing on the front porch. Neal got out a pack of cigarettes. "Smoke?"
Jack said, "No, thanks."
Neal shook loose a cigarette and lit up. He said after a pause, "Anything about the scene speak to you?"
Jack shook his head. "Nothing yet. Weird. Maybe they had some kind of palace coup or something."
Neal said, "That's as good a theory as any. But where did they all go? And why?"
The admin building was bracketed by a pair of identical one-story, cabin-style structures. Neal indicated first the one on the south, then the one to the north. "That's the men's barracks, the other's the women's barracks. Prewitt doesn't just have some funny ideas about economics, he's got some about sex, too."
Jack said, "Do tell."
"To be in the leadership cadre at Red Notch, you've got to dissolve all previous relationships. A married couple can't stay married, at least not to each other. The men bunk in one barracks, the women in another. From time to time Prewitt, or more precisely Ingrid, pairs up a couple to, as they put up, gratify their natural physical urges. There's a couple of smaller bungalow units for their conjugal visitations. That's usually a reward for some meritorious service to the cult. They keep juggling the partners around to keep any permanent relationships from forming. The only permanent relationship that's allowed is to the cause of Zealotry."
"No children in the compound?"
"No, they're too smart for that. With all that partner-swapping going on, they won't risk any charges like child endangerment or contributing to the delinquency of a minor. Prewitt's done time on a tax rap once; since then, he's been damned careful not to give the law any cause to go poking around in the compound."
Jack said, "Until now, on the eve of the Round Table."
Neal said, "Even then, there's not much to go on. The Zealots are free to come and go as they please, there's no law against that. The broken glass and bloodstains and bullet holes are suspicious, but a good lawyer could probably explain them away. Especially with the Zealots cooperating in the cover-up. They could always claim a party got out of hand or somebody had a shooting accident or something. Luckily the Sky Mount conference gives us a loophole to go poking around on national security grounds."
Neal finished his cigarette. He stubbed it out and put the butt in his jacket pocket, explaining, "Force of habit. Even though the criminalistics crew has been over the site, I can't bring myself to litter the scene."
Jack said, "You mentioned bullet holes. You mean the light shot out here?"
Neal said, "There's some more at the men's barracks. Looks like they had a pretty good brawl over there. Want to take a look?"
"Yeah."
Jack and Neal went down the stairs to ground level and started toward the barracks. Jack glanced back over his shoulder at the admin building, all ablaze with electric lights. He said, "I feel funny about not shutting off the lights."
Neal said, "When we're done here I'll shut off the generator. That'll kill all the lights. Personally I'm happy with as much light as we can get."
"You've got a point there."
The men's barracks was a large single room shoebox-shaped cabin with whitewashed wooden walls and a peaked shingled roof. It fronted east, like all the buildings in the compound, as if they were deliberately turning their backs on the alien otherworldliness of the sandstone piles. Its long axis was east-west, so that its short side faced front.
The upper half of the front door consisted of four framed glass panes; they were all broken. A horizontal line of a half-dozen bullet holes pierced the wall to the left of the door below the window cell. On the right side, one of the windowpanes displayed a bullet hole with a corona of spidery cracks. Jack said, "Some party."
He and Neal went inside. An open central aisle was flanked on either side by rows of double-decker bunk beds set at right angles to the center space. Gray metal wardrobe cabinets like gym lockers stood against the far walls between the bunk beds. The floor was uncarpeted wood.
There were bloodstains on the floor and some of the mattresses. A couple of the bunk beds were askew and some of the walls were pocked with bullet holes. Neal said, "Yeah, some party."
He and Jack went back outside. Some bats flitted out from under the eaves, whirling and pinwheeling aloft in seemingly random, zigzag patterns. Jack's hand was under his coat, touching the butt of the gun holstered under his arm. Neal saw it and grinned. Jack grinned, too, a bit sheepishly, bringing his hand out empty and letting it fall to his side.
Neal pointed out the next buildings south of the barracks. "The near one's the mess hall and the barn next to it is the garage where they kept the blue bus."
They started toward the mess hall. Neal said, "Here's where the Zealots get their chow. Not Prewitt and Ingrid, though. They've got a private chef to rustle up their meals in the kitchen in the admin building."
Jack said, "Rank has its privileges."
The mess hall was a shedlike structure whose long side fronted east. Jack and Neal were closing on it when rattling sounded in back of the building.
The two men froze. Silence reigned for a few heartbeats, only to be broken by a soft metal clangor. Jack whispered, "Don't tell me that was bats."
Furtive rustling and rattling sounds came from behind the mess hall. Jack's semi-automatic pistol, a 9mm Beretta, was in his hand; he didn't even remember drawing it. Neal's gun was drawn, too, a .357 magnum revolver with a shiny metal finish. He said, low-voiced, "It could be a bear."
Jack's face must have reflected his skepticism. Neal said, "No kidding, the mountains are overrun with black and brown bears. Hunts have been curtailed for years because of environmental politics, and the bear population is out of control. Familiarity breeds contempt, and they're not afraid of men."
Jack said, "Let's find out. I'll go the long way around the mess hall, so give me a minute to get in place. If it's a man, we'll take him."
Neal said, "I hope it's a man. If it's a bear, for God's sake don't shoot unless you absolutely have to, if he's charging you. Fire some warning shots, maybe that'll scare him off. Believe me, with all the red tape and paperwork involved, it's less hassle to shoot a man than a bear."
Jack grunted, an acknowledgment that he'd heard what the other had said but that committed him to nothing. He peeled off from Neal, light-footing it at quick time south along the front of the mess hall, down to the southeast corner. Neal rounded the northeast corner, vanishing from sight.
Jack edged along the short south face of the building, keeping close to the wall and crouching low to avoid the oblongs of yellow light shining out through the mess hall windows. More rattling sounded from behind the back of the building.
Maybe it was a bear. Jack's Beretta was armed with cartridges that were made up with a hot hand-loaded powder mix he had on special order. Each round was a potent man-stopper. Would it have the same effect on a charging bear? He'd hate to have to find out. He had no relish for reporting such an encounter to Ryan Chappelle.
Jack halted at the southwest corner of the mess hall, back flat against the wall. He peeked around the corner.
The back of the mess hall wasn't as well-lit as the front. There were fewer windows to let the light shine through. The scarcity of electric light was compensated for by the moonlight. A concrete loading platform jutted out at the midpoint of the building's rear. A Dumpster and a clump of garbage cans stood nearby. A stooped, shaggy figure stood swaying upright among the garbage cans, rummaging around inside them.
Neal stepped out from behind the building's north face into view, holding his gun leveled at the indistinct shape that stood reeling on two legs.
Jack stepped into the open in the moonlight so Neal could see he was in position. Neal shouted, "Freeze!"
The shape started, knocking over some garbage cans, stumbling over them, raising a racket as it tried to get clear of them. It fell, crawling on all fours. Jack and Neal closed in from both ends.
The figure scrambled upright and started to run. It wasn't a bear, it was a man. A shaggy man. Jack and Neal moved to intercept him.
The shaggy man started across the open space toward the fence. Jack double-timed at a tangent to cross his path. The shaggy man's hands were empty. If he had a weapon he hadn't drawn it.
He was big, even running stooped forward as he was, big and thick-bodied. Jack neared him. The other looked like the last of the mountain men, with dark shoulder-length hair and a full beard. He was clumsy, unsteady on his feet.
Jack plowed into him sideways, slamming his right shoulder, upper arm, and elbow into the shaggy man's left side, knocking him off balance. The shaggy man fell sprawling into the dirt, crying out in terror.
He was still in the game. He rolled and got his legs under him, standing on his knees. His hand darted to his right side, drawing a knife worn there in a belt sheath. A hunting knife with a wickedly curved and gleaming eighteen-inch blade.
Jack's foot lashed out in a front snap kick to the shaggy man's wrist, sending the knife flying from his hand.
Neal came up behind him and laid his gun barrel behind the back of the shaggy man's ear, rapping his skull hard enough to stun him but not so hard as to knock him out. The shaggy man fell forward face-first into the dirt.
Neal's mouth was open, he was breathing hard. Jack said, "Damned funny bears you grow out here!"
Neal said, "That's no bear and no Zealot, either. Who in the hell is he?"
"Let's find out."
##
## THE FOLLOWING TAKES PLACE BETWEEN THE HOURS OF 4 A.M. AND 5 A.M. MOUNTAIN DAYLIGHT TIME
Red Notch, Colorado
The shaggy man wore a flannel shirt, overalls, and work boots. He lay facedown in the dirt. Neal stood on one knee beside him, holding the muzzle of the .357 against the back of his skull. He said, "Keep still."
The other grunted something that could have been an affirmative. He remained motionless while Neal's free hand gave him a pat-down frisk, searching him for weapons, finding none.
Jack's gun hand hung along his side. He held the knife that he'd picked up in his free hand. The ball of his foot still throbbed from where he'd delivered the front snap kick to disarm the shaggy man. The knife had deer antler plates inset in the grip, a hilt to keep the hand from slipping, and a long, sharp-pointed blade. He held it up to the moonlight, turning it so that moonbeams glimmered off the steel.
Neal rose, saying, "He's clean. Of weapons, that is. He smells like he hasn't had a bath in a long time. No wallet, keys, or identification of any kind." He nudged the shaggy man in the ribs with the toe of his shoe, none too gently. "Get up. And no tricks. Try anything funny and I'll shoot you in the knee."
He said to Jack, "I don't like guys with knives."
Jack said, "I don't blame you. That's some knife, too. A real pigsticker."
Neal's shoe toe prodded the shaggy man's ribs again, harder. "Come on, get up."
The shaggy man got on his hands and knees, shaking his head to clear it. Neal had him covered with the .357, so Jack holstered his pistol, fitting it into the shoulder sling. He still held the knife.
The shaggy man groaned, rubbing the back of his head where Neal had clipped him with the gun barrel. He rose unsteadily to his feet, swaying. Neal came up alongside him and put the arm on him cop-style, using his free hand, the one not holding his gun, to grip the other firmly just above the elbow, steadying and steering him.
Jack flanked the shaggy man's left side but otherwise let Neal handle the play. He was a visitor, a guest, while this was Neal's home territory. Let Neal have the credit, if any, for bagging a suspect, if the shaggy man should prove to be one. Neal was right about one thing, though; the man was no Zealot. The Zealots' dress code ran to jackets and ties for the men, obedient to their guru Prewitt's admonition that they should always be mindful of making a positive appearance of neatness and cleanliness on the public at large. The shaggy man looked like a tramp, a hobo.
Neal said, "Come into the light and let's see what we've got." He and Jack hustled the shaggy man around the corner of the mess hall, across the north face, and around to the front of the mess hall. The captive lurched forward with a shambling, shuffling gait.
A wooden platform something like a plank sidewalk made a kind of apron along the east face, serving as a kind of unroofed front porch. A lamp mounted over the front door shed a yellow cone of light.
Neal sat the shaggy man down on the platform under the light. The shaggy man rested his elbows on his knees and buried his face in his hands. His long hair fell over the front of his face like a curtain. It was the color and texture of a steel wool scouring pad.
Jack eyed the hunting knife under the light. Its finish was dulled with dark patches, not of dried blood but of rust. He pointed it out to Neal, quietly and out of the hearing of the shaggy man.
Neal nodded. Then he went to work on the captive. "Look up when I'm talking to you."
The shaggy man lifted his head up out of his hands. His raggedy iron-gray beard reached down to his collarbone. It had the texture of a bird's nest. He didn't show much skin between the hair on his head and his face, and what did show was seamed, weathered. Bloodshot watery gray eyes were tucked into baggy pouches between a wide, flat-bridged nose. A threadbare flannel shirt was so grimy that its original red-and-black-checked pattern could barely be made out. A pair of denim bib overalls hung in place by a single shoulder strap; the other was broken.
Neal said, "Who are you? What's your name?"
The shaggy man said, "Lobo..."
"Lobo? What kind of a name is that? Lobo what? What Lobo?"
"Just—Lobo. That's what everybody calls me. That's been my name for as long as I can remember." A confused look came over his face. "Which ain't all that long..."
Lobo rubbed his face with his hands like he was scrubbing it, trying to rub some feeling into it. Fear stamped his features. He looked up at Jack, Neal, said, "Don't! Don't kill me!"
Jack and Neal exchanged glances. Neal said, "What's all this talk of killing? Nobody mentioned killing but you."
Jack chimed in, "You're the one with the knife."
Lobo said, "I was scared, I didn't know what I was doing!" He rubbed and chafed his right wrist and forearm. "You like to busted my wrist when you kicked it, mister. It hurts awful bad."
Neal was unsympathetic. "You're lucky you didn't get shot pulling a stunt like that. Anyway, the hand's still working. I don't see that anything's broken."
Jack said, "You pulled that knife quick enough."
Lobo said, "To defend myself. I thought you were some of Them." The way he said "Them," you could practically hear it being capitalized.
Jack said, "Them?"
"The devil men!"
Neal scoffed, "That crazy talk won't buy you anything. You're sane enough, so talk sense. And make it quick."
A shift came over Lobo's features, firming them with stubbornness. He looked down, not looking Jack or Neal in the face. He muttered, "I know what I saw..."
Jack said, "What did you see?"
Lobo looked up now, staring Jack in the face, studying him. He blinked repeatedly, his watery eyes glimmering. He came to a decision. "Nope. You ain't one of Them."
Jack pressed, "One of who?"
Lobo stared Neal in the face, coming to a quick conclusion. "And I know you ain't one of Them. You got a mean face, but not as mean as they got."
Neal said, "Who's Them? Damn it, man, speak out plain!"
Lobo said, "Them devil men." Tension fled from his face, his expression sliding into slack-jawed relief. "Huh! Maybe you ain't going to kill me after all?"
Jack said, "We're not killers."
Lobo pointed out, "You got guns."
"To defend ourselves. Like you with your knife. You're not a killer. You just wanted to protect yourself. Against the devil men."
Lobo grinned, bobbing his head in agreement. "That's right! Now you got it. So can I have my knife back?"
Neal said, "Don't get ahead of yourself."
Jack said, "You don't need a knife, Lobo. We'll protect you against the devil men."
"So you say. But it's easier said than done. They got Satan's power working for them."
"Remember the Psalm, Lobo: 'I will fear no evil.'"
"You would if you seen what I saw. That's why they want to kill me."
"What did you see?"
Lobo shook his head, sadly. "Can't tell."
"Why not?"
"If I do, they'll have to kill you, too. Nobody is safe who knows the truth."
Neal, fighting down impatience, said, "We'll take our chances."
Jack said, "They'll want to kill us anyway for siding with you, Lobo. So you might as well tell us. The more we know, the better we'll be able to help."
Lobo tilted his head to the side, as if listening to unheard voices. "You may just have something there...'Course, bad as they are, the devil men ain't the worst. Oh no." He leaned forward, with an air of one about to impart some great truth. "It's those hog-faced demons you really got to worry about!"
Neal, dangerously calm and soft-spoken, said, "Hog-faced demons, is that right?"
Lobo nodded vigorously. "The gospel truth. The devil men, they look just like us. Like anybody, only more mean-faced. That's how they can walk among us. Two of Them have been dogging me all day, back in the hills. That's where I live, all by myself, in a little hidey-hole I got fixed back up there." He gestured toward the sandstone formations. "Ain't nobody can find me in the rocks if'n' I don't want 'em to, devil men included.
"But I got hungry. I ain't had nothing to eat for two days. There's a hole under the fence that I slip right through sometimes at night. I sneak up in back of the kitchen here and raid them Dumpsters for what I can find. Lawd! The food that these here camp folk throw away would feed an army! Perfectly good food, meat, taters, bread, vegetables, sometimes even cake!" He smacked his lips at the conclusion of the recital.
Jack prompted, "Camp folk? You mean the folks living here in the compound?"
Lobo nodded. "The very same. It's like a church camp, an old-time revival meeting, the way they're always getting together and listening to that ol' preacher of theirs. He'd come on the loudspeaker and jaw to 'em for hours at a time and they'd just be a-setting there in the campground, taking it all in. Hoo-whee, how that man could talk! 'Course, it was way over my head, I couldn't make no never mind of it. But they seemed like decent enough folks, what I seen of them."
His face fell, becoming despondent. "Not that it did 'em any good in the end, though, not when those hog-faced demons came to drag 'em all off to hell last night."
Jack said, "It happened last night, you say?"
"Yes, sir! As I live and breathe. It was only by the purest luck they didn't get me, too! I come sneaking around when the moon was low, like I always do when I plan to do me some Dumpster diving. I was up in the rocks when I seen it, a green fog coming out of nowhere and covering the whole camp."
"Green fog?"
"Green as pea soup, sonny! Damnedest thing I ever seen. Right off I knew it wasn't natural, wasn't nothing that comes from God's good earth. It rose up out of the east and in less time than it takes to tell, it grew into a great big green cloud that rolled right over the whole danged camp and just set on it. Like to froze me in my tracks at the sight of it! I stayed up in the hills to marvel at it and a good thing, too. Else it would have got me, along with the rest of them poor souls."
"The camp folks, the people in the compound."
"None other. They was all sleeping, I reckon, tucked up tight in their bunks when the green cloud come up on 'em. Like a thief in the night, just like the Good Book says. That's when them hog-faced demons showed. Lawd, they must've been vomited up straight out of the gates of hell! You never seen nothing like it, nobody ever did—and pray that you never do!
"Hog faces they had, big ol' long snouts sticking out and big bug eyes a-goggling and staring! Hog faces and the bodies of men! And here's something for you to think on: they didn't come a-riding Satan's lizards or flying in on bat wings, no sir. They drove up in cars! Cars, mind you, just like normal everyday folks out for a moonlight drive! Now, don't that beat all?
"Then all hell broke loose. Them hog-faced demons fell on those poor folks like badgers on a warren of baby rabbits. It was something awful. They just waltzed right into their cabins and houses and carried 'em away. There was screaming and shouting and shooting and all kinds of unholy racket going on. I'm surprised they didn't hear it over to the next county."
Neal said, "You saw all this?"
Lobo's expression was patronizing, almost pitying. "I'm telling it, ain't I? The green cloud covered most of it at first, which was a mercy, but it thinned out and broke up pretty quick, so I seen most of it. The worst of it, sure, when the demons dragged those poor souls out into the open, herding them like cattle to the slaughterhouse. Some they killed straight off, gunned 'em down—seems funny, don't it, Satan's minions using firearms to do the Devil's work here on earth? Guns and cars? Makes sense when you think about it, though. Who better than Lucifer to make use of the modern ways of destruction? No back number him, he's up-to-date!
"The demons loaded every last one of 'em, man and woman, onto that blue bus. A blue bus! Them hog-faces got back in their cars and the blue bus and they all drove off straight to hell! Not before they almost got me, though. Like I said, that green cloud lifted mighty quick and I got antsy to see what was happening. I got a mite careless and showed myself up on the rocks and one of them hog-faces sees me and starts taking potshots at me! Came mighty close, too, but my guardian angel must've been working overtime and the demon missed.
"After that I faded back into the hills and made myself scarce. 'Course, Satan don't give up that easy. That's why he sent two devil men into the hills today to ferret me out. I reckon those hog-faces can't walk around in the daylight. The devil men, though, they look like anybody else, only meaner than most, like I said. That pair was a couple of two-legged rattlesnakes in tandem. Not that it done 'em any good. Trying to track me down in my own hills! Shoot, I seen 'em coming from a country mile away. They were combing the rocks all day until they got tired and went away."
Nobody spoke for a moment. Neal broke the silence at last, saying, "That's some story."
Lobo said, "Every word of it is true. The proof's in the pudding. Look around you. Where'd everybody go? If I'm lying, where'd they all git gone to? Tell me that!"
He looked Neal in the face, then Jack. "No answer, huh? I didn't think so. Well, that's about all of it there is to tell...Say, you boys wouldn't happen to have anything to drink on you, would you? I was stone-cold sober last night and I'll swear to it on a stack of Bibles. But I sure could use a drink right about now and I don't mean the nonalcoholic kind, neither. Something with a kick to it. This talking is mighty thirsty work."
Jack said, "Sorry, no."
Lobo said, "Figgers. That's my fool luck working against me." He brightened. "Still, it was working for me pretty good last night, to keep me from getting tooken!"
Neal took a cigarette from the pack and put it between his lips. Lobo looked up hopefully, said, "You wouldn't have a smoke to spare, would you?"
Neal gave him a cigarette. Lobo said, "Thank you kindly." Neal flicked on his lighter, holding the flame to the tip of Lobo's cigarette until it got going, then lighting up his own. Jack instinctively looked away while they were lighting up, to avoid totally canceling out his night vision.
Jack said, "Didn't you see the police searching the compound all day, Lobo?"
"Sure, I seen 'em."
"Why didn't you come down and tell them what you saw?"
"Mister, I make it a practice to keep as much distance between myself and the law as possible. I got no hankering to go back to the state hospital again so the doctors could treat me like I was sick in the head."
Lobo took a long draw on his cigarette, exhaled a cloud of smoke. "Some of them cops had pretty mean faces, too. Could've been devil men for all I know. I sure wasn't going to put myself in their clutches. I laid low until they packed up and went home and I stayed low until that pair that was dogging me in the hills got tired and went away, too.
"Even then I didn't show myself for fear of the hog-faces coming back by night. It was getting late and the moon was low and they still hadn't shown, so I took a chance on breaking cover and coming down into camp to see what I could scrounge up. I was getting powerful hungry, my belly was all twisted into knots. I made my move and that's when you fellows showed up and turned on the lights. I ducked down among all them garbage cans to hide. I was afraid you was gonna search back there and I wanted to get away before you did only I made too much noise and gave myself away."
Lobo smoked his cigarette down to the nub and tossed it away, the bright orange-red tip making a tiny splash of embers when it hit the dirt. He looked up at Neal. "Could I trouble you for another of them smokes?"
Neal gave him a fresh cigarette and held the lighter until Lobo got it going. Lobo said, "Much obliged. You fellows cops?"
Jack said, "No."
Lobo nodded, as if confirming a previously held notion. "Thought not. Cops would've already been whomping on me, beating the piss out of me for drawing my knife, even though I was scared and just trying to defend myself."
Jack said, "We're government men." Neal looked at him sharply, unsure of where Jack was going. Jack went on, "We're part of a top secret outfit set up to investigate satanic crimes."
Lobo cackled, "I knew it! Like the Men in Black."
"We're the Anti-Beast Brigade." Jack was straight-faced, serious. "You're an eyewitness to what happened here, the only eyewitness. We're going to take you to a safe place where the devil men can't get you and you can tell your story. You'll also be able to get cleaned up and get a hot meal."
"I ain't so big for cleaning up but the hot meal sounds all right. You think maybe I could get me a drink or two?"
"I can't make any promises but we'll see when we get there."
"You'll put in a good word for me, won't you? About that drink. After all I seen last night, I sure could use one!"
"I'll see what I can do."
"Let's get out of here then. I don't mind telling you that being bird-dogged by those two devil men day and night kind of got me spooked. I won't mind putting some distance between me and Them."
Lobo rose, standing up. His sudden movement undoubtedly saved Jack's life. Shots cracked; Lobo pitched forward, slamming into Jack, knocking him off his feet.
Jack was still holding Lobo's knife in his right hand and he twisted sideways to keep Lobo from impaling himself on the blade as the other lurched into him. He needn't have bothered because Lobo was already dead, killed by that first shot. But things were happening too fast for Jack to make sense of it all.
They both fell tumbling in a tangle of limbs to the mess hall's wood-planked porch. Jack lay on his left side, with Lobo sprawled half across him.
Jack glanced up in time to see the top of Frank Neal's head explode, spraying blood, bone, and brain matter. It meant instant neural extinguishment, the cessation of all thought and reflex motor action. The body dropped like a stone.
A bullet hole showed in Lobo's upper back between the shoulder blades, marking the shot that had brought him down. His dead weight pinned Jack to the boards. Jack let go of the knife and started wriggling out from under him.
Lobo's body spasmed violently under the impact of a second round thudding into it. The shot had been meant for Jack but hit Lobo instead. Jack clawed out his pistol.
Two figures stood in front of the men's barracks north of the mess hall, barely a stone's throw away. One had a rifle and the other a handgun. A patch of gun smoke like a small, puffy ash-gray cloud hung in mid-air in front of the duo. The rifleman stood with the weapon held at his shoulder, swinging the barrel to get a clear shot at Jack.
Jack fought down the urge to jerk the trigger, squeezing it instead several times to place a couple of rounds into the rifleman's middle. The rifleman went over backward like a tin duck in a shooting gallery.
Jack and the rifleman had had fairly clear firing lines on each other. Jack had been fortunate in that Lobo had been unlucky enough to stand up in time to catch that first bullet that had been meant for Jack. No such luck for Frank Neal. The rifleman had tagged him with a head shot. Jack and Lobo had gone crashing down to the planks together, and Lobo had caught the rifleman's hasty third shot.
The rifleman had done all the damage; his partner must have been more of a spotter and backup. Now he returned fire with the handgun, loosing a fusillade in Jack's direction. Neither he nor Jack had much in the way of sightlines on each other.
He had a semi-automatic pistol and he must have pumped out a dozen shots. He made a lot of noise, but none of the rounds came close to Jack. He shot out a mess hall window and punched holes in wooden walls, spraying a lot of wood chips, splinters, and sawdust. He was less interested in getting his man than he was in covering his retreat.
He ran north, angling his flight to put the men's barracks building between him and Jack. Jack raised himself up on his elbows, wiping the back of his free hand across his eyes, trying to clear them of the blood that had sprayed his face when Lobo had been tagged. He got his feet under him and hunkered down beside Lobo, feeling his neck for a tremor of a pulse. He knew it for an exercise in futility but went through the motions anyway, confirming what he'd been certain of, that the man was dead. That first shot had done for him, the one that had been meant for Jack.
Neal lay on his back, face upturned to the night sky. It was like the top of his head had been scooped out with a shovel. The rest of his face below the brow line didn't look too bad. Jack knew Neal had put the truck keys in his front pants pocket but he couldn't remember which one so he patted them both down, feeling the keys through his right front pocket.
It's not so easy to pick a dead man's pocket. Jack knelt beside the corpse, twisting his hand at an odd angle to get it inside Neal's pocket. Neal's body was warm with the life that had just been let out of it. Jack's fingers fastened around the keys and fished them out.
A figure darted out from between the women's barracks and the blockhouse holding the generator. The rifleman's partner. He could have done some mischief if he'd thought to disable the Toyota, but the only thing on his mind was escape. He burst out into the open, running east across the oval toward the front gate.
It was a long shot for a pistol, too long, so Jack didn't even bother trying. He started north, double-timing it. Caution and curiosity compelled him to pause to give the rifleman a quick onceover, drawing him to a halt beside the body.
The shooter was middle-aged with a lanky runner's physique. He had short wavy hair, bushy eyebrows, and a mean face. His expression was one of intense irritation, as though he was extremely annoyed at having been shot dead. He wore no flak jacket, no bulletproof vest. Jack's rounds had shattered his chest, one penetrating the heart, negating the need to deliver a coup de grâce to the head.
His weapon was a hunting rifle, a scoped thirty ought-six. A standard telescopic sight, not a night vision rig. Jack snatched up the weapon, shouldering it, but the fleeing gunman was already below the crest of the rise. He put it down and got moving, running to the Toyota.
The triggerman was unknown to Jack, a stranger. No mean feat, since Jack's access to information as SAC of CTU/L.A. made him cognizant of most of the top pro shooters currently active in the milieu.
He must have been one of Lobo's devil men, part of the team that'd been searching the hills for him. The other half of the duo was fleeing the compound. That much of Lobo's story had been true. And the rest?
Jack reached the pickup truck, jumping behind the wheel and starting it up. He made sure to fasten his seat belt harness, he was going to need it. He drove east, fast, toward the front gate.
Neal knew the area and had said there were no roads into or out of the sandstone piles west of the compound. The two killers couldn't have driven into the compound without having been seen by the CTU agents. Therefore they must have parked their vehicle outside the front gate and entered on foot.
Jack tore across the short axis of the oval, making a beeline for the exit. He paused for an instant at the edge of the top of the slope, scanning the landscape. There weren't too many places where another vehicle could be. It had to be on the access road or the blacktop road, or parked somewhere just off either road.
A pair of headlights flashed on behind a clump of brush on the shoulder on the east side of the blacktop road, north of where the access road met it. A dark-colored boxy sedan emerged from behind a screen of foliage. Jack thought it might be a Subaru from the quick glimpse he got of it, but that was only a rough guess. The sedan fishtailed along the shoulder and onto the blacktop, flashing north along it in a big, big hurry.
Jack took off after it. He first had to get to the bottom of the hill. He toyed with the idea of saving time by quitting the road and plowing straight down the hillside but discarded it. A big enough rock could bust a tire or an axle and stop the pursuit before it got started.
The pickup's nose tilted downward as he began descending the dirt switchback road, whipping the steering wheel left and right, standing on the brakes at times, sliding into some of the hairpin turns, whipping through others, laying down fat, feathery plumes of dust as he powered his way down the dirt track.
A couple of heart-stopping instants threatened to see the pickup truck go sailing off the edge, but each time luck or skill or both saw him through, enabling him to thread the twisty course in a speedy blur.
There was a bounce and then a liftoff at the bottom of the slope as all four wheels left the dirt road. Jack felt like a paving stone had been dropped into the bottom of his belly.
A timeless swooping interval came to an abrupt end as all four wheels touched down on the pavement of the two-lane blacktop. The vehicle bottomed out, banging its underside on the roadway with a bone-jarring thud that set Jack's teeth to rattling, but the shocks absorbed the impact and the tires held up without any of them suffering a blowout.
The wheels bit, gaining purchase, squealing as Jack whipped the steering wheel around to make a hard left, then burning rubber as he stomped the gas pedal and the machine bulleted northward, taking up the chase.
##
## THE FOLLOWING TAKES PLACE BETWEEN THE HOURS OF 5 A.M. AND 6 A.M. MOUNTAIN DAYLIGHT TIME
Nagaii Drive, Colorado
The road ran north-south, hemmed in by a river on the east and mountains on the west. It ran not in a straight line but in broad, sweeping curves molded to the contours of the river valley.
The pickup truck's engine was well-tuned and possibly customized for speed; there was a lot of power under the hood, as Jack was happy to discover. It handled well on the curves, too.
The sedan ahead knew it was being chased and was doing its best to widen the distance between it and its pursuer. The driver had an advantage over Jack in that he presumably knew the terrain, while to Jack it was all unknown territory.
Jack was a veteran of the Los Angeles Police Department, and while the bulk of his term there had been spent on the SWAT team, he was no stranger to hot pursuit driving. No matter what the locale, a road's a road, and he was a quick study.
The fugitive had been handicapped by having to flee Red Notch on foot, scrambling down the hillside and across the road to reach the place where he'd hidden the sedan. That had cut down considerably on his lead. He was trying to increase it now, while Jack labored to whittle it down.
The scenery shot by in a blur. The river lay east of the road, to the right of Jack's northward course. A thin line of trees stretched along the top of the embankment. Gaps in the tree line afforded glimpses of the river and the terrain beyond it, a long, shallow slope slanting upward for several miles to the ridgeline. The river was about an eighth of a mile wide and flowed southward. It looked fast, frothy and churning with latent power.
The sun was a long way from rising but the sky was lightening in the east, fading from purple-black to royal blue.
This predawn effect was suddenly negated as the pickup truck left Red Notch behind. The notch was just that, a gap between the mountains. The looming bulk of Mount Nagaii appeared to the north of it, a towering rock rampart that rose up and up to dizzying heights. The mountain blocked the low-hanging moon in the west, shutting off the moonlight and throwing the river valley into deep shadow.
A positive result was that the thickened darkness caused the sedan's taillights to stand out more brightly, a pair of hot red dots gliding above the winding roadway.
Jack switched on his high beams, expanding his view of the road ahead. It was a help at the speed he was traveling. There were no crossroads or intersections as far as he could see, no place where the sedan could turn off in another direction.
Events had happened so quickly that Jack hadn't dared risk losing a precious second to the distraction of communicating with Central. He could do something about that now that he was settling into the rhythm of the chase.
The mobile comm unit mounted on the dashboard was a variation of the standard model used by CTU/L.A., so he could work it without too much trouble.
He switched it on, its power light brightening to a glowing green bead.
He steered with his left hand, holding the hand mic in his right. He thumbed down the transmit button, said, "Central, this is Unit Three. Over."
Central acknowledged the transmission. Civilian and military police authorities generally use a numerical code system as a kind of verbal shorthand for their radio communications. A "ten-ten," say, means that the unit is going temporarily out of service. Other number codes stand for such things as robbery in progress, shots fired, officer down.
CTU wasn't the police and except for specialized operations, such as those carried out by tactical strike forces, relied on plain speaking and the technological sophistication of their hardware systems to ensure the security and privacy of their communications.
Jack spoke into the hand mic. He had to shout to be heard over the sound of the engine as it ground out the RPMs necessary to keep pace with the sedan. "This is Agent Bauer. This unit was attacked by two shooters at Red Notch. Agent Neal is dead. So is a civilian informant we discovered at the site. One of the shooters is dead, too. His partner is fleeing north in a dark-colored sedan along the road at the foot of Red Notch. I don't know the name of the road but it runs west of a river."
The dispatcher started shouting too, out of excitement. He identified the road as Nagaii Drive. Jack said, "I am in pursuit of the vehicle. The driver is armed with a handgun. He may have other weapons. We're about five miles north of Red Notch and proceeding northward."
The dispatcher had all kinds of questions about the incident that he wanted answered right away. The road started shifting into a series of tricky S-curves that required Jack's undivided attention. The sedan had widened its lead while Jack was talking. He signed off, took hold of the wheel with both hands, and concentrated on driving.
The road snaked around outcroppings and concavities of the mountain, an undulating ribbon unwinding at high velocity. Anxiety gripped Jack each time the sedan's taillights whipped around a curve and out of sight, lessening only when those two red dots came back into view.
Jack drove with one foot on the gas pedal and the other poised above the brake pedal. He put more weight on the accelerator. The engine noise wound up to a higher pitch. He glanced at the temperature gauge on the instrument panel. The needle held steady at the midpoint between the two extremes, right where it was supposed to be. That was good, the engine wasn't running hot.
The chase had been run on the flat with no real downgrades or slopes to speak of. The road skirted the base of the mountains, avoiding even the foothills. That was okay with Jack, he had his hands full keeping up the pace on this course. The road was bare of all but these two vehicles, the sedan and the pickup truck.
Jack sat hunched forward over the steering wheel like a jockey leaning into the saddle. His skull pounded with a splitting headache, the king-sized killer headache of all time.
No mystery about that. It was the altitude. Jack came from Los Angeles, he was a flatlander. But Denver was a mile above sea level and the Red Notch area was higher than that. He wasn't acclimated to the elevation. The experts said that flatlanders should take it easy for their first day or two in the heights to avoid altitude sickness. It wasn't a matter of conditioning, a trained athlete from the lowlands was as likely as an overweight, lazy layabout to suffer ill effects from initial exposure to the rarefied air of the mountains.
Jack's head felt like a railroad spike was being hammered into the center of his skull between the twin cerebral hemispheres. Then he remembered Neal with the top of his head shot off. Jack decided he was damned lucky to have a head to suffer headaches with. He'd tough it out, let the adrenaline rush of the hunt help power him through it.
The curves started to smooth out, flattening into a long straightaway. Jack floored the gas pedal. The pickup shook from the engine vibration, but it was manageable. The engine roar almost but not quite drowned out the transmissions of the frantic dispatcher at Central as he kept firing off demands for an update on the situation.
The situation was that Jack was closing in on the sedan. It was only a couple of hundred yards or so ahead, and the gap was steadily decreasing. Whatever the sedan had under the hood, it lacked the muscle of the pickup truck, and that lack was inexorably telling.
The sky was lightening. A trestle bridge spanning the river came into view on the right. A gap opened opposite it on the left, where Mount Nagaii ended. It was a crossroad that cut Nagaii Drive at right angles. A handful of buildings stood at the junction.
The bridge was a railroad bridge, inaccessible to vehicular traffic. Railroad tracks stretched from the west end of the bridge, crossing Nagaii Drive and continuing into the gap between Mount Nagaii and a mountain to the north of it.
The tracks that crossed the road at right angles were sunken, the twin rails inset in slotted grooves in the asphalt. There were no cross ties. A black-and-white-striped bar and a set of signal lights marked the crossing. No train was using the line so the signal lights were dark and the barrier gate was raised to permit free passage.
A small town was clustered around the crossing. Town? It wasn't even a village. A hamlet, maybe. There were a gas station, a diner, a strip lined with a couple of convenience stores, a post office, and a handful of houses.
The sedan blew through the crossing with no slackening of speed. The pickup truck did the same, flashing over the sunken railroad tracks with nothing but a slight change in pitch in the whirring drone of the racing wheels on the roadway to mark their presence.
Jack glimpsed in the corner of his eye a tiny café fronting the east side of the road. A police car stood parked beside its north wall, facing the road at right angles. Its lights were dark, but he could make out what looked like two figures in the front seat.
He passed them doing about eighty, eighty-five miles per hour. He'd been going faster but had slowed down a hair just to be on the safe side when crossing the railroad tracks. They proved to be no obstacle, so once he'd cleared them he pushed the pickup back up to ninety.
There was a pause while the occupants of the police car woke up or got over their stupefaction at seeing a high-speed chase zip right by them. Then the police car swung out of the lot into the northbound lane, turned on its headlights, switched on the emergency flashers of its rooftop light rack, and took off after the sedan and the pickup truck.
Jack glanced in his rearview mirror, seeing the police car's light rack flickering bright blue and white. They looked bright and happy, like party lights. They were a long way off. The sedan was much closer, the gap between it and the pickup truck closing up.
Telephone poles lining the roadside went by in a blur. Road signs whipped by so fast there was no time to read them.
The road started to slope upward, beginning a long gentle incline that curved slightly to the west, rounding the southernmost limb of Mount Zebulon, the next peak north of the gap beyond Mount Nagaii.
Jack was so close to the sedan that he could make out the outline of the driver's head and shoulders. How to take him? He'd like to take him alive if he could, but at these speeds that would be a tall order. He didn't intend on getting killed himself trying it. The pickup was bigger than the sedan, had more muscle. He could run him off the road, if it came to that. If he came alongside the sedan, he could shoot him. The fugitive had a gun, too, though, and Jack didn't fancy the idea of trading shots with him at ninety miles an hour. No, best to bull him off the road. If the other should survive the crash, so much the better.
The sedan was nearly at the crest of the long incline. A peek in the rearview mirror told Jack that the police car was still a long way behind.
The sedan topped the summit, disappearing down the other side. The hilltop zoomed ahead, and for a split second Jack was looking down at the far side of the slope.
There was a village at the bottom of the hill. There wasn't much to it but it was a metropolis in comparison to the whistle-stop at the railroad crossing. A bridge spanned the river here, too, but this one was for cars and trucks.
A strip of stores lined both sides of Nagaii Drive at the village's center. Jack guessed that was what passed for Main Street, the business district. A dozen or so two-and three-story brick buildings were grouped around both sides of the main drag. A couple of blocks of one-family houses stood on the west side of town.
The intersection of Nagaii Drive and the road to the bridge formed a square, complete with traffic lights. The lights flashed amber.
A police car came into view in the western arm of the crossroad, rolling eastbound toward the square, its emergency lights flashing.
The sedan got there first, flying through the intersection and continuing north on Nagaii Drive. The police car halted, partially blocking the square.
A second police car appeared, coming from the east branch of the crossroad, rolling west. Its flashers were on, too. It halted in the middle of the square, nose to nose with its twin, the two of them forming a roadblock that walled off Nagaii Drive.
Jack was in a tight spot. He thought about driving up on the sidewalk and swerving around the roadblock, but the sidewalk looked too narrow to accommodate the pickup. It didn't look doable even if the sidewalk had been wide enough, not at the speed he was going. At that speed it looked suicidal. He wasn't sure that even without trying any fancy tricks he could stop in time to avoid crashing into the roadblock.
The cops must have thought so, too, because they jumped out of their cars and hustled to the sides. There were two of them, one per car. One was carrying what looked like a rifle.
Jack's calculations were carried out in split seconds. They weren't so much calculations as reactions. He knew that if he stomped on the brake pedal the brakes were likely to seize up and cause him to lose control of the car. He pumped the brakes instead, manhandling the steering wheel to minimize the inevitable slide.
The tires howled, leaving twin snaky lines of burnt rubber on both sides of the street's painted yellow centerline as the pickup shimmied, fishtailed, and skidded.
The machine slid sideways a good part of the way down the hill, leading with the driver's side. Multiple collisions would have been inevitable if any cars had been parked on either side of the street. Jack needed all the space on both sides of the street to wrestle some kind of control into the pickup.
It was close, very close. The pickup skidded sideways toward the twinned police cars, lurching to a halt less than six feet away from them. The engine stalled out.
The radio still worked, though. Every now and then it squawked out another frantic, near-unintelligible query from the dispatcher at Central.
Jack felt like he'd left his stomach somewhere back on the downgrade, probably at the point where he'd first started working the brakes. The stench of burnt rubber and scorched brake linings was overpowering, stifling. He felt like he could barely draw a breath.
He could see now that what he'd thought had been a rifle in the hands of one of the cops who'd jumped clear of the roadblock was actually a shotgun. It was pointed at his head.
The cop who was wielding it stood on the passenger side of the truck cab. He looked unhappy. He gave off the impression that pulling the trigger might make him happy.
Jack raised his hands in the air, showing they were empty.
A second cop appeared on the driver's side of the truck, brandishing a long-barreled .44 magnum. Both cops wore Western-style hats that heightened their resemblance to cowboys.
The cop with the handgun was shouting at Jack to get out of the truck. Jack stayed where he was because in order to comply with the command he'd have to use his hands, and he was afraid that if he moved them one or both cops would think he was reaching for something and use that as an excuse to open fire on him.
The cop with the gun used his free hand to open the driver's side door. Pale gray eyes were wide and bulging in a flushed, angry red face. He said, "Get out! Get out of the vehicle!" He pronounced it "veehickel."
Jack said, "I can't—the seat belt."
The cop shook his head in seeming disbelief as though this was some new, undreamed-of height of criminal audacity. He stuck the gun muzzle against the underside of Jack's chin and said, "If I see you reaching for anything but that seat belt fastener, I'm gonna see your brains all over the inside of this truck cab."
Jack said, "I'm going to unfasten it now."
"You do just that, mister."
Jack moved very slowly, like he was in a sequence filmed in slow motion. He lowered his arms and worked the seat belt release. It came undone with a click.
The cop grabbed him by the back of his collar and hauled him out of the cab, flinging him out on the street. Jack hit the pavement sprawling, skinning his hands and knees.
The cop with the shotgun circled around the front of the pickup, holding his shouldered weapon so that it pointed down at Jack.
The pale-eyed, red-faced cop said, "Lie facedown on the street and don't you move, boy; don't you even breathe."
Jack did as he was told. He could see that the cop with the .44 wore cowboy boots under his tan pants. The boots had sharp, pointy toes and lots of fancy leatherwork and embossing. They looked expensive.
The cop twisted Jack's arms behind his back, wrenching them as though he'd like to tear them out of the sockets. Steel bracelets encircled Jack's wrists, biting deep, cinching tight.
He grabbed Jack by the back of the neck and hauled him one-handed to his feet. Not by the back of his collar but by the back of his neck. He was strong. Jack stood there with his hands cuffed behind his back.
He looked across the police roadblock, north up Nagaii Drive. There was no sign of the sedan, not even a glimmer of its taillights. It was long gone.
The cop with the shotgun held it pointing muzzle-down. The pale-eyed, red-faced cop was holstering his sidearm. His gun belt was fancy and hand-tooled.
A third police car was on the scene, halted in the middle of the street behind Jack's pickup. It must have been the one that had been beside the café and chased Jack along Nagaii Drive into town.
It yielded two more cops, a male officer and a female one. They both wore Western-style hats. The woman wore her hair pinned up in a bun at the back of her head, below the hat brim.
Her partner was a big, hulking specimen, about six-four with shoulders as wide as an axe handle is long. He was in his mid-twenties, with hair so black it had blue highlights. His hair seemed long for a police officer's. He was clean-shaven, with smooth, bright pink skin. He looked more enraged than the two cops from the roadblock who'd actually made the arrest. That might have been because he had a big, dark patch of wetness staining the crotch and upper thighs of his trousers.
The cop with the shotgun flashed a wolfish grin, showing a lot of teeth and little mirth. He said, "Holy cow, Fisk! What'd you do, piss your pants?"
Fisk said, "Never you mind about that, Cole Taggart! I spilled a cup of coffee on myself when we took off after this lawbreaking son of a bitch!"
"Sure you did."
Fisk indicated the female officer. "It's true! Ask her—"
Taggart said, "Of course Trooper Stallings will cover for you, her being your partner and all." Taggart was the type who obviously liked working the needle, at least on Fisk.
The pale-eyed cop said, "That must've been some hot coffee, Fisk, from the way you're walking around all hunched over and bowlegged, like a little old man."
Fisk said, "Hot? I'll tell the world it was hot! I like to scalded my—"
"Spare me the details. Save it for the medical report."
Taggart chuckled. "That should be some report. Good thing Bryce and me was here to catch this speed demon."
Fisk said, "We'd have caught him. No way he was getting away after causing me to ruin a perfectly good pair of pants!"
Taggart said, "Let's hope that was all that was ruined. Ain't that so, Sharon?"
The female officer said coolly, "You're the one who's interested in what's in his pants, not me, Cole."
Taggart said, "Ouch! That's one on me. Though I guess it's Fisk who's the one who ought to be saying ouch."
The pale-eyed cop, Bryce, said, "All right, can the back chat." The others fell silent. Bryce was in charge.
Jack thought that this was hardly the time or place to try explaining that he was a CTU agent who'd been chasing an accomplice to murder. He said to Bryce, "You'll find my ID in my wallet."
Bryce said, "Shut up." There was no rancor in his tone, which was the same as when he'd told the officers to can the chatter.
Fisk sidled up alongside Jack, peering narrow-eyed at him. "He's got a gun, Lieutenant." He was speaking to Bryce.
Bryce said, "Is that a fact? That's a fine piece of detective work, Fisk. Keep it up and you'll make sergeant in no time." His voice drawled with mild sarcasm. He reached under Jack's coat, freeing the pistol from the shoulder rig. He turned it over in his hands, eyeing it. "Nice piece." He held it under his nose and sniffed it. "Been fired recently, too."
Taggart said, "Looks like we got us a real desperado."
Jack said, "I can explain—"
Bryce said, "That'll take some pretty tall talking, stranger. But you'll get your chance." He handed the pistol to Fisk, saying, "Here, hold this."
Taggart said, "That there's what we call evidence, Trooper."
Fisk took the gun. His heavy-lidded eyes were smoldering, resentful. He called Jack a dirty name and slammed the flat of the gun against the side of Jack's face. Jack went down.
##
## THE FOLLOWING TAKES PLACE BETWEEN THE HOURS OF 6 A.M. AND 7 A.M. MOUNTAIN DAYLIGHT TIME
Mountain Lake, Colorado
The pale-eyed cop said, "Is my face red!"
It was just a figure of speech. His face wasn't red, not really, not the way it had been earlier when he'd held a gun to Jack's head. That had been a product of the heat of the moment, the adrenalized rush of apprehending a suspect. His complexion had since reverted to its normal color, the rugged bronze tones of one who spends much time exposed to the elements in the out-of-doors. His eyes were still pale, though, with clear gray irises that accented the prominence of dark pupils.
His name was Bryce Hardin, and he was a lieutenant in the state police, the head of a Mobile Response Team that had been formed as a troubleshooting unit for the Sky Mount Round Table. The MRT consisted of Hardin; his second-in-command, Sergeant Cole Taggart; and troopers Sharon Stallings and Miller Fisk.
They were all state police officers who'd been detached from their regular duties for this special temporary assignment. They operated out of a substation at Mountain Lake, a site on the lower slopes below Sky Mount.
The substation was a tan brick blockhouse, a minimalist single-story structure with a low, peaked roof. It contained a front desk area, a couple of detention cells, a squad room, Hardin's office, and several back rooms. It was only in use during the warm weather months; winter's heavy snows closed all but the main roads for weeks at a time, making it impractical to keep the substation open throughout the icy season.
Hardin's office was a modest-sized rectangle whose entrance was in one of the short sides.
A window in the rear wall opened on a spectacular view of the eastern foothills and the river valley. Hardin sat with his back to it, facing the office door from behind a golden oak desk. A handsome reddish-brown leather couch stood against one of the long walls. The wall space above the couch was decorated with honorary plaques and citations awarded to Hardin for various achievements in law enforcement. A row of gray metal filing cabinets was lined up along the opposite wall. The space above them displayed framed photographs of Hardin posing with important-looking personages, presumably politicians and suchlike dignitaries.
A pair of armchairs stood at tilted angles facing the front of Hardin's desk. Jack Bauer sat in one of them. It wasn't particularly comfortable, but even if it had been, Jack was in no mood to appreciate it. He noticed that both chairs were short-legged and set close to the floor, forcing those who sat in them to have to look up at Hardin. Hardin's desk was flanked by a pair of flagpoles mounted in floor stands. The flag of the United States stood on the left and the state flag on the right.
Hardin was in his fifties, with wavy dark hair gray at the temples framing a thick-featured, square-shaped face. He was thick-bodied, heavy in the chest and shoulders. His manner expressed sincerity and frankness. It occurred to Jack that Hardin was something of a politician himself.
Hardin said, "As the saying goes, when I make a mistake it's a beaut! I pulled a real boneheaded move when I apprehended you, Agent Bauer. I had no idea that we were both on the same team, you being a Federal officer and all. Quite frankly—I goofed."
Jack sat holding a towel-wrapped ice pack against the left side of his face, where Fisk had smacked him with his own gun. Jack had seen the blow coming and rolled his head with it, deflecting some of the impact. It had knocked him down and stunned him despite his evasive response. He'd never actually lost consciousness but he'd seen stars for a while. Fisk had slapped him with the flat of the gun and Jack had caught most of it on his left cheekbone. Nothing was broken and his teeth on that side were all intact as far as he could tell. The side of his face was numb and swollen with a purple-brown bruise about the size of a man's palm. The altitude headache he'd been suffering from earlier was as nothing compared to the colossal, king-sized headbanger he was experiencing now.
Hardin had picked Jack up off the pavement and half carried, half dragged him to his patrol car and thrown him into the backseat. The first setback to the MRT's certainty that they had snagged a hot one came when they looked inside the pickup truck and saw its sophisticated dashboard-mounted comm system. Sergeant Cole Taggart had explained it away by saying, "He probably stole the vehicle. We'll get it all straightened out at the substation."
The quartet formed a convoy to the substation. Hardin took the lead, with Jack in the back of his car, a wire safety grille separating the lawman in the front seat and the suspect in the back. Taggart drove the pickup truck. Sharon Stallings drove Taggart's car, while Fisk drove the car that had been assigned to him and Stallings.
Jack had been groggy and his ears were still ringing, so he kept his mouth shut during the drive. He learned later that the town where he'd been stopped was named Random. That seemed fitting somehow.
The MRT convoy climbed Rimrock Road to reach the Mountain Lake substation. Rimrock Road was built on a stony ledge; a cliffside loomed on its west, while an ever-increasing drop over an empty void opened on its east. The road leveled off for a mile-long stretch, at the end of which the ledge widened into a large circular outcropping that was a scenic overlook point.
The substation was firmly hunkered down on that outcropping. It presented a spectacular view, but Jack's interest in sightseeing was nil.
Taggart had had time during the drive to work the pickup truck's comm system and make contact with the CTU Central dispatcher at Pike's Ford. It was a tossup as to which of the two was more startled, the dispatcher or Taggart. Central managed to convey something of the reality of the situation before the convoy reached the substation.
Taggart was unable to communicate directly with Hardin through the pickup's comm system without going through Central, something that he was not minded to do in any case now that he realized there had been a screwup of major proportions. Taggart started playing it cagey, his responses to Central becoming vaguer and more evasive before he finally signed off by saying that someone from the MRT would get back to them as soon as the issue had been clarified.
The convoy pulled into the substation parking lot. Taggart was out of the pickup fast, scurrying over to Hardin, who was still in his car. Hardin rolled down his window to allow Taggart to stick his head inside for a hurried urgent consultation. Taggart did most of the talking, or whispering rather, buzzing in Hardin's ear. The more Taggart talked, the redder became Hardin's ears and the back of his neck. Hardin turned around in the driver's seat to look back at Jack, staring at him through the wire mesh grille of the protective barrier. He listened to more of Taggart's whisperings, at one point blurting out, "Impossible!"
Taggart said, "I'm not so sure, Bryce—"
Hardin got out of the car and opened the back door. He said, "Let me give you a hand." He gripped Jack under the arm, helping him out of the police car. He said, "Careful you don't bump your head." Jack gave him a dirty look.
Hardin held Jack under one of his handcuffed arms and Taggart held him under the other as the two cops walked Jack across the lot and into the station, their manner a lot more solicitous than it had been. They took him to the front desk where a suspect would normally be booked. A phone bank and two-way radio were part of the desk's complement of equipment.
Taggart emptied out Jack's pockets, placing their contents on the countertop. They included a couple of spare clips for his gun, several sets of keys, a notepad and pen, some loose change, a cell phone, and a wallet. His handset transceiver had been in the truck but Taggart had brought it inside.
Taggart went through Jack's wallet. It didn't take him long to find Jack's CTU ID card. It bore a thumbnail photo of Jack. Taggart and Hardin stood side by side, alternately looking at the ID and at Jack. Taggart said, "Jack Bauer, that's the name they gave me over the radio, and believe me, Bryce, they weren't giving much."
Hardin said, "Uh-oh." Stallings and Fisk had been standing off to the side, watching the proceedings. They didn't know what it was all about but they knew that something was up.
Taggart looked at Hardin. Hardin scratched the side of his head, cleared his throat. He said, "Well. Ahem. Er, Mr. Bauer, I'm afraid that we're all the victims of a terrible mistake."
Jack said, "You think?" The numbed side of his face gave him some difficulty in forming the words.
Hardin said, "Cole, take the cuffs off him."
Taggart unlocked Jack's handcuffs. Jack's wrists bore angry red grooves where the tightly fastened cuffs had bitten deep in the skin. Loss of circulation made his hands feel clumsy and oversized, like he was wearing a pair of oven mitts.
Hardin said, "Maybe you'd like to sit down." He indicated a chair in the front desk area. Jack sat down in it, resting his hands on top of his thighs. He flexed them, clenching and unclenching his fists. Electric needles of sensation pierced his hands as feeling began to return. His face was pale, waxen, except for where the bruise had flowered on his left cheek.
One of the front desk phones rang. Taggart answered it. Squalling sounded from the earpiece where he held it to his head. The words were unintelligible but their tenor was unmistakable. Taggart winced, handing the receiver to Hardin. "You better take this, Lieutenant."
Hardin got on the phone. He barely had time to identify himself before receiving an earful. He did a lot of listening and not much talking. His few responses were limited to such phrases as "an honest mistake...in the heat of the moment judgments had to be made...can't be too careful, with the conference on...mistakes were made, yes...dreadfully sorry...the department regrets...I deeply regret...you have my full apologies..."
He held out the phone to Jack. "They want to speak to you."
Jack rose, took the phone, holding it to the right side of his face. A voice on the other end of the line said, "Hello, Agent Bauer? Anne Armstrong here."
Anne Armstrong was one of Garcia's top staffers at CTU/DENV, one of the special agents overseeing the handling of the Sky Mount assignment out of the Pike's Ford command post.
Their conversation was naturally circumscribed by its being carried on an unsecured phone line and could be conducted in only the most general terms. That didn't prevent her from asking, "What have those idiots gone and done?"
Jack looked at Hardin and Taggart. "Let's call it a case of mistaken identity."
"Are you all right?"
"I've been better, but I'll live."
"I'm on the way. I'll be there in twenty minutes."
"I'll be here. I'm not going anywhere."
The call completed, the connection was broken. Jack handed the phone to Hardin, who placed the receiver back on the hook. Hardin said, "No charges will be filed, of course."
Jack said, "By me or by you?"
"Ha-ha. These things happen, you know. With the Round Table opening today, and those crazy Zealots dropping out of sight to get up to who knows what, I'm sure you can understand that we're all a bit on edge, keyed up as it were, so there may have been a tendency to overreact."
"If you'd acted a bit sooner to catch the guy I was chasing, you would have nabbed a hot lead."
Hardin mustered a sickly smile. "Reckon that makes me the goat. I'll take full responsibility for it. We were just a few seconds off in closing that roadblock. But how were we to know that you were a Federal agent chasing a fugitive? If you or your people had communicated with us in time...As it was, though, we didn't know what we were dealing with."
Jack said, "I know the feeling."
"Well, you can see how it is then." Hardin indicated Jack's belongings laid out on the desktop. "You'll be wanting your stuff back."
Jack started picking up items, putting them in his pockets. Hardin said, "When you're done, step into my office and make yourself comfortable."
Jack followed Hardin to a closed office door whose upper half was a translucent pane of pebbled glass. Hardin's title and name were stenciled on it in black letters, along with the legend: "Private."
Hardin opened the door and ushered Jack in, following him and closing the door behind them. He said, "You'll be wanting to clean up. You can use my private washroom."
Jack said, "That's big of you, Lieutenant. Mighty big."
Hardin chose to ignore the sarcasm. "Just a little interagency cooperation. After all, we're all on the same team."
A connecting door in the long wall with the filing cabinets opened on a small bathroom. Jack eyed his reflection in a mirror mounted over the sink. He hoped it was the overhead fluorescent lighting that made him look like death warmed over. The left side of his face was bruised and swollen but not as badly as he'd expected it to be from the way it felt. He ran some cold water and rubbed it on the parts of his face that weren't sore. He soaked a washcloth with hot water and held it against the left side of his face. He patted himself dry with a hand towel and stepped out.
Hardin did his best to make himself agreeable. He offered Jack a cup of coffee. Jack passed on it. He offered Jack a drink from a bottle of whiskey he kept in a desk drawer. That offer was more tempting but Jack declined. He didn't want to meet Anne Armstrong with liquor on his breath.
A discreet knocking sounded on the office door; Hardin said, "Come in." It was Sharon Stallings with a towel-wrapped ice pack. Jack accepted with thanks. She went out. Jack sat in one of the armchairs holding the ice pack against the left side of his face.
Hardin said, "We put out an all-points bulletin on that car you were chasing. Too bad we don't have a license plate number to go on. Maybe something'll come of it. Mind telling me what it's all about?"
Jack said, "The driver's wanted in connection with a shooting."
Hardin showed interest. "You don't tell me! Who got shot?"
"We'll get back to you on that later."
"Top secret stuff, eh? Sure, sure. I understand. Any information you can extend to me will be greatly appreciated. We're both after the same thing, making sure that the Round Table goes off without a hitch." Hardin's chair creaked as he leaned forward in it. "This suspect—he one of Prewitt's crazies?"
Jack shrugged. "That remains to be seen."
"They're a bad bunch, a bad bunch. Them going missing right as the conference kicks off, well, it can't be a coincidence. Or a good thing."
A knock sounded on the door; it opened and Taggart stuck his head in without waiting for Hardin's acknowledgment. He said, "Bauer's people are here, Lieutenant."
Hardin said, "Okay."
Jack got up, placed the towel-wrapped ice pack on top of a filing cabinet, and went out the door. Hardin pushed back his chair and hurried after him. Jack went into the squad room. Fisk and Stallings were talking but fell silent when he entered. Fisk had put on a clean pair of pants since Jack had last seen him. Taggart had regained his seat behind the front desk.
Jack crossed toward the front desk without looking at anyone. His path took him in front of Fisk and Stallings. He stepped on Fisk's foot. That was to pin him in place. Jack pivoted on the spot, driving a left-handed spear thrust at Fisk. The fingers of his hand were held together, the tips slightly curled inward.
He thrust the fingers into the top of Fisk's belly, just below the bottom of the rib cage. He turned in toward Fisk as he struck, leaning into him, putting some weight behind the blow. His body screened Stallings and Hardin from seeing the strike. His curled fingertips went in deep.
Fisk jackknifed, going, "Whoof!"
Jack stepped back, said, "Excuse me."
Fisk folded up, almost doubled over. His eyes bulged and his mouth was a round sucking O, gasping for breath. His pink face whitened, going green at the edges. He hugged his middle with both arms. A trembling right hand drifted toward his right hip where his weapon was holstered.
Jack said, "Reach for that gun and I'll kick your teeth out."
Fisk decided against it and went back to hugging himself with both arms and sucking air, hating eyes glaring out of an anguished face.
Jack said, "Not so much fun when the other guy isn't handcuffed, is it?"
Hardin said, "Here now, what's all this?"
Taggart, elaborately nonchalant, said, "I didn't see anything, Lieutenant."
Hardin got a knowing look on his face. He said to Jack, "Okay, that evens things up. You happy now?"
Jack said, "Happier."
Hardin said, "Fisk, if you're going to be sick, you'd by God better not do it out here."
The station's front door opened and two people walked in, CTU's Anne Armstrong and Ernie Sandoval. Jack nodded to them, said, "I'll be right with you, there's just one more detail I need to get straightened out."
He crossed to Taggart at the front desk. Taggart eyed him warily. Jack held out a hand and said, "My gun."
Taggart opened a drawer in the desk, reached in, and pulled out Jack's pistol and a magazine clip, setting them both down on the desktop. He said, "It's not loaded."
Jack picked up the pistol, examining it, making sure the chamber was empty. It was. He fitted the clip into the slot on the gun butt, slapping it with the heel of his palm to send it on home.
He didn't bother jacking a round into the chamber. He'd already made his point. He fitted the gun into the shoulder holster, letting the flap of his jacket fall to cover it.
He faced the two CTU agents and said, "Let's go."
Anne Armstrong had a primly disapproving look on her face, like a schoolteacher who stepped out into the hall for a minute and found on her return that the pupils were acting up.
Ernie Sandoval indicated Fisk, who stood bent double with one arm extended, clutching the wall for support. He said, "What happened to him?"
Jack said, "Too much coffee."
##
## THE FOLLOWING TAKES PLACE BETWEEN THE HOURS OF 7 A.M. AND 8 A.M. MOUNTAIN DAYLIGHT TIME
Sky Mount, Colorado
It was a beautiful morning, and that was part of the problem as far as Jack was concerned. All that bright sunshine pouring out of a cloudless blue sky up here in the heights was dazzling and made his head hurt. He put on a pair of sunglasses. That cut down on the glare, but the pressure of the sunglasses on the bruised left side of his face added to his discomfort.
He sat in the front passenger seat of a car being driven by Anne Armstrong. She was tall, lean, with short blond hair and a long, narrow, high-cheek-boned face. She wore a tan blazer, light blue blouse, sand-colored skirt, and brown low-heeled loafers. She also wore a snouty, short-barreled semi-automatic pistol in a clip-on holster attached to her belt over her right hip.
She drove north on Rimrock Road, leaving behind the Mountain Lake substation. Their destination was Sky Mount. Ernie Sandoval had taken the Toyota pickup truck, driving it to the CTU command post at nearby Pike's Ford.
Anne Armstrong said, "Our people found the bodies of Frank Neal and a civilian who fit your description of Lobo at Red Notch, but no dead shooter."
Jack said, "Damn, they work fast."
"Who?"
"The other side, whoever that is."
"The Zealots?"
"I wonder."
Armstrong thought that over for a minute. "No results on finding the Subaru you were chasing."
Jack said, "I'm not surprised. There must be thousands of places to hide a car in these mountains."
"No car of that description has been reported stolen. And with no license plate number..."
"Those killers were pros. They wouldn't use their own car. The license plates were probably lifted from another car to further muddy up their tracks."
A mile went by. Anne Armstrong said with a touch of frostiness, "What was the purpose of that macho display at the station?"
Jack said, "Equilibrium."
Her face tightened, a network of fine lines showing around her eyes. "I don't follow."
He said, "Hardin's boy pistol-whipped me when I was handcuffed. Can't let him get away with that kind of thing. This is an aggressive business with a lot of high-testosterone characters who're always testing the limits to see what they can get away with—and that's just the ones who're supposed to be on our side. You don't want the word to get around that a CTU agent can be roughed up without any consequences. Otherwise our guys lose respect with the other agencies we have to work with. It's bad for morale. By paying that thug cop back in kind, proper balance is restored. The word gets out that our guys can't be pushed around without some kind of comeback."
Her pursed lips parted to speak. "I see. So it was all for the benefit of CTU. There wasn't any personal animosity involved." She didn't bother to mask the disbelief in her tone.
Jack said, "Personal feelings aside, I did it for the good of the service."
Anne Armstrong said a dirty word. "You'll be going back to Los Angeles in a few days but the rest of us will be staying here. Try to remember that we have to work with the local authorities."
"Hardin will get the message. I used to be on the LAPD. I know how cops think because I used to be one myself. By the way, what's the story on the MRT?"
"Bryce Hardin is a power in state law enforcement circles. He's a highly decorated officer with numerous commendations for valor and high-profile busts. He's got a lot of pull with the governor's office at the capital. The MRT is his and the governor's way of injecting themselves in Sky Mount doings and increasing their profile and political prestige."
The cliff wall on the west ended, opening into a box canyon whose centerpiece was a lens-shaped lake. The lake was the color of the sky. The picturesque landscape had a gravel parking lot and was dotted with picnic tables scattered among the trees surrounding the lake.
A metal signpost identified the area: MOUNTAIN LAKE STATE PARK. A chain barred the entrance. A printed cardboard sign fixed to it said, TEMPORARILY CLOSED.
Jack said, "So that's Mountain Lake. I was wondering where they were hiding it."
Armstrong said, "It's closed for the duration of the Round Table. The authorities don't want a lot of unauthorized civilians up here during the conference. It's one less variable for them to have to deal with."
They drove past the space and the cliff walls returned. Jack said, "Something else has been bothering me, something that might be a possible lead. It's a long shot but it could be worth following up. The compound at Red Notch should be checked for traces of chemical weapons."
Armstrong's cool demeanor gave way to outright surprise. "Chemical weapons? Where do you get that?"
"Something Lobo said about the compound being covered by a green cloud. It could have been some kind of CW, a toxic gas attack. Or maybe only a smoke bomb."
"Or the demented ravings of a half-mad homeless drunk."
"Somebody was worried about Lobo enough to have him killed by a team of assassins. Cultists and CW isn't so much of a stretch, either. Look at the Aum Shunrikyo doomsday cult that set off sarin nerve gas bombs in the Tokyo subways some years ago."
Anne Armstrong looked worried. "The Zealots and chemical weapons—the idea alone could set off a panic."
Jack said, "There could be traces of residue remaining in the compound. For that matter, it might be worthwhile to have Lobo checked for the same in a postmortem. He might have been exposed to some of the stuff, and it's possible that whatever it is could be retained in organic matter."
Armstrong used her in-car comm system to contact Central. She relayed the message that Red Notch and Lobo should both be examined for possible exposure to airborne chemical weapons. She also noted that this was the suggestion of her colleague, Agent Bauer.
A nice touch, thought Jack. That way she got it on the record that the idea had originated with him. If it failed to pan out, it was his bad idea, not hers. He held no resentment against her for the gambit. That was how the game was played.
Several miles of mountain scenery unrolled in silence. The throbbing in Jack's head worsened as the car continued to climb. He said, "You wouldn't happen to have any aspirin on you, would you?"
She said, "Headache?"
"A little bit."
"You should get checked out by a medic, make sure you're not suffering from a concussion."
"I'm fine. Just a touch of altitude." Jack didn't want to provide any pretext, medical or otherwise, that might result in him getting pulled off this duty. The violent deaths of Frank Neal and that strange hermit Lobo had given him a personal stake in the mission. It wasn't about keeping Chappelle happy, it was about cracking the case, finding the killers, and solving the mystery of the Zealots' disappearance. He now felt that there was a direct and legitimate threat to the Round Table and its array of high-powered, high-finance invitees.
Armstrong said, "Yes, the height can get to you flatlanders, can't it? That's what happens when you're out of your element."
That could have been a veiled crack about his being an outsider who'd been forced on CTU/DENV through power politics. Jack couldn't blame her for feeling that way, but it didn't stop him from saying, "I haven't done too badly so far."
She said, "You're still alive."
A long pause followed, then she said, "I think there's some aspirin in my pocketbook." Her pocketbook was on the transmission hump between their seats. She steered with one hand and opened the pocketbook with the other. She reached inside it, rummaging around.
The road was no longer straight but twisty, winding around a succession of blind curves. Armstrong drove at a quick pace with no reduction in speed, glancing alternately at the road ahead and the interior of her pocketbook. It made Jack a shade anxious, since the road on his side had only a few feet of shoulder and a knee-high metal guardrail standing between him and a thousand-foot drop.
She said, "I know it's in here somewhere..."
Jack was on the verge of telling her to forget it, that he could get along fine without the aspirin. The car rounded a curve, coming face to face with a two-and-a-half-ton truck coming in the opposite direction. The truck was a foot or two over the centerline and Armstrong had to swerve to avoid it, the two right-side wheels crunching the loose dirt and stones of the shoulder.
She said, "Jerk!" She passed the truck and swung back into the lane so all four wheels once more gripped solid pavement.
Jack had a hollow feeling in the pit of his stomach. He forced himself to adopt a conversational tone. "That was a catering truck. Must be coming from Sky Mount."
She said, "Oh, there's a regular caravan of suppliers going up and down the mountain for the whole time the conference is on. Nothing but the best for the guests, you know. I wish some of those truckers would learn how to drive."
She went back to rummaging through her pocketbook, finally coming up with a bottle of aspirin. "I knew it was in there." She handed it to Jack.
Jack said, "Thanks." He took his time uncapping the container and shaking out two tablets. His mouth was dry from that recent near miss and he needed to work up some saliva. He popped one pill in the back of his mouth, giving his head a toss to get it started down his gullet, then repeated the process.
Armstrong said, "Swallowing them without any water? My, you are tough."
He said, "Can you spare a few extra for later?"
"Keep the bottle if you like."
"No, that's okay, I just want a couple in reserve." He shook four pills into his palm, dropping them into the breast pocket of his jacket. He capped the bottle, handed it back to her.
A gap opened on the west side, revealing a road sloping up a long incline. Armstrong turned left, entering the road. She said, "Masterman Way. That'll take us up to Sky Mount."
Jack said, "This is the first CTU vehicle I've ever ridden in that was a Mercedes-Benz. How'd you manage to work that with the bean counters?"
"Operational necessity. We needed it for protective coloration to blend in with all the other highline models at Sky Mount. Otherwise we'd have stuck out like a sore thumb."
They climbed the slope. The road split into two branches at the summit. A checkpoint had been established there, manned by two deputies from the county sheriff's department. Their car was parked in the middle of the road. Each branch of the road was blocked by a set of wooden sawhorses.
One of the deputies approached the CTU car on the driver's side. Anne Armstrong presented her credentials, including her ID and a pass to enter Sky Mount. The deputy took the documents to his car and radioed in to Sky Mount to verify them. They must have checked out okay because he returned to the car a moment later and gave Armstrong her paperwork. His partner moved the sawhorse out of the way and waved them through, moving it back to block the road after they had passed.
The road switchbacked up the side of a mountain, unwinding in a series of hairpin curves that topped out on a plateau. High mountain valleys in the Rockies are known locally as parks. This park was a vast circular meadow that was open on the east and ringed the rest of the way around by three mountains: Mount Nagaii, Mount Zebulon, and Thunder Mountain. It created an amphitheaterlike effect, with the park being the floor and the mountains being the semicircular tiers that soared up and up toward the zenith.
An amphitheater of the gods. A fit setting for Sky Mount itself. Sky Mount was the name of both the park estate and the fabulous structure that crowned it. The building was a unique creation, part Gothic castle, part Tudor-style manor house, and part chateau. It was an architectural folly on a grand scale, a magnificent white elephant that could be compared only to such equally monumental efforts as the du Ponts' Winterthur estate, Hearst's San Simeon, and the baroque nineteenth-century castles of Ludwig, the Mad King of Bavaria.
The edifice occupied the flattened top of a rise in the park. It fronted south, its long axis running east-west. Its central portion suggested a medieval keep, with a facade loosely modeled after the church of Notre Dame in Paris. Long, multistoried wings extended east and west from it, garnished with balconies and terraces. The spiky roofline bristled with spires, towers, turrets, and battlements. It had been built in the late 1800s, the Gilded Age, and sought to render the intricate architectural "gingerbread" decor of the period not in woodwork but in stone.
The mansion stood at the center of intricately landscaped grounds, a complex of gardens, fountains, galleries and arcades, patios and pavilions. The rise on which it sat had been cut into stepped terraces that were hanging gardens. The rest of the estate spread out from it in a pastoral vista of gently rolling green fields, woodland groves, and sylvan ponds, honeycombed with winding paths and decorated with statuary.
It was one of the damnedest things Jack Bauer had ever seen. He said, "Is that really there or is the altitude getting to me?"
Anne Armstrong said, "It is something, isn't it?"
"It makes Neverland look like a country shack."
Cresting the edge of the plateau and suddenly coming upon Sky Mount had created a visceral impact. Now that Jack had had time to process the big picture, he began to pick up on significant details, the telltale signs of modernity.
The mansion's roofline was studded with satellite dishes, looking like white toadstools that had sprouted out of the crevices of a gnarly rock formation. A helicopter landing pad stood on the flat south of the rise, toward the west end of the park. Two helicopters sat there. A large field in the southeast sector had been turned into a parking lot. It was filled with scores of luxury cars, high-end SUVs, and limousines, all arranged in neat, orderly rows. A line of trucks and delivery vans stretched along a driveway that curved around to the rear of the mansion.
Big black limos and shiny new cars followed the main drive up the rise to the front of the building, disgorging passengers and their luggage. Groups of people, guests, swarmed the grounds, wandering among the arcades, galleries, and gardens. The scene was alive with activity, vibrant color, motion.
The site was well covered by a large number of security personnel, some in uniforms, others in civilian clothes. Groups of guards patrolled the estate in golf carts. Jack thought that was a nice touch.
The open, eastern end of the park was ringed by a black iron spear fence ten feet tall. The sections of fence were interspersed with stone pillars. The park had a single entrance, a double-gated portal that controlled access to a two-lane drive into and out of the estate. The guardhouse inside the gates looked like a Tudor mini-mansion. Jack noted with a pang that it was bigger than his own house back home.
Twin guard shacks stood outside the gates, as did a half-dozen uniformed guards all equipped with sidearms. The county sheriff's department and the state police each had several cars in place, standing well off to the edges of the property, away from the front drive and main gate.
Jack said, "Looks like the local law's been shoved over to the sidelines."
Anne Armstrong nodded. "That's about the size of it. The police are good enough for keeping citizens, protestors, reporters, and other pests off the heights, but they're barred from the sacred precincts, too. Sky Mount itself is guarded by the Brand Agency, a private security firm hired by the Masterman Trust, which runs the estate and the Round Tables."
She drove up to the main gate, halted a dozen paces away from it by a guard. The gate was closed.
The guard came around to the driver's side of the car. He wore a gray cruising cap with black patent leather brim, a long-sleeved gray shirt and black tie, and gray trousers with black vertical stripes on the sides. Blazoned on his left breast was a badge-shaped emblem embossed with the words "Brand Agency." He wore a Sam Brown black patent leather belt and hip-holstered sidearm. All the uniformed guards were identically attired.
He said, "Good morning, ma'am."
She said, "Anne Armstrong and Jack Bauer to see Don Bass, please."
"Is Mr. Bass expecting you?"
"We have an appointment."
"May I see your ID, please? Both of you."
Jack and Armstrong handed over their CTU ID cards. The guard studied Anne Armstrong's photo, comparing it with the driver. He did the same thing with Jack but he spent a lot more time doing it. Jack removed his sunglasses to facilitate the identification. The guard's expression was dubious. He walked around the front of the car to take a better look at Jack through the passenger side window. He still seemed unhappy. It occurred to Jack that his misadventures since arriving in Red Notch had left his appearance somewhat disreputable.
The guard returned their ID cards. He said to Armstrong, "I'll have to contact Mr. Bass at the mansion. Please pull over to the side so you're not blocking the gate."
He crossed to the guard shack on the right and went inside.
Jack said, "I don't think he liked my looks. I've got a feeling I might be underdressed for the occasion."
Anne Armstrong said, "You can always say you're working undercover." She put the car in reverse and backed up along the roadside so the Mercedes was out of the way of any incoming traffic.
She said, "Don Bass heads the security for the conference. Dealing directly with him will cut through a lot of red tape. Among other things, we won't have to check our sidearms at the gate.
"I'm sure you'll like that," she added.
Jack just grinned.
Five minutes passed before the guard returned. He walked briskly to the driver's side of the car, said, "Mr. Bass is unavailable at this time. Mr. Noone will be coming down instead. He's Mr. Bass's assistant."
Armstrong said, "Yes, I know him."
"He'll escort you to the mansion."
"Thank you."
"You're welcome, ma'am. Have a nice day." The guard said nothing to Jack, not even looking at him. He rejoined the other guards outside the gate.
Anne Armstrong said to Jack, "Larry Noone is Bass's number two man. He'll be just as good for facilitating our entry."
Ten minutes later a golf cart rolled down the hill and halted just inside the gate. The driver was a uniformed guard, the passenger a heavyset, bearish man. The latter hopped out of the cart, went through a swinging door to the right of the gatepost, and hurried over to the car.
He was in his mid-fifties, about six feet, two inches and 220 pounds. He wore a canvas duckbilled cap, navy-blue blazer, green open-neck sport shirt and khaki pants. He was balding with a fringe of short blond hair and pale blond eyebrows. Clean-shaven, with a ruddy complexion.
He went to the driver's side and reached in to shake hands with Anne Armstrong. His jacket fell open when he leaned forward, and Jack could see that he wore a short-barreled revolver in a shoulder holster under his left arm. He flashed a big toothy grin like he was glad to see her and said, "Hi, Anne."
She said, "Hello, Larry."
"Don was in conference with Mr. Wright and couldn't get away. Sorry to keep you waiting."
"No problem. Larry, this is Jack Bauer. He's on loan from our Los Angeles division and will be working with us during the conference. Jack, this is Larry Noone."
Noone came bustling around to the passenger side of the car. He flashed another big grin and thrust out a big right hand. "Pleased to meet you, Agent Bauer."
Jack shook his hand. Noone's grip was solid but he didn't overdo it. "Glad to know you. Call me Jack."
"Okay, Jack. Call me Larry."
Noone climbed into the backseat of the car. "Go ahead, Anne, they'll let you through."
The main gate was already opening. It was powered by an electric motor that caused the gate to slide sideways. One of the guards waved her through, and the car drove into Sky Mount.
##
## THE FOLLOWING TAKES PLACE BETWEEN THE HOURS OF 8 A.M. AND 9 A.M. MOUNTAIN DAYLIGHT TIME
Sky Mount, Colorado
Larry Noone escorted Jack Bauer and Anne Armstrong into a reception area where they were met by Marion Clary. She was a gatekeeper for Cabot Huntington Wright, the man in charge of running the Sky Mount Round Table, among his many other responsibilities. Wright's suite of offices was on the ground floor in the southeast corner of the mansion's east wing.
The reception area, an anteroom to the suite, was itself an imposing space, expansive and high-ceilinged, its wood-paneled walls hung with ornate-framed paintings and tapestries. Jack's wife, Teri, was a graphic artist and designer with an art history background, and Jack had absorbed enough from her through osmosis to recognize the paintings as being in the style of Italian and Northern Renaissance masterworks of landscape and portraiture. He knew that Sky Mount's creator, tycoon H. H. Masterman, had been a celebrated collector of the works of the Old Masters and had no doubt that these were not copies but originals worth several million dollars.
Marion Clary occupied a mahogany desk the size of a compact car. She rose and came around it to meet and greet the newcomers.
She was a handsome woman, sixtyish and well-preserved, with carefully coiffed blondish-white hair, fine features, and dark, bright eyes set in a porcelain-colored complexion.
The porcelain was webbed with a network of fine lines when seen close up. She was slim, straight-backed, with good posture. She wore a tailored jacket and pleated skirt, both charcoal-gray; a white blouse with a thin red and yellow paisley kerchief, and black pumps with chunky three-inch heels.
She was already acquainted with Anne Armstrong and greeted her warmly. Noone introduced her to Jack. They shook hands. Her palm was dry, her grip firm.
Noone's handset radio squawked, prompting him to excuse himself for a moment. He stepped a few paces away and held the transceiver to the side of his head, taking a message and responding to it.
He said to the other three, "I'm needed at the guardhouse to iron out some business. Nice seeing you again, Ms. Clary. I'll see you later, Anne—Jack." He went out.
Marion Clary said, "Mr. Wright's meeting with some of the event planners is running a little long. Please excuse the delay."
Jack said, "I thought he was meeting with Don Bass."
"He was, but Mr. Bass was called away unexpectedly a few minutes before you arrived and the planners seized the opportunity to see Mr. Wright for a few minutes. He's scheduled to deliver the opening keynote address at ten and there were one or two last-minute details to finalize."
"Mr. Wright is going to speak today?"
"Oh yes, he always delivers the opening address to the conference. It's a tradition and a high point of the Round Table, if I say so myself. Of course, I'm hardly in a position to be objective, knowing him as well as I do. His talk should be especially interesting this year, what with all the turmoil in the global markets."
"I'm sure," Jack said. He was thinking that if Wright and the high-finance attendees knew of the short-selling bets being made against their companies, there'd be some real turmoil right there in the conference room. But that information was being closely held by Chappelle and a handful of others. Chappelle was as tight at disseminating confidential intelligence as a miser would be in handing out dollars. Which was one of his good points as far as Jack was concerned.
The pattern of shorting had of necessity been made known to CTU/DENV head Orlando Garcia, since it was the wedge that had gotten Jack involved in the local operation. Jack didn't know how far down the line Garcia had passed the intel. He didn't know if Anne Armstrong was aware of it. She hadn't mentioned it, and he wasn't about to volunteer anything on the subject until he was sure she had an irrefutable need to know.
Marion Clary said, "While you're waiting, may I offer you some refreshments? Coffee, tea, or some other beverage?"
Jack said, "Coffee would be fine, thanks."
Anne Armstrong said, "Yes, I'd like some, too, please."
The process was nothing so simple as pouring a couple of cups from a coffee urn. Marion Clary spoke into her desk intercom, issuing a summons. A white-coated server appeared within less than two minutes, wheeling in a serving cart. It held silver pitchers, china cups and saucers, and an assortment of muffins, buns, and pastries. One pitcher held coffee, another held decaffeinated coffee. Jack had the full-octane coffee, black.
It was good coffee, rich, aromatic, flavorful. His stomach growled at the sight of the pastries, but the left side of his face still felt too sore for much chewing so he reluctantly passed on them. Anne Armstrong had the decaf coffee with plenty of cream and sugar. Marion Clary had a cup of tea. The server exited, wheeling away the cart.
Jack's eye was caught by a picture that looked out of place among the Old Masters creations. It was a full-length portrait that hung high on the wall behind the mahogany desk. Its subject was a man dressed in the garb of the late nineteenth or early twentieth century. He had a shock of white hair, a hawklike predatory face, and a white walrus mustache that failed to disguise a self-satisfied smirk. His eyes were hard, narrow, and bright, boldly, contemptuously staring out at the viewer with a go-to-hell directness. He stood in a posture of dominance, hands thrust in his jacket pockets with the thumbs hanging out over the edges, narrow feet spread shoulder-length apart.
It was a masterpiece in its own way, the painter certainly having captured the personality of his subject.
Marion Clary noticed Jack's interest in the picture. She said, "That's a portrait of old H. H. Masterman himself, founder of the trust which bears his name, and the builder of Sky Mount."
Jack thought that if the likeness was an accurate one, the H.H. in his name should have stood for "Hard-Hearted." He looked like a money-grubbing skinflint who would have thrown widows and orphans out in the cold if their eviction would have earned him an extra dime. He settled for saying, "He looks like a pretty tough old bird."
Marion Clary said, "He was a self-made man who started with nothing. He struck it rich with a silver mine near Cripple Creek and expanded into banking, railroads, and real estate. And he did it in the days before income tax. Even in an age of robber barons he was considered something of a pirate." She spoke of him with a kind of proprietary pride.
She said, "His financial interests were centered in Denver and in his later years he built Sky Mount as a vacation home and retreat from city living. It was originally planned as a hunting lodge, but as you can see, it developed into a far more grandiose vision."
A faraway look came into her eyes. "'In Xanadu did Kublai Khan a stately pleasure-dome decree'—according to the poet Samuel Taylor Coleridge. But in Sky Mount, H. H. Masterman built his own dream castle. And unlike the poem, which Coleridge never finished, Mr. Masterman finalized his creation, an architectural poem wrought in stone and timber and furnished with some of the greatest masterpieces of the Old World."
Anne Armstrong said, "You certainly know your subject."
Marion Clary said, "I should. Actually I'm the curator of the estate, in charge of overseeing everything from the upkeep and restoration of the art treasures to making sure the lawns get mowed and the garbage collected."
Jack said, "Sounds like a big job."
She said, "I love it. I live here all year round. Sky Mount is open to the public, except when the Round Tables are being held. It's a major tourist attraction and draws thousands of visitors annually."
Anne Armstrong said, "I shouldn't wonder. It's like a fairyland castle come to life."
Marion Clary beamed. "I can say without exaggeration that it's one of the most fantastic realms in all the world."
"No doubt. But you live here? I think I'd find that somewhat overwhelming, making a home in a setting as stupendous as this. Even intimidating."
Marion Clary shook her head. "It's not as if I live here all alone. There's a permanent party of over a dozen staffers who live here full-time, too. That's not including the tour guides, guards, chambermaids, handymen, gardeners, and all the others who are here during working hours. It takes a small army to keep Sky Mount functioning properly.
"I've lived here for over ten years, and even when I think I know every nook and cranny of it, I'm always discovering new and wonderful things about it. To me it's an old friend. I couldn't imagine living anywhere else. There's so much history here, so many art treasures at every turn...There's a greatness of spirit here that seems to have gone out of today's modern world. Of course, I'm an antiquarian or I couldn't do my job properly. An antiquarian and something of an antique."
Jack said, "Hardly that, Ms. Clary."
"You're gallant, Mr. Bauer." She smiled piquantly, a bit wryly, as if shaking off her visionary mood and returning to the business at hand. "During the run-up to the Round Tables and the conclaves themselves, my role becomes more that of a personal assistant to Mr. Wright. It's the one time of the year that I do see him. His responsibilities as chairman of the board of trustees take him all over the country—the world, really—and he spends very little time at Sky Mount except during the Round Tables."
A stirring of muffled motion sounded from behind the tall set of double doors accessing Wright's inner sanctum. The doors opened outward, allowing the exit of a handful of staffers, young men and women. Some carried portfolios, others briefcases and oversized loose-leaf binders. They looked sleek, well-groomed, fit, competent, energetic, and enthusiastic. They weren't more than a few years Jack's junior, but they made him feel old by comparison.
They crossed the anteroom and exited. A man, thirty, dark-haired, with tortoiseshell glasses, stood in the doorway. He said to Jack and Anne Armstrong, "Mr. Wright will see you now."
The two CTU agents crossed to the portal and entered the space beyond. The man with the glasses closed the double doors behind them, following. The office space was immense, the walls lofty, the windows tall and arched, the ceiling vaulted. The decor was suggestive of the period of Louis XIV, the Sun King, a mélange of neo-classical formalism and rococo ornamentation. The walls were white with golden trim, the deep-pile wall-to-wall carpeting was royal-blue decorated with white fleur-de-lis, emblem of the Bourbon dynasty. There were paintings by Watteau, Fragonard, even a Poussin.
Alcoves held marble statuary and portrait busts with neo-classical themes, Greek gods and goddesses, nymphs and warriors.
Richly ornamented drapes screened the windows, filtering out the morning sunlight. A crystal chandelier hung down from the ceiling, its radiance augmented by strategically placed floor lamps and indirect wall-mounted pinlights and spotlights.
Glass-fronted cabinets contained shelves lined with rows of volumes handsomely bound in gold-embossed leather bindings. There was an antique desk the size of a pool table. Standing in front of it with his hands held behind his back was Cabot Huntington Wright.
Wright's age was somewhere in his fifties. A leonine head was mounted on a pair of broad shoulders. His square-shaped torso hung straight down from those shoulders, presenting a solid, wall-like front. A superbly tailored summer-weight dark blue suit could not disguise the fact of his spindly legs, giving him a top-heavy appearance. His feet were small and narrow.
Lead-gray hair was brushed straight back from the forehead, giving his sleek hair the aspect of a metallic cap. His face was spade-shaped, with the hint of double chin. His upper lip sported a neatly clipped silver-gray mustache of the type that Jack associated with old-time bank presidents and district attorneys.
Wright was the director of the Masterman Trust, a philanthropic foundation with a billion in assets that were disbursed to a variety of do-good organizations, from cultural centers to soup kitchens. He was president of the executive committee in charge of holding the trust-funded Round Tables, and a multimillionaire in his own right.
He crossed to meet Jack and Armstrong as they entered. He said, "I am Cabot Wright." His voice was deep, resonant—rich. Like him.
Jack said, "I know, I've seen your picture in the papers."
He said, "You must be Agent Bauer. Don Bass told me to expect you."
Anne Armstrong said, "Agent Bauer is on loan to us from the Los Angeles branch."
Wright said, "Glad to have you aboard. We can use all the help we can get."
He and Jack shook hands. Wright's palm was smooth, uncallused, but there was strength in his grip. Wright said, "Good to see you again, Ms. Armstrong." He indicated the man in the tortoiseshell glasses, said, "This is Brad Oliver, my executive assistant."
Brad Oliver had a thicket of oily, wavy black hair parted on the side, pale waxy skin, and a cleft chin. He made no move to shake hands. He said, "Hello."
Wright said, "Please accept my apologies for keeping you waiting. Sky Mount is an absolute madhouse today, buzzing with activity. Everyone on my staff seems unable to do without an urgent last-minute consultation with me and they all want to see me at the same time."
Anne Armstrong said, "That's quite all right, Mr. Wright. We appreciate that you're a busy man."
Wright said, "I'm sure you're busy too, with far greater responsibilities." He gestured toward a group of club chairs facing his desk. "Please sit down and make yourselves comfortable."
Jack and Armstrong seated themselves. Jack's chair was straight-backed and thickly cushioned, so comfortable that he wouldn't have minded having one in his living room at home.
Wright said, "May I offer you some refreshments?"
Jack said, "Thanks, but Ms. Clary has already seen to that."
"Ah, one can always trust Marion to observe the amenities. She's a pearl."
"She certainly seems to know her Sky Mount."
"She's our ultimate authority. I go to her when I need to know any esoterica about the layout." Wright went behind his desk and stood in front of his chair without sitting down. "Well. I understand you've got some updates for me this morning."
Jack said, "Shouldn't we wait for Don Bass? That'll save us from having to do a double briefing."
Wright said, "Quite so. Oliver, go see what's keeping Bass."
"Yes, sir." Oliver turned, exited via the double doors, closing them behind him.
Wright sat down. He picked up what looked like an antique letter opener from the desktop and toyed with it, weighing it in his hand. "Are you a history buff, Agent Bauer?"
"Some."
"Then perhaps this should interest you. This letter opener was once the property of Marshal Fouché. Do you know of him?"
Jack nodded. "He was Napoleon's spymaster."
Wright smiled, pleased. "And before that the spymaster of the French Revolution. One of the greatest intelligence officers of all time. He survived both the Terror and the Empire, living to see Robespierre go to the guillotine and Bonaparte go into exile at St. Helena."
He handed the letter opener to Jack. It was sharp-pointed and slim-bladed, as much dagger as letter opener. Wright said, "Imagine, if you will, the secret correspondences numbering in the hundreds, the thousands, all laid bare to Fouché's inquisitive eye by that instrument; the missives of kings and queens, popes and generals, royalists and revolutionaries."
"It's a real collector's item." Jack handed it back to Wright.
Wright said, "Perhaps you recall Fouché's famous maxim: 'The art of the police is in knowing what not to see.'"
"That might have served him well in the Napoleonic Empire, but it's not so apt for today."
"I'm keenly interested to know what you have seen."
Oliver returned with Don Bass in tow. Bass headed the Brand Agency security presence at Sky Mount. He was middle-aged, beefy, with short, curly brown hair topping a head shaped like a cured ham. He had baggy spaniel eyes and a meatball nose; his face was jowly and his wide mouth turned down at the corners. He wore the standard outfit sported by plainclothes Brand operatives, a blue blazer with the company emblem blazoned on the left breast and khaki pants. Big feet were encased in extra-wide, thick-soled shoes. His blazer was rumpled and his pants needed creasing. He carried a dog-eared brown leather briefcase.
He knew Anne Armstrong; he and Jack were introduced. He pulled a club chair up to Wright's desk and plopped himself down in it.
Brad Oliver hovered around the edges of the scene, notepad and pen in hand. Wright said, "That will be all for now, Oliver; you may go."
"Yes, sir." Oliver went out.
Wright said, "So. What are the latest developments in the Prewitt affair?"
Wright already knew about the abandonment of the Red Notch compound and the disappearance of the Zealots. Jack told of his and Neal's night trip to the site; of the discovery of Lobo; of the shooting deaths of Lobo, Neal, and the rifleman; and of the rifleman's partner's getaway. Those were facts. He said nothing about Lobo's tale of hog-faced demons and the green cloud. That was hearsay, and he didn't want to whip up a storm of excitement and possible hysteria on an as yet unverified account, especially his suspicions that some kind of chemical weapons might have been used on the night of the vanishment. Time enough to open that can of worms if and when CTU forensics turned up actual evidence of such substances. He was keeping quiet until then to avoid stirring up a panic.
That went double for Chappelle's SIU detecting the sinister short-selling pattern. That would stay secret until events necessitated otherwise, and he saw no sign of that need yet. The intelligence was an ace in the hole, a trump card that might precipitate the final denouement, and he would keep it well hidden up his sleeve in readiness for the showdown.
Jack finished his self-redacted account of the proceedings. That was the time for Anne Armstrong to speak up if she wanted to surface the possible CW involvement but she remained mum on the subject. If she was in on the secret intel about the recent shorting on the market, she kept it to herself.
Don Bass said, "How do you read it, Jack?"
"I think that the Zealots' disappearing act was accompanied by violence. Maybe there was a schism in the sect, some doctrinal or procedural disagreement that led to a falling out between two or more factions."
"Two or more? How do you figure?"
"Notice the timing. It's surely no coincidence that the disappearing act came on the eve of the Round Table. It's possible that one faction of the Zealots was in favor of a violent action against the conference, another was against it, and a third was neutral, just sitting on the fence not wanting to take sides. Things came to a head and the disputants settled it with a Night of the Long Knives. It doesn't necessarily follow that the pro-violence group took the initiative against the dissidents, but that's the way to bet it. It's not likely that the anti-violence bunch took action against the pro-violence crowd to forestall them. It's possible, but not probable."
Jack went on, "The violent ones did in some of the opposition, maybe all. That would have cowed the faint hearts and the fence-sitters. They all loaded up in the blue bus and whatever other vehicles they had, abandoned Red Notch, and went underground."
Bass said, "What about the ATF agents, Dean and O'Hara?"
Jack said, "They were outside the compound keeping it under surveillance. They had to go to keep from spoiling the Zealots' getaway. It's possible that Zealot assassins took them by surprise and did them in. Dean and O'Hara were monitoring the cult, but nothing on the record shows they were expecting any rough stuff. Up to now, the only thing the Zealots have shot off are their mouths. The killers drove away with the ATF agents' dead bodies in their own car and hid it wherever they hid the rest of the cadre, living or dead, and their blue bus."
Don Bass nodded. "Makes sense."
Cabot Wright said, "Where does this Lobo character fit in?"
Jack said, "Near as I can figure it, he was a homeless guy, a derelict who was living in the sandstone hills above the compound. He saw something on the night of the disappearance. What's more, somebody saw him and sent a kill team to silence him. They didn't know what if anything he told Neal and me so they decided to make a clean sweep."
"They being from the violent faction of the Zealots."
"Possibly." Jack wasn't so sure that that was the case, he had his doubts that the killers were Zealots at all. He even had an alternate theory of the case but for now he was keeping it to himself.
Anne Armstrong said, "We're running a trace on Lobo to determine his true identity. When we know that some more pieces of the puzzle might fit."
Cabot Wright said, "But this is astounding! Where could the Zealots be hiding?"
Jack said, "This isn't my home turf but from what little I've seen of the terrain around here, there's a lot of places where two dozen people and a couple of vehicles could find a hole to hide in. Canyons, gorges, abandoned railway tunnels, ski lodges that've closed for the summer or gone out of business. The cult might not all be hiding in the same spot, either. They could have split up into cells and be hiding in a half-dozen spots, waiting for the go signal to greenlight whatever action they're planning."
Armstrong said, "Search planes and helicopters would be a big help. We could get county and state police pilots to start combing the region."
Wright said flatly, "I can light a fire under them to make that happen."
"But it's got to be done discreetly. A mass panic is the last thing we want."
Wright blanched. "My God, no! That would ruin the Round Table!"
Jack said, "We don't know what the Zealots are up to. Whatever it is, it's vital that they not get into Sky Mount to carry it out."
Don Bass said, "That's something we can do something about! Sky Mount's greatest strength is its defensibility. It's protected by concentric rings of security cordons. The only viable approach is from the east. The mountains provide a natural defense barrier on the other three sides. We've got shooters posted on the high ledges just in case any strike force is mountain goat enough to scale those peaks.
"That leaves us open on the east. We've got the county and state cops controlling all access roads to the park. From there Brand takes over and our cordon is even tighter. Nobody can get in or out of the main gate without proper ID. Between our men on the gate and the police auxiliaries outside, we've got the firepower to repel any mass attack."
Jack said, "Suppose they get close enough to crash the gate with a truck loaded with explosives?"
"The inside of the drive is rigged with a bed of concealed spikes a dozen yards long. If the gate goes down the spikes come up and they'll rip to pieces the tires of any vehicle before it gets more than a couple of feet inside."
"How about if they skip the gate and crash through a section of the fence?"
"The reinforced stone fence pillar posts are strong enough and too close for any truck to get through."
"What about cars?"
Don Bass stroked his chin. "You might have something there. We'll post some extra snipers and run roving patrols of crash cars along the inside of the fence line to harden the targets. We'll pay a double bonus to the crash car drivers—I don't think we'll lack for volunteers."
Jack said, "Sounds good. What about an air assault, a private plane that's a flying bomb designed to crash into the building?"
Anne Armstrong said, "We're ahead of you there, Jack. You came into the middle of the movie on that score. We've got the Air Force and the Air National Guard posted to forestall just such an attempt. The air space for a several hundred square miles around has been declared a restricted no-fly zone for the conference. Any unauthorized aircraft entering the zone will be forced down or shot down. Besides which, it would take a hell of a pilot to be able to fly through these peaks to make the approach."
Don Bass added, "But in case some hotshot should get through, confidentially, we've got an anti-air-craft nest set up in the heights armed with a couple of Stinger missiles as a last resort."
Jack said, "Glad to hear it."
Cabot Wright shook his head sadly. "Lord! The precautions that must be taken merely to hold a peaceful and positive gathering whose purpose is the betterment of society and the national—and global—economy! It's enough to drive one to despair..."
Jack said, "That's the way we have to live today."
The double doors opened and Larry Noone entered, purposeful, grim-faced.
Don Bass said, "What is it, Larry?"
Noone said, "Those ATF agents have just been found."
##
## THE FOLLOWING TAKES PLACE BETWEEN THE HOURS OF 9 A.M. AND 10 A.M. MOUNTAIN DAYLIGHT TIME
Lone Pine Gorge, Colorado
The car was at the bottom of Lone Pine Gorge. The gorge was a narrow, rocky, V-shaped cleft in the foothills of Mount Nagaii.
Jack and Anne Armstrong had to approach it by a dirt road that turned west off Nagaii Drive, traversing several miles of woodlands before curving north to run along the bottom of the slope. The rutted road was in bad shape, and it was a rough ride for the Mercedes.
The road skirted the gorge, bypassing it. The mouth of the cleft was obscured by a lot of brush and would have been easy to miss had it not been for the cars and emergency vehicles parked outside it.
A woodland path branched off the dirt road, leading into the gorge. The path was too tough for the Mercedes. The two agents had to get out and walk. They were challenged by a county sheriff's department deputy posted at the foot of the path to keep out civilians and other unauthorized personnel. They showed their CTU ID cards and were allowed to proceed.
Trees grew on both sides of the gorge entrance, meeting overhead to form a canopy of foliage. The path was little more than a trail, accessible only to heavy-duty SUVs rigged for off-road running. The overhanging trees formed a tunnel through a hundred feet or so of greenery. It was cool and dim under the trees except where sunbeams slanted through gaps between the boughs.
The tunnel ended, opening into a steep-sided ravine bright with sunlight. A thin trickling creek ran through the middle of the bottom of the gorge. Tufts of dry, weedy grass sprouted in clumps along its length.
The rocky terrain otherwise supported little in the way of vegetation. The north side featured a projecting ledge about two hundred feet above the ground on which stood a single tree. A long-dead tree weathered silver-gray, its twisted branches bare of any foliage. Jack guessed that this was the lone pine that had given the gorge its name, although as far as he could see there was nothing about it to identify it as a pine.
Jack was feeling better, his headache had lessened, possibly because of the aspirins or being at a lower attitude or a combination of the two. The left side of his face where he'd been struck still felt stiff and swollen, though.
A few vehicles—a tow truck and two police SUVs—had managed to bull their way up the trail path and into the gorge. A knot of people was centered around a wreck at the bottom of the ravine.
The wreck had been a dark green sedan; now it looked like a piece of metal that had been wadded up into a ball and thrown away. Jack looked up to see where the car had gone off the edge of Rimrock Road some eight hundred feet above. He had to tilt his head far back to see it, so that he was looking almost straight up.
Police and emergency personnel were clustered around the wreck. A few paramedics stood off to one side, waiting; there wasn't much for them to do until the two occupants were freed from the wreck. They'd have little more to do when that time came than to declare them DOA, dead on arrival.
Some mechanics from the tow truck were wrestling with a Jaws of Life device to pry open the collapsed metal, but the wreck was so crumpled up that they were unsure of where to begin applying the pressure and had already gotten off to a few false starts.
Jack stood at the edge of the group, craning to see inside the wreck. A pulpy mass of flesh and tangled limbs was sandwiched inside the collapsed heap, in such a condition that it was impossible to tell if it comprised one body or two.
A man in a pair of gray twill coveralls who'd been laboring in vain to pry open a compressed metal flange looked up and said, "This ain't working. We'll probably have to cut 'em out with a torch."
A county deputy said, "Can't do that here, too much risk of fire."
The mechanic said, "No gas in the tank. It busted on the way down and spilled the contents all over the gorge. Lucky it didn't catch fire and burn."
"Yeah, lucky."
"You don't want to start fooling around with a torch with all that spilled gas around here. Might start off a real blaze."
"Best tow it into town then."
"How? Got to have at least two working wheels on it to give it a tow and there ain't none of them. Nothing to tow."
A man in a short-brimmed hat and a dark suit who'd been listening to the conversation put himself forward. "You can't just leave them out here, for God's sake."
Anne Armstrong told Jack, "That's Inspector Cullen of the Denver branch of the ATF." She spoke in a low voice so that only he could hear it.
The mechanic said, "My advice is to hook it up to the tow truck winch and drag it out of here to the dirt road. Get a flatbed truck out there. Flatbed couldn't get into here but it should be able to handle the access road. Hoist the wreck on the flatbed and take it to town where we can open it properly with the right tools."
Cullen said, "Do it, then."
The mechanic looked him up and down. "And who might you be, mister?"
"Cullen of the ATF. Those are my men in there."
"Oh. Sorry. You'll sign the authorization? I got to know who to bill for it, the county or the state or whoever—"
"The Federal government'll pay for it. Give me the paperwork and I'll sign it and you can get the show on the road."
"Coming right up, mister. Again, sorry about them fellows of yours. These mountain roads are a tricky proposition in even the best of weather." The mechanic went to the tow truck to get the paperwork.
Jack and Anne Armstrong went over to Cullen. She and Cullen were professionally acquainted, having worked joint operations in the past. Cullen had a wedge-shaped face with narrow slitted eyes, a knife-blade nose, and a thin horizontal slit of a mouth. Armstrong and Jack expressed their condolences.
Cullen said, "Mountain road my eye! Dean and O'Hara have been working this territory for years. They were both expert drivers. If that is them in the wreck."
Jack said, "Do you have any reason to doubt it?"
"The condition they're in, their own mothers wouldn't recognize them. But I'm sure it'll turn out to be them, worse luck. It was no driving mishap that did them in, though."
"I'd say that's a sure bet."
Cullen turned his narrow-eyed gaze full on Jack. "You know that or are you just guessing?"
"We lost a man at Red Notch last night and it was no accident. He was shot dead."
"Who was it?"
"Frank Neal."
"Too bad. He was a good man. So were O'Hara and Dean. And it happened last night?"
"Yes."
"Dean and O'Hara went missing the night before, Wednesday. I figure that that's when whatever happened to them happened. The car wasn't found until today. Somebody reported a gap in the guardrail up top yesterday, but the wreck couldn't be seen from up there so it wasn't followed up on. A Boy Scout troop hiking in the area found it early this morning.
"Neal was killed last night, eh? That compound's a death trap even after it's been abandoned. Who did it? Zealots?"
Jack shrugged. "No proof on that either way yet."
Cullen shook his head. "They were always a screwball outfit, but nothing compared to some of the other groups on our list. No history of any real violence apart from minor scuffles at demonstrations, breaking windows, resisting arrest, that sort of thing. We monitored them more as a preventive measure than anything else, to make sure they kept out of trouble.
"Well, they're in it now, right up to their necks. Too bad killers don't hang anymore. Lethal injection is a whole lot less satisfying somehow. But I'll settle for it when we get the bastards."
Anne Armstrong said, "Were your men working on anything specific on Wednesday night?"
Cullen shook his head. "Routine monitoring, maybe stepped up a notch on account of this Round Table meeting. What about your man?"
"Just doing a follow-up, checking out the compound."
"The Zealots must've gone kill-crazy. Maybe Prewitt had a divine revelation that the time had come for him to take up the sword."
"He's not the type for divine revelations. More likely he reasoned that events required him to seize the world-historical moment."
"We'll ask him before he's wheeled into the death chamber on a gurney."
Jack said, "Got to find him first."
Cullen said, "We'll find him."
The mechanic approached with a clipboard with a stack of papers attached. Cullen said, "Excuse me, I've got to take care of this."
Anne Armstrong said, "We've got to be going, too. I'm sorry about your men."
Cullen said, "That goes for me, too. I liked Neal."
"I'd appreciate it if you could send me a copy of the autopsy reports on O'Hara and Dean."
"Will do. Keep me posted on anything you get."
"Of course."
Cullen went into a huddle with the mechanic, scowling as he scanned the estimate of charges. The CTU pair drifted away.
Jack said, "I'm sure a postmortem will show that Dean and O'Hara were dead before they went over that cliff. I'd also like to have them tested for traces of CWs in their bloodstream."
Anne Armstrong said, "It can be arranged, but this wasn't the time and place to bring it up. That aspect will have to be handled with extreme delicacy."
"But quickly. The Round Table is already in session."
"You don't need to remind me of that," she said. "I think we've seen all there is to see here."
Jack nodded. They went back down the ravine and through the arcade of overhanging trees to where their car was parked. Jack said, "There's a familiar face."
He was referring to the MRT's Cole Taggart. Taggart and a county deputy were having words with two bikers. The bikers looked like the real thing, hard-core outlaw motorcyclists. "One-percenters," as they were called, their own mocking self-description to distinguish themselves from the "ninety-nine percent of respectable, law-abiding motorcyclists" that industry spokesmen and proponents for responsible biking enthusiasts routinely invoked to polish up the public image that in their view had been tarnished by the fringe outlaw element.
Not so unusual a sight in the West, where biker gangs were more numerous and firmly established than in the more urbanized areas east of the Mississippi. Denver and its surroundings had more than their fair share of renegade motorcycle clubs.
These two specimens were emblematic of the type. Each sat astride a heavy-duty Harley customized with extended front forks and all the trimmings. The duo were down and dirty in greasy, well-worn denims, but their machines were in top shape, their gleaming streamlined shapes marred only by a coating of dust picked up while cruising the dirt road. The machines weren't dirt bikes built for off-roading but rather muscular cycles designed for high-speed highway long hauls. One thing outlaw bikers can do is ride, handling their machines with the facility of a Cossack on horseback, taking them to the streets or the back trails as they pleased.
Jack's activities in the past had caused him to work undercover operations among outlaw motorcycle clubs with a penchant for gunrunning and operating meth labs, so he eyed these two with a professional interest.
One of them was medium-sized, with long, greasy black hair slicked back and a hipster goatee. His eyes were banded with oversized sunglasses that looked like the kind worn by patients recovering from cataract operations. Jack figured there was nothing wrong with the cyclist's eyesight and that he sported the shades because they provided a kind of effective half mask, obscuring his features. His face above and below the dark glasses was wizened, sharp-featured, and weasely.
The other was big, hulking, pumped up with that comic book superhero physique that comes from steroid use. Reddish-gold hair was combed up in a pompadour and hung down the back of his neck in a classic mullet. His nose was crooked from having been broken several times, and he had a wide, jack-o'-lantern mouth.
The smaller of the two was saying, "We saw that some joker must've gone off the high side but we couldn't see nothing from up there so we came down for a better look."
The deputy said, "There's nothing to see so you can go back the way you came."
The big biker said, "That's some drop. How many people got killed?"
Taggart said, "You can read about it in the papers."
The big biker snickered. "Reading? What's that, man?"
His buddy laughed, said, "That's telling him, Rowdy."
The deputy said, "You can practice by reading a few traffic summonses if you like."
Rowdy said, "Hey man, what're you picking on us for? We ain't doing nothing."
Taggart said, "Go do it somewhere else."
The deputy said, "We don't rightly care for your kind hereabouts. Make yourself scarce, unless you'd like to spend ninety days as a guest of the county."
Rowdy turned to his buddy, said, "You heard the man, Griff. No point hanging around where we're not wanted."
Griff said, "I can take a hint."
The dirt road was narrow and the bikers had to maneuver their machines to turn around. Their backs were to Jack and for the first time he could see their colors, the emblem of their club that was sewn to the backs of their sleeveless denim vests.
Their insignia depicted a demonic, quasi-humanoid Gila monster straddling a souped-up cycle on two stumpy legs. It bore the legend: "Hellbenders M.C."
Hellbenders Motorcycle Club. Jack had heard of them. A tough outfit, very tough. They'd been in the headlines about six months ago when some of their leaders had been swooped up in a high-profile gunrunning bust.
One area of equipment where their bikes came up short was in the muffler department. The choppers took off with an earsplitting crack of iron thunder. The machines churned up dust clouds as they vroomed east on the dirt road, heading for Nagaii Drive.
The deputy and Taggart watched them go. The deputy muttered, "A-holes. You know if you search them bikers you'd find a half-dozen violations easy. And you know what'd happen if I did that?"
Taggart said, "No, what?"
"The sheriff'd have me on the carpet for a royal ass-chewing, for diverting precious departmental resources on them hog-riding fools when we're already stretched thin providing security for the Round Table."
Taggart laughed. "That's why he's sheriff. He's got his priorities right. Nothing's more important than making sure that nobody crashes that private party for Richie Riches."
The deputy said, "Soon as they haul that wreck with those two stiffs in it out of here, I got to go back to patrolling Sky Mount."
"You and me both, brother."
"I don't know what the big deal is. It ain't like that heap was going anyplace."
"It had a couple of ATF guys in it, so that makes it Federal."
"Big deal."
Taggart joked, "Maybe they were drunk when they went over the edge."
That got a laugh out of the deputy. "That's what I'm going to do when the conference is done—get drunk. And not before then. They've got us all pulling double shifts while it's on. All leaves and days off canceled for the duration."
Taggart said, "Times are tough all over."
Jack and Anne Armstrong had to cross the road to get to where their car was parked. Their path crossed that of Taggart and the deputy. The deputy had seen their credentials when they first arrived so he let them pass without comment.
Jack and Taggart made eye contact. Jack said, "Small world."
Taggart smiled. "Miller Fisk is mad at you."
"He can have a rematch anytime he wants."
"He ain't that mad. Anyhow, Hardin's got him pulling roadblock duty way up in the hills right now. He's so teed off at Fisk that Fisk is lucky he's not cleaning latrines at the station instead."
"Is Hardin mad at him for abusing a prisoner or for getting chopped down to size?"
"There's a question. You'll have to ask Bryce the answer to that one."
"And you?"
"Far as I'm concerned, that overgrown plowboy got what's been coming to him for a long time. 'Course, I ain't related to him, like Bryce is."
"Is that right?"
"Fisk is Hardin's nephew. You don't think Fisk made the MRT because he's a regular Sherlock Holmes, do you?"
Jack said, "I'm going to try to not think about it at all."
Taggart said, "Not a bad idea. See you around."
Jack nodded to him. Anne Armstrong was already in the car, waiting for him. She looked pleased. She said, "I just finished talking with Central. Good news for a change."
Jack said, "What've you got?"
"A lead, maybe. They've turned up somebody who's seen the blue bus."
##
## THE FOLLOWING TAKES PLACE BETWEEN THE HOURS OF 10 A.M. AND 11 A.M. MOUNTAIN DAYLIGHT TIME
Dixon Cutoff, Colorado
Cletus Skeets said, "Is there going to be any reward money in this?" He pronounced it "ree-ward." He was of medium height, reedy, with muddy eyes, a three-day beard, and a prominent Adam's apple.
Anne Armstrong said, "It's possible, Mr. Skeets."
Skeets indicated Ernie Sandoval. "Because that's what he told Mabel. That there was a reward."
Ernie Sandoval, a CTU/DENV investigative agent in his mid-thirties, was short, chunky, moon-faced, with close-cropped dark hair, dark brown eyes, and a thick mustache. He'd been doing some good old-fashioned legwork all morning, canvassing stops along some of the back roads in the Red Notch area. He'd found a possible lead at the Pup Tent, a greasyspoon diner located on the Dixon Cutoff, a pass between Mount Nagaii and Mount Zebulon that was used by local drivers and long-haul truckers.
Jack Bauer, Armstrong, Sandoval, and Skeets were standing in the parking lot of the Pup Tent, a roadside eatery on the north side of the east-west running Dixon Cutoff. The diner was a white wooden-frame building that looked like what it was, an overgrown hot dog stand. A hand-painted marquee on the roof depicted a cartoonish hot dog in a ten-gallon hat and cowboy boots firing off a pair of six-guns. The legend beneath it read: "Ask about our famous foot-long Texas Wieners!"
The structure sat in the middle of an elongated gravel parking lot, extra-sized to accommodate big-rig trucks whose drivers wanted to grab a bite on this side of the mountains. There were no big rigs in the lot now, just Armstrong's Mercedes, the Toyota pickup that Sandoval had been making his rounds in, and a couple of cars belonging to diner patrons and personnel.
Sandoval had warm brown eyes and an engaging smile. He said, "That's not entirely accurate, Mr. Skeets. What I told your employer was that we are prepared to pay a modest sum in the event that the information you supply helps us to locate the people we're looking for."
Skeets said, "That's a reward, ain't it?"
"Call it what you like. The information has to be verified and significantly useful in discovering the whereabouts of the persons of interest. In other words, if your tip pays off, we pay off."
Skeets licked his lips. "How much?"
"That depends on how useful the information is. We won't be able to assess that until it's been properly evaluated and followed up on."
"A couple of thousand bucks?"
"A couple of hundred bucks, maybe." Sandoval was starting to get irritated. "We're not exactly buying the plans for an atomic bomb here, we're just trying to find some missing persons."
Skeets got a shifty look in his eyes. "Well, I don't know about that. I had to come down here on my time off. I work nights and I ain't had my proper sleep. Could throw off my recollection, that is if I did see anything at all."
Jack, impatient, decided to play bad cop to Sandoval's good cop. He said, "Maybe a stretch in jail will improve your memory."
Skeets tried to tough it out. "You got no call to arrest me. I ain't done nothing. I got my rights!"
"You're a possible material witness who's impeding a Federal investigation. That's grounds for holding you in custody for forty-eight hours. For starters."
Skeets's eyes bulged and his Adam's apple bobbed. "Now hold on a danged minute—"
Jack had an inspiration. "Maybe you know a state cop named Miller Fisk?"
"Who don't? Everybody around here knows him. They ought to, he throws his weight around enough. He's a real mean SOB."
"A session with Fisk might help you get your mind right. Why don't we give him a call and tell him to come on down?"
Skeets held up both hands palms-out in a gesture of surrender. "You don't have to do that! It's all coming back to me now."
"Okay—give."
Skeets said, "I'll tell you what I told Pedro."
"Who's he?"
"The dishwasher on the night shift. We was both working on Wednesday night. Thursday morning, actually. Mabel, she goes home at midnight, so there's just me and Pedro holding down the fort. I do the cooking and he does the cleaning and we get by. Don't get many customers between midnight and dawn, 'cept for some long-haul truckers and night owls with a load on who want to get something in their bellies to help them sober up. So the two of us is plenty.
"Anyhow, pretty late in the shift, it was dead quiet so I went out for a smoke. Mabel used to be a heavy smoker but she quit and now she don't allow no smoking inside nohow. Not the customers or nobody. She knows if anybody's been smoking in the diner when she ain't there, she's got a nose on her. I tried it once or twice and sure enough, as soon as she comes in, first thing at six o'clock in the morning, she wrinkles up her nose and sniffs around and says, 'Cletus, you been smoking.' She told me off but good both times and after you've been told off by Mabel, you've been told. So when I want a smoke I go outside, which is what I done that night."
Jack said, "What time was that, Mr. Skeets?"
"Well, I went out a couple of times, but the time we're talking about was four-thirty in the ay emm. I remember that 'cause I looked at the clock and said to myself, Just another hour and a half to go and I'm out of here. I went outside and sat down on the front stairs and lit up.
"No sooner do I fire up a smoke than I seen a pair of headlights coming. There ain't much traffic at that hour and I said, Dang, don't that beat all? Soon as I take a break, a customer rolls along. Figured it was a customer because there ain't hardly no traffic at all at that hour."
"Which direction was the vehicle coming from, Mr. Skeets?"
"East, from the east. Only it wasn't no vehicle, it was a bunch of them. A regular convoy. A pickup truck, a couple of cars, and a bus, all riding together in a line. They didn't stop, neither, but kept right on going."
"They went west?"
Skeets nodded. "Yep. West, toward the pass. I wouldn't have thought nothing much about it, 'cept for the bus."
"Why is that?"
"It was a school bus. Just struck me funny somehow. I mean, here it is the middle of summer. Ain't no school in session. Summer school, maybe, but they don't need no bus for that and even if they did, they don't run at four-thirty in the morning. Ain't even no schools around here, for that matter. Sure ain't none west of the pass. So I said to myself, What-all do they need a school bus for in July?"
Skeets went on, "Another thing that struck me about it was the color. It was a funny color. Every school bus I ever seen was yellow. Not this one, though."
"What color was it?"
"Blue. It was blue. The diner's all lit up at night so truckers can see it from a long way off and they'll have plenty of time to slow down their rigs and pull in. So I could see the bus nice and clear and sure enough, it was blue."
"Did you notice anything else unusual about it?"
"Nope."
"Were there any people inside it?"
Skeets shrugged. "Danged if I know. It was dark—the bus, I mean. I couldn't see inside it."
"What happened then?"
"It drove by, along with the rest of the convoy. They was heading west, toward the pass. That's all I know. I finished my smoke and went inside. I told Pedro what I saw. It's funny to see a school bus in summer, a blue one at that. Shoot, you can't get these no-account kids today to go to school even when it's in session. They'd rather be going off skylarking and cutting up capers...I went back to work and didn't think no more about it. Not till today, when I got roused out of a sound sleep by a phone call from Mabel telling me to get my butt down here to talk to you folks. Which I have now done."
Sandoval said, "And we appreciate it, Mr. Skeets."
"Any chance you could show that appreciation with some folding green?"
"If your information proves to be instrumental in locating the missing persons, there may be some financial remuneration forthcoming."
"I don't suppose you could let me have a couple of twenties now, just on account, say?"
Sandoval shook his head. "It doesn't work that way, Mr. Skeets. Sorry."
Skeets looked glum but resigned. "Reckon I got to trust you folks then."
Sandoval reached for his wallet. Skeets's eyes brightened for an instant, only to dim again when Sandoval took out a card and handed it to him. Sandoval said, "If you should think of anything else in connection with the sighting, you can reach me at that number any time of the day or night."
Skeets dropped the card into the left breast pocket of his T-shirt. He assumed a conspiratorial manner and said, "Hey, who's missing anyhow? You can trust me to keep my mouth shut."
Sandoval said, "We're not authorized to release that information at this time."
Skeets nodded, not surprised. "You done? I can go?"
"Yes, sir. Thank you for your cooperation."
"Reckon I'll go inside. Mabel can't begrudge me a cup of coffee, after getting me out of bed on my time off to come down here. She will, though."
Skeets started toward the diner, halting after a few paces. He turned and looked at Jack, said, "You won't say nothing to Fisk about me calling him a mean SOB? I'd hate to get on his bad side. Not that he's got a good one.
"Er, forget that I said that, too," he added.
Jack said, "You can rely on our discretion, Mr. Skeets. Just like I'm sure we can rely on you not to discuss this conversation with anyone else."
Skeets said, "I'm a closed book. I won't crack to nobody about nothing."
"Fine."
Skeets took a few more steps, paused, looked over his shoulder. "If there is any reward..."
"We know where to find you."
"I won't have to split it with Mabel and Pedro, will I? After all, I done the seeing, not them."
Sandoval said, "We'll be in touch."
Skeets crossed to the diner, entered it, the front screen door banging shut behind him. Jack said, "There goes a public-spirited citizen."
Sandoval said, "You forgot something, Jack."
"What?"
Sandoval indicated the sign on the roof. "You didn't ask about their famous Texas wieners."
"Yeah, I missed that one."
"I didn't. I had one while I was waiting for you and Anne."
"How was it?"
Sandoval held his stomach with both hands, looking slightly bilious. "Don't ask."
Anne Armstrong said, "Never mind about that. What do you make of Skeets's story?"
Jack said, "He knew about the blue bus without being told. The Zealots' disappearance has been withheld from the media so he didn't pick it up there."
Sandoval said, "I doubt friend Skeets knows the difference between the Zealots and a basketball team."
Jack said, "The time of the sighting works, too."
Sandoval said, "It's also significant that he didn't come to us. If he were a walk-in I'd be suspicious that he was a plant because the public doesn't know CTU is looking for the Zealots. No, he told Pedro the dishwasher, who told Mabel, who told me when I came in two hours ago. I was canvassing the area, asking around if anybody'd seen anything unusual early on Thursday morning. She volunteered the information, and when I expressed interest she phoned Skeets and told him to come on down."
He added, "Skeets was right about that, too. When Mabel tells you to do something, you get told."
Jack looked westward, where a ribbon of road stretched across the flat before disappearing in a gap between Mount Nagaii and Mount Zebulon. "Where does that pass lead?"
Sandoval said, "Straight through to the western slopes of the mountains."
Anne Armstrong, thoughtful, said, "Shadow Valley is in the pass."
Sandoval said, "That's right!"
Jack looked from one to the other. They both looked intrigued. He said, "What's Shadow Valley?"
Armstrong said, "A canyon complete with its own ghost town, Silvertop. There used to be a big silver mine there before the lode played out. It's been abandoned for years—decades."
Jack said, "You can hide a lot of things in an old mine."
##
## THE FOLLOWING TAKES PLACE BETWEEN THE HOURS OF 11 A.M. AND 12 P.M. MOUNTAIN DAYLIGHT TIME
Shadow Valley, Colorado
Shadow Valley is a canyon that cuts deep into the foothills of the southern slopes of Mount Zebulon. The canyon's main trunk runs north-south, with numerous side canyons branching off to the east and west.
A Toyota pickup truck and an Explorer SUV turned right off Dixon Cutoff, entering the canyon. The pickup held Anne Armstrong and Jack Bauer; the SUV held four well-armed members of a CTU/DENV tactical squad.
Jack, Armstrong, and Sandoval had earlier gone from the Pup Tent to the CTU mobile command post at Pike's Ford, located at the foot of Sky Mount. Various bits of business had to be taken care of at the CP. Armstrong had changed clothes, shedding skirt and loafers for khaki pants and hiking shoes in preparation for probing the canyon ghost town. Jack had taken the opportunity to check his pistol, making sure that it worked properly. It had been out of his hands from the time he'd been busted in Random until it was returned to him at the Mountain Lake substation. It was basic tradecraft: he needed to make sure that it hadn't been tampered with during the time it had been out of his possession. He didn't think it had been but he wasn't going to stake his life on an assumption by going back out in the field with a potentially defective weapon. He'd examined it before going to Sky Mount and everything had seemed to be in order, but he hadn't been able to give it the acid test by firing off some rounds.
He did so at Pike's Ford at a crude but effective firing range that had been set up well away from the group of mobile home trailers that comprised the temporary command post. One of the pluses of this mountain locale was that there was plenty of empty land and no neighbors to kick about guns being discharged in the vicinity. The weapon fired effectively and accurately with the original ammunition that had been in the magazines, which had also been confiscated by the MRT during his arrest. He also fired off some rounds from the Pike's Ford ammo, with equally positive results. He would have liked to break down the weapon and clean it but there wasn't time for it now. He armed himself with a half-dozen magazines of fresh ammo and a couple of handfuls of loose rounds that he put in the side pockets of his jacket.
He checked his cell phone for voice mail. A number of messages had been left for him by Ryan Chappelle, offering no new information but instead first requesting and then demanding updates and progress reports on the Sky Mount situation. Jack left them unanswered for the time being. He was covered; he was in the middle of an investigation and had no time to spare answering Chappelle's queries. Chappelle could stew in his own juices for a while longer, while Jack prowled Shadow Valley to see what if anything could be found. Jack grinned to himself at the image of Chappelle on the boil waiting for a response back in L.A.
He spent the time more productively by grabbing a few sandwiches from a vending machine in an area of one of the trailers that served as a galley. He heated them in a microwave and wolfed them down with a couple of cups of coffee. They tasted like cardboard, and the coffee was no prize, either, but it felt good to have some food in his belly to restore his strength and energy.
CTU/DENV head Orlando Garcia was off-site but was represented here by his assistant SAC, Dirk Vanaheim. Vanaheim conveyed Garcia's directive that the searchers would be outfitted in protective gear and escorted by a four-person tac squad. This precautionary measure was a result of the attack last night at Red Notch.
Ernie Sandoval had some inquiries he needed to follow up on so he remained behind at the CP. Jack, Armstrong, and the tac squad mounted up in their two vehicles and headed out. The Explorer was a regular battle wagon, with bulletproof glass, armor plating, solid rubber tires, and a souped-up engine with big muscle to propel the machine at high speeds. The pickup truck was customized with some extras but wasn't bulletproof. The Explorer took the lead, the pickup following. Armstrong drove, Jack riding in the cab's passenger seat.
Twenty minutes' motoring put the two-vehicle convoy at the mouth of Shadow Valley canyon. The canyon's floor of hard-packed dirt was crisscrossed with countless tire tracks. Anne Armstrong said, "Off-road enthusiasts use it for dirt biking and ATVs, and teens come out here to park and get drunk and generally party."
Jack said, "Might not hurt to see if any of the locals have been reported missing in the time from early Thursday morning to now. Someone might have seen something that somebody else didn't want seen and been disappeared for it."
Armstrong radioed in to Central requesting information on that score. Central replied they'd look into it and get back to her.
The canyon was seemingly unoccupied today with no other humans in sight. Arroyos, gullies, and gorges branched off to the east and west of the main trunk. A couple of miles deeper into the ravine brought the CTU team in view of a sugarloaf-shaped bluff on the western side of the passage. Its slope was honeycombed with vertical and horizontal trails and speckled with bits and pieces of old weathered wooden structures: here a half-collapsed platform supported by a framework of trestles and cross-braced timbers, there a section of diagonal sluice spillway leading to nowhere, and scattered throughout, the ruins of shacks and sheds.
This was the one area of the canyon where an attempt, however feeble, had been made to restrict public access. The base of the slope was bordered by a chain-link fence; a gate barred the way to a dirt road stretching to the summit.
Armstrong said, "That's Silvertop. The bluff held a rich lode of silver ore and was mined extensively in the 1890s. It gave rise to a boom town, but when the deposit played out, the town did, too."
The SUV and the pickup halted at the gate at the foot of the road that climbed the hill. The long, gentle rise was stepped with terraces that had been carved out so that different veins of the lode could be mined.
The occupants dismounted from the vehicles. The tac squad consisted of four men, Frith, Bailey, Holtz, and Sanchez. All were fit, well-conditioned, clear-eyed, and cool-nerved. They were action men—trigger pullers. Frith and Holtz were armed with M–16s, Bailey and Sanchez with M–4 carbines. They were all equipped with sidearms, too, semi-automatic pistols in hip holsters.
CTU wanted to minimize its footprint in Sky Mount and environs so the squad members wore civilian clothes, utility vests, and T-shirts, baggy cargo-pocketed pants, and hiking boots. They, like Jack and Armstrong, were all outfitted in flak jackets. The squad men openly carried their weapons. Clandestinity only went so far before being superseded by the demands of security.
Frith, long-faced and lanky, was in command of the squad but subordinate to Jack and Armstrong. There was a sensitive issue of protocol here. Jack expected to be obeyed when he gave an order but he was the outsider here, an unknown quantity to the others. He would let Armstrong take the lead, couching his commands where possible in the form of suggestions. The key phrase: where possible.
It was hot out here in the open, hotter still because of the flak jackets. Jack consoled himself with the thought that it was nothing compared to how hot things could get without the protection of the Kevlar vest if trouble happened. Shooting trouble.
Shadows were short under a midday sun that stood almost directly overhead, pouring its rays straight down into the canyon. Sun-baked flats and rock walls threw back the heat. The CTU team all wore headgear. Frith and Sanchez wore baseball caps, Holtz and Bailey wore soft, shapeless fabric hats. Armstrong and Jack wore unmarked baseball caps that had been supplied from tac squad stores at Pike's Ford. Jack's headache had subsided in recent hours but the pounding heat made his temples throb, a portent of returning discomfort.
The fence enclosing the bottom of the hill was old, rusted. It was hung with a number of NO TRESPASSING signs, all of which were nearly illegible due to being bullet-riddled by high-spirited sportsmen. Gaps opened in the barrier where whole sections had been trampled flat by dirt bikers and ATV riders, the tracks of whose vehicles had worn clearly marked trails up the sides of the bluff.
The gate blocking the main access road was made of stronger stuff and stood solid and intact. It was secured by a length of padlocked chain. Both chain and lock were shiny and bright. Jack said, "They look brand-new, like they were put here yesterday."
Anne Armstrong said, "Maybe they were."
"But by whom?"
The Explorer's rear hatch was opened, revealing a variety of gear. There were hard hats and flashlights, coils of rope, picks and shovels and other tools. Bailey pulled on a pair of work gloves, picked up a pair of bolt cutters, and went to the gate. He snugged the open pincers against one section of a link of shiny new chain and squeezed the handles. He had to put some muscle into it, his face reddening and veins standing out on his forehead.
The section parted. He did the same thing to the other half of the link until it parted, too. The chain fell away from the gate. Holtz opened the gate all the way. Bailey stowed the bolt cutters and the work gloves in the rear of the SUV and closed the hatch.
Jack and Armstrong got back into their vehicle, the squad men into theirs. The SUV climbed the main road up the long, low-angled slope, the pickup following. They crested the summit, rolling to a halt on a spacious, flat-topped expanse.
Anne Armstrong radioed Central, informing them that the team vehicles would now be going out of service. Central acknowledged, their response scratchy with static. Armstrong said, "Radio reception is spotty here due to interference from the canyon walls and the mountains, Jack. Our portable handsets lack the power to send or receive messages to Central. The truck radio is stronger because it works off the vehicle's battery, but even it's barely adequate for communication purposes. The same goes for the SUV. So when we go out of service here, our outside comm is really closed down."
Jack nodded. They got out of the pickup. The tac squad piled out of the Explorer. The squad men shared one trait in common: ever-alert eyes that were constantly scanning the surroundings, always in motion, never lingering for too long on any one fixed point. The eyes of hunters. Jack recognized the behavior pattern because he was the same way.
The top of the bluff was a rough oblong the size of several football fields put together, its long axis running north-south. Its western edge bordered the foot of a long, low ridge beyond which could be seen lines of wooded hills, rising in tiers to southern spurs of Mount Zebulon.
The flat-topped mound was littered with remnants of what had once been a thriving mining town at the end of the nineteenth century. It didn't fit in with Jack's notions about a western ghost town. It looked more like a war ruin. Some of the buildings were made of brick or stone and might have been factories or warehouses. Others were rows of wooden frame buildings that had collapsed into heaps. Not a single structure was fully intact. One consisted of fragments of two stone walls that met to form a corner, another was a heap of plank board rubble with part of a stone chimney remaining. That was the pattern.
The weed-grown site had been plucked, pillaged, and otherwise deconstructed by generations of vandals, looters, and troublemakers. No standing section of wall or foundation was without layers of spray-painted graffiti, no pane of glass was unbroken. Mounds of ashes and charred timbers marked the spot where houses had been burned down.
It was a popular site with the locals, judging by the remains of bonfires and the profusion of broken bottles and empty beer cans. Tire tracks from two-, three-, and four-wheeled vehicles overlaid the ground. The raggedy fence and gate below had proved no deterrent to the many who'd driven their machines up and down the sides of the bluff.
Jack and Anne Armstrong each had a set of field glasses. The entire CTU team was equipped with headset communicators, miniaturized transceivers consisting of an ear bud with a flexible plastic frame that fitted around one ear, extending into a curving plastic tube the width of a pipe cleaner that terminated near the wearer's mouth in a condenser microphone. They all now donned the transceivers, running a comm check to make sure each unit was properly sending and receiving.
The receiver bud buzzed in Jack's ear as he and the others sounded off. Audio quality was good, the signal strong and clear now that they were all grouped together. Whether it would remain so once they were scattered around the mound and no longer in one another's line of sight had yet to be determined.
Anne Armstrong addressed the group. "Watch your step. The town was built on top of the mine and is shot through with vertical air shafts. Some are boarded over and posted with warning signs, others are open holes in the ground that go down a hundred feet or more."
She had accessed the maps and diagrams of Silvertop's inner workings that were archived in the computers at the Pike's Ford CP and printed them out, bringing them along in a folder. They were crude and incomplete but workable as a rough guide to the sugarloaf butte's underground world. They would be of little use, however, unless a starting point was first found.
Sanchez said, "What exactly are we looking for?"
Jack said, "Anything that might indicate the presence of some or all of Prewitt's two dozen Zealots in the recent past or right now. For all we know, they could be holed up in some nest up here or nearby. Or down below, in one of the abandoned tunnels. Some of them could be watching us right now."
A ripple went through the others, causing the squad members to spread out so they wouldn't be bunched up in one tight target group. Bailey said, "That's a happy thought."
Holtz scanned the landscape and shook his head doubtfully. "It sure looks deserted."
"So did Red Notch last night but there were two killers there, and now a CTU agent is dead." Jack paused to let that sink in before continuing. "Zealots might be using this as a base, a rendezvous, or a staging area. They might have stored hardware or vehicles here. They might have left a cell behind for security while the others moved on. They might have come here for some unknown purpose on Wednesday night and moved on."
Holtz said, "They might not have come here at all."
"It's possible. But even then this won't be wasted effort. If Silvertop comes up clean, that's one possibility we can cross off the list and narrow the search perimeters in the hunt for Prewitt and his crew."
Anne Armstrong said, "A final word of caution. Silvertop is what's called an attractive nuisance. It's a hangout for high school kids and might also harbor squatters and hoboes. So if you see someone suspicious, make sure you know who you're shooting at before opening fire. We don't want to accidentally shoot some teens who came up here to get high or make out."
The team split up into two search groups, one consisting of Jack and Frith, the other of Armstrong, Holtz, and Sanchez. Bailey stayed behind to guard the vehicles and keep watch on the canyon below.
Jack and Frith would start at the southern end of the hilltop and work their way north, Armstrong's group would begin at the northern end and work south. Jack and the squad leader crossed on a diagonal toward the southwest, a path that skirted the southernmost of the ruins.
The air was still, with barely the breath of a breeze. The sun was moving toward its zenith. Jack hadn't gone very far before breaking out into a sweat. His face was slick with wetness, and beads of perspiration trickled between his shoulder blades and down his back.
Frith said, "What do you reckon our chances are of finding something?"
Jack said, "I think it's worth a look or we wouldn't be out here. We've got a witness who saw Prewitt's blue bus and some other vehicles heading for this vicinity around the time of the disappearance early on Thursday. There've been no reported sightings of the convoy west of Dixon Cutoff. The Zealots may not be here now but they might have been here and left evidence that'll point toward where they went."
"The area was covered yesterday by search planes. They didn't turn up anything."
"As an old GI ground-pounder, I believe there's no substitute for on-site recon to see things the flyboys might have missed."
Frith grinned. "I've got to agree with you there. I'm ex-infantry myself."
Jack said, "I'm also a firm believer in taking the high ground." He pointed to the ridge at the western edge of the bluff. "That should be a good spot for surveying the terrain."
Bare dirt gave way to weeds that soon reached mid-calf height. Frith said, "Watch out for snakes."
Jack looked to see if he was kidding. Frith was dead serious. He said, "Rattlesnakes like to prowl the tall grass for field mice and other varmints."
Jack was careful from then on to keep an even warier eye on the ground he trod. Ten minutes' hiking put him and Frith at the foot of the western ridge. It was a short walk to its low, rounded summit. The far side of the ridge dropped steeply into a deep hollow with a thin trickle of a creek running along the bottom. A higher, more heavily wooded slope rose on the other side.
Jack and Frith stood on the near side of the ridge, below the ridgetop to avoid skylining that would more readily reveal their presence. They faced east toward the ruins on the bluff. Jack took off his sunglasses and slipped them in the left breast pocket of his jacket. The jacket was thin but it still added to the oppressiveness of the heat. He would have liked to have shucked it off but it held his spare clips and loose rounds, and it was worth putting up with a little additional discomfort to have the extra ammo ready to hand. He reminded himself that compared to summer in Baghdad or the Sudan—he'd gone on missions in both—this was brisk, crisp weather.
Sweat stung his eyes and he wiped them against his sleeve in the crook of his arm. The field glasses hung from a strap around his neck. He tilted back the lid of his cap, raised the binoculars to his eyes, peered at a row of ruins, and adjusted the focus, sharpening it to clarity.
He mentally divided the landscape into grid squares and methodically scanned them one by one, working his way along the line of structures from south to north. He saw empty window frames with weeds growing behind them, blackened timbers that were the skeletal remains of a house's framework, ash heaps, and piles of rubble. He saw nothing out of the ordinary, no deviation from the pattern of abandonment and neglect.
Jack said, "Nothing." He removed his cap, freed the binocular strap from around his neck, and handed the field glasses to Frith. "Maybe you'll spot something."
Frith scanned the scene, studying it long and hard. "Nope."
"Let's try it a little further to the north."
They went north along the ridge. Jack contacted Anne Armstrong via his transceiver headset. She reported that so far her results were negative, too.
Jack and Frith halted some fifty yards north of their first position. Jack pointed the optics at a new section of the scene and resumed his methodical grid square survey of the ghost town.
More of the same unrolled itself through the twin lenses until he came to the shell of a long, shedlike structure whose long axis ran east-west. Its short, western wall was mostly intact but slanted inward at a forty-five-degree angle. The part of its north wall he could see was also tilted inward at an acute angle. The southern side was no wall at all but a heaped-up woodpile that was holding up a section of the collapsed roof.
There was something vaguely off and out of place about the ruin's outline that caught his attention, prompting him to give it a closer study. The roof, what there was of it, which wasn't much, was broken into sections that stuck out of the heap at odd angles.
There was a hole in the south side of the roof. It was covered with what looked like a canvas tarp. The fabric was a tan, sandy-gray color. It would have been hard to see from ground level on the bluff, and even from the elevated vantage point of the ridge, he had to look twice to make sure what he was seeing.
He looked a third time and still saw it. He handed the field glasses to Frith. "Take a look at the roofline on the south side and tell me what you see."
Frith peered at the shed, the lower half of his face impassive below the binoculars. He fiddled with the focus knob and looked some more. "There's a covering on the roof...a tarpaulin of some sort."
Jack said, "Who puts a tarp on an old ruin? Somebody who wants to hide something inside, maybe."
Frith lowered the field glasses and looked at Jack. Jack said, "Let's go see."
He took the field glasses and slipped the strap on around his neck. He and Frith started downhill, angling toward the shed. Jack said, "Let's make sure it's not a false alarm before alerting the others."
Frith nodded, said, "Right." He'd been carrying his M–16 so that its barrel pointed at the ground; now he held it level but off to the side. Jack reached inside his jacket to give his gun butt a little nudge, adjusting it in the holster so it would come free easier if he needed it in a hurry.
They came down on the flat and made for the shed's southwest corner. Each step closer made it more evident that a tarp was fixed to part of the roof. Its tan, sandy color was much like the terrain at the top of the bluff—surely no coincidence. The ground around the shed was churned up with a lot of tire tracks, ruts, and broken earth.
Frith suddenly made a wide detour around a patch of ground. Jack froze, said, "Snake?"
Frith shook his head, showed a toothy grin. "Bear scat."
Jack took a closer look. The ground was littered with animal droppings. A sizable pile, not human. He said, "You can tell they're bear?"
Frith said, "Hell, yeah. I grew up in these parts and I live here now. The bear population has been allowed to grow until now they're a real nuisance. They're not afraid of humans and they like the taste of people's garbage better than the food they can forage in the woods."
Jack joked, "I'm warning you in advance. Zealots and killers are one thing but if I see a bear, I'm running."
"Won't do you any good. Bears're fast. They can run faster than you can." Frith eyed the pile with an outdoorsman's discernment. "The spoor's at least a day old so we're probably in the clear."
Jack noticed that they were both talking in low, hushed voices. He said, "I'll tell you this: it wasn't a bear that put that tarp on the roof."
They continued onward, closing on the shed. Planks in the tilted west wall were cracked and splintered at about midbody height and bore fresh gouges and scrapings. The ground on the west side of the shed was noticeably torn up.
The south wall was a massed rubble of broken boards and beams. The edges of the canvas tarp hung down over the top of the pile. Football-sized rocks had been placed along the rim to hold it down and pin it in place.
Jack said, "I want to see what that tarp is covering up." The heaped rubble on the south was too unsteady to climb. The east side, the building's front, was in similar condition, a junk pile.
The long north wall looked more promising. The northwest corner of the shed was its most intact section. Part of the roof there was solid. Much of the wall was broken into slablike sections. There were a couple of empty window frames but the roof had fallen in, blocking a view inside.
Jack tackled the northwest corner. He took off the field glasses and set them down. He pulled his hat down tight on his head. A beam end protruded from the wall at about chest height. Jack hung on to it with both hands, testing it with his weight. It seemed solid enough.
Frith gave him a boost, allowing Jack to scramble up the side of the tilted wall and stand on the beam end. Jack reached up, grabbing the overhang of the roof with both hands, steadying himself. He chinned himself up to the top of the wall, booted feet scrabbling against the boards. He grunted and panted as he heaved his upper body onto the roof.
The wood creaked and groaned under him, giving him a bad moment, but it stayed in place. He got his feet under him and rose into a half crouch, ready to jump clear at the first sign of an imminent collapse.
He could see where a line of nails the size of railroad spikes had been hammered into the wood along the edge where the roof had broken off and fallen in. They anchored the near end of the tarp in place. They looked new. He really wanted to see what was underneath that tarp.
He dropped to his knees and lay prone on the roof. It seemed solid underneath him. He thought that if it had held the weight of whoever drove the nails it could hold his weight. He bellied his way to the edge.
The tarp was tough and nailed down tight. If he only had a knife...But the tarp wasn't nailed down on the south side of the shed, it was held in place by rocks. He clawed at the canvas, trying for a handhold. The tarp sagged in the middle, there was some play in it. He grabbed a double handful of a fold in the fabric and started pulling it toward him.
The tarp was heavy and didn't want to move. He tugged the fold over the edge of the roof. Now he could rest his arms on the roof and pull the canvas down toward him. He had the advantage of gravity and his weight working for him. He heaved and pulled.
The tarp yielded, folding toward him. There was the sound of rocks falling down the other side of the shed. Jack kept pouring it on. More rocks fell until there weren't enough of them to hold the tarp in place.
The tarp fell through the hole in the roof, except where it was nailed down on Jack's side. It didn't fall far. Something underneath was holding it up.
Jack stuck his head over the edge and looked down. The tarp was draped over a whalelike shape that filled the collapsed shed, nearly reaching what was left of the roof beams. He reached inside the hole, heaving the tarp toward him with both hands, slowly uncovering what lay beneath.
He said, "Huh!" He was too out of breath to say anything else. He sounded part surprised, part triumphant.
Frith stood with his head tilted back, looking up, but he couldn't see what Jack saw. He called, "What is it?"
Jack said, "The blue bus."
##
## THE FOLLOWING TAKES PLACE BETWEEN THE HOURS OF 12 P.M. AND 1 P.M. MOUNTAIN DAYLIGHT TIME
Silvertop, Colorado
There was no stopping Jack, he had to see the thing through. Now that he knew what lay within the collapsed shed, he had a better idea of how to proceed and the tolerances of the structure he was scrambling around on. He was avid for clues but not at the cost of breaking his neck.
The bus was fitted into the shell of the shed so that its front was at the east end. Jack toed the edge of the roof and jumped down through the gap to the top of the bus. It was a short drop, only a few feet; the shed was low and the bus was tall. He came down toward the rear of the vehicle, the impact of his landing making a hollow booming sound. He landed on his feet, knees bent to absorb the shock.
The shed had been knocked down around the bus. Debris hemmed it in on all sides. He kicked the rest of the tarp off the roof; it hung down like a curtain from where it was nailed to the roof. Sunlight streamed in through the hole in the top of the shed.
The roof of the bus was slightly curved but provided solid footing. He knelt facing the right side, thinking to hang over the edge so he could look through the windows. But there was nothing to hang on to and he didn't want to risk sliding off the edge headfirst.
He walked across the roof to the front of the bus. Rubble was piled up to the top of the hood but no higher. The right front door was blocked by too much debris to allow him to open it. He stepped down on to the hood and hunkered down there, facing the windshield. It was opaque with a coating of dust. He rubbed his sleeve against it to clean it and peered through the pane.
The interior was dark, thick with shadows. Jack put his face up close to the glass, holding a hand against the side of his face to screen out the glare.
The bus was empty.
That surprised him. He couldn't see much through the gloom, but as far as he could tell the bus contained no bodies. As far as he could tell.
He had to know for sure. He sat down on the hood. He was able to reach down and pick up a rock from the top of the pile of debris crowding the vehicle. He picked up a big one and brought it down hard on the center of the windshield. The glass puckered where the leading edge of the rock hit it, a spiderweb of cracks radiating out from the point of impact.
He struck again, harder. The spiderweb expanded, the pane becoming translucent as if frosted in the center. He bashed it a few more times, turning his head away from it to protect his eyes in case of flying glass shards.
The windshield was made of safety glass. It held until it reached its breaking point and then it came apart all at once, disintegrating into a mass of crystal cubes that looked like several shovelfuls of miniature ice cubes. They went crashing into the bus, clearing out the windshield frame.
A wave of heat and stink came pouring out through the opening. The bus had been shut up tight, all windows closed, causing massive heat to build up inside. The stink was the smell of decay. Jack caught a whiff of it. He felt his gorge rise and he had to fight to keep from gagging.
The gloom inside the bus was not static but dynamic, flowing, pulsing—buzzing. Its source was a horde of flies, much of which came pouring out of the hole. Jack climbed on top of the bus's roof and walked to the rear of it, filling his lungs with fresh air.
He waited a few minutes for the worst of it to clear before returning to the front of the bus and squatting on the hood. The smell was still pretty rough. He'd have covered his mouth and nose with a handkerchief or piece of cloth if he'd had one but he didn't, so he had to make do. He held his breath and stuck his head through the windshield frame.
The stench came from masses of dried blood on the inside of the bus. The central aisle was smeared with it. So were the three steps leading down to the front door. The bus had seemed sealed tight but the flies had gotten inside. They always do, somehow.
The safety glass had come apart in cubes that looked like rock salt. There were no jagged, razor-edged shards. Jack brushed aside the fragments on the hood. He crouched almost double, sticking a foot through the frame and stepping down to the driver's seat. The driver's area was free of blood.
Jack eased himself through the frame into the bus. It was like stepping into a baker's oven. Sweat sprang out from every pore. He breathed through his nose as shallowly as possible. A cloud of flies buzzed around him. He waved his hand in front of his face, batting them away, but they kept coming back.
He took out a small flashlight from one of his pockets and switched it on to dispel some of the murky shadows. He made his way down the aisle toward the rear of the bus, the flashlight beam gliding over rows of seats, the floor and walls. Some seats were bloodstained but most were not. A few side windows were cracked but none had been broken. No bodies were in view. He ducked down to shine the light under the seats but there were no bodies there, either.
He worked his way to the back of the bus. It lacked an emergency rear door. A mass of dried blood stained the floor and back panel. It was reddish-brown and several inches thick. The evidence seemed to indicate that there had been a number of bleeding bodies at the back of the bus, that they had been dragged to the front and out the door.
Jack figured he had seen all there was to see for now. The forensics team could take it from here. He wanted out.
He went to the front of the bus, using the driver's seat as a stepping stone to climb through the windshield frame and out on to the hood. He hopped up on the vehicle's roof and went to the rear. He boosted himself onto the top of the shed and jumped off. He landed on the ground with knees bent, rolling on his shoulder to absorb the impact.
Anne Armstrong, Holtz, and Sanchez had joined Frith and they were all waiting for him. Bailey was still back at the vehicles keeping watch. Sanchez said to Jack, "You look pretty shook, man."
Jack took some deep breaths, filling his lungs with clean air. He could still taste the blood reek in his nostrils and at the back of his throat. Holtz had a canteen. Jack took a mouthful of warm water, swished it around in his mouth, and spat it out. He drank some more before returning the canteen.
Anne Armstrong said, "What did you find?"
Jack told them. Armstrong said, "What do you make of it?"
Jack said, "I'm only guessing based on what I saw. The Zealots didn't just pull a disappearing act on Thursday morning. There was a purge, too. One faction cleaning up on a dissident element, say. The victims were killed or wounded at Red Notch. Maybe some were killed and some only wounded. The entire compound cadre cleared out in the blue bus and some other vehicles. Our witness Skeets said there was a convoy of a couple cars and trucks along with the bus. The victims were in the bus.
"Somewhere along the way but most likely here at Silvertop the bodies were disposed of. The bus was backed into this shed, which was a wreck already. The killers finished the job, probably by battering it down with one or more of their other vehicles. It wouldn't take much to bring the walls down considering the age and state of disrepair of the shed. You could do it with a pickup truck or SUV. The west wall is broken in at just about the right height for a truck bumper and there are fresh scrapes and gouges on the boards.
"What the collapsed shed didn't hide was concealed under the tarp. It's the same color as the surroundings and would blend right in with the scenery. Especially to any air searches doing a flyover."
Sanchez said, "I don't get it. Why go to all that trouble?"
Jack said, "The blue bus was a liability. Too big, too obvious, and too well-known. And one more reason: the surviving Zealots didn't need it anymore. They were able to leave in the other vehicles that made up the convoy. Which tells you another thing—there couldn't have been too many Zealots left out of the original two dozen or so."
Frith said, "That must've been some purge."
Jack nodded. "A real Night of the Long Knives."
Armstrong said, "It sounds plausible but one aspect puzzles me. After taking such pains to hide the bus, why not leave the bodies inside?"
Jack said, "I picked up on that, too. It's a key question. Why not leave the bodies inside? It suggests that discovery of the bus by the authorities is less important than the discovery of the bodies. For some reason, the bodies must not be found. Why not?
"One answer comes to mind. Whose bodies are they? What if Prewitt himself was one of the ones purged? Suppose the cult leader and his loyalists were eliminated by an upstart faction. That development would electrify the rest of his crowd, namely the hundreds of rank-and-file members outside the inner circle. Many of whom are known to reside in this state to be close to their guru.
"The usurpers could be the ones planning a strike against the Round Table. Prewitt and his loyalists opposed it so they had to go. But the plotters still require the assistance of Zealots outside the Red Notch cadre to carry out their plan. True believers who'd jump to obey the commands of their grand exalted leader Prewitt would balk if the orders came from someone else, some upstart who's trying to take over the whole works."
Armstrong frowned, stroking her chin. "Prewitt's death—murder—would have the members scrambling like an overturned anthill if it were known."
Jack went on, "Or it could work the other way. Maybe Prewitt's in favor of a strike and liquidated all those who opposed him. That would split the cult, too, at a critical time when unity is required for a Sky Mount action."
He smacked a fist against his palm. "All of which makes it vital that those bodies be found—and quick!"
Armstrong said, "Yes, but how?"
Jack said, "I think I've got a lead. A clue. If I'm right we won't have to look very far." He indicated a gaping hole in the ground about a hundred feet east of the shed. "See that ventilator shaft?"
The others turned to look at where he was pointing. He went on, "There's something different about it from the other holes in the ground on top of the bluff. I noticed it when I was up on the roof of the shed. It stands out when you see it from above. What it is, is that the soil around the hole is a different color from the rest of the terrain. It's darker. Like maybe somebody raked it up to cover their tracks."
Armstrong said, "It's worth a look."
The group crossed toward the shaft. The sun was a bit past the zenith, and the team members cast blobs of shadow that slanted slightly east.
The mouth of the shaft was an unnaturally regular circle of blackness gaping in the middle of the ground. It was not boarded over or fenced in. It was about thirty feet in diameter and was ringed by a brown band of soil. Beyond the ring the ground was light brown streaked with tans and grays.
Sanchez said, "It is a different color."
Jack said, "There's no tracks running through it, either."
Holtz said, "That doesn't mean anything. Nobody's going to ride a dirt bike or off-road vehicle too close to the edge."
Jack said, "Not many weeds or bushes, either. And no trash, bottles, beer cans, and the like."
The group fanned out in an arc bordering a section of the dark band. There was a clear line of demarcation between it and the surrounding lighter-colored soil. Jack dug his heel into the light-colored soil, gouging out a patch several inches deep. The soil that he uncovered was the same dark color as the ring bordering the shaft. He said, "How close can you get to the edge here anyway?"
Anne Armstrong said, "I wouldn't get too close."
Jack stepped into the ring of dark soil and moved toward the rim of the shaft. He moved slowly, carefully, halting about four feet away from the edge. He could see the edge of the rim opposite him on the other side of the shaft. The shaft was a hole bored straight down through the ground.
He took off his baseball cap, folded it in two, and stuck it in his back pocket. He got down on his knees and lay flat on the ground, feeling the warmth of the earth beneath him. Frith and Sanchez crouched behind him, each holding one of Jack's ankles—a safety precaution in case the ground at the rim should give way.
Jack stuck his head over the edge and looked down. The shaft plunged more than a hundred feet straight down. The sun was almost directly overhead, allowing him to see most of the bottom of the hole, except where a fingernail sliver of shadow edged the western rim. A mound of loose dirt and rubble lay at the bottom of the pit. He couldn't tell what color it was.
He eased back from the edge and had Frith hand him the field glasses that the latter had been holding. Jack took another look, this time through the binoculars. He could see now that the bottom of the pit was a junction point with four tunnel mouths opening on it. The tunnels were set ninety degrees apart. The dirt mound covered most of the tunnels so that only their arched tops showed above it.
The binoculars brought the dirt mound into clearer focus but he was still unable to draw any conclusions from it. It looked the same color as its surroundings. Maybe it was a trick of the light, maybe not.
Something sticking out of the dirt mound might have been a tree branch or it could have been a half-buried body part, an arm or a leg. Maybe it, too, was a trick of the light, maybe not. He couldn't tell from up here.
Jack withdrew from the edge and once more felt the relief of having his feet planted on solid ground. He said, "I want to see what's at the bottom of that pit."
The maps that Anne Armstrong had printed out earlier indicated the location of the tunnel leading to the shaft in question. Guesswork and a degree of uncertainty were involved because the original maps were old and not definitive. But the tunnel she and Jack selected seemed to fit the bill.
Silvertop's north, east, and south faces had had so many tunnels drilled into them that they looked like Swiss cheese. The entrances had all been sealed up a long time ago. It would be necessary to break into the desired tunnel, but the team had come well prepared for such contingencies.
The tunnel mouth was located on the southern slope about a hundred and twenty-five feet below the summit and a hundred yards or so east of that face's western edge where it met at right angles a ridge running north-south, a lower section of the same ridge that Jack and Frith had climbed earlier to survey the top of the bluff.
The pickup truck and SUV were moved close to the southern edge of the hilltop overlooking the tunnel. Hard hats, flashlights, pry bars, crowbars, a pair of bolt cutters, and other tools were unloaded from the rear of the Explorer and distributed among the team members, except for Holtz. Holtz would remain behind to guard the vehicles and keep watch on the canyon as Bailey had done earlier.
Frith and Holtz had the two M–16s, and Frith wanted someone armed with that weapon to stand sentry duty. The M–16 was better suited for long-distance shooting than the M–4s wielded by Bailey and Sanchez. The squad leader had not thought it necessary earlier for the posted sentry to be so armed, but that had been before the blue bus had been found. That finding upped the potential threat level. Frith was going down to the tunnel so Holtz would stay up top.
Armstrong radioed the news of the discovery to Central at Pike's Ford. Central replied that it would be sending a forensics team out to the site. The lab crew had to come out from headquarters in Denver and would reach Silvertop in roughly two and a half hours. Armstrong informed Central that her team would now attempt to access the shaft through the tunnel. Central acknowledged and Armstrong signed off.
Jack, Armstrong, Frith, Sanchez, and Bailey climbed down the slope to the tunnel. The face's low grade made for an easy descent.
The south face, like those of the east and north, was terraced. The tunnel accessing the shaft was aproned by a fifteen-foot-wide ledge. The entrance was a rounded arch ten feet high with a base of about the same width. It was boarded over. A metal sign nailed to the planks warned that trespassing was a Federal offense and that violators would be prosecuted to the full extent of the law. It bore the seal of the U.S. Bureau of Mines.
Jack and Bailey went to work on the barrier with crowbar and pry bar. The boards were old and flimsy and came apart with little trouble. There was a squeal of rusty nails giving way and the splintering of planks. A few minutes' hard work was all it took to open up a man-sized gap in the wooden wall. One of the last boards to give way had the sign on it; it hit the dirt with a thud.
Sanchez said, "Now you've done it. Wait till the Bureau of Mines hears of this."
Bailey said, "I'll never tell. It's top secret."
Frith said, "Let's widen that hole for easier access and exits." He and Sanchez took their turn with the tools and soon doubled the size of the gap.
Jack, Armstrong, and Bailey were going inside, Frith and Sanchez would remain behind at the entrance. The three would-be tunnel probers donned hard hats and equipped themselves with flashlights. Jack hefted a pry bar. It was five feet long with a pointed tip and a wedge at the opposite end. He said, "Might come in handy for poking into that mound at the bottom of the shaft."
Bailey said, "So will this." He picked up an entrenching tool. It had a three-foot-long shaft with a fold-out sharp-pointed spade at one end.
Frith said, "Leave your weapon here, you won't be needing it underground."
Bailey handed him his M–4. "I hate to be without Baby."
Frith said, "I'll take good care of it."
"Here's the ammo pouch." Bailey unslung an olive drab canvas pouch that he wore suspended over one shoulder by a strap. It bulged with loaded clips for the M–4. He gave that to Frith, too.
Sanchez said. "Hey, I hope none of you have claustrophobia."
Bailey said, "This's a hell of a time to be asking but no, not me."
Armstrong smiled and said, "I'm fine."
Jack said, "I'm used to being in tight spots."
Frith saluted them with a half wave. "Have a nice trip."
Jack stepped through the gap in the boards and into the tunnel, Armstrong and Bailey following. It felt noticeably cooler once he was out of direct sunlight. Light flooded through the hole in the barrier, shining about a dozen feet into the interior. Beyond that the tunnel got dark in a hurry.
The tunnel had been carved out of living rock and was shored up on the walls and roof at regular intervals by timbers and crossbeams that were thick with dust and dark with age. The floor was covered by a layer of dirt. No human footprints marred it but it was marked by tracks made by small varmints, most likely marmots and field mice. There was a flinty smell in the air underlaid with a trace of dampness and moisture; otherwise it seemed fresh and cool.
The hard hats had built-in flashlights above the center of their rounded brims. Jack reached up and switched his on, the others doing the same. He said, "The air seems pretty breathable."
Anne Armstrong said, "That's those air shafts working, providing natural ventilation. There's a slight draft blowing from deeper in the tunnel."
Jack said, "Yes, I can feel it."
She said, "Silvertop has no history of methane pockets or other gases, so we won't need respirators."
She went a dozen paces into the tunnel, blackness engulfing her except for the beam from her hard hat's built-in light. She took a spray can from one of her pockets, removed the cap, and turned to face the wall on her left. She thumbed down the nozzle, spraying an arrow on the rock at shoulder height. The arrow, pale green and glowing in the dark, pointed back toward the tunnel mouth. She said, "Luminous paint for trail markers. Just in case."
Jack said, "Good idea."
She switched on her handheld baton flashlight, sending a beam deep into the tunnel's interior. Jack and Bailey did the same.
Bailey said, "Before we get started—I'm the most expendable so I should go first." He looked Armstrong in the face, then Jack. He grinned. "What, no arguments?"
Jack would have liked to take the point but what Bailey said made sense. Armstrong must have felt the same; she nodded in agreement.
Jack said to Bailey, "You take the pry bar, it'll be useful if you need to probe any doubtful footing." He swapped Bailey the bar for the entrenching tool.
They fell into a file with Bailey at the head, Armstrong in the middle, and Jack bringing up the rear. Bailey gripped the bar at mid-shaft with his right hand, holding it horizontally at his side. He said, "Here goes nothing," and started off, the others following.
The tunnel drove north deep into the guts of Silvertop bluff. The trio came to the first branching at fifty yards in and halted. Mouths for side pocket excavations opened to the left and right of the main tunnel.
Jack looked back. The exit was a dot of brightness a long way off. Armstrong sprayed a green arrow on the main tunnel wall. Jack ran a comm check on his headset transceiver. Frith's reply was mushy with interference but intelligible.
They continued onward along the main tunnel. It was much cooler here, sending a chill along Jack's spine. He hoped it was from the coolness. There was something primal about venturing deep into the bowels of the earth, an instinctual aversion to having all those thousands of tons of solid rock between oneself and the open air. A nagging anxiety that poked up from somewhere deep in the psychic basement and sought to override logic and cool-nerved competency.
Jack was mission-oriented, though, and the fear of failing to carry out his duty was far greater than any emotional dreads or apprehension could ever be. The confidence born of hard training and self-mastery asserted itself, narrowing his mental focus to the job at hand. He had to admit, though, that being a miner must be a hell of a way to make a living.
The trio kept on moving forward. There was blackness ahead, beyond the reach of their electric torches and hard hat lights; and blackness behind, the point of light that was the exit having long since been swallowed up by darkness. The three of them were encapsulated in a glare of artificial brightness from their electric lights that glided through the tunnel like a glowworm inching along a sunless pipeline.
Jack reminded himself that the more distant the exit, the closer they were to their objective. Twice more they came to junctions where side passages branched out from the main tunnel. Armstrong marked the rock wall with a green glowing arrow each time. The comm check at the first such junction found Frith's reply breaking up into a garbled word jumble of meaninglessness. The next comm check reduced Frith's transmission to a crackling burst of static. They were out of communication with the outside world.
Walls remained upright, the ceiling unrolled seamlessly, and the tunnel floor continued rock-solid. Bailey had no need of the pry bar to probe doubtful patches of footing; there were none.
There was a change in the air now, a taint of rottenness that rode the current of cool air coming from deep within. It evoked another primal response, raising the hairs on the back of Jack's neck. He knew that smell: it was the scent of death.
Bailey halted, causing the duo in his wake to also fall still. He said, "Whew! Get that?"
Armstrong's nostrils crinkled with distaste. "And how!"
Jack said, "It won't be long now."
They started forward. The darkness must have heightened their other senses because it was some time before they could make out a fuzzy patch of grayness far ahead. The reek of rot and decay had grown with every step and was quite strong now.
It was the herald of the vertical shaft and the mound that lay at its bottom. The trio hurried forward toward the light, drawn to it.
Bailey started coughing, deep hacking coughs that he managed to suppress with difficulty as they neared the end of their quest. Armstrong fastened the hook at the end of her flashlight to her belt, freeing her hands so she could tie a handkerchief over her nose and mouth.
Jack envied her the handkerchief; he wished he had one so he could follow her example. He breathed through his mouth as much as possible, panting as though he were on the final lap of a marathon.
Daylight loomed ahead, not much of it, but what little there was seemed neon-bright after the blackness of darkness through which they had come. The glare was minimized because the mound at the bottom of the pit reached almost to the top of the tunnel's rounded archway where it met the vertical shaft. The dirt and rocks at the top of the heap could be glimpsed through the narrow space left unfilled.
Bailey stopped short so suddenly that if Armstrong's reflexes had been any slower she would have bumped into him. The pry bar slipped from his hand, striking a hollow rattling sound against the tunnel's rock floor.
Rage battled revulsion with rage winning, allowing Bailey to overcome a fit of gagging in order to choke out an obscenity. Armstrong reeled as if from a physical blow.
Jack knew what was coming, had known for a long time, having first guessed the truth up on the hilltop in what now seemed an eternity ago. He was taken aback by the extent of the devastation, though. The slaughter.
The mound was nothing more nor less than a mass grave, a heap of bodies piled high. Many bodies, male and female. The mound would have been higher except that some of the bodies had rolled into the tunnel. No doubt the same thing had happened at the other three tunnel mouths at the junction of the shaft. The overflow had lowered the pile's height.
The corpses had been thrown into the shaft and a mass of loose dirt and rocks and rubble thrown on top of them to cover them up. Enough dirt and debris had been shoveled into the pit to mask the atrocity when seen from the surface but not nearly enough to hide the pathetic remains when seen from below.
Armstrong said, "My God! How many of them?"
Jack said, "Twenty? More? Most of the Zealots, if not all. They're not missing anymore." He took a certain pride that his voice was able to maintain a steady, even tone.
Something fell with a thud on top of the mound. It had fallen a long way and hit the dirt pile with a loud, thwacking slap. It sat there emanating a sizzling sound like bacon frying on a griddle.
It was a bundle of dynamite, sticks of dynamite held together by several loops of tape. The sizzle came from the length of fuse cord that curled out of one end of the bundle. A short length that grew shorter with every eye blink.
Jack grabbed Armstrong by the shoulders, picked her up bodily, and turned her around, giving her a shove that propelled her a half-dozen paces deeper into the tunnel. He shouted, "Run!"
Bailey was already in motion, spinning and leaping forward away from the shaft.
Armstrong ran, Jack close at her heels. She broke into a sprint, arms and legs pumping, rising on the balls of her feet, accelerating with a burst of speed.
Jack and Bailey were right behind her, running neck and neck. The tunnel was wide enough to accommodate both of them.
There was a chaos of pounding footfalls and their resounding echoes, a blur of hard hat lights and flashlight beams flickering over rock walls as the trio fled, racing to put some distance between themselves and the bomb in the pit.
The dynamite exploded.
##
## THE FOLLOWING TAKES PLACE BETWEEN THE HOURS OF 1 P.M. AND 2 P.M. MOUNTAIN DAYLIGHT TIME
Shadow Valley, Colorado
Out of the frying pan, into the fire. Out of the bomb blast and into a firefight. Jack Bauer, already brutalized by the effects of the explosion, now found himself in a raging gun battle with a ruthless strike force.
It was a blessing in a way. It meant that he was alive and now faced a foe he could come to grips with.
The wraithlike nature of the opposition, up to now as hard to get hold of as a fistful of smoke, had resolved itself into flesh-and-blood attackers who were trying to kill him and what remained of the CTU team. Flesh could be made to bleed, and Jack ached for a reckoning with the enemy.
He ached, period. But he knew it could have been worse. To ache is to be alive, and to live offers the prospect of a righteous revenge.
It had been a lucky break that the bundle of dynamite had landed on top of the pile instead of rolling down its side to fall into the tunnel. That had been the second lucky break, actually. The first had been that the bundle survived the long fall without detonating the blasting caps and triggering off the sticks of TNT on impact. Dynamite is relatively stable; it's the blasting caps that are fluky, fickle, and chancy. It was blind fate that had caused the bundle to hit the mound in such a way as to avoid touching off the caps. The soft dirt at the top of the heap must have cushioned the fall to prevent premature detonation.
The fuse had been short but long enough to give the trio precious time for a good running start. Time? Time had seemed to stand still during that nightmare interval of mad flight away from the shaft.
More luck: the shaft and the pile of bodies at its bottom had absorbed most of the force of the explosion. What got through was devastating enough.
The blast came like the Trump of Doom at the End of Days, rocking all creation with a shock wave that mingled light, heat, and noise in a rush of pure force. Jack was lifted up and catapulted bodily by a senses-shattering pressure that wiped everything blank.
He came to in a howling torrent of darkness. The darkness was incomplete, lacking the utter black of totality. He saw as much as sensed a whirlwind of smoke and dirt streaming over him. That he could distinguish gradations in the murk meant that at least one light was still working.
It wasn't his. His hard hat lamp had gone dark and the flashlight had fallen from his hand. He was unsure whether he even still wore a hard hat. He felt around the top of his head but numbed fingers and stunned senses were unable to feel the difference between his skull and the protective headgear.
Some of the murk thinned as reality returned with each heartbeat. He was on his hands and knees and a light dangled back and forth, swinging pendulum-like in front of his face. Was it the sun? The moon?
Hands were tugging at him, hooked under his arms and urging him to his feet. The chaos was strangely silent, drowned out by the ringing in his ears. The picture came into tighter focus. The shining globe waving before his eyes resolved itself into the lens of a blazing flashlight. The lens was cracked but the beam still shone.
The flashlight hung from where it was hooked to Anne Armstrong's belt, hanging down from the side of her hip. She was crouching over him, trying to help him stand up. He knelt on the tunnel's rocky floor, smoke and dust roiling all around him.
It was hard to breathe in the murk-laden air. Mouth and nostrils seemed filled with dirt. He coughed, choked, spat, managing to clear his throat. It helped him draw a breath, then another.
Armstrong's face was close to his, weirdly underlit by the flashlight's glow, her eyes wide and staring, her features harsh and angular. Her mouth was moving but Jack was unable to hear what she was saying due to the roaring in his ears. More imperative than words was the pull of her hands urging him upward.
He said, "I'm all right!" He mouthed the words but couldn't hear them. He reached out with his right hand, touching the tunnel wall. He braced himself and rose to his feet, lurching into a half crouch. A wave of dizziness overswept him and he stumbled sideways, bumping his head against the wall. A chinstrap throttled him; that's how he realized he was still wearing his hard hat.
He got his back against the wall and stood there with legs bent at the knees until the dizziness passed. Armstrong tilted the flashlight at her hip so that it shone at a forty-five-degree angle. It was pointed at something to Jack's left and behind him. His gaze followed the direction of the beam.
It shone on Bailey, crawling forward head-down on his hands and knees. He was bareheaded, having lost his hard hat in the blast. The view wavered as banks of dust and smoke rolled by. Bailey raised his head, looking up when the light hit him. The whites of his eyes stood out in a dirt-smeared face. Blood trickled from his nose and the corners of his mouth. He lowered his head and continued crawling forward.
Jack staggered to one side of Bailey and Armstrong the other. They each took hold of an arm and tried to lift Bailey to his feet. Jack could barely stay on his own. He tottered, almost falling before managing to right himself. A fall would be disastrous; he was unsure whether he'd be able to get back up. He spread his feet wider for balance; it helped stabilize him.
He and Armstrong somehow managed to get Bailey up and standing. Bailey got his arms across their shoulders as they propped him upright. Armstrong tilted the baton flashlight level so that it shone forward, pointing the way ahead.
The dust and smoke clouds were drifting away and ahead of them, seeking an exit at the far end of the tunnel, wherever that might be. They must be going in the direction opposite that from which the blast had come. That was the way the trio must go, too.
That was good. Jack's sense of direction was scrambled and he would have been just as likely to go the wrong way as not. The streaming airborne debris was a signpost showing the way out.
The three agents started forward. Jack and Armstrong had to half carry, half drag Bailey at the start. Each lurching pace forward was a win. Jack's thoughts flashed back to Army days, to forced marches with full field battle gear where trainees were pushed to the limits of endurance and beyond, to prepare them for combat conditions that would demand that one must march or die. There had been times later during his term of service when that became literally true and survival depended on the ability to put one foot forward and then the other, slogging along until you were all used up and continuing to keep on going after that.
Now as then he concentrated all his thoughts and energies on forward motion. The passage became a torturous nightmare, a seeming treadmill to oblivion. But he kept on going, reaching down deep somewhere to find something to pick up those feet and put them down.
Bailey began carrying more of his own weight. That helped. Jack was in top condition, and his tremendous endurance began to reassert itself. That helped more.
Armstrong kept the flashlight hooked to her belt. That was smart, keeping it safely tethered so as to avoid having it slip free from a betraying hand to fall and shatter, blacking out their sole source of light. Jack had his pocket flash but he didn't know if it still worked. It took all the energy he had to continue moving ahead and propping up Bailey; there was none for checking on the pocket flash, not when Armstrong's light served their purposes. She needed to be able to use her right hand to tilt the light to point the way, so Jack did what he could to shoulder more of Bailey's weight to enable her to do just that.
They came to a junction point. That was tricky because the murky clouds streamed into all three tunnel mouths in search of an exit. It would be easy to take the wrong branch. Jack steered Bailey to a tunnel wall and put both their backs against it and held him up in place, freeing Armstrong to search for the green arrow marking the correct tunnel.
She found it and returned, got Bailey's other arm across her shoulder, and the trio continued their onward march. It was like one of those textbook-case nightmares where life depends on fleeing deadly danger but the sleeper is slowed to a maddening tortoise pace, creeping along while swift-winged Death swoops in for the kill. But this was no dream, it was real, appallingly real.
How many junction points had they passed on their way in toward the shaft, two or three? Jack couldn't remember. No matter. All that counted was picking up one foot and putting it down, one after the other. Don't bother counting them; waste of time. Just keep going.
His ears popped and the constant roaring in his ears was replaced by intermittent bursts of sound. Rasping breath that was his own sounded like wood being sawed. The shuffling tread of his footfalls, more felt than heard, as each successive step sent tremors from his feet through bones all the way to the top of his head. Groans and inarticulate mumblings from Bailey. Maybe they were articulate but Jack couldn't make them out.
Armstrong's panting gasps were sometimes interrupted by shouted remarks that reached Jack in bits and pieces like garbled transmissions.
He thought of that hoary comedy cliché, the one where a couple of drunks hold each other up as they stagger and reel from lamppost to lamppost. That's what he, Armstrong, and Bailey would have looked like to a stranger's eyes. Not so funny when you were living it.
Another junction point. Armstrong seemed to take forever to find the green arrow marker while Jack kept Bailey on his feet. Jack told himself that that was okay, better she should take her time and get it right rather than make a mistake that would send them up a blind alley. The trouble was that Bailey's legs kept folding at the knees and he'd start sliding down the wall, forcing Jack to expend more effort to keep him upright.
It was a relief when Armstrong took her place beside Bailey, lessening some of the weight on Jack. They lurched forward, resuming their stumbling, swaying stagger along the tunnel.
Jack's hearing was returning, the roaring muting down to a continuous low murmur like surf breaking on an unseen shore. It counterpointed the sounds of their passage, the heavy breathing, gasps, and groans, the foot-dragging shuffle of their forward march.
An eternity of slogging and heaving brought them to a third junction. Jack shook his head, stifling a groan. It wasn't fair to have come so far only to have to continue the ordeal. What of it? Fairness was irrelevant, only facts mattered. The fact was that they must go on.
Wasn't there some character in Greek myth whose punishment for offending the gods was something like this? Only he was condemned to roll a rock up and down hills for all time. Jack had it better than him, he told himself. At least he didn't have to climb any hills.
Pick them up and put them down. Jack couldn't help resenting Frith and Sanchez. They must know what had happened. Where were they? Why hadn't they come to help?
Never mind about that. Just keep moving.
The airborne murk was starting to thin out as it streamed ahead. It was like a river made up of many currents of different hues, some light, some dark, all of them intertwining and writhing, coming together and splitting apart on their journey to the same destination.
There was light ahead, a pale glow shimmering off in the dim distance. The tunnel brightened to a kind of ashen sooty dusk. A pair of figures were outlined at the far end of the milky, smoke-strewn brightness.
It took a minute for Jack to realize that the duo was Frith and Sanchez and the glow was, literally, the light at the end of the tunnel.
Armstrong called out to them, a wordless cry. She was answered by a series of flat muffled cracks from outside. Jack's hearing had recovered to the point where he was able to recognize them as gunfire.
Frith and Sanchez turned at the sound of her voice. They crouched sheltering behind the tunnel's rock walls, pointing their weapons at the gap in the wooden barrier. Smoke and dust poured steadily out through the opening. A large number of holes in the barrier let in beams of sunlight. Bullet holes.
More flat cracks sounded from outside, punching fresh holes through the boards and ricocheting off rock walls.
Frith said, "Get down! We're under attack!"
Jack heard that. He got down. So did Armstrong and Bailey. All three of them more or less collapsed at the same time, tumbling to the tunnel floor. Which was just as well. It was safer down there. For the moment.
Frith gave the trio a rundown on the situation: "Holtz said, 'That's funny.' That's all, just 'That's funny.' That's the last message we received from him on the headset. He didn't respond after that. He must've noticed something wrong but too late to do anything about it. There was no sound of a shot being fired. We found out why later.
"The blast came a few minutes later. A cloud of smoke and dust came out of the tunnel. It got so thick in here that we couldn't breathe so Sanchez and I got out. We went out on the ledge. That smoke saved our lives. It caused the shooter up top to miss his shot. Or shots. We don't know how many he fired before he tagged Sanchez."
Sanchez glanced over his shoulder at Jack, Armstrong, and Bailey where they sat on the tunnel floor with their backs to the wall. His face and clothes were blackened with dirt and soot, just like the rest of the group. He grinned, said, "I got hit in the back and knocked down but my Kevlar vest saved me from worse. Gave me a hell of a jolt, though."
Frith went on, "Some falling rocks almost hit me. The shooter must have knocked them over the edge while he was angling for a better shot. I looked up and saw him. He was leaning way over trying to get a bead on me. He made a nice fat target. I got him first. He fell off. You can see his body sprawled out on one of the ledges down there. But don't try. Stick your head out and his playmates will try to shoot it off."
Anne Armstrong said, "How many are there?"
"About ten I'd say, at least to start with. They popped up in the rocks at the bottom of the hill right after the shooter fell. I guess they'd been there for a while but the first we knew about it was when they opened fire. I don't mind telling you that it got pretty hot out on that ledge! We ducked back in here for cover. It's hard to breathe with all this crap in the air but it beats not breathing at all."
Sanchez said, "We got two of them, so that leaves about eight, give or take a corpse."
Jack was checking his pistol to make sure that it was undamaged by the blast. It seemed to be working fine. He was eager to put it into action.
Frith said, "That first shooter had a silenced rifle, of course. That's why we didn't hear anything when Holtz was shot or when he was using Sanchez for target practice. The others down below don't give a good damn how much noise they make. Why not? Nobody around to hear them."
Bailey sat slumped against the wall, hugging his middle. His chin rested on top of his chest. He raised his head a little, said, "The one up top...How—how'd he get there?" His voice was a harsh croak.
Frith said, "He must've been up there all the time. He sure as hell didn't drive or walk up, Holtz would have seen him coming from a mile off."
Jack said, "That's why Holtz had to go first. The Zealots probably had a man in place the whole time serving as a spotter. There's plenty of places on the hilltop to hide out. He was probably watching us the whole time we were searching the site."
Frith nodded. "That's how I read it. He must've sent for reinforcements when we found the bus."
Bailey said, "Our radios don't work in the canyon. Why his?"
Anne Armstrong said, "Probably because it's not a radio. A satellite phone could do it."
Bailey said, "That's a good one. He's got a sat-phone and we don't. He can call for help but we can't. Those CTU budget cuts will be the death of us yet. What a joke—a sick joke." He laughed without mirth, his laughter a harsh crow's caw. His face contorted with pain; he bit down on his lip to keep from crying out until the spasm passed.
Sanchez said, "Write a letter to your congressman to complain."
Bailey forced a weak grin. "I would...if I thought he could read."
Frith said, "We didn't know what happened to you three. Thought the blast might have got you. Glad you made it back in one piece."
Jack said, "Somebody—most likely the shooter who got Holtz—tossed a bundle of TNT down the shaft to cover up the evidence."
Sanchez said, "What evidence?"
"A mass grave at the bottom of the pit. We found the missing Zealots. There had to have been twenty bodies there, maybe more."
Sanchez frowned, puzzled. "Then who are the guys shooting at us now?"
"That's the big question."
Bailey said, "Why don't you go ask 'em, Sanchez?"
Frith lay in a prone firing position on the tunnel floor, pointing his rifle downhill. The figure of an armed man darted out from behind a boulder and ran toward the slope. Frith squeezed the trigger. The figure fell sprawling and lay motionless on a piece of open ground. Frith put another round in him to make sure. He said, "Got one!"
An instant later the dead man's teammates loosed a fusillade at the tunnel mouth. The shooters were scattered in a loose arc among a jumble of boulders and slabs at the bottom of the hill. Some were armed with rifles and others with machine gun pistols of the Mac–10 variety. They had a lot of firepower.
Frith and Sanchez wriggled backward, covering behind rock ribs and outcroppings. Rounds ventilated the remnants of the plank barrier. The real danger came from ricochets that bounced off the inside of the tunnel. The shooters didn't have the proper range and the rounds angled away into the tunnel leaving the defenders unharmed.
The death of one of their own provoked a prolonged outburst. The racketing rattle of assault rifles and machine pistols on autofire sounded like a street crew of jackhammer operators at work.
The shooting subsided, falling silent except for an occasional potshot. Frith and Sanchez bellied back into position and scanned the slope. Frith was on the east side of the tunnel mouth, Sanchez the west. A flicker of motion in the corner of his right eye caught Sanchez's attention and he swung the gun muzzle toward it but held his fire.
He said, "One of them's working his way up the west ridge. He took cover before I could get a bead on him."
More shots popped from below. Frith said, "They use that covering fire to change position, move in and up. A couple of shooters are higher up the slope now."
Jack noticed an M–4 standing propped up against the wall near Sanchez. It was Bailey's weapon, the one he'd left behind before going on the tunnel probe. The ammo pouch stood beside it. "That M–4 functional?"
Frith said, "Should be."
Jack said, "Let's get some more firepower into play."
Bailey rested a clawlike hand on Jack's shoulder. He said, "You take it. I'm not much good now—all busted up inside."
"Hang on. We'll get you to a medic."
"Sure." Bailey smiled with his lips.
Jack low-crawled across the tunnel floor to the other side. He sat with his back to the wall with legs extended. He slung the ammo pouch over his shoulder, picked up the M–4, and examined it. It checked out okay. He said, "What's the plan?"
Anne Armstrong said, "We can't just sit here and try to wait them out. It'll be a long time before help arrives. Central doesn't know we're in danger. We've been relatively safe so far because the attackers are too far down to do much damage. The angle of their line of fire is too steep. But if they get higher up on the slope and flank us they'll be in a position to shoot us to pieces with the ricochets. If we retreat deeper into the tunnel to avoid it, they'll be able to come even closer."
Frith said, "They know it. They use each burst of covering fire to climb the ledges."
Sanchez said, "They've definitely got a couple of guys on the west ridge. I see 'em moving but they take cover before I can get a shot at 'em."
Jack said, "Another thing. Sooner or later the forensics team from CTU/DENV will be arriving here. We can't let them come unsuspecting into a massacre."
Armstrong said, "What do you suggest?"
"It's been my experience that the aggressor usually has the advantage. We need to regain the initiative. Let's make a breakout."
"I agree."
Frith said, "I'm always in favor of taking the fight to the other guy. But what about Bailey?"
Bailey's harsh laugh sounded, followed by a coughing fit. Blood flecks splattered his mouth and chin. A half minute passed before he was able to speak. "Forget about me. I'm excess baggage. If we stay put, I'm sunk anyway. If you make a break I've got a chance. We all do."
Silence fell. A short-lived silence that was quickly broken by a ragged series of shots from below. Sanchez squeezed off a quick burst at a figure breaking cover on the west ridge but missed. The figure ducked down behind a rockpile that was a bit higher up.
Bailey looked Jack in the face, then Armstrong. He said, "Like I said before we went in the tunnel, I'm expendable. More now than ever. These are the cold, hard facts so don't argue. Leave me with an extra pistol and some clips. If any of those bastards make it this far, I'll give them a warm welcome.
"Don't delay. The longer you wait the less your chances get."
He was right and they knew it. Armstrong said, "It might already be too late if they've got people on the hilltop. They might be there if the shooter who got Holtz and the dynamiter are two different people."
Jack said, "We'll just have to risk it. Otherwise they will get some shooters up there for sure and have us caught in a high-low crossfire that'll kill any chance of a breakout."
Frith said, "This being pinned down works both ways. They've got us holed up here but Sanchez and I did the same thing to them. None of this bunch has been able to circle around to the east. I tagged the one who tried."
Sanchez said, "There's plenty of cover we can use on the ledges. Woodpiles, fallen rocks, tumble-down shacks, all kinds of crap."
Jack said, "Best use it while we can, before they do."
It was decided. All that remained were the tactics. Jack and Armstrong took quick turns scanning the slope, noting strong points and weak links. A stack of old timbers stood on the ledge a stone's throw east of the tunnel mouth. The next stepped tier below this ledge featured a rusted ore bucket, a steel-wheeled hopper the size of a compact car. It lay on its side on the west side of the terrace.
A quick plan of assault was made, finalized. Jack and Sanchez would make for the ore car, Armstrong and Frith for the timber stack. Jack and Sanchez had M–4s; Frith had the M–16, and Armstrong armed herself with a second pistol, Frith's sidearm, in addition to the one she was carrying. The thinking was that the M–16's capacity for shooting at long range and close quarters would compensate for the lesser firepower of the pistols. Bailey had his own pistol and Sanchez's. He and Armstrong equipped themselves with pocketfuls of spare clips.
Jack said, "Something else we've got working for us. They think all they've got to buck is two guns, Frith and Sanchez. They don't know about Anne and me. That gives us an element of surprise plus added firepower."
Frith said, "When do we go?"
"After their next burst of covering fire. I'll go first."
Armstrong and Frith dragged Bailey to the cover of the east side of the tunnel mouth near the entrance. Bailey's face was dead white where it showed between the soot and grime streaking his face. He grinned through gritted teeth, gave them the thumbs-up sign.
Frith and Sanchez loosed a few rounds down the slope to stir up the opposition. The enemy returned fire in earnest, burning off another sizzling fusillade. Shooters popped up from behind rocks and out of ditches to sling lead at the tunnel, betraying their location by doing so. They had advanced a lot higher uphill; the nearest were only two ledges below. A pair scaling the west ridge were nearly level with the ledge below the tunnel.
Some laid down covering fire while others worked their way farther up toward their objective. The shooting fell off once the climbers took cover.
Jack readied to make his move. The bomb blast and the tortuous trek back from the shaft had taken their toll on him, but his innate vitality and peak physical condition had helped restore some of his energies. So did the prospect of immediate action.
Frith and Sanchez opened fire, Frith selecting the nearest shooters on the eastern half of the slope, Sanchez focusing on the pair on the west ridge. Their targets had already gone to cover and there was next to no chance of hitting them. The purpose of shooting at them was to keep them pinned down so they couldn't fire at Jack when he made his breakout.
Frith said, "Go!"
Jack jumped out of the hole in the barrier, bent almost double as he angled west across the ledge with the M–4 in his hands. His appearance took the enemy by surprise. He reached the point on the ledge above the ore cart on the next terrace down before somebody took a shot at him. It missed.
Others recovered their wits and began popping away. Shots cracked, rounds whizzing through empty air around Jack to bury themselves in the hillside.
The slope to the terrace below declined at about a thirty-degree angle. Jack jumped off the ledge feet-first, throwing himself over the side. He slid down the hillside like a runner sliding into home base in a race to keep from being tagged out. It was a long slide. He set off a mini-landslide of falling rocks and dirt during the descent.
The ore cart lay on its side with its open hopper facing the slope. It was orange-brown with rust but the sides of the hopper were several inches thick and its base was about twelve inches thick, not including the undercarriage with its trucks and sets of grooved wheels.
One of the duo on the west ridge was so provoked by Jack's ploy that he rose up from behind a rock to point a rifle at him to line up a shot. Sanchez had been waiting for just such an opportunity and squeezed off a three-round burst, chopping the rifleman before he could fire.
Jack's feet hit the ledge below and then he went into a roll, a shoulder roll that took him across the terrace toward the shelter of the ore car. Lines of lead zigzagged the slope behind him, kicking up dust and cutting down small bushes and scraggly dwarf trees growing out of the hillside.
He scrambled into the hopper, thinking for an instant that it was the kind of shady retreat that a rattlesnake might prefer. If that was the case it would be too bad for the rattlers. He rolled to a halt, his shoulder slamming into the now-vertical bottom of the hopper. Happily the car was unoccupied by any other life form than himself.
Jack pulled in his feet and hands, curling up inside the hopper so that no part of him was showing. The ore car vibrated with a metallic clangor as slugs began smashing into its underside. Wheels, trucks, and undercarriage proved a formidable shield, the rounds flattening themselves into lead smears against the cart.
Shooting from below burst out with renewed intensity but a different target. Jack knew that that meant that Anne Armstrong was making her break. She was lightly armed and would be relying on the covering fire laid down by Jack, Frith, and Sanchez.
Sanchez had the surviving shooter on the west ridge covered so Jack didn't have to overly concern himself with that direction. He peeked around the east side of the ore car seeking targets. A shooter on the next ledge down huddled behind a stone wall three feet high, all that remained of a long-gone building. He had a machine pistol in each hand and was streaming lead at Armstrong as she dashed for the timber stack. A regular Two Gun Kid, thought Jack. Two Guns was pretty well covered and Jack's chances of scoring on him were slim.
A rifleman came into view much farther down near the bottom of the slope, springing up from behind a rock slab, exposing himself from the waist up. Jack triggered a short burst at him. The downhill angle was tricky and Jack's rounds passed harmlessly over the rifleman's head. It threw a scare into him and he ducked down out of sight behind the slab before Jack could correct his aim for a second try. But it stopped him from shooting at Armstrong for the moment.
Two Guns seemed to take that as a personal affront and turned his attention toward Jack. He squatted behind the woodpile, gun hands resting on top of it as he turned to squirt bursts of lead at Jack, alternating between one machine pistol and the other. He had maximum firepower and minimum accuracy. The rounds flattened themselves against the ore car, sounding like someone was tap-dancing against it.
Two Guns's change of position put him in Jack's line of fire. His head was raised above the woodpile so he could see what he was shooting at. Jack squeezed off a triple burst that blew apart the other's skull above the eyebrows.
Armstrong staggered, breaking stride. Had she been hit? She stumbled forward, falling behind the timber stack, dropping out of sight.
How many of the enemy were left? Frith had estimated ten to start with. He and Sanchez had each bagged one before Jack and the others emerged from the tunnel. Frith had since tagged another at the bottom of the hill, Sanchez had gotten one of the duo on the west ridge, and Jack had just neutralized Two Guns.
That made five. Frith's estimate might have been off because Jack thought that there were more than five shooters still in play, maybe six or even seven. It was hard to tell for sure because they moved around a lot while rarely showing themselves for more than a brief blur of motion and a burst of gunfire.
Say six shooters remained. Six versus five CTU members. Three of the CTU team had heavy firepower, the other two had pistols. Pistols were for close quarters combat, not much good in this kind of firefight. Jack was a crack marksman with a handgun but he knew their limitations in such an encounter. There was also doubt whether Bailey would be effective at any range. He'd looked weak, shaky, on the verge of passing out. The bomb blast had inflicted serious damage on him, maybe internal injuries, maybe a concussion, maybe both. He needed medical attention as soon as possible.
Jack didn't know if Anne Armstrong had been tagged or not. There was no sign of her behind the timber stack but then there wouldn't be whether she'd been hit or not. The smart way to play it was to keep the foe guessing until the optimal moment for intervention.
Three CTU shooters versus six, maybe seven of the enemy. Not bad odds. Jack meant to do what he could to improve them.
Now Sanchez showed himself at the west side of the tunnel mouth. He immediately ducked back in, taking cover. The attackers opened fire, shooting at where he'd been. Jack scanned the landscape. He thought there were seven shooters left.
The shooting stopped almost as soon as it started as the foe realized that Sanchez's ploy had only been a feint, a ruse to draw their fire to force them to reveal their position. A knot of two or three of them were clustered on the ledge below Jack's, behind a massive old boiler that nestled in a collapsed framework of thick-beamed trestles and cross braces. The cylindrical boiler lay on its side. It was fifteen feet long and six feet wide. It and its shattered frame provided plenty of cover.
Sanchez's move had exposed their presence but failed to lure them out from behind their cover. But the gambit was a double-feint. Frith ducked out of the eastern side of the tunnel a few beats after the shooting stopped. He ran for the timber stack.
Pistol fire cracked from behind the stack. Armstrong had made it and was still in the game, firing steadily to help cover Frith. A succession of shots popped as she emptied one magazine, almost immediately following it up with another volley from her other pistol.
Gunfire blazed from three places around the boiler, tearing up the hillside, trying to intercept Frith before he reached cover. That was the heaviest concentration of firepower. Triggermen opened up from three other separate spots on the slope.
A seventh man was on the west ridge. He took advantage of Sanchez's momentary absence to step out from behind his rock and train his weapon on the back of the running Frith.
Jack was ready for him. His burst cut the other down before he could fire. The shooter staggered backward, bumped into a boulder, and pitched forward headfirst. He looked like he was taking a bow. He kept on going, rolling and tumbling down the ridge. The ridge was steeper than the Silvertop bluff and he picked up a fair amount of speed on his way down, arms and legs flailing until he hit an outcropping and bounced off, falling straight down to land in a heap at the foot of the ridge. He was motionless after that.
One down, six to go. Jack withdrew into the ore car's protective shell an instant before drawing heavy fire from the attackers. The ore car shuddered, raining a shower of rusty flakes down on Jack. But it held, impervious and bulletproof.
The crack of an M–16 told him that Frith had reached the timber stack and was responding in kind. Bullets spanged against the boiler and splintered timbers, quelling the onslaught from the three gunmen sheltering behind it. Armstrong's pistol chimed in, cracking away as she fired.
Sanchez's M–4 barked, adding its voice to the chorus. The other three shooters spread out among the rockfalls east of the boiler returned fire.
Sanchez would be making his move next. His firepower joined to Jack's would make a potent and lethal anchor for the western half of the planned crossfire. Frith's M–16 backed by Armstrong's pistols would supply the eastern component. Together they could begin clearing the slope of the rest of the enemy.
Shouting sounded from below. Jack couldn't make out what it was but it sounded like someone giving orders to the others, perhaps to unleash a counterstrike of their own.
He squirmed around in the hopper, changing position to cover the boiler and points east. He was shaggy with fallen rust flakes from head to toe. They dusted him like a coating of orange snowflakes. He ejected an empty clip and inserted a fresh one in the M–4.
The three shooters among the rocks concentrated their firepower on the timber stack. Those beams had the dimensions of railroad ties and there was a waist-high cube of them. The rounds could chip away at them but Frith and Armstrong were safe behind them, though their weapons were stilled for the moment while they took cover.
One of the shooters behind the boiler fired an assault rifle at Jack, snapping shots at him each time he stuck his head out from behind the ore car looking for a target of opportunity. A second shooter was trading bursts with Sanchez. The time was not yet right for Sanchez to make his move.
There was a lull in the gunfire directed at Jack. He'd been peeping out from behind the side of the hopper looking for a shot. He now changed tactics, unexpectedly popping up from behind the top of the overturned car.
He sprang up just in time to see a third shooter who'd been sheltering behind the boiler do the same. The other was a big man with a platinum-blond crew cut and clean-shaven face wielding an assault rifle with a tubular attachment underslung to the bottom of the barrel.
Jack knew it for a grenade launcher. He swung his gun muzzle toward its wielder but the man with the platinum hair fired first, instantly dropping out of sight behind the boiler.
The grenade launcher went off with a thump, a hollow crumping sound. It was immediately followed by a burst fired at Jack by the shooter behind the boiler who'd previously been busy trying to nail Jack.
Jack had seen the shooter take aim at the same time that the man with the platinum hair ducked. Jack dropped behind the hopper, a hot round smacking the hillside behind him.
The grenade described a tight lobbing arc, hitting the slope between the tunnel mouth and the timber stack. It bounced off, falling like ripe fruit on the ledge. It detonated not with an explosive blast but with a juicy wet splat like a fat pumpkin dropped from a height to smash apart on hard ground.
Masses of green fog erupted from it, blossoming, expanding into a monstrous cloud that squatted and heaved across the upper ledge.
The cloud was the color of mint mouthwash, a harshly unnatural green that was shot through with myriads of tiny iridescent yellow-green particles. The cloud seemed almost as much liquid as gas, like the smoke that comes boiling off dry ice.
The green fog that Lobo had told of, the toxic cloud that fell on Red Notch. Jack shouted, "Poison gas! Run! Run!"
He was up and running as he shouted. He didn't have to worry about the enemy because they were running, too, fleeing down the slope and onto the flat for all they were worth. He couldn't run downhill because they would get him. He couldn't go up because that's where the green cloud was massed.
He ran across the ledge toward the western ridge. He ran all-out, sprinting, legs pumping. The landscape zoomed past him. There was nothing wrong with his hearing now, the huffing of his lungs, the creaking of his gear and the pounding of his footfalls all reverberating in his ears.
Jack could hear the green cloud, too. It made a hissing sound suggestive of the effervescence of a freshly opened bottle of pop.
He did not look back but he couldn't help looking up. The western arm of the cloud raced along with him, lazily uncurling itself to overspread the ledge above. Streamers and tendrils extended from its underside, slithering off the rim of the ledge and reaching downward.
The viscous, semi-liquid nature of the stuff worked in Jack's favor. It drifted down the hillside but slowly, lazily, its buoyancy keeping it afloat. It rolled across the upper ledge but took its own sweet time doing so, sometimes pausing to curl in on itself and thrust upward to climb the hillside, only to resume its inexorable westward thrust.
Jack's ledge ran out and then there was the steeper slope of the western ridge. Jack jumped on to it and began scaling it toward the top. He now heard only the beating of his own heart pounding in his eardrums. His heaving lungs burned, his limbs felt heavy, leaden.
The footing was treacherous and he fought to keep from slipping. He grabbed the trunks of small bushes growing on the ridge and used them to pull himself up. He scrabbled at rock outcroppings to haul himself higher.
He could see the ridgetop. A gauzy green tendril brushed the back of his hand. He jerked it away, his flesh tingling from the contact.
Jack kept moving solely by instinct. The summit was a dozen feet away—but the green cloud was already there. A thin curtain of it shimmered above him. He held his breath and scrambled upward on his hands and knees.
Green mist enveloped him, its touch like cobwebs against his bare flesh. His tortured lungs could withstand no more; he gasped for breath. The mist was cool and damp, he could taste it in his nostrils. Its scent was part medicinal, part chemical.
Jack Bauer threw himself over the ridgeline and down the other side. The far side was steeper than the near one. It was covered with weeds and a layer of short, dry, colorless grass. He tumbled downward, falling, sliding, rolling, the world pinwheeling around him.
##
## THE FOLLOWING TAKES PLACE BETWEEN THE HOURS OF 2 P.M. AND 3 P.M. MOUNTAIN DAYLIGHT TIME
Pine Ridge, Colorado
Jack Bauer wanted to put as much distance as possible between himself and the green cloud. He didn't fight the fall, he went with it. It was no straight drop, of course; no man could have survived that. It was a skittering, sliding tumble that he helped along as much as possible down a fifty-degree-angled, weed-and brush-covered slope. His descent slowed at times, not often, but sometimes, and when it did, he did what he could to speed up the process, scrambling and rolling, anything to keep moving downward.
He had no time to think during that frantic downhill slide. He was too busy trying to keep from breaking his neck or anything else. Extensive martial arts training in judo had trained him in handling rough, violent falls but this was a marathon ordeal.
Jack reacted by reflex and instinct, dodging rock outcroppings and darting toward open, weedy spaces. Everything zipped by him in a blur that was punctuated by sudden, jarring shocks and the flailing of thorny bushes as he tore through them. He took a brutal pounding.
The downgrade began evening out, becoming less steep, slowing his plunge in the process. He slammed to a halt on a level piece of ground.
He lay on his back, gasping, panting. His head swam. His motion had stopped but the world kept moving, wheeling past him in a dizzying whirl. He shut his eyes for a few beats, and when he opened them the world had caught up with him and stopped moving, too.
Jack felt like he'd been worked over from head to toe. His heart hammered, his pulse raced. Above was blue sky and a yellow dancing sun. He took several deep breaths. His ribs ached but nothing felt broken. They were particularly tender on his left side where his gun in its shoulder holster had banged against them.
He still had his pistol. That was something. The M–4 was long gone. He couldn't remember if he'd had it with him when he went over the ridgetop or if he'd dropped it before then. The ammo pouch with its extra clips for the weapon was still with him, its strap tight against his neck as though trying to choke him. He got his fingers under the strap and tugged it to give himself some breathing room.
There was a chemical taste in his mouth and the back of his throat. The realization of it gave him a surging jolt of adrenaline that washed away the last of his stunned confusion and brought his awareness into sharp focus.
The green cloud!
Jack sat up, the action tormenting his aching body and forcing a groan from him. He'd been exposed to the gas externally and internally; externally where it had touched his skin and internally from the whiff of it he'd breathed before he escaped it.
Had he escaped it? He looked up and to the east, scanning the ridge. Its summit was several hundred feet above him, part of it obscured by the scalloped edge of green cloud. The cloud had crept a few dozen yards down the near side of the slope but its progress was arrested as though it had snagged itself on the jagged crest.
The stuff was heavier than air but not much. Its viscous quality had kept all but its westernmost arm penned on the far side of the ridge. The still air of high noon had since been replaced a slight breeze blowing out of the west that buffeted the cloud, pushing it back. The gas was thinning out, too, dispersing itself into the upper air.
Jack quickly calculated the icy equations of survival. His depended on the nature of the unknown substance to which he'd been exposed.
Was it poison gas or a nerve agent? Lethal gas had to be inhaled to do its work, while a nerve agent could kill on contact with the skin. Modern varieties of either could kill by exposure to a single microscopic particle, but he'd both breathed and been touched by the green gas. He was still alive, though, the prime factor in his high-speed mental calculus.
The bodies he'd seen in the mine shaft bore the marks of death not by gas but by violence. Some had been shot, some had had their throats cut, others had bloody, broken bodies. Corpses don't bleed. Lobo had said that the Zealots at Red Notch had been exposed to the green cloud. One could deduce from that that its purpose was not to kill. What was its purpose?
Most CW gases are invisible, odorless, and tasteless. The green gas was none of these. It was something outside Jack's experience of chemical weapons. Its high visibility, taste, and smell suggested that at least part of its purpose was to terrify, as the mustard gas of World War I had panicked troops into abandoning their trenches and retreating when they saw its characteristic yellow cloud approaching their position. Mustard gas was an irritant, not necessarily fatal, doing its work by attacking the eyes, the tender membranes of the nose and throat and the lungs of its victims. Tear gas, a milder variant on the same principle, is often used for crowd control.
Jack was experiencing none of the symptoms of an airborne irritant. The green gas was not an irritant, then. What was it?
A nonlethal weapon for crowd control? It was unlike anything he'd ever heard of in the arsenals of the military or law enforcement agencies of any nation in the world. A nauseant? He didn't feel sick. Knockout gas? He was wide awake. His limited exposure might have cut down the gas's effects in either of these two cases. The stuff could be slow-acting, by nature or by dosage, its effects delayed while it continued working on his system.
The chain of reasoning flashed lightninglike through Jack's mind. He wasn't just sitting there while he puzzled it out, either. He'd been in action from the start, on his feet and moving.
He'd landed on the east bank of a shallow creek that ran north-south along the center of the bottom of a valley. Both banks were thatched with thick, dark green grass, perhaps because of their proximity to the water. The greenery stood in contrast to the short, dry, yellow-brown turf that matted the rest of the valley.
The west bank stood at the bottom of a long, low slope that stretched for a hundred yards or so before rising into a high, rounded hill. The hill was taller but less steep and jagged than the one Jack had descended. It was dotted with stands of timber, unlike the opposite side with its scant sprinkling of low, scraggly brush. The far hill was crowned with an abundance of the shaggy blue-green foliage of pine trees.
The hilltop provided the only real cover in the valley. It was a place where Jack could hide and shelter if he reached it before the green gas took effect, whatever that might be. He had little doubt that hostiles would come looking for him. The Silvertop strike force was unlikely to leave any surviving witnesses to the raid. And that was the optimistic view, assuming as it did that Jack would survive the green gas.
He was alive now and would continue to act on that premise unless and until circumstance proved otherwise.
The bank was a few feet above the creek bed. He hopped down off it, raising a splash in the shallow water. A scattering of silver droplets fell back to the surface in what seemed to be slow motion.
Jack started across the creek, stepping carefully to avoid slipping on the smooth, rounded stones lining its bottom. Water milled around his ankles, rising to slop over his boot tops in mid-stream.
He clambered up the side of the muddy west bank and began jogging across a long stretch of open ground that rose only a few degrees on its way toward the base of the hill. The green grass of the bank quickly yielded to short, dry, yellow-brown turf. It felt springy under his feet.
He angled across the tilted flat instead of crossing it directly, making for a spot at the bottom of the hill that was in line with a clump of trees higher on the slope. He wanted to make use of what minimal cover was available as soon as possible.
These first trees were about twenty yards above the bottom of the hill. Jack climbed to them. They were skinny pines about fifteen feet tall with a few sparse, forlorn-looking boughs. The trunks at their thickest were the width of a thin man's leg. Jack grabbed one to help pull himself up.
The wood writhed in his grip.
Jack pulled his hand back, recoiling. He thought for an instant he'd grabbed a snake that was curled around the trunk and had wriggled away at his touch. He looked at the tree he'd grabbed and the ground surrounding it. He saw no snake. He circled the tree carefully, looking at its far side and the ground at its base.
No snake.
He held out his hand palm-up, the one that had done the grabbing, and looked at it. It was the hand of a stranger.
Perhaps he'd never looked at it properly before. He stared at it. It seemed to swell and grow, then to dwindle and shrink.
A fascinating phenomenon. He continued to watch it. The hand continued to expand and contract in size. He realized that the cycle of expansion and contraction was in synchronization with the beating of his heart. It was an awesome revelation.
Awe turned into anxiety. He must be crazy to waste time staring at his hand when he should be climbing the hill as fast as he could!
Jack turned his face toward the slope and readied himself to continue. He examined the ground ahead to make sure there was no snake there. It came up clean and he strode briskly forward, mounting the hillside at a quick pace. He made a point of not looking at his hands as he climbed.
He was one-third of the way up the hill when he realized his error. He'd planned to use the trees for cover and here he was walking out in the middle of the open where anybody could see him!
He shook his head at his own carelessness, changing course toward the right and another clump of trees. They had some waist-high bushes at their bases. He hurried toward them but slowed as he neared them, just in case they harbored snakes.
There was a snake! He stopped short. No, it was only a dead tree branch.
Jack stood still but the hillside kept moving. Solid ground seemed to flow like water racing uphill and away from him. Earthquake!
They didn't have earthquakes in Colorado, did they? He blinked and the illusion vanished. The earth was solid and motionless beneath his feet.
Jack made the conceptual breakthrough: the illusion vanished. That's all it was, an illusion. Unreal, like the other phenomena that had been bedeviling him throughout the climb.
It was all in his head and the reason it was in his head was because the green gas had put it there. The green gas was a hallucinogen.
The breakthrough was thrilling and alarming. Thrilling because it gave him a handle on the weird things that were happening to him. Alarming because of its implications.
His awareness snapped into focus with an almost physical lurch. He was Jack Bauer and he was experiencing the effects of exposure to a hallucinogenic gas. An airborne psychedelic whose effects were something like LSD only stronger.
His training as an agent had included advanced courses in resisting hard interrogations. Drugs were frequently used to break a subject's will and the trainees had been dosed with a variety of psycho-chemicals to strengthen their resistance and give them a sampling of the techniques that might be used against them in the event of being taken alive by the enemy.
Now that Jack knew what was happening to him he could fight it. No, that was wrong. The drug had him in its sway and its effects could not be willed away, no more than a swimmer in the sea could will away a giant wave that was about to come crashing down on him.
He couldn't fight it but he could surf it, ride that wave out until it had spent itself and washed him safely up on shore.
He looked back the way he came, across the valley to the ridge bordering Silvertop. The sky above it was blue, free of any taint of the green.
The green was in his head now. He had to stay on top of the wave, maintaining his sense of self and purpose, stay flexible and adapt to whatever came his way until the drug worked its way out of his system.
The creek wound its way through the valley bottom like a giant snake, sunlight glistening on its surface in countless myriad swirls.
All illusion, the trick of a drugged mind.
Jack pointed himself toward the hilltop and resumed his climb. It was not unlike escaping the tunnel after the dynamite blast. You took one step forward and then another and kept on doing it until you reached your goal.
The weird physical and visual effects were Alice-in-Wonderland trimmings on bedrock reality. A rock was still a rock and a tree was still a tree, even if the rocks were made of jelly and the trees were swaying, with their branches wriggling in snaky motion.
The ground leveled out as he crested the summit of the hill. He was on a plateau, a wooded flat that spread out for miles in a sprawling pine forest. These were real pines, tall and towering with thick trunks and abundant foliage.
The pine scent was heady, intoxicating. The edge of the forest was a solid wall but when Jack approached it the trees spread apart in a maze of paths and trails. Not manmade trails but game trails.
He entered the woods. They were filled with pools of cool shadow and hot sunlit glades that alternated in a checkerboard pattern if one had the wit to see it. They were never silent but quick with life and motion: birds flitting, pine cones dropping, boughs creaking.
Jack followed a trail into the depths of the forest. It wound through the trees, around mossy boulders, down into hollows carpeted with dead pine needles and up into rises that broke into columns of sunlight shafting through spaces in the canopy of trees. It was a place of mystery and enchantment where time lost all meaning...
Somewhere, somewhen, somehow the scene began to stabilize and come back into focus. The initial, overpowering rush of the drug, a physical onslaught of raw sensation that sent Jack reeling among the big trees, reached its peak. The wave crested, broke, and ebbed.
The trail went up a low rise and into a clearing, a broad grassy open area about fifty feet in diameter that was enclosed by a thicket of trees.
Jack slipped through a wall of foliage to enter the glade. The grass was emerald-green, and above the treetops stood a circle of blue sky. It was an idyllic nook, like a woodland scene depicted in an illustration for a children's storybook.
He was well into the clearing before he realized he was not alone. There was a purposeful rustling motion in the bushes at the glade's opposite end. Rustling and scratching.
This did not come as a total surprise. He'd been aware for some time of noises of movement in the woods around him, but they'd been following in his wake. These noises came from the opposite direction, though: in front of him.
He'd felt more bemused than fearful ever since the drug's physical rush had lessened, and he felt no anxiety now as he moved to one side to investigate the source of the disturbance.
He saw through a gap in the brush what first looked like a huge black dog. It stood on all fours, as tall at the shoulder as a pony and weighing between three hundred and four hundred pounds. Incredible beast!
It wasn't a dog, though, it was a bear. A bear with rounded ears and a muzzle-shaped face and fur so brown it was almost black. It was tearing at a fallen log with its front paws, clawing away rotted and pulpy wood to get at the grubs and insects that infested it.
A mini version of the bear, a cub, stood nearby watching its parent take apart the log. The adult lifted a front paw swarming with insects to its snout, licking them off with broad swipes of its tongue and swallowing them down.
It could have been a scene from a TV nature special, a charming vignette of animal life in the wilderness.
A twig snapped somewhere behind Jack, sounding loud in the sudden stillness. The big bear froze. The cub was the first to notice Jack. It made a cute bawling cry.
The big bear turned its gaze toward Jack. It growled. The growl was low, muttered, and reverberant. It touched something in Jack that must have been hardwired into the human brain since caveman days, triggering a sense of full-body fear.
The bear growled again, snarling, baring gleaming yellow-white fangs all curved and dripping fat gobbets of saliva. Its fur stood on end, electric with sudden menace and aggression. It lunged forward, charging.
Jack jumped to one side at the same instant that a gunshot sounded, detonating in the glade like a thunderclap.
The bear changed course on a dime, swerving to meet this newly perceived threat. It whisked past Jack toward the other side of the glade.
Two men stood there, crouched in postures of fear and stupefaction. One was a broad-shouldered hulk with a platinum-blond crew cut. The other was short and round-faced with a stubbly beard, dark eyes, and a slack-jawed, gaping mouth. The latter held a leveled rifle with smoke curling from its muzzle.
The image of the duo was engraved on Jack's brain with the clarity of a photograph. He could see the platinum-haired man's cold blue eyes and the jagged scar that split his left eyebrow. He could see the short man's dark eyes bulging like black olives stuck in the sweaty white pudding of his fear-ridden face. He realized the bullet had been meant for him but had missed because of his sudden lunge to avoid the charging bear.
The bear went for the short man. He fired again but too late; the bear was on him and he was bowled over backward, the gunshot zipping harmlessly through the trees.
The bear knocked him to his back on the ground and tore at him with tremendous swipes of its clawed front paws, ripping him apart as it had done to the log but with much greater ease.
He screamed, "Oh Gawd, Reb, help!"
Reb, the man with platinum hair, did not help. He was too busy running at top speed in the opposite direction.
Jack came to himself again. He knew who he was, where he was, and what he had to do.
He got the hell out of there. Fast.
##
## THE FOLLOWING TAKES PLACE BETWEEN THE HOURS OF 3 P.M. AND 4 P.M. MOUNTAIN DAYLIGHT TIME
Silvertop, Colorado
Some people love a mystery. Dirk Vanaheim hated them. It was a trait he'd had since boyhood days. He was intensely irritated by unsolved crimes, locked room murders, unexplained disappearances, and the like. He took them as a personal affront. He had no belief in supernatural intervention in human events. Every crime must have a solution, it was only lack of information on the players and the scene that prevented its solution.
This aspect of his personality had served him well in his chosen profession in the fields of first counterespionage and currently counterterrorism. He had risen to the number two post of Assistant Special Agent in Charge of CTU/DENV. He now had the responsibility of securing and managing the crime scene at Silvertop.
He was thin with lead-colored hair worn brushed straight back from a high forehead. It lay on his scalp like a metal skullcap. A long face featured a pair of horizontal eyebrows over deep-set eyes with dark rings around them. The ever-present fatigue-born rings had deepened during the run-up to the Sky Mount Round Table. They could only worsen as a result of his having to handle the Silvertop mess. He had the feeling that before the Sky Mount conference was done he'd look like a raccoon.
Silvertop mess? Debacle was the word for it. Five CTU agents were dead and a sixth missing. This was more than a crime scene, it was a battlefield. The forensics team from CTU/DENV had arrived to do a thorough examination of the Zealots' blue bus in Silvertop's ghost town. They had discovered a chaos of carnage on the bluff's south slope and the grounds below it. Special Investigator Anne Armstrong and three members of the tac squad, Frith, Sanchez, and Bailey, were dead. CTU/L.A.'s SAC Jack Bauer, here on temporary duty, was missing, as was tac squad member Holtz. The corpses of a half-dozen unknown assailants had been found strewn about the south face along with the CTU dead. The bodies of the unknowns had been left mutilated to thwart a quick identification.
Vanaheim was grateful that the forensics team had arrived after the battle was over and the victors had departed, otherwise the body count would have been far greater. CTU/DENV's ranks had already been decimated as it was.
Further investigation had revealed the presence of tac squad member Holtz on top of the bluff near the team's vehicles. He'd been shot through the head by a high-powered rifle. It was a black day for CTU/DENV and for the entire unit as a whole.
CTU/DENV chief Orlando Garcia had assigned Vanaheim the task of securing the site and the situation. Vanaheim and an eight-man tac squad had raced to the site. CTU/DENV's people were already spread thin by a variety of duties connected with protecting the Round Table conference. The most recent losses had only exacerbated the problem. Vanaheim was faced with a delicate situation in controlling the site with the limited number of personnel now available for him to draw on.
CTU's original mission charter specified that one of its goals was closer cooperation and sharing intelligence with other agencies. All government agencies are traditionally turf-conscious and jealous of their prerogatives, none more so than those involved with the national security sector. The CIA/FBI rivalry is well-known. The events of 9/11 had fostered a greater sense of unity of purpose between them as well as with the Department of Homeland Security. But this mutual amity and concord had its limits.
SAC Garcia had been quite specific: "Those are CTU dead out there. This matter is going to remain in our hands until the perpetrators are found and brought to justice."
Vanaheim was not without other auxiliary support to draw on. The U.S. Army had become involved due to certain features of the Red Notch incident. Army Intelligence officers who had a close working relationship with the CIA, CTU's parent organization, had been called in to help. They were able to supply much-needed manpower required to secure and properly investigate Silvertop.
Shadow Valley had been cordoned off. Its sole entrance where the canyon opened on Dixon Cutoff was closed and guarded by an Army Military Police detachment. The MPs were dressed in civilian clothes but their weapons were Army-issue M–16s, M–4s and sidearms. The Army, too, wanted to minimize its footprint in the affair where possible. The entrance was also guarded by a covert fifty-caliber machine gun nest, a precaution prompted by the unknown enemy's use of heavy firepower. MPs in Humvees patrolled the valleys to the east and west of the canyon to contain and detain unauthorized personnel.
The Sky Mount area had been declared a no-fly zone for the duration of the conference. Now Shadow Valley and environs had been added to the restricted list. Any snoopy reporters who somehow got wind that something big was cooking at Silvertop would be unable to overfly the site to satisfy their curiosity.
The south face of the bluff swarmed with activity. Teams of forensics experts and special investigators went about their business of methodically cataloging the carnage. The criminalistics crews included CTU/DENV agents working in conjunction with their opposite numbers from the MP's Criminal Investigation Division, CID. They took photographs, diagrammed the disposition of bodies on the slopes, made plaster moulages of tire tracks left by vehicles at the scene, and collected a wide variety of evidence, all of which were properly sealed and labeled in protective envelopes. It was too early yet for the bodies to be bagged, tagged, and taken away, but that, too, would eventually be part of the process.
Vanaheim was also able to draw on the resources of the Denver field office of the ATF. They had a dog in this fight, too. ATF agents Dean and O'Hara had been among the first casualties of the Red Notch incident. Vanaheim was working closely with ATF Inspector Cullen, now also present at the site. The two were much of a type: grim, hard, sour-faced man hunters.
They were on top of the bluff, where investigatory efforts also continued. The mass casualties on the south face had absorbed the lion's share of resources, but important activities also continued on the summit. Holtz's corpse was being examined by several specialists while a second group was covering the blue bus.
A handful of persons stood clustered around the mouth of the air shaft near the collapsed shed. They included Vanaheim, Cullen, and several of their administrative aides. They stood close to the edge—but not too close. A thin film of brown dust rose steadily from the hole.
Some hardy souls had already stretched prone on the ground to peer over the rim. They'd reported that the bottom of the pit was heaped with fresh mounds of dirt and rock that bore every sign of having been brought down by an explosive blast.
Vanaheim said, "Our last report from Armstrong and her team reported that they were going to enter the mine in order to investigate the shaft to see if it'd been used as a body dump. It had to have been blown up after that to block the inquiry."
Cullen said, "The rubble will have to be cleared and the pit examined. A big job. We'll have to get a crane hoist out here. They'll need an earthmoving bucket and enough heavy duty cable to reach to the bottom. It'll have to be cleared one bucketful at a time."
"An order's already been put in for one but it'll be hours before it gets here."
"And hours more before the pit is cleared."
Vanaheim shrugged. "What can you do? That's the way of it." He knew Cullen hated delay as much as he did.
Cullen scowled. "This Prewitt character is shaping up like another Jim Jones." He was referring to the infamous leader of the People's Temple cult who'd orchestrated a mass suicide of the nine hundred believers who'd followed him to the hellhole colony of Jonestown in South America. Those who'd refused to drink the poisoned Kool-Aid that the guru had prescribed for the mass self-extinction orgy had been murdered.
Vanaheim said, "Jones mostly killed his own. Prewitt's killing mine and yours."
Cullen said, "Jones had his death squads, too. They eliminated defectors and mass-murdered a congressman and his entourage and some reporters who went down to Guyana to investigate the cult."
"Looks like Prewitt has his own death squads as well. They're no pushovers, either. They outshot my tac squad, and they were all top men."
"At least they went down fighting. My guys Dean and O'Hara never knew what hit them. For that matter, neither do I."
One of Vanaheim's aides appeared, jogging across an open space toward the pit. He spotted his chief and changed direction, coming toward him. Vanaheim turned to face him. "What's up, Murphy?"
The aide said, "Message for you from Pike's Ford, sir."
"What is it?
Murphy glanced at Cullen standing nearby. Vanaheim said, "It's all right, he's with us. Shoot."
"Sir, they've found Jack Bauer."
There was a new addition to the Pike's Ford command post complex. It was a recreational vehicle that had been turned into a mobile laboratory and clinic. Nothing in its exterior coloring, design, or identification numbers indicated that its true owner was the United States Army. Its driver and crew wore civilian garments. It had arrived at Pike's Ford a few hours earlier that afternoon.
Its forward section housed the clinic, its rear the infirmary, and its open center space served as a kind of informal office/day room. It was hooked up to a portable generator that powered its lights, air conditioner, and various appliances and equipment.
The center space held a workstation and several chairs that folded out from the side walls. It was occupied by CTU/DENV head Orlando Garcia and Dr. Fenton Norbert.
Garcia was heavyset with salt-and-pepper hair, a craggy face, thick dark eyebrows, and a neatly trimmed mustache. He said, "How soon can I see him, Doctor?"
Norbert said, "Just another couple of minutes. My nurse is finishing up the last of his treatment." He was tall and slightly built, with a few strands of dark hair combed over his shiny scalp. He wore an open white lab coat over a white shirt, tie, and trousers. He was an active duty colonel in the Army with high-level connections with military and civilian intelligence. He had a black security rating, allowing him access to all but the most stratospheric levels of top secret material.
Garcia's brown eyes were so dark that they seemed to blend in with the pupils, giving the impression of a pair of large black dots. They were intent and intense as they fastened on the medic. "He needs to be able to talk, Doctor. That's vital."
Norbert's hazel eyes, deceptively mild, did not flinch from Garcia's gaze. "He will."
"But will he be making sense?"
"Yes. I gave him a Thorazine derivative that neutralizes the effects of the drug, as well as a stimulant to counteract the sedative."
"Good. I need to know what he knows."
"As do I."
A narrow passage along the driver's side wall connected with the clinic, which was partitioned off. A door opened on the right and a nurse emerged, brown-haired, round-faced, and solidly built. She carried a clipboard with some documents attached. She was Army, too, and her security classification was almost as high as Norbert's.
She went into the center space. "All through, Doctor."
"Thank you, Nurse."
She stood to one side to let Norbert and Garcia pass, then sat down at the workstation and began processing the documents. Norbert and Garcia had to proceed single file through the passage, Norbert leading. He opened the door, and he and Garcia entered the clinic.
It was a windowless rectangular cubicle lit by overhead fluorescent lights and smelling sharply of alcohol and disinfectants. Its design maximized the available space much in RV style but with a medical slant. A foldout examining table stood lengthwise along the vehicle's passenger side wall. The rear wall had a stainless steel sink with a cabinet above it. A locked glass-fronted cabinet stood in the corner between the rear and driver's side wall. It contained rows of glass shelves stacked with medical instruments and supplies.
Jack Bauer sat on the examining table facing the door with his legs hanging over the sides. He was stripped down to his shorts. His athletic form was mottled with a variety of bruises and abrasions, the worst of which were covered with taped gauze patches. The left side of his face was still swollen where Trooper Fisk had hit him with the flat of the pistol. His face was scratched and cut in a number of places and pasted with adhesive bandages in several places. His glittering eyes were calm and clear.
The two men came in, Norbert closing the door behind them. A lock clicked into place. He and Garcia had to do some careful jockeying to avoid bumping into each other in the cramped confines.
Norbert indicated a round-seated metal stool with tubular legs in the corner between the sink and the glass-fronted cabinet. "Why don't you sit there while I attend to my patient?"
Garcia said, "I thought you were through with him."
"A doctor's work is never done." Norbert gestured toward the chair. "Please."
Garcia, wide-bodied and thick-armed, sidestepped between table and wall and seated himself on the stool. He sat leaning forward with thick forearms resting on meaty thighs. He said, "I never expected to see you again, Jack."
Jack said, "For a while I had some doubts on that score myself."
Norbert said, "How do you feel, Jack?"
"I feel fine—all things considered."
Norbert took a silvery pen flashlight and shone it into Jack's eyes. "The pupils are dilating normally. That's good. It means that the drug's effects have been neutralized."
Jack said, "What drug is that? You said you had some answers for me, Doctor."
Garcia thrust his head forward aggressively. "I've got some questions I'd like answered myself. What happened out there on Silvertop? What happened to you and why are you the only one of the team to come back alive?"
Jack's face fell. "None of the others made it?"
"Only you."
"That's a damned shame. I'm sorry."
Garcia pressed, "Time won't wait. I need answers now. The doctor can answer your questions when you're done." His manner was surly, skeptical, and suspicious. Jack didn't blame him. He'd have been the same way himself had their positions been reversed.
Jack said, "You sure you want the doctor to hear all this?"
"Dr. Norbert is U.S. Army Colonel Norbert. He's not only a medic, he's Army Intelligence. The Army is working hand in glove with CTU on this."
Jack said, "Okay."
Norbert stood leaning against the forward wall with arms folded against his chest. Garcia said, "Just give the highlights that I can use for immediate action. You'll be fully debriefed later."
Jack summarized the team's actions on Silvertop after going into the mine. He told of finding the mass grave at the bottom of the shaft.
Garcia said, "How many did you say there were?"
Jack replied, "I'd say at least twenty, maybe more."
"How can you be sure of that?"
"I can't, not totally. It's an estimate. The killers didn't bother covering them up with too much dirt. I've seen mass killings—too many—out in the field before, in the Balkans, Darfur, Iraq...You know the litany. I've got the experience to make a pretty accurate guess."
Garcia looked disturbed, dubious. "But that would be pretty much all of the Red Notch cadre!"
Jack said, "Yes. I find that to be particularly significant, don't you?"
"If true. Maybe they weren't all Zealots."
Jack shrugged. "I suppose one-half of the cadre could have killed the other and then thrown in some other folks they didn't like, but that's not how I'd bet it."
Garcia demanded, "What about Prewitt? Was he one of the dead? Or Ingrid Thaler?"
"I didn't see them. But conditions at the bottom of the shaft weren't conducive to making any positive identifications."
"So either one or both could still be alive."
"It's possible. You'll know for sure when you dig them up and haul them into the light of day."
Garcia shook his head like a bull tossing its horns. "That won't be for some considerable time. That dynamite blast brought a lot of the shaft down on top of them."
Jack said, "Which was the purpose of the exercise."
"Eh? How so?"
"To delay identification of the bodies until the Round Table is over."
Garcia challenged, "Who'd want to do that and why?"
"Who, I don't know. Why—because that delay is key to someone's plans."
"What plans?"
Jack said, "Look at the time element, that's the critical factor. The Red Notch disappearance took place early on Thursday morning the day before the start of the conference. This morning a hit team liquidates the lone witness to the disappearance and tries to do the same to Neal and me because maybe we heard something. Today a team goes out to Silvertop and finds the blue bus and a mass grave. A strike force shows up almost immediately to eliminate the evidence and its discoverers. It was so important to the plotters that the evidence be concealed that they must have left a spotter in place to sound the alert if anybody got too close. The spotter killed Holtz, blew up the shaft, and sent for the kill squad, not necessarily in that order.
"The blue bus and mass grave were found out. Killing our team and blowing up the shaft only underlined the fact that something important is hidden in that grave. It's only a matter of time before it's unearthed and exposed. How much time? A day, maybe two at the most? That's when the Round Table ends.
"I'm saying that a plot is aimed at the conference. A deadly plot that needs the facts to be concealed while Sky Mount is in session."
Garcia knitted his two fists together and leaned forward, putting his weight on his thighs while he thought it over. "It's possible. Prewitt's crowd hates the Round Table members like poison."
Jack said, "I'll tell you something else. That strike force was no motley crew of cultists turned shooters. They were professional guns."
Garcia had been looking down at the tops of his shoes. He raised his head, turning those big-bore, gun-sight eyes on Jack. "Which brings up another question: how'd you escape when no one else on the team did?"
Jack said, "A fair question and I'll give you the answer. Because I was the only one who knew what was going to happen when Reb fired the gas grenade."
"Reb?"
"The leader of the strike force. A big, humongous dude with a dyed platinum-blond crew cut."
"How come you know his name?"
"That part of the story comes later."
Dr. Norbert took out a pocket digital recorder and set it down on a countertop. "The rest of this concerns me and my part of the operation. I'm going to record it if you have no objection."
His last remark was directed at Garcia. Garcia said, "Go ahead. We're in this together."
Nobody asked Jack if he objected to being taped. He didn't and let it pass. He went on, "The grenade exploded, releasing a cloud of green gas. I knew it was dangerous because of what Lobo had said about a gang of hog-faced demons loosing a green cloud on the compound. As soon as I saw it, I knew the only chance was to get away from it."
Garcia accused, "You ran and left the others to die!"
Jack took it without flinching. He was more than a little contemptuous. "Don't be childish. You know how the game is played. Better that one should escape to tell what happened than all should die.
"The gas grenade detonated on the tunnel level where Anne and the others were. I was on the next level down where the gas hadn't reached yet. They were done for. What good would it have done for me to make a heroic last stand and throw my life away in vain? If I escaped then at least somebody would know the truth.
"By the way, there's an ore cart on the level below the tunnel that I used for cover as my shooting point. If you check it you'll find fifty or more slugs smeared across it. In case you were wondering where I was while the shooting was going on."
Garcia looked away, rubbing the lower half of his face with his hand. After a while he said, "You're right, you're right, of course. You did what you had to do, what I would have done if I'd been in the same position. It's just that I'm so damned pissed about what they did to Anne and the others!"
Jack said steadily, "How did they get it?"
"Shot in the head at point-blank range. Cold-blooded murder!"
"I'm sorry. I liked Anne, liked them all. They were good teammates, good agents."
"I've known them all for years. I'm the one who'll have to tell their families."
There was nothing to say to that. Jack went on, "The killers gave them the coup de grâce. It was as easy as shooting a sitting duck. Easier, because the duck's not drugged up. The team would have been totally out of it, helpless as babies from the green gas."
Garcia said, "The strike force did worse to their own. Blew their heads apart with shotgun blasts. No facial or dental identification there. Then they cut their hands off and took them away with them. No fingerprint ID. We don't know if they did it just to their dead or if they killed the wounded, too."
Jack said, "Delay. There it is again. Some if not all of the bodies will eventually be identified by their DNA, height, weight, and age, distinguishing body marks such as scars and tattoos, but it'll take time. Time enough for them to accomplish their purpose."
"Which is?"
"Something massive with the Round Table on the receiving end."
Dr. Norbert cleared his throat, said, "You were lucky to get out alive, Jack."
Jack nodded. "I thought the green gas was some kind of knockout gas but it's not. It's a hallucinogen. I got a whiff of it and it sent me rocketing clear out of this world for a while. Anybody that got a lungful of it would've been knocked flat, too tripped out to do anything but lie there and look at the pretty color.
"That must be what happened at Red Notch. A strike force—maybe the one at Silvertop, maybe another, I don't know—bombed the compound with gas grenades and rounded the cultists up while they were helplessly tripping out. Some might have tried to resist or been too crazed to control so they were killed there. That explains the seemingly random dispersion of bloodstains at Red Notch. I don't think they were all killed there, not enough blood, but who knows? Dead or alive, the Zealots were loaded onto the blue bus and taken to Silvertop where they were finished off. They were thrown down the shaft and covered with dirt, not to be found until after whatever is supposed to happen at Sky Mount happens. That was the plan, anyway."
Garcia said, "You're making a lot of sense for a guy that was blitzed with a psychedelic bomb not too long ago."
Jack grinned. "You should've seen me earlier, I was flying like a moon bat. Before that I made my break by going over a ridgetop and tumbled into next valley. I climbed the next hill and went into a forest."
"Pine Ridge."
"If that's what it's called. The strike force must've seen me get away because the leader and one of his sidemen came after me. I wandered around the pines in a daze, not knowing what I was doing. I didn't know Reb and his buddy were dogging me. The stuff started to wear off but I was still pretty wasted. I stumbled into a clearing and came across a mama bear and her cub. I guess it was a mama bear but I don't know for sure.
"That's where Reb and his pal found me. I didn't even know they were there. One of them stepped on a twig and broke it. Mama bear charged me and I dodged right when Reb's pal tried to shoot me in the back. He missed, but the gunshot spooked the bear into going for the shooter instead." Jack shook his head. "Frith was right."
Garcia said, "The tac squad leader? Where's he fit in with all this?"
"Earlier he said that bears were fast. He was right. That bear moved like an express train. It knocked the shooter down and ripped him up like he'd fallen into a threshing machine. He cried for help to his partner—once. Called him Reb. Reb was busy hightailing it out of there.
"I got out, too. The bear didn't bother with Reb or me. It had what it wanted and was slicing and dicing him with those wicked claws. Seeing that pretty well straightened me up and brought me to my senses. I followed Reb out of the woods. He didn't know I was there. Once he started running he never looked back. I lost sight of him but could hear him in the distance up ahead, crashing through the brush. Good thing, too. I'd lost my bearings, and without his lead I'd have had a tough time finding my way out.
"He emerged from the pine forest and went down the hill. I hung back under cover, watching him. A pickup truck was cruising the valley looking for him and picked him up. They drove south out of the valley and must have come out on Dixon Cutoff. I stuck to the cover of the tree line, making my way south along the hilltop, figuring to make my way to the highway.
"I was in the valley near the roadway when a Humvee turned in and saw me. I didn't know if it was the killers coming back to look for me or not, but when they got closer I saw they were too military to be part of the gang. Even with civvies on you can't disguise the look. I know; I was Army myself.
"They picked me up and brought me back here to Pike's Ford. The rest you know. Dr. Norbert gave me some shots that neutralized what was left of the drug in my system, and he and his nurse patched me up."
Jack turned his face to the doctor. "That's my story. Now it's your turn, Doc. You knew what that stuff was in me and had a hypodermic full of the antidote before I said a word. What's it all about? What is that green gas? And where do you and the Army come in?"
##
## THE FOLLOWING TAKES PLACE BETWEEN THE HOURS OF 4 P.M. AND 5 P.M. MOUNTAIN DAYLIGHT TIME
Pike's Ford, Colorado
Dr. Norbert said, "The drug to which you were exposed is called BZ."
Jack said, "Never heard of it."
The medic nodded. "I'm not surprised. Few people have, and for good reason. BZ is an incredibly potent psychedelic. It's a byproduct of the LSD craze of the 1960s. BZ is like LSD on steroids. In its unadulterated form it can produce 'trips' lasting up to a week. Luckily for you, you got it in a diluted form. One that was diluted by close to forty years. I'll explain why in a moment.
"Even in the sixties with its 'anything goes' mentality, BZ never caught on with the drug-taking public. It's too strong for the most hard-core 'recreational' drug user. Which is why it was of interest to the military. The research was, in a sense, benignly motivated. The thinking was that BZ could be used as a nonlethal weapon to incapacitate enemy forces, allowing our troops to achieve a bloodless victory.
"Before judging the attempt too harshly, keep in mind the spirit of the times. It's no secret that the CIA conducted its own extensive research into the use of LSD for interrogation, hypnosis, and mind control. See the files on Project Artichoke if you're interested. The Army experimented with BZ for possible battlefield use. It was found that BZ could be delivered in the form of a gas. The drug itself was present in microscopic amounts in an inert aerosolized carrier format of highly compressed vaporous gas. The gas would be contained in artillery shells, grenades, and canisters."
Jack said dryly, "Obviously the research progressed beyond the theoretical stage."
Norbert said, "Ah, quite. The BZ experiments reached a dead end for the same reason as CIA's work with LSD as a mind-controlling drug. It doesn't work, not in that way. Psychedelics are too unpredictable in their effects on human subjects to be depended on. They work on different people in different ways, depending on the individual's psychological makeup and the setting in which the drug is administered. Some react violently, others are incapacitated. The same subject can have wildly different reactions to identical doses taken at different times.
"The hope was that enemy troops hit with a BZ bomb would be rendered pacified and incapable of resistance. The reality was that it was just as likely to transform them into a horde of raving maniacs, maniacs with guns."
Dr. Norbert tilted his head, the overhead lights reflecting off the lenses of his spectacles to render them temporarily opaque. "Apart from the practical side, there were political considerations weighing against BZ's use in combat conditions. It would have been a propaganda coup of the highest magnitude if the other side could prove that we used a psycho-chemical gas bomb in the field. That's why it was never used by either side, since the Cold War Soviets conducted similar lines of research and could have disseminated their versions of BZ bombs to their client states for use against our troops.
"The BZ research was filed under 'Project Canceled' and forgotten as the world moved on. Unfortunately a number of prototype delivery systems had already been made, including a BZ gas grenade. Grenades are used in relatively close combat conditions, so the gas incorporated a green coloring agent that made it highly visible. Our troops would have been wearing gas masks or nose filters when using the weapon; the green coloring would allow them to see where the grenades had landed on a battlefield and react accordingly in real time.
"When the project was canceled, the BZ weaponry was ordered destroyed. The vast majority of stocks were. However, a certain number escaped destruction due to bureaucratic oversight, misfiling, snafus, and just plain human error. The BZ gas grenade was produced in 1970 and a handful of crates of it have been sitting in chemical warehouse arsenals for almost forty years now.
"This year during an exhaustive inventory of existing stocks of all CWS a quantity of BZ gas grenades came to light. Their extreme age created the added danger that the container vessels, the grenades, had weakened through corrosion and might be susceptible to leakage. They were given a top priority for immediate destruction in an incinerator specifically designed to handle CWs.
"This particular facility is in Texas. Somehow, somewhere along the way, a crate of BZ weapons was stolen from the consignment. Army CID is still investigating how it was done and by whom. Whoever commissioned the theft has a very thorough and murderous organization behind them because all the suspects in the theft were eliminated early on, killed in such a way as to look like fatal accidents or suicide."
Jack said, "Well, Doc, it looks like your wandering boys have come home."
Norbert looked vaguely embarrassed. He said, "Ahem. Er, yes."
Garcia said, "We followed up on your suggestion that Red Notch be investigated for traces of CWs, Jack. The compound itself, cabins and sheds and so forth, came up clean. But bloodstains at the site tested positive for the presence of an unknown molecular complex that was subsequently identified as BZ."
Norbert said, "The BZ in the gas grenades was designed to have an extremely short life once released. Exposure to the oxygen in open air breaks down the BZ compound, rendering it inert and harmless within approximately ten minutes. When ingested by humans who've breathed the gas, traces of the compound remain for a time in the blood and tissues."
Jack stirred uneasily. "I've been exposed. Where does that leave me?"
Norbert held his hands palms out with fingers spread in a placating gesture. "Not to worry. The antidote you were given neutralized the drug's effects. The molecular remnants are inert, harmless. They'll be broken down naturally by your body processes within forty-eight to seventy-two hours."
Jack said, "So there's no danger of a recurrence, Doctor?"
"No. Absolutely not."
"Good. I'm not eager for any flashbacks of that experience."
"Remember the gas in those grenades is close to forty years old. Apparently it's lost a great deal of its potency over time. Otherwise even your minimal exposure to it would have had much more deleterious consequences."
"I'll tell you this, Doc: it still packs a hell of a punch."
Garcia said, "You might be interested to know that the hermit Lobo tested negative for the presence of BZ. He hadn't been exposed to the gas. The same applies to the bodies of ATF agents Dean and O'Hara. Whatever happened to them happened before the gas grenades were used. Their skulls showed indications of blunt force trauma. My guess is that they were taken at gunpoint, knocked out, and taken away to be disposed of later.
"If not for the BZ traces in the bloodstains, we wouldn't have known what we were dealing with. Our forensic pathologists didn't know what they were looking for but were able to identify the molecular complex by computer analysis. I never heard of BZ myself, and they had to tell me what it was. References to it in the reports we filed with headquarters were flagged at Langley and sent up a red flag on the seventh floor. They contacted Army Intelligence, who contacted me. Dr. Norbert and his staff and mobile lab arrived here a few hours ago."
Jack said, "You were unaware of the missing crate of BZ grenades?"
"I was until earlier today." Garcia's face hardened into stubborn lines and he glared out of the corner of his eye at Norbert.
Norbert said, "Naturally the Army isn't eager to advertise the loss of a dangerous psycho-chemical weapon for fear of triggering a mass panic."
"Oh, naturally."
The doctor ignored Garcia's sarcasm and said quickly, "What's important is that it has been found. Or at least a lead to it, the only one we've had since it went missing. We'll be working closely with CTU to locate the rest of it."
Jack said, "Did your investigators turn up any links between Prewitt's group and the stolen BZ grenades?"
"Frankly our investigation hit a blank wall. But the Zealots never surfaced in any of it, not a hint. If it had, we'd have been all over them."
"How does a crackpot cult turn up in the middle of a plot involving death squads and exotic psychochemical weapons?"
Garcia shrugged massive shoulders. "You tell me."
Jack said, "I hope to do just that." He hopped down off the examining table, reached for his clothes, which were hanging on a wall hook, and pulled on his pants.
Garcia stood up, nearly overturning the metal stool. "What do you think you're doing, Jack?"
"Putting on my pants."
"I can see that. But what have you got in mind?"
"I'm going back to work. I've got things to do."
Garcia shook his head. "Oh no. You're in no condition to go back on duty—"
Jack said, "What do you say, Doc? Any reason why I can't get back in harness?"
"Considering what you've been through, a few days' rest is highly advised—"
"Come on. You yourself said that the drug has been neutralized in my system and that there's no danger of a recurrence."
"That's true, but the antidote is a powerful depressant."
"And offset by the stimulant you gave me which you said counteracts the antidote's effects. So I'm good to go."
"I wouldn't put it that way."
"You don't have to. I did." Jack reached for his shirt, made a face. "Whew! I'm going to need to round up some clean clothes. Where can I get some around here?"
Garcia said, "You won't be needing them. You're not going anywhere, Agent Bauer—except maybe back to L.A. You've done your part here and more. Now let the rest of us do our jobs."
Jack said, "You won't be rid of me that easily. I bought a ticket for this ride and I'm not getting off before the last stop."
"Your last stop is right here."
Norbert edged toward the door. "I'll just step outside for a moment to give you gentlemen some privacy."
Jack said, "Stick around, Doc, this concerns you, too. You want your stolen BZ back, don't you?"
"Yes, of course."
"I'm the best chance you've got of getting it back."
Garcia scoffed, "You better sit down and take a rest. That BZ is going to your head."
Jack said, "Let's get real. If I wanted to, I could go to Chappelle. He's got enough of a legitimate stake in this mission to go to Langley for a directive to keep me on assignment here."
Garcia's nostrils flared, whites showing around his black-bore eyes. "You think so, huh?"
"I don't want to go crying to Chappelle. I'd prefer to convince you by the logic of my position."
"That'll take some doing."
"Fact: I'm the only who's seen the face of Reb the strike force leader and is still alive to put the finger on him. Whoever's behind him will know that, too. You need me around in a high-profile position if only to serve as bait. The plotters know who I am but not how much I know. They'll want to get rid of me, and to do that they'll have to tip their hand, which gives us a chance of getting a hot lead."
Norbert said thoughtfully, "He's got a point."
Garcia snorted in disgust. "You're as bad as he is!"
Jack said, "Tell me you'd do anything different if our positions were reversed and you were in my shoes." Garcia fumed silently. Jack pressed, "Go on, tell me you'd quit in the middle of a mission."
"I can't. But all that means is that we're both a pair of damned fools. Happy now?"
"I'm still in?"
"You're in."
"Great. Now where can I find some clean clothes? And my gun."
##
## THE FOLLOWING TAKES PLACE BETWEEN THE HOURS OF 5 P.M. AND 6 P.M. MOUNTAIN DAYLIGHT TIME
Pike's Ford, Colorado
Lila Gibbs said, "Is that your Reb?"
She and Jack Bauer were occupying a cubicle in a section of one of the mobile home trailers that housed CTU/DENV's command post at Pike's Ford. They sat side by side facing a flat-screen computer monitor that was set on a countertop.
Jack was freshly showered and wore a change of clothes that had been lent to him by an agent of similar height and build who had a spare set of garments in a locker in another trailer that served as a barracks for the on-site team.
He still wore his own gun and shoulder holster. Weapon and harness both had taken a lot of punishment lately but were still operable. The same could be said of Jack. He'd forgotten that he even had a gun during his BZ-induced fugue, but it had remained in his possession throughout the experience. He told himself it was a good thing he had forgotten it during his altered state because otherwise he might have been tempted to look down the barrel to see where the bullet comes out of or done some other thing that would have gotten his head blown off permanently.
He'd repeated the now-familiar routine of test-firing the gun at the outdoor range, this time to determine that it had suffered no internal damage when he'd tumbled down the ridge. It checked out fine. So did he. His eye was sharp, his hand steady, and his aim true. The shoulder harness was scratched and sweat-stained but unbroken. He stuck with it because he was used to it and didn't want to risk breaking in a new rig whose unfamiliarity might slow down the speed of his draw. Especially since he was going back out in the field to serve as live bait.
But not yet. There was still an important task to be carried out here at Pike's Ford. That's why he was now working with Lila Gibbs.
She was an expert in the use of facial morphology software for identification purposes. Her role was like that of the old-time police sketch artist who draws a suspect's picture based on eyewitness testimony. The software was essentially a twenty-first-century update of the classic police Identi-Kit that uses a variety of facial features to create a composite image of the suspect's likeness.
Jack had described Reb to her: "He's between his mid-thirties and forty in age. Height about six-four; weight anywhere from 220 to 240 pounds. He's all pumped up like a pro wrestler or bodybuilder, even his muscles have muscles. I'd say look for heavy steroid use in the profile because nobody can get that kind of build without getting on the juice. Platinum-blond crew cut, a flattop. That hair color isn't found in nature and must have come out of a bottle. His left eyebrow is split by a diagonal scar a couple of inches long. Square-shaped face with a lot of jaw and chin. Clean-shaven. No identifying marks or scars that I could see, except for that scar over the left eye. He's a mean-looking dude, too, if that's any help."
Lila Gibbs was in her forties, matronly, with curly brown hair, green eyes, and a heart-shaped face. She worked the keyboard, inputting the specifications and searching the archives. The computer was linked to the CTU data net, itself able to draw on a multiplicity of sources among law enforcement, intelligence, the military, and other governmental agencies.
She said, "The name Reb could be a help or hindrance depending on whether or not it's a longtime alias or one that was recently assumed. If the latter, it may not be in the files or it could be a name he's taken to deliberately mislead the authorities and hide his real identity. But we'll include the alias with the first search. If it hits, so much the better, and if not, we can rule it out and proceed from there."
Her fingers deftly manipulated the keyboard, calling up the data. Somewhere in an unknown location massive CIA supercomputers processed the request, winnowing through oceans of binary zeroes and ones to find the desired droplets in the cyber sea.
There were thousands of "Rebs" in the archived United States police, military, and national files, more in the international ones. A hundred fit the general description; a dozen or so had facial scars in the vicinity of their left eye. Three of that twelve were described as having scars that split the left eyebrow.
Lila Gibbs pulled up their facial photos one at a time. Jack selected the third, said, "Try that one."
The screen was filled with a police mug shot containing two views of the suspect, one full-facial and the other a profile. Jack said, "I didn't see him in profile, just full-on."
Gibbs minimized the profile and maximized the frontal. It depicted a man with shoulder-length dark hair and a full beard; a cold-eyed, glowering thug with a scar across his left eyebrow. "Is that your Reb?"
Jack said, "Could be. It could be. It's hard to be sure with all that foliage covering the face, but definitely maybe."
"I could search for other photos of the subject but this is the most recent one. There's an easy way to get rid of that mess, though."
She worked more keys and a mouse, and after a pause the subject's image broke up only to be immediately reformatted. "This is how he'd look without the hair and beard."
Jack said, "Bingo! That's him. That's Reb."
She did some more manipulations. "Just to be sure, that's how he'd look with a crew cut."
"That's him all right."
The subject was identified as one "Weld, Gordon Stuart; aka Reb, The Rebel, Gordy, Gordo," and a number of other aliases that were mostly variations and combinations of his first and middle names.
Gordon Stuart Weld, thirty-seven, born in Atlanta, had an extensive criminal record throughout the South and Southwest. He had a high IQ, a hatred for authority, and a propensity for ultra-violence.
His early years included several stays in a state reformatory and six months' confinement in a mental hospital for stabbing a schoolmate with a penknife. His psychiatric record featured frequent use of the terms "sociopathic," "narcissistic," and "paranoid." He became heavily involved in gang activity during middle school, a pattern that would continue into his adult life. He was an avid motorcycle enthusiast, a skilled rider, and an expert mechanic.
He'd enthusiastically embraced the world and lifestyle of violent biker gangs, belonging to several such outfits in the South. His lengthy arrest record showed numerous counts of assault, illegal possession of firearms, drug dealing, and theft. He was arrested for rape several times but released when complainants refused to press charges due to intimidation by his fellow gang members.
His size, strength, and ruthlessness won him a spot as gang enforcer, dealing out beatings and brutality on a businesslike basis. He freelanced as a collector for loan sharks and a hired gun for drug dealers. He served three and a half years in a state penitentiary for manslaughter and five years in Federal prison for gunrunning. His arrest record fell off after that, largely because the witnesses to subsequent crimes were found slain or simply disappeared.
He became a member of the Hellbenders Motorcycle Club, an outlaw biker gang with chapters throughout Texas and the Southwest. He rose fast through the ranks and was a major player in the gang's rackets that included methamphetamine manufacturing and distribution, forced prostitution of topless and strip club dancers, gunrunning, extortion, and murder. He was rumored to be part of the gang's elite squad of executioners.
Weld had had a falling-out with his associates in the past year following an arrest in Texas for illegal gun dealing. The mug shots in his computer file had been taken during his booking on those charges. He turned informant to avoid a lengthy prison sentence for this second Federal term. He set up his fellow biker gang partners for a bust, at the same time absconding with the loot from the racket and dropping off law enforcement agencies' radar. He was now a wanted fugitive sought by police and the Hellbenders M.C., the latter having posted an open murder contract on his head with a fifty-thousand-dollar bounty collectable by anyone who could produce same. Literally.
Jack Bauer said, "Two Hellbenders were at the gorge today where the AFT agents were found. They must be looking for Reb. His being on the run explains the platinum hair dye job, too. It's such an obvious giveaway that it could only have been done to draw attention away from his previous appearance."
Lila Gibbs said, "He should have changed his name along with his hair color."
Jack grinned. "He never planned on any outsiders hearing it and living. I got a lucky break."
"Sometimes that's all it takes to crack a case wide open."
"Let's hope this is one of those times."
Now the CTU machinery would go to work on Reb Weld, vacuuming up every speck of data relating to his life and habits, criminal career, known associates, friends, and enemies, anything that might be of use in tracking him down. The manhunt would begin in earnest, putting the Big Heat on Weld. Jack described Rowdy and Griff, and Lila Gibbs added the data to the search.
Somewhere in that mass of facts was the answer to the big question: what was the link between outlaw biker thug and killer Reb Weld, a stolen case of BZ gas grenades, a crank-kook cult of Zealots, and a multimillionaires' conclave at Sky Mount?
A discreet rapping on a cubicle partition wall caused Jack and Gibbs to look up from the monitor screen. The knocking had been done by a soft-faced man in his mid-twenties, a staffer at the Pike's Ford CP.
Gibbs said, "Yes, Charlie?"
Charlie said, "Excuse me, Lila. Jack Bauer?"
Jack said, "Yes?"
"There's a phone call for you."
"For me?"
"Yes, sir."
Jack rose from his chair. "Thanks, Lila. Thanks a lot."
She said, "My pleasure. I'll start our mill wheels grinding on the Rebel and friends."
"Grind them fine."
Charlie said, "This way please, sir." Jack followed the youth along the central corridor to a cubicle at the far end of the trailer. Charlie indicated a satellite phone on the counter. "It's a secured phone, sir."
"Thank you." Jack picked up the receiver and Charlie made himself absent.
"Bauer speaking."
A voice at the other end of the connection squawked, "What the hell are you trying to pull, Jack?"
There was no mistaking the strident tones of Regional Division Director Ryan Chappelle in high bad humor.
Jack sighed. "Hello, Ryan."
"Are you insubordinate or just plain neglectful?" Chappelle's voice tended to get screechy when he was angry, and it was screechy now. "You were told to report to me regularly on the Sky Mount situation. That was an order, not a request. I haven't received a single report from you since you got there, not one. I've been calling you every hour and all I keep getting is your voice mail. My messages to you to contact me have been piling up with no reply. I've had to get all my information on the scene from Garcia's people and I can only imagine what they're leaving out. Even with the minimal amount I'm getting it looks like things have gone to hell in a handbasket. What's going on out there?"
Jack knew from long experience that when Chappelle was in one of these moods there was nothing to do but wait until he paused for lack of breath. "Ryan, in the last fourteen hours I've been shot at, pistol-whipped, dynamited, ambushed, gas-bombed, and attacked by a bear. I've been too busy trying to stay alive to report back to you and for that I apologize."
There was another pause while Chappelle digested the import of Jack's words. His voice was lower and more modulated when he spoke again. "Then there is an ongoing conspiracy against the Round Table."
"That's correct, Ryan."
"Thank God for that!"
Jack couldn't help but grin wryly to himself at the heartfelt relief in Chappelle's statement. Everything was okay. Ryan Chappelle was on the record as having been proven right in his conjectures about an anti–Sky Mount plot. That would look good with the top brass at CIA headquarters in Langley, Virginia.
Jack gave Chappelle a brisk summary of the salient facts: the mass murder of the Zealot cadre, the BZ gas grenade connection, and Reb Weld's kill squad. Chappelle, once again the resolute in-charge authority that he saw himself as, said, "You can figure all that out later. Right now I've got something important for you to do. Top priority, immediate action."
"What's that, Ryan?"
"Apprehend Brad Oliver."
"Brad Oliver?"
"Confidential secretary to the great Cabot Huntington Wright himself."
"I know who he is, I've met him."
"Arrest him. My fiscal analysis team finally broke through the web of shell companies and dummy corporations to finger the mystery man who's been short selling stocks to bet on catastrophe. It's Oliver."
Jack knew better than to ask Chappelle if he was sure of the information. Chappelle would never have put it out there if he were unsure.
Chappelle's tone was confidential, intense. "This is hot stuff, Jack. I only found out about it minutes ago myself. I wanted you to have it first so we can steal a march on Garcia."
"I'm working with the man, Ryan. He's going to have to know about it."
"Of course. My people are contacting him now with the intelligence. But this way you'll be right in on the kill so he can't freeze you out and grab all the credit for himself."
"You think of everything, Ryan."
"I try. One more thing, Jack, and this is strictly for your ears only: Oliver's just the tip of the iceberg. Behind him is something a hundred times bigger. The find was serendipitous—my analysts came across it while cracking Oliver's manipulations. His short selling is the opening wedge of a far greater financial conspiracy being conducted in the world markets. It's the same pattern of betting on disaster only a quantum level higher. There could be as much as a hundred million dollars' worth of shorted stocks gambling on an imminent economic meltdown."
Jack's eyebrows lifted. A hundred million dollars! He said, "That kind of money requires a major player. A hostile nation, maybe."
Chappelle said, "We're working on it. It's top secret until we've got it nailed down for sure. For now, though, grab Oliver. Sweat him. Make him talk. I'm counting on you, Jack. You know how these things are done."
"Yes, I do."
##
## THE FOLLOWING TAKES PLACE BETWEEN THE HOURS OF 6 P.M. AND 7 P.M. MOUNTAIN DAYLIGHT TIME
Sky Mount, Colorado
Long shadows fell on Masterman Way as Jack Bauer and Ernie Sandoval drove up to the gates of Sky Mount estate. The midsummer sun was still high in the sky but the mountains were tall, bathing the grounds in blue shadow. Lights winked on in the mansion and its surroundings.
The two agents were in the Mercedes. Sandoval was behind the wheel and Jack sat on the front passenger side. Jack had felt a sharp pang of regret and sorrow earlier when he first realized that they were going to be traveling in the Merc; the last time he'd been in it, Anne Armstrong had been his partner.
The car was trailed by a dark green SUV carrying a well-armed backup unit of four CTU action men. No unpleasantness was expected on this visit but it was good to expect the unexpected.
Anything involving the Round Table conclave had to be handled with kid gloves. Its host and attendees comprised a significant slice of America's moneyed and powerful elite; their feathers were not to be ruffled without good cause.
This explained the security arrangements at Sky Mount. Its rich and powerful guests had reasons of their own for maximizing their personal and professional privacy. They didn't want the place swarming with FBI agents and intelligence operatives who might conceivably ferret out secrets about their businesses and private lives. Knowledge is power; no one knows that better than those at the top.
That was why security at the estate itself was being handled by the Brand Agency, a private firm known for secrecy and discretion where the ultra-rich are involved. The fact that a controlling interest in the firm was held by the Masterman Trust was a further guarantor that what happened at Sky Mount would stay at Sky Mount.
Similar thinking lay behind the national intelligence establishment's decision to keep the events and revelations of Silvertop hidden behind a wall of secrecy, a directive handed down from the highest levels in Washington, D.C. It was believed that disclosure of the truth about the Zealots' mass grave and the killer strike force would trigger a panicky mass evacuation of Sky Mount.
No real, tangible evidence that those dealings involved a plot against the Round Table had as yet been unearthed. There was nothing to be taken in hand to Cabot Huntington Wright and associates to prove to them that the gathering must be gaveled to a premature and disastrous close.
Ruining the conference without good cause would create an avalanche of bad publicity and ill-will that would bury any officials rash enough to take it on themselves to cause it to be canceled simply to be on the safe side.
People resent having their lives disrupted by a false alarm. The master and guests of Sky Mount had ways of making their displeasure felt by those who'd sounded the alert because of a fire somewhere way off in the distance when the conferees hadn't even smelled the smoke.
It was the old one about the boy who cried wolf. The wolf had better be at the door, or let the crier beware.
The BZ connection was an additional complicating factor, one that would never surface if the Army had its way, and there was no reason to expect that it wouldn't.
That was a national security nightmare and potential public relations debacle that was best kept hidden from the power brokers at Sky Mount, to say nothing of the average citizens and taxpayers who're generally kept in the dark as a matter of policy.
What they didn't know couldn't hurt all parties concerned—unless the worst happened and disaster struck. So it had better not happen.
Anything else was unthinkable.
Brad Oliver's arrest would be handled with a maximum of discretion. Jack Bauer and Ernie Sandoval would make the pinch, quietly whisking Oliver out a side door and off the premises without the guests suspecting that anything was amiss.
The CTU agents were casually but correctly attired to blend in with the surroundings. They were armed only with their guns and wore no protective bulletproof garments. Each man was equipped with a pair of nose filters and a half-dozen slapshot ampoules containing an antidote to BZ, gear that fit comfortably in their jacket pockets and had been supplied to them earlier at Pike's Ford by Dr. Norbert.
This precaution had been taken not because of any danger that might threaten at Sky Mount but in anticipation that an attempt might be made against them in transit while they were taking Oliver to the command post.
That was also the reason for the presence of the backup unit. They would wait outside the gates while Jack and Sandoval apprehended Oliver.
No advance notice had been given to the conference's hosts or guardians to avoid Oliver's learning by accident or design of his imminent arrest.
Jack and Sandoval had to endure the tedious admittance process necessary for the uninvited to gain entry to the estate.
They could have pulled a power play by using their Federal authority to bull their way in but chose not to do so for fear of prematurely alerting Oliver or any accomplices he might have inside the estate.
Oliver's status as a wanted man was a tightly held secret known only to Chappelle, Garcia, and the two agents in the Mercedes. The backup crew knew that an arrest would be made but were unaware of the suspect's identity.
Fifteen minutes passed before someone came down from the mansion to escort the agents beyond the gate. They were met this time by Don Bass, head of the Brand Agency's presence on the estate.
The first of the conference's day-long sessions had apparently not been without its rigors for the security chief, who looked considerably more rumpled and frazzled than he had early that morning. His forehead was corrugated by worry lines, his eyes were tired, and his jowly face exhibited a glum, hangdog expression.
He summoned up a cheerful grin as he climbed into the back of the car. "Hi fellows, what's up?"
Sandoval said, "A routine visit. How goes the gathering of the high-and-mighty?"
"Hectic!" Bass settled back into the seat cushions as the Mercedes rolled through the open gate and up the long curved driveway toward the mansion.
Jack turned in his seat so he could look Bass in the face. He said, "Actually, we're here to make an arrest."
Sandoval added dryly, "A routine arrest."
Bass reacted like he'd been zapped by an electric cattle prod. He bounced upright in his seat so abruptly that the top of his head barely missed hitting the roof. "What? You're kidding!"
Jack said, "No."
Bass sat leaning forward, his thickset upper body rigidly tilted at an acute angle. "There's no such thing as a routine arrest here."
Sandoval said, "We'll try and keep it that way any how."
Bass said eagerly, "Who's the pigeon? Anybody I know?"
Jack said, "Brad Oliver."
Bass's broad face creased in lines of wonderment and disbelief. "Masterman's stooge? What's he done?"
"We just want to have a little chat with him."
Bass's expression took on a wise and knowing look. "Can't tell, huh? More cloak and dagger stuff. Must be something big if you boys are putting the arm on him."
Sandoval said, "Naturally we'll be relying on your discretion, Don."
"You'll have my full cooperation, of course, and the entire firm's as well."
Jack said, "We'd prefer that this be kept between the three of us. Private and confidential."
Sandoval said, "We don't want to rile up any of the guests. Might cause them to choke on their caviar and truffles."
Bass said, "I understand completely. You call the signals and I'll play them."
"Thanks, Don, I knew we could count on you."
"You bet!" Bass was happy and excited, like a young baseball fan with a ticket to a big league game. "Boy oh boy! This is really something. I never had much use for the pussyfooting little creep, but who'd have thought that Oliver had it in him to run afoul of Uncle Sammy? This'll really knock old Huntington Wright back on his heels."
The car pulled over to the side of the main drive and rolled to a halt a few lengths short of the front entrance. Sandoval said, "The three of us will handle it, Don. Please don't mention this to your associates."
"I've been to the fair before."
Sandoval switched off the car and the trio got out. A uniformed Brand guard came hurrying over to wave them away. "Hey! You can't park here—oh, sorry Mr. Bass, I didn't know it was you."
Bass said, "That's okay, they're with me. See that nothing happens to this car."
"Yes, sir!"
Jack, Sandoval, and Bass made for the front entrance, an elaborately pillared portico. Bass said in an aside, "Once you've got your man we'll have the car brought around to the side and you can take him out that way. Makes less fuss all around."
Sandoval said, "Sounds good."
Bass strode a pace ahead of the two others, his mere presence assuring that they breezed past all potential obstacles of the Brand guard variety. There were few uniformed guards in the building, most of the indoor security being handled by plainclothes operatives with the agency's emblem on the left breast of their navy blazers.
The conference's events had ended for the day, giving the guests an hour or two to freshen up and dress for tonight's formal dinner banquet.
Knots of attendees stood lingering in the grand hall, chatting and socializing. The women were mostly beautiful and on the thin side; the men displayed a far greater variety of age, height, weight, and physical attractiveness or the lack of it.
Jack recognized a number of familiar faces from the financial pages of the news and the TV business channels. There was a California aerospace tycoon, a Seattle software titan, a maverick oil wildcatter from Texas with a big stake in alternative energy sources, a deputy assistant to the Secretary of the Treasury, and a Manhattan real estate magnate, to name a few.
He caught glimpses of them as he, Sandoval, and Bass made their way through the grand hall and down a corridor leading into the east wing. Their progress was brisk but without urgency to avoid attracting any undue attention. All the in-house personnel, guards, staffers, and service persons, moved at a similar pace; only the guests lounged, ambled, or lingered.
The trio bypassed a cordon of security guards before arriving at the grandiose anteroom outside Cabot Huntington Wright's office suite. Marion Clary still occupied her post at the reception desk, glancing up as the visitors entered and greeting them with a warm smile as she recognized them.
She said, "Good afternoon, or good evening, I should say. One loses track of the time out here."
She looked down at an open ledger on her desk, scanning the entries. A frown creased her smooth, shining forehead as she looked up with mild vexation and puzzlement. "I'm sorry, but I don't seem to have you gentlemen down in my appointment book—"
Bass interceded. "This is a special matter, Marion, one that's come up rather suddenly. It's somewhat urgent. I'll take full responsibility."
"It's all very strange to me but if you think it's important—"
"It is. Is Mr. Wright in?"
Her face showed signs of strain. "I'm afraid he's away from his office."
Jack stepped forward, a pleasant smile masking inner urgency. "Actually it's Mr. Oliver we're interested in. May we see him, please?"
Marion Clary said, "Brad Oliver? He left here some time ago."
"I'm sorry to trouble you, but do you know where he went?"
"No, I don't. But he certainly was in a hurry."
Jack and Sandoval exchanged glances, Jack wondering if he looked as crestfallen as the other did.
Marion Clary went on, "Yes, he rushed right out of here. Probably one of those little minor emergencies that always seem to come up during a conference and has to be fixed immediately if not sooner."
She reached for her desk phone. "Shall I have him paged for you?"
Jack said, "No, don't do that!" The receptionist was somewhat taken aback by his sudden burst of vehemence and he added quickly, "Thank you, but that won't be necessary."
Sandoval and Bass had already stepped off to one side for a quick, low-voiced exchange. Bass got out his handset and started talking into it.
Marion Clary fretted. "I'm quite sure I don't know what all the fuss is about."
Jack assumed a cheeriness at odds with the sinking feeling in the pit of his stomach. "Just one of those little emergencies you were talking about."
Sandoval was giving him the high sign, gesturing for Jack to join them. Jack said, "Excuse me."
He went to the others. Bass's handset fell silent as a crackling transmission ended. Bass said, "That was the gatehouse. Brad Oliver signed out and drove out of here a half hour ago. Like a bat out of hell, the guard said."
Jack was out of words for the moment. He had nothing to say. Neither did Sandoval.
Don Bass looked from one to the other. "Tough break, fellows. Looks like you missed him. Gee, if you'd just contacted me ahead of time, I could have picked him up and held him till you got here!"
##
## THE FOLLOWING TAKES PLACE BETWEEN THE HOURS OF 7 P.M. AND 8 P.M. MOUNTAIN DAYLIGHT TIME
Rimrock Road, Colorado
Brad Oliver had had a short run.
Sandoval said, "This is one of those good news–bad news situations. The good news is that we don't have to tell our bosses that Oliver got away from us. The bad news is that he's dead."
Jack Bauer said, "If that is Oliver down there." The two CTU agents stood in front of the Mercedes, which was parked on the shoulder of the road about twenty-five yards away from a gaping hole in the guardrail on the east side of the roadway.
The hole was several car lengths wide. It was bracketed by the twisted ends of a severed rail section. They had corkscrew shapes and were bent back so they thrust out into the empty air over the chasm.
This ordinarily empty stretch of Rimrock Road now bustled with lively activity.
A hundred-yard length of the east lane with the hole in the rail had been blocked off at both ends by patrol cars from the county sheriff's department. The two-way, two-lane road had been turned into a one-lane, two-way road in the area where Oliver's car had gone off the cliff.
Deputies with baton flashlights stood at opposite ends of the closed lane directing traffic. Southbound vehicles were temporarily halted at the north end to allow northbound vehicles to pass the accident site, then northbound vehicles were halted at the south end to allow some of the southbound vehicles to go on their way. The direction alternated every few minutes. The stop-go system caused vehicles to collect at both ends, creating a mini traffic jam.
A complicating factor was that everybody who drove by wanted to gawk and rubberneck at the spectacle, even though there was nothing much to see except the hole in the rail. Deputies shouted at the drivers of creeping vehicles, telling them to "Keep it moving! Keep it moving!"
Jack said, "Funny how an accident can draw a crowd even out here in the middle of nowhere. Before Oliver went over the side, there probably wasn't a car coming along this way more than once every five or ten minutes. Now it looks as busy as Main Street."
Sandoval said, "If it was an accident."
"And if that's Oliver."
The sun was behind western peaks, leaving the eastern slopes thickly shadowed with purple-blue gloom. Flashing lights on top of the police cars created a kind of carnivallike atmosphere. Most of the civilian vehicles had their headlights on. Emergency flares had been placed on the pavement at both ends of the closed east lane, throwing a lurid red glow in their immediate vicinity.
The closed lane and surrounding shoulder were reserved for parked patrol cars and their complement of officers who now stood around surveying the damage—all except those handling traffic control chores.
Sandoval's CTU ID card had placated the deputies who tried to shoo away the Mercedes and its accompanying SUV with the backup crew when they first arrived at the site and pulled over at the side of the road. The backup men stood grouped around their vehicle, doing what everybody else at the scene was doing; namely looking down over the edge into the gulf below.
It was a long, long way down. Vertical cliffs alternated with angled wooded slopes, stepping down for many hundreds of feet to the bottom of a rocky chasm. A handful of small fires lit the shadowy murk at the foot of the precipice. They looked like candle flames when seen from Rimrock Road. Other lights twinkled in the same general area, the lights of police and emergency vehicles that were gathered at ground level. They had to stand off some distance away from the fires because the car had fallen where the road below did not reach.
Jack said, "We'll have to go down there and take a look at the body for ourselves, even though it may be burned beyond recognition."
Sandoval said, "There's ways of identifying a burned body."
"But not immediately. It buys time."
Sandoval raised his gaze from the chasm to look Jack in the face. "You think Oliver pulled the old switcheroo and had someone else's body thrown off the cliff in his car?"
Jack shrugged. "Who knows?"
"He didn't have much time to pull off a tricky fast one like that."
"He had time enough to evade us."
"Yeah, I don't like that angle myself. It smells of a tipoff."
A state police car rolled into view driving along the shoulder of the southbound lane. A deputy halted traffic so the newcomer could edge across the lane into the closed northbound section. It halted and two uniformed officers got out: Lieutenant Bryce Hardin and Sergeant Cole Taggart.
They crossed to a knot of deputies at the side of the road. Sandoval said, "There's your buddies from the MRT, Jack."
Jack nodded. "I'd like to have a little chat with them, see what they know about the wreck. They reported it in first, according to the radio chatter."
"Why don't I go on ahead and check out the body while you do that? It'll save us some time."
Sandoval added, "Besides, it looks like a long hike over rough ground to get to the crash site and there's no need for both of us to it. You've done enough walking for today."
Jack waved it away. "That's all right."
"Why not? I'll ride down with the backup boys and you can join me later in the Merc."
Jack thought it over. "Okay."
Sandoval said, "I'll see you down below then." He started toward the SUV, paused, and turned around to face Jack again. He said, "Um, that's all you're going to do with Hardin and Taggart, right? Just talk?"
Jack said, "Just talk."
"Because you still might be sore about last night—I wouldn't blame you if you were—and Garcia'd raise holy hell if you get physical with the MRT again."
"Don't worry about it, we're all chums now."
"I bet." Sandoval's laughter was a little shaky. "See you later."
"In a bit."
Sandoval crossed to the backup crew clustered around the SUV. They all got into the vehicle, which started up, made a K-turn, and eased into the closed section of the southbound lane. Jack absently rubbed his swollen left jaw as he watched a deputy halt the traffic flow so the SUV could exit the scene.
Hardin stood off to the side talking to some deputies. Taggart stood by himself, hands on hips as he gazed into the abyss. Jack went to him.
Taggart looked up at the sound of approaching footfalls, turned, and saw Jack. He grinned tightly, said, "Well, well." He raised his hands in an I-surrender gesture, said, "Don't shoot!"
Jack said, "Ha-ha. I've already bagged my quota for today."
Taggart's toothy grin widened. "Just joshing you. No hard feelings?"
"What's done is done." Jack kept his expression blandly noncommittal. "Here we are again at another cliffside high dive."
Taggart said, "It's only been since this morning and already it seems like old times."
"Same road, different victim."
"'Cept the ATF boys going over was no accident. They were already dead when they were put in their car and shoved over the side."
"That's the consensus." Jack indicated the hole in the guardrail with a tilt of his head. "You think this was an accident?"
Taggart pushed back his hat brim to scratch his head. "Beats me. I didn't see it."
"I thought your MRT unit was the first to call it in."
"That's right, we did. Sharon Stallings over to Mountain Lake got the call. She's working dispatcher on the front desk. Some citizen phoned in to report that he'd seen a car go plowing off the cliff and she broadcast the alert."
Jack said, "That citizen have a name?"
Taggart shook his head. "Anonymous call, I do believe."
"Was the caller male or female?"
"I don't know, but Sharon could tell you. You know how these civilians are, they don't want to get involved. Afraid they're going to be called in as a witness and lose a lot of unpaid time in court waiting to testify. Can't say as I blame 'em much. So they phone in the tip and figure they've done their civic duty."
He looked shrewdly at Jack. "Any reason to think it wasn't an accident?"
Jack said, "Two in one day on the same road seems like more than coincidence."
"It's a dangerous road and by all accounts that boy was flying when he hit the rail. Must've been doing sixty, seventy miles an hour the way that rail is all torn up. Say, who was it, anyhow?"
"Brad Oliver."
"Never heard of him."
"He worked for one of the big shots at Sky Mount."
"Uh-oh. That'll raise a big stink. Means a whole lot more paperwork for everybody."
A second state police car entered the scene, approaching from the northbound direction. It halted beside Taggart and Hardin's car, which was parked a dozen yards north of where Jack and the sergeant were standing. The new arrival held one trooper behind the wheel and two shaggy-haired figures in the backseat. Taggart smiled slyly, said, "There's your buddy Miller Fisk."
Jack failed to rise to the bait. "Looks like he's got two prisoners."
"Looks like."
Fisk tapped the horn a few times, tooting lightly, causing Hardin to look away from the deputies with whom he was chatting. Hardin crossed to Fisk's car, approaching it on the passenger side. The passenger side window was rolled down, and Hardin rested his forearms on top of the door as he bent down and leaned forward to speak with Fisk.
Jack peered at the duo in the backseat. They were Griff and Rowdy, the two Hellbenders outlaw bikers who'd appeared earlier at the site of the ATF car wreck. They looked much the worse for wear, like they'd had a pretty hard time of it.
Hardin did more telling than listening and he didn't do much of either, engaging in a quick exchange with Fisk before straightening up and taking a step back from the car. Fisk put the car in drive and eased away, creeping up to where a deputy stood directing traffic.
Hardin turned, looking around. He spotted Taggart, scowling when he recognized Jack. He motioned to Taggart, gesturing for him to come over. Taggart said, "Oh well, back to work. See you around."
Jack said, "So long."
Taggart crossed to Hardin. They stood talking for a brief exchange, Hardin glancing again at Jack, the same scowl still on his face. He and Taggart turned and went to their vehicle, Hardin getting in on the front passenger side and Taggart on the driver's side. The car drove away, following in Fisk's wake.
Jack crossed briskly to the Mercedes and got behind the wheel. The keys were in the ignition where Sandoval had left them. Jack started the car and pointed it northbound.
He had his CTU ID ready for use in getting through the tie-up ahead but didn't need to use it. The deputy working traffic control duty must have assumed that any vehicle that'd been parked in the closed section had priority and automatically halted the traffic flow to let the Mercedes proceed.
Jack drove on the shoulder to the right of the northbound lane to get clear of the civilian vehicles clustered beyond the restricted area. Gravel crunched under the tires as his car snaked its way along the curved path until breaking through to the open road.
He had solid blacktop under his wheels now and he stepped on the gas, the Mercedes accelerating smoothly with a hum of power. The state police cars were out of sight, but this stretch of Rimrock Road led to the Mountain Lake substation a few miles ahead, and Jack guessed that that was their destination. He'd keep going if it weren't, in the hope of picking them up later, but he thought he wouldn't have to.
He reached for the hand mic of the dashboard radio comm set to contact Central to let them know what he was doing but stopped. Earlier events had given him pause. Brad Oliver had known that he was going to be arrested, and that tip could only have come from someone in CTU, someone high up in the command structure here or in L.A.
He didn't want the MRT members to know he was tailing them, and he feared that if he gave Central the information, the troopers would know. He stopped reaching for the mic and put both hands on the wheel and concentrated on closing the distance between himself and the state police cars.
A straightaway loomed ahead and Jack used it to pass several civilian vehicles poking along in the northbound lane. They were doing the legal limit but it seemed like they were crawling along to Jack, who was in a hurry to close with his quarry.
He zoomed into the opposite lane when it was clear in order to pass a few cars and trucks ahead of him. He cut it pretty close, narrowly avoiding a southbound car by swerving back into the northbound lane. The other car honked loudly in protest at the near-collision, its blaring horn Dopplering away as he speedily left it behind.
A road sign swam up in the headlights indicating a curve ahead. Jack rode the brakes, tires yelping as he cut a curve too closely before getting back on track. The blackness of empty space beyond the guardrail yawned but the car held the road as it rounded the turn.
The road straightened out again. Jack slowed, thinking that Brad Oliver must have been driving somewhat like this when he'd made the fatal plunge.
The route passed a couple of turnoffs on the left, west side of the road that were cuts in the mountainside. The Mercedes entered a lonely stretch of empty road. An oddly shaped rock formation thrusting out above the road looked familiar to Jack, who recognized it from his previous trip to Mountain Lake.
He estimated he was halfway to the substation. A pair of red taillights winked far ahead, swerving left and disappearing as they rounded a curve.
Jack eased up on the accelerator. He didn't want to show himself if that was an MRT car. He slowed to let the other vehicle gain some distance.
He rounded the curve. A scenic lookout area bordered the road's eastern shoulder where a knob of rock jutted out from the cliffside, leaving enough space for a gravel parking area and a grassy patch studded by a boulder faced with a metal plaque tourist guide.
A pickup truck sat in the parking area facing north, its lights dark. The Mercedes zipped by it. A pair of headlights flashed their sudden bright, dazzling glare in the rear window.
The pickup truck zoomed out of the parking area and into the northbound lane with its high beams on. It was a big machine and the sound of its engine was loud as it took off after Jack in a hurry.
It ate up the distance between itself and Jack's car, quickly closing the gap. It had a high suspension and its headlights were correspondingly raised so that they seemed to shine directly into the Mercedes, flooding its interior with white-hot glare.
The road hit a series of curves, forcing Jack to slow still further. The pickup was only a length or two behind him. The Mercedes handled beautifully but the pickup's greater weight compensated for its height and Jack couldn't shake it.
The curves were long, lazy, and looping but the pace was frantic. The pickup truck bumped the back of Jack's car, jostling it. Jack had to fight to keep from losing control of the wheel as the right side tires slid on the shoulder but managed to whip the Mercedes back on to the pavement.
The pickup's front was bolstered by a piece of solid steel plate that covered it from bumper to hood. Holes were cut in the plate to allow the headlights to shine through.
The pickup lunged forward, slamming the Mercedes again, delivering a bone-jostling thump to Jack. His belly knotted at the thought that another such blow might trip the car's air bag safety device, a development that could prove fatal in this lethal game of high-speed bumper cars.
Trouble was that the pickup was doing all of the bumping and the Mercedes all of the catching. Jack could do nothing but thread the curves, riding both lanes and hoping no oncoming vehicle lay around each blind corner.
Another hit destabilized the car, causing it to weave crazily and slide sideways toward the guardrail and the abyss. The Mercedes fishtailed as it took the curve but it took it, tires digging in and biting deep into the pavement.
The pickup nudged the car, snugging its steel-plated front against the vehicle's rear at a tilted angle. The truck lunged with a snarl of power and shoved the Mercedes sideways.
The car would have been swept off the cliff if the tilt were angled outward. But the tilt was angled inward, causing the car to slide sideways toward the rock wall on its left.
The rocks loomed up in the driver's side window, their craggy surfaces harshly lit by the intense glare of the pickup's high beams. The car rushed sideways to meet them.
There was the crump of collapsing metal and an explosion of shattered glass as the Mercedes plowed into the mountainside under the pickup truck's impetus.
A stunning impact followed, setting off a massive fireworks display inside Jack's head.
Then, blackness.
##
## THE FOLLOWING TAKES PLACE BETWEEN THE HOURS OF 8 P.M. AND 9 P.M. MOUNTAIN DAYLIGHT TIME
Mountain Lake, Colorado
Jack Bauer's awareness flickered, sputtering like a TV set with a loose connection. Bursts of sound and vision alternated with patches of darkness.
Rough handling jarred him into wakefulness, restoring his sense of self. He could think but not move. His seat belt harness was open and he was being hauled out from behind the air bag, which had mushroomed out of the top of the steering column to fill the driver's side with a big white balloon. He was dragged across the passenger seat out the open door and dropped to the ground.
The fall jolted Jack into opening his eyes. He was bathed in white light streaked by red and blue flashes. He couldn't see their source. He lay on his side on the shoulder of the road. His field of vision encompassed two pairs of legs and feet. Both sets of feet wore cowboy boots under loose-fitting trousers. One pair had sharp-pointed toes and fancy hand-tooled leatherwork, the other was squared off at the toes and unornamented.
A square-toed boot stepped on his upright shoulder and shoved him on his back, setting off fresh fireworks in Jack's head. A body wash of aching soreness kept him from blacking out.
The boot's owner straddled him and bent down. It was Taggart. He pulled Jack's gun from the shoulder holster, said, "You won't need this." He straightened up and stepped away, tucking the gun into the top of his waistband. "Maybe I'll keep it for a souvenir."
He stood on one side of Jack. Hardin, the owner of the fancy boots, stood at the other. A third figure stood at Jack's feet. The stranger was a grotesque, short, skinny, and bowlegged. He had a bony, close-cropped scalp and wore round wire-rimmed glasses that made his orbs look like those of a popeyed frog. He wore a thin vest, a dark T-shirt decorated with an elaborate skull emblem, and skintight jeans tucked into oversized combat boots with steel toes and three-inch soles.
Red and blue lights flashed on the trio, splashing them with weird highlights and color accents. The stranger gave a start, said, "Here comes a car!"
Hardin said, "What of it? We're supposed to be here, we're cops. Wave 'em on, Cole."
"Right." Taggart walked away out of Jack's vision, his footsteps sounding on hard pavement. A car approached, its headlight beams sweeping across the scene. Its engine noise was loud as it slowed to a crawl and drew abreast; the noise lessened as the car passed and drove away.
Hardin said, "That's all there is to it, Mr. Pettibone. You're nervous in the service."
Pettibone, the third man, was restless, fidgeting. One of his legs shook, vibrating to an invisible rhythm. He said, "I ain't got all night. You, neither." His nasal voice had a Western twang with a bite as sharp as a crosscut saw. He said, "The Rebel wants things done quick!"
Hardin's expression turned ugly. "I don't take orders from Weld."
Pettibone fired back, "You both take orders from the same fellow—"
"Yeah, and you ain't him, so no more of your lip."
Taggart rejoined them. "What's the problem?"
Hardin said, "Pettibone's got ants in his pants, that's all. He's scared of Weld."
Pettibone said, "Never mind about that! You do your job and I'll do mine."
Taggart stood beside Hardin. "We're doing it."
Pettibone said, "Take him to the station." By "him" he meant Jack Bauer. "I'll get rid of the car and be by directly to pick him up and take him to Winnetou."
Taggart said, "Sure you can handle that car by yourself?"
Pettibone said, "Hell, yeah! I got me a set of chains and binders in the back of the truck. I'll hook one end up to the car's rear axle, drag it to the other side of the road, and push it off the cliff. I know what I'm doing, I used to be in the wrecking business."
Taggart laughed. "Used to be, he says."
Hardin cautioned, "No fires. We don't want to call attention to this one for a while."
Pettibone shook his head. "Ain't gonna be no fires. The engine's off so there won't be no spark to touch off any spilled gas."
Hardin said, "Get to it, then. We got things to do, too." He turned to Taggart. "You take Bauer's arms and I'll take his feet so we can carry him to the car."
Taggart said, "You would leave me the hardest part of the work."
"Rank has its privileges," Hardin said, chuckling. He stood at Jack's feet while Taggart stood at his head. Taggart hunkered down, getting his hands under Jack's arms and clasping them on top of Jack's chest. Hardin grabbed Jack's feet by the ankles, holding them together and getting them under one arm. The lawmen straightened up, lifting Jack off the ground. He choked back a groan as the movement sent new pain waves shivering through him.
The scene came into view from a different perspective. The battered Mercedes slumped against a rocky mountainside. The pickup stood near it, facing north on the shoulder to the west of the road. The MRT car stood a dozen feet away, facing south in the southbound lane and blocking it. Red and blue lights flashed in the rooftop light rack.
Hardin and Taggart hauled Jack to their car. Hardin said, "Hold him up—I'll get the door." He was breathing hard, huffing and puffing. He set Jack's feet down on the pavement. Taggart stood crouched holding up Jack's upper body while Jack's legs rested on the macadam. Hardin opened the vehicle's rear door and helped Taggart heave Jack across the backseat.
Pettibone called, "Remember, Reb wants him alive and able to talk."
Taggart said, "We'll treat him like an egg wrapped in cotton." Hardin didn't say anything, he was still trying to catch his breath. Taggart slammed the rear door shut. It was a patrol car so there were no handles on the inside back doors, and a wire cage separated the front seat from the back.
Taggart got behind the wheel and Hardin got in the front passenger seat. Hardin wheezed, "Man! I got to get in shape some of these days."
Taggart said, "You're in shape, you've been exercising those table muscles."
Hardin told the other what he could do to himself. Taggart laughed, made a K-turn into the opposite lane, pointed the vehicle north, and drove away.
Hardin said, "No emergency lights. We don't need them."
Taggart switched them off, said, "Like I told the man, it's a dangerous road."
Hardin said, "For some folks, yeah."
A few cars passed them going in the opposite direction on the way to the Mountain Lake substation. The drive took less than ten minutes, each precious second giving Jack Bauer more time to collect his wits and gather what stores of energy remained to him. He could move now even though it hurt to do so. Every heartbeat was like a giant fist squeezing him, wringing him out like a wet washrag. His head pounded and he felt sick to his stomach.
The MRT car turned right into the drive leading to the substation. Hardin said, "Pull into the motor pool so we don't have to carry him so far."
The motor pool was a two-car garage attached to the substation. Its rollup door was open and its overhead lights burned bright. The car rolled to a halt inside one of the bays. A pair of chopped, heavy-duty Harley-Davidson motorcycles with extended front forks occupied the other bay.
Taggart switched off the headlights and engine. He and Hardin got out and went to one of the rear doors. Hardin opened it. Jack lay sprawled on his side across the backseat, legs bent at the knees. Hardin used the pointy toe of one of his boots to kick the sole of Jack's shoe. He said, "Get up."
He stepped back, unfastening the flap at the top of his sidearm so he could get at it more quickly. His hand rested on the wood-handled gun butt of the big .44. "Weld wants you alive but he didn't say nothing about putting a bullet through your kneecap and that's what I surely will do if you try any of your fancy tricks. Do you read me, mister?"
Jack said, "Yes."
"On your feet then."
Jack swung his legs, sitting up and putting his feet on the garage floor. Colored dots of light showered over him, dimming his vision. He clutched the rim of the open door with both hands to keep from falling. The dizziness passed, the scene brightening as the stream of colored dots thinned and receded.
Jack pulled himself out of the car and stood up. He lurched and staggered, putting out a hand on the rear fender to steady himself.
Taggart said, "You look a mite unsteady, Jack. Let me give you a hand." He gripped Jack's arm above the left elbow.
A closed door stood on the left side of the garage's rear wall. Hardin backed up to it, hand resting on his gun butt, steadily eyeing Jack. He reached around behind him with his free hand, grabbing the doorknob and turning it, opening the door. The door opened outward and he had to step forward out of its way to open it fully. He backed through the doorway into the station, watching Jack all the while.
Taggart said, "Here we go," the pressure of his hand on Jack's arm urging him forward. Jack advanced with slow shuffling steps, making out that he was weaker than he was in hopes of misleading the others about his condition. He made a point of staring straight ahead, not even glancing at Taggart's sidearm holstered on the right hip or his own gun stuck into the top of Taggart's waistband on the left side.
Taggart said with great good humor, "Bet you'd like to get your hands on one of these heaters, eh, Jack?"
Hardin said, "Try it. Just try."
Jack didn't try for it. He went through the doorway into the main room of the substation. Sharon Stallings sat behind the dispatcher's front desk drinking coffee. Miller Fisk was on the other side of the space, sitting with his feet up on the squad room desk and reading a hunting magazine.
The two bikers, Griff and Rowdy, were both penned in the same single detention cell. Three steel-barred cage walls met a building wall of solid concrete blocks painted pale green like the rest of the building's interior. A metal plank bunk covered by a thin fabric pallet jutted out from the stone wall. The big biker, Rowdy, sat on the bunk with his head tilted back. The smaller one, Griff, stood leaning against the front wall of the cell. Their faces were bruised, cut, and swollen from a recent beating.
Fisk looked over the top of the magazine, eyes widening as saw Jack in the custody of Hardin and Taggart. His lips puckered in a soundless whistle. "Well, looky here! So you got him!"
Taggart said, "Yeah, we picked him up for reckless driving."
Hardin said, "Get your feet off that desk, Fisk. You're not back home in the barn now."
Fisk put his feet on the floor fast and stood up. "Sorry, Uncle Bryce—"
"That's Lieutenant Hardin to you and don't you forget it!"
"Yessir!" Fisk fastened his eyes on Jack, staring at him with abject fascination. His eyes got a glazed look in them. He licked his lips. "Yes, sir!"
Taggart guided Jack across the room to the squad room desk. He said, "You don't look so good, amigo. You better sit down." He indicated the chair that Fisk had just quit, an office chair with four roller-mounted legs. Jack sat down in it. Hardin circled around to the other side of the desk so he'd have a clear line of fire if Jack tried something. His hand was still on his gun.
Taggart took out a pair of cuffs from a leather case clipped to his belt. "Gun hand first." He circled Jack's right wrist with one of the cuffs and clamped the other one to the chair arm on that side. "Now the other. Let me have your cuffs, Fisk."
Fisk was quick to comply, his eyes glimmering and his face shining. Taggart repeated the process, this time cuffing Jack's left hand to the chair's left arm. He said, "Jack's trouble with either hand. Ain't that right, Fisk?"
Fisk colored. "Never you mind about that!"
Hardin took his hand off his gun. Taggart grinned, stepped back. "There you go, Jack. Now you don't have to worry about falling out of your chair."
Jack said sourly, "That's right neighborly of you, pardner."
Taggart laughed out loud. "That's the spirit. Keep your chin up. You know, believe it or not, I kind of like you, Jack. That's the hell of it."
Fisk said, "Well, I don't!"
Taggart rested a hip on the edge of the desk. "Tell me something, Jack, just to satisfy my own curiosity. How'd you get on to us?"
Jack looked steadily at him. "The ATF men, Dean and O'Hara. They were pros. They weren't drugged. I couldn't see the cultists sneaking up on them and catching them unawares. But the MRT, fellow cops they knew and thought could be trusted, you could have walked right up and gotten the drop on them. They didn't know the truth until it was too late."
"That's not bad figuring."
"After that it was just a matter of the way things went down. You people were always in the right place at the right time to do some damage. When I found out that your outfit reported the Oliver crash first, it all added up."
"Too bad you didn't know that we had a crash all planned and ready for you."
"Yeah, too bad. I didn't think the whole unit was dirty, either."
"For what we're getting paid, we can't afford not to be."
Hardin said, "That's enough, Cole. You're talking too much."
"Why? What difference does it make?"
Jack said, "Dead men tell no tales, eh?"
Taggart nodded. "Not this side of the grave."
Hardin made a dismissive gesture and crossed to the front desk. "Any messages for me, Sharon?"
"Yes, sir. Sheriff Mack called to remind you about that confab over to Sky Mount tonight."
"Damn! I most forgot about that." Hardin glanced at his watch. "Still got time to make it. Let's go, Cole, we got to saddle up. We got that meeting with the county boys to map out security arrangements for tomorrow's Round Table."
Taggart's laugh was a short, humorless bark. "That's a good one."
"We don't show, some folks might get the crazy idea that we thought there wouldn't be a session tomorrow."
"I see what you mean. Can't have that."
Hardin spoke to the dispatcher, "Sharon, you're in charge here while Cole and I are gone." He turned to Fisk. "You hear that, boy? Trooper Stallings is in charge, and if she gives you an order it's the same as if I did, so you hop to it and do like she says, savvy?"
"Yes, sir."
"Mr. Pettibone will be by to pick up that one," Hardin said, indicating Jack. "Don't take any chances, Sharon. Make sure his hands are cuffed behind his back when you make the transfer. I want you to supervise it personally, you hear?"
"You can count on me, Bryce."
"He's got to be able to talk and he can't be too busted up. You listening, Fisk?"
"Yes, sir. Uh, sir? How much is too much?"
"He can't look like he's been beaten half to death when he's found later. Otherwise, have your fun. I know you're going to anyway."
"Aw, Uncle Bryce, you know I wouldn't do nothing without your say-so."
"Lieutenant Hardin, boy."
"Yes, sir. Lieutenant Hardin. Sir."
Taggart went to the front desk and handed Jack's gun to the dispatcher. "Add that to your collection, Sharon."
She opened a drawer on the side of her desk and placed the gun inside. Hardin frowned, said, "That's another thing, Cole. That weapon's evidence that Bauer's been here. You can't keep it."
Taggart said, "Good point. I'll get rid of it when we get back."
"Good." Hardin fastened cold eyes on the bikers in the holding cell. "We got some more house cleaning to do when we get back. Take out all the trash."
Sharon Stallings said, "When'll you be back?"
"An hour or so, no more." He and Taggart crossed toward the garage door. Hardin went out.
Taggart paused in the doorway, turning to look back. He said, "Adios, Jack. No hard feelings—at least, not on my side. I wouldn't blame you if you had some, considering. That's your prerogative."
Hardin called to him from the garage. Taggart said, "Coming." He went into the garage, closing the door behind him. After a pause came the sound of an engine starting up and then the car driving away.
Fisk licked his lips. They were already wet and glistening. His eyes were focused and intent above a loose, sloppy smile. He made a big show of cracking his knuckles. He said to Jack, "You don't look like so much now."
Jack was silent. Fisk went on, "Where's all your smart remarks?"
Sharon Stallings rolled her eyes. "For Pete's sake, Fisk, quit jawing and get on with it."
"I'm taking my own sweet time. I'm gonna enjoy this." Fisk squared off, looming above Jack in the chair. "Uncle Bryce said not to leave you half killed. He didn't say nothing about no three-quarters, though."
He punched Jack in the face. The impact snapped Jack's head back and sent the roller-mounted chair with Jack in it wheeling backward until it crashed into a wall.
Jack's face was numb where he'd been struck but he could feel something leaking from his nostrils, and the taste of blood was copper-tangy in his mouth. Rowdy and Griff crowded the front of their cell, clutching the bars and staring. Sharon Stallings watched, chewing gum. That was the detail that stuck with Jack: her chewing gum.
Fisk ruefully eyed his big fist. "Dang, I like to've skinned a knuckle on that one!"
The smaller biker, Griff, said, "You dirty dog! You've got to have a guy tied down before you beat on him..."
Fisk grinned wetly, waving the other's complaint away. "Shut up, runt. You had your hands free when I gave you your whomping."
"You hit me with your gun first!"
Rowdy said, "Let me out of here and try me on, farm boy!"
Fisk said, "You already had your turn. I'm just getting warmed up on this one."
He clouted Jack with a vicious backhand to the left side of the face. Jack saw it coming and tried to roll with it. It was a swivel chair so he was able to rotate the seat away from the blow, but even so it rocked him from head to heels. The chair toppled over, falling on its right side to the floor with a crash. Fisk giggled. "Whoops!"
Sharon Stallings stood up. She showed some animation now, spots of color burning in her cheeks. "Fisk, you better not break that chair—"
Fisk grabbed a chair arm in each hand and by main strength yanked it and the man chained to it upright. He drove a hard right into Jack's belly, burying it deep. Jack doubled up as the chair zoomed backward, crashing against the front of the cell.
Fisk crossed to Jack, grabbing the chair and spinning it around a half circle so that he stood facing Jack with his back to the cell. A thin line of spittle drooled down the corner of Fisk's mouth, wetting his chin. His hot, moist breath was on Jack's face. He launched an uppercut that collided with Jack's chin. The chair wheeled backward into the side of the squad room desk.
Sharon Stallings was outraged now. "Fisk! The desk!"
Fisk stalked Jack, closing on him, his hulking form looming larger. Griff and Rowdy stood pressed against the bars of the cell. Jack made eye contact with Griff and tilted his head in a slight but perceptible nod. Griff blinked, eyes narrowing.
Fisk never saw the nod because he was too busy leaning over the chair and winding up to deliver another haymaker. Jack kicked him between the legs.
Fisk went, "Whoof!" He doubled up and grabbed his crotch with both hands. He wavered, swaying. Cold sweat filmed his leaden, gasping face.
Jack got both feet on the floor and spun the swivel seat around so that he was facing the side of the desk with his back to Fisk. He brought his feet up, bending both legs at the knees, and pushed off from the desk as hard as he could.
The chair with Jack in it went slamming into Fisk, knocking him off balance. Fisk backpedaled to keep from falling. Jack dug his heels into the floor and kept working his legs, driving the chair into Fisk and pushing him backward.
Fisk collided with the front of the cell. Griff and Rowdy were ready. Rowdy thrust his right arm between the bars and hooked it around Fisk's neck. Griff was shouting, "Get him, bro! Get him!"
Rowdy pulled back hard. He was a big man, too, almost as big as Fisk, and with plenty of muscle. The back of Fisk's head fetched up against the bars with a clang that sent them ringing. Rowdy grabbed his own right wrist with his left hand and got Fisk in a choke hold.
Jack kept his feet working, scrabbling them on the linoleum floor to keep slamming the top of the back of the chair into Fisk's middle. Rowdy got a knee up against the bars where the small of Fisk's back was. The big biker leaned back, putting his weight into the choke hold.
Fisk's eyes were like soft-boiled eggs floating in a purple face. He had both hands up clawing at Rowdy's forearm where it circled his throat but he couldn't break the other's grip.
Griff said, "Get him, bro, get him!" He'd stopped yelling and was calling out in a breathy whisper, like a crapshooting gambler urging the dice to come through for him on a long-shot roll.
Griff grabbed the top of Fisk's gun in its hip holster. It was a .357, held down not by a flap but by a leather strap. Griff's fingers tore at the strap, loosing it.
Sharon Stallings already had her gun out. It was a .357, too. She came out from behind the front desk, angling for a clear shot. Fisk was in her way, causing her to hesitate.
Griff yanked Fisk's gun clear of the holster and leveled it at Stallings. Gunfire cannonaded as he cut loose, shooting the middle out of her. She came apart and fell down in a heap.
Griff shoved the gun's smoking snout against Fisk's side but before he could pull the trigger there was a cracking sound like a crisp breadstick being snapped in two. Only instead of a breadstick it was Fisk's neck that was being broken as Rowdy pivoted his upper body and twisted Fisk's head to an angle beyond the human design tolerance limit.
##
## THE FOLLOWING TAKES PLACE BETWEEN THE HOURS OF 9 P.M. AND 10 P.M. MOUNTAIN DAYLIGHT TIME
Mountain Lake Substation, Colorado
Griff said, "Keep holding him up—don't drop him!"
Rowdy said, "I ain't dropping nothing." He still stood with his arm through the bars holding Fisk in a chokehold. Fisk was all dead weight now with nothing to hold him up but Rowdy. His bulging-eyed, slack-jawed head lolled at an unnatural angle.
Rowdy said, "I could do this all day."
"We ain't got all day, man."
Jack said, "You're so right."
Griff's eyes lit up. "Hey, you're still with us! Stay awake, dude. Don't pass out!"
"I won't."
Griff first checked Fisk's handcuff case on his belt because sometimes cops keep their handcuff keys there in a small compartment but the search came up empty. He now stood with his hand between the bars reaching into Fisk's right front pants pocket. He stood on tiptoes, standing sideways with the side of his face pressed against the bars for a longer reach. His face was hot and the bars were cool. He was reaching in and down. He said, "I think I got his key ring—"
Rowdy said, "Don't drop it, man."
The pocket tore at the seams as Griff closed his fingers around a chunk of loose metal and fished it out. He pulled his closed fist back to his side of the bars before opening it. A key ring lay in his palm. He said, "I ain't dropping nothing."
Rowdy sagged in relief, allowing Fisk to slump a bit lower. Griff, alarmed, said, "Whoa! Keep hold of him until I'm sure the handcuff key is on this ring."
Rowdy said, "Step on it. This slob weighs a ton."
Griff's hands shook with eagerness as he held the ring up to the light and began flipping through the keys until he came to a pair of tiny black keys that looked like something out of a child's play set. "All right! Handcuff keys if ever I saw 'em—and believe me, I've seen plenty!"
"Stop bragging. Can I let this pig loose?"
"Yeah, but don't let him fall too far in case these keys don't work and we got to check him for others."
Rowdy eased Fisk down the bars to the floor, the body folding up as though it were as boneless as a bag of dirty laundry. The corpse sprawled at the foot of the barred cell door, an inert lump.
Now the bikers could see Jack Bauer, the back of him anyway. Jack's head tilted forward, chin resting on his chest. He breathed slowly, deeply.
Griff said, "Shit! He's passed out!"
Jack raised his head. "No...I was just resting."
"Rest later. Fight it, man! You need to stay awake and alert. We need you. You need us. We need each other to get out of this scrape."
"You should set that to music."
Griff grinned tightly, below the eyes. "You can still joke, huh? That's good. That'll keep you going. Turn around so I can see you."
Jack used his feet to rotate the chair's swivel seat in a half circle so that he was facing the cell door. He had to step on one of Fisk's outflung arms to do so. It bothered him not a bit.
Jack's right eye was blackened and his left was swollen half-shut but he could still see out of it. His nose was bloodied and his lips were smashed and split, bleeding on the inside where they'd been cut against his teeth. He felt around with his tongue; his teeth seemed to be all there. He couldn't tell if any of them were loose or not. His jaws ached at the hinges. His ribs were bruised and his belly ached.
Griff said, "What's your name, dude?"
"Jack."
"I'm Griff and this is my buddy Rowdy."
"Hi."
Rowdy said, "Pleased to meet cha', Jack."
"Likewise."
Griff said, "You handle yourself okay, Jack. That was nice work softening up that big pig for us."
Jack glanced at Sharon Stallings, a corpse in the center of a still-expanding pool of blood, then eyed Fisk. It was the first time he'd taken a good look at Fisk dead. On him it looked good. Jack said, "You men didn't do too badly yourselves."
"Thanks. Now that we've done the mutual admiration bit let's focus on something really important, like getting out of here." Griff held a handcuff key between thumb and forefinger so it was separated from the other keys on the ring. "Wheel that chair over here so I can reach you."
Jack used his feet to propel himself on the rollers so his left side was against the bars. Fisk's body blocked his progress until Rowdy collared the corpse by the shirt at the back of its neck and dragged it to one side.
Griff knelt so he was at eye level with the chair arm. He reached through the bars with both hands, the left holding Jack's cuffed hand steady while the right fitted the key into the equally tiny slotted keyhole and turned it until something clicked.
The cuffs unlocked, Jack freeing the metal bracelet from his left wrist. The flesh was marked with angry red grooves where the cuff had bitten into it. He bent his arm at the elbow and raised his hand, flexing it to restore the circulation. Numbness was succeeded by a tingling wave of pins-and-needles sensation that momentarily took his breath away.
Griff said, "I don't know if the same key will work on another cop's cuffs but it should. It'd be a hassle for them to keep track of different sets of keys for each pair of cuffs."
Rowdy said, "If it don't work you can blast the chain loose with the .357."
Jack said, "Let's try the key first." He jockeyed the chair around so that its right side pressed against the bars. It was easier to get around now that he had one hand free.
Griff fitted the key into the lock and jiggled around with it. "Wait a minute—wait a minute—there, I got it!" There was a click and the cuff opened, falling away from Jack's wrist.
Jack flexed both hands, clenching and unclenching his fists as the feeling returned to them. The sharp edge of it chased away the fog of haziness that shrouded his awareness and sought to pull him down into the sweet, pain-free oblivion of unconsciousness.
An oblivion that might prove permanent if indulged in, he reminded himself. Mr. Pettibone was coming.
Griff urged, "Don't fade on us now, dude. We're so close to making the breakout."
Jack said, "I'm good."
"Okay. The key to the cell is on a big ring hanging on a hook behind the front desk. You can't miss it."
Jack grabbed the cell bars with both hands and pulled himself out of the chair to his feet. He lurched, almost losing his footing but regaining it before he went over. He stood there clutching the bars. Griff's mouth was moving but the words seemed to come from the bottom of a deep well—or was it the top? They were hollow and echoing in any case, mixing with the sound of crashing surf rising in his ears that threatened to overwhelm him.
Jack stood there until the spell of weakness passed and he could make out Griff saying, "Are you okay? You okay, man?"
Jack said, "Yeah." He put his hand against the wall to steady himself as he walked step by step from the cell to the front desk. He was stiff-legged and halting at first but grew surer and more certain with each step. The front desk was on a kind of dais that nearly tripped him when he stepped up onto it but he staggered to the desk and rested both fists on the desktop and leaned forward until the pounding in his head went away.
He went behind the desk and tore open a drawer. Griff called urgently across the room, "Not there, Jack! The keys are on a hook on the wall behind you!"
Jack said, "Hey man, chill." That surprised Griff so that he shut up for a minute. Jack reached into the drawer, reaching for his gun. It lay on its side on top of a pile of hardware that included guns, knives, brass knuckles, blackjacks, and other goodies.
The gun felt good in his hand, he liked the heft and weight and balance of it. He seemed to drew strength from it, like a parched plant soaking up moisture.
He fitted the gun in his shoulder sling. It felt nice nestled down below his left arm.
The key ring was where Griff had said it would be, on a ring on a peg sticking out of a plaque mounted on the wall behind the desk. The oversized ring was a steel hoop as wide in diameter as a pie tin, and when Jack saw the key he thought that little had changed over the centuries when it came to keeping prisoners penned in cells because the key looked like it could have unlocked a medieval dungeon. It had an eight-inch-long bolt with a notched and grooved rectangle at the tip and a solid steel loop at the end.
Jack went with the key to the cell. His progress was better. He never came close to blacking out and he staggered and almost fell only once.
Griff and Rowdy eyed his approach with silent wariness. Griff had a sharp-featured face with long, narrow, slitted green eyes, a beaky nose and pointed chin. Rowdy's forehead was as wrinkled as an elephant's knee, the result of deep thought. He said, "You ain't no cop, Jack. What are you—a hit man?"
Jack realized that coming from Rowdy that was a compliment. He said, "No, I'm a secret agent." It was more complicated than that but he gave them a short version they could wrap their heads around. They didn't necessarily believe him but at least they could understand him.
He stuck the key into the cell lock. The steel loop at the end was big enough for him to fit his fingers around and use for a hand grip. He needed it, too, to turn the lock and unseal its reluctant internal mechanism. Bolts and tumblers fell into place with a thud and then the door opened.
Jack got out of the way to avoid being trampled by the two bikers in their eagerness to be free of the cell. They whooped it up. Jack figured it was best to let them get it out of their systems before he made his pitch.
Griff wasted little time on euphoria, hurrying to Sharon Stallings's corpse to pry the unfired .357 from her hand. He hefted it, saying, "Nice piece! The gun, I mean."
Jack said, "There's an open drawer in the front desk that has some of your stuff in it, I think."
Griff and Rowdy went to it, pulling the weapons out and laying them on the desk. Rowdy picked up a snub-nosed .38 special and eyed it with the tender regard of a schoolgirl for a warm puppy. "I never thought I'd see this again!" He explained, "It's got sentimental value. I took it off some plainclothes cop I beat the shit out of."
Griff brandished a commando knife the size of a small sword. It was razor sharp with a grooved blood vein on either side. "My Arkansaw toothpick!"
Jack thought that if they didn't know it already, when they found out their bikes were in the garage they'd be like a couple of kids on Christmas morning. Now was the time to start working his pitch. He said, "Why fool around with that when you can raid the substation's armory? There's sure to be one here and the keys are probably somewhere in the desk."
Griff stopped waving the knife in the air to eye Jack with crafty calculation. "I don't get you, Jack. You look like a straight citizen but you're a stone killer. What's your angle? I mean, where do you fit in with all this anyway?"
"I told you, I'm a secret agent."
"Bullshit—"
"I'm also your key to Reb Weld."
That got them. Saying the name was like invoking the magic word. Griff and Rowdy stopped dead in their tracks, exchanging poker-faced glances. The mood in the air was delicate, hanging by a thread. Their hunting instinct was on full alert and one wrong word, one misstep could trigger a mutually slaughterous gun-down.
Griff said, too casual, "I don't believe I caught that name."
Jack said, "Now who's bullshitting? Reb Weld. A name I'm not likely to forget because he's tried to kill me several times today. I've got a feeling it means something special to you, too. You boys better stick close and make sure nothing happens to me because I'm your one way of finding the Rebel and cashing in on that fifty-thousand-dollar bounty on his head."
Griff and Rowdy stood there poised on the razor's edge not knowing which way to jump, torn between greed and suspicion. Jack worked on their greed. "I'll give you the short version. You may have heard that there's a millionaires' convention being held not far from here."
Rowdy said, "Yeah, we heard of it. Impossible not to with all the cops it draws; they've really been cramping our style."
"Reb's being paid to wreck the party. My job is to stop him. You want him, too. He betrayed your outfit and sold out his club brothers so he could skip with the proceeds on their gunrunning racket."
Griff got huffy. "That's not a job to us—it's a sacred trust."
Jack said, "Yeah, with a fifty-thousand-dollar payday. We both want Reb Weld chopped for different reasons. I know how to find him. What's more, I can square this cop-killing beef so that you'll never catch any heat for it."
"You talk big."
"I can deliver. We're all in on this double kill together. You know there's not an undercover cop in the U.S. that could tie into that kind of action and ever testify about it in court."
Rowdy said, "That's right, Griff—"
"Shut up and let me think. Who the hell are you, dude?"
Jack said, "I could show you a card that identifies me as a member of the Counter Terrorist Unit but anybody can get a fake ID. You've seen me in action, I've seen you in action. I'm not asking you to take it on faith. If I can't deliver Weld, there's two of you and one of me. You're not afraid, are you?"
Griff tsk-tsked. "That's low, Jack. No need to get insulting."
Rowdy said piously, "The 'Benders fear no man!"
Jack said, "That's what I'm counting on. We can talk about the deal while we're cracking into the armory. We're going to need some heavy firepower quick and time's running out."
Pettibone wouldn't talk. He was the stubborn type. Also maybe a little bit stupid because there was no mistaking that Jack Bauer and his two biker allies meant business.
Jack said, "The trouble is he's more afraid of Weld than he is of us."
Griff said, "I'll fix that."
"Remember he's our prime lead to finding the Rebel. We need him alive and talking."
"He'll live—unless he's got a bum ticker."
Rowdy said, "He's a speed freak. If he gets off on meth there's nothing wrong with his heart. That shit's a rocket ride."
Griff said, "You should know, bro."
"Look who's talking."
The object of their attention sat tied to a straight-backed wooden chair. Jack had decided to use a wooden one instead of a roller-mounted office chair for the simple reason that wood doesn't conduct electricity.
He was not unaware of the ironies present in the reversal of fortune that had seen him transformed in less than an hour from the subject of torture to the inflictor. This turnabout troubled him not at all, considering that it was Pettibone who'd delivered him to the tender mercies of the MRT. There was no way around the hard fact that Pettibone had to be made to talk, to spill his guts about the plot against Sky Mount. Hundreds of innocent lives and perhaps the fate of a great nation depended on it.
Pettibone had walked unaware into the lion's den less than fifteen minutes earlier. He'd arrived at the Mountain Lake substation to pick up Jack for delivery to Reb Weld. He parked the pickup truck with the steel-plated front behind the back of the building where it couldn't be seen from the road. He went through the garage door into the substation, his knowledge of the site suggesting that this was a familiar routine with him. Jack wondered how many others Hardin and his crew had handed over to Pettibone for a one-way ride.
Pettibone stepped through the door only to discover the muzzle of Jack's pistol being pressed against his skull. He froze except for his eyes, which looked like they were going to pop out of the sockets. His thick-lensed glasses magnified his already bulging orbs as they took in the dead bodies of Fisk and Stallings.
Jack said, "That's right, Mountain Lake is under new management."
A quick search relieved the captive of a gun, switchblade knife, several sets of keys, a wallet, a packet wrapped in tin foil, and some pocket litter. The wallet yielded a state driver's license issued to one Arthur Conley Pettibone. Jack couldn't tell if the license or the bearer's name or both were phony but it didn't matter now that he had possession of the man himself. The tin foil packet contained several grams of a grainy white powdery substance. Rowdy put some on his forefinger and tasted it. "Crystal meth," he said. "Pretty good shit, too."
Jack folded up the packet and pocketed it. Rowdy said, "Hey!—"
Jack said, "I need you with a clear head and a steady hand for the next couple of hours."
Rowdy started to do a slow burn. "You're taking a lot on yourself, dude."
Griff clapped him on the shoulder. "Forget it, man. Jack's right. You don't know what that shit's cut with or what it might do to you. Besides, you don't want him thinking that you're one of those shooters who gets his nerve from a noseful of crank."
Rowdy decided to let it go. "I better not catch you tweaking any, Jack."
"No worry about that."
Griff said, "You know, Jack, I think I'm starting to believe your story after all."
Jack didn't know how much the bikers believed of what he'd told them, which was nothing but the truth: that he was a counterterrorist agent on a mission to stop a plot spearheaded by Reb Weld. They did believe he could help them get Weld, and that was enough for now. That and the fact that he wasn't a cop. Griff and Rowdy hated cops, as they declared at some length and with feeling. Jack actually had been a cop, a member of the LAPD SWAT team, but he saw no reason for burdening the bikers with unnecessary details that might derail the start of a potentially productive alliance.
That's how it is in the field, you work with what's at hand. Griff and Rowdy were choirboys compared to some of the warlords and cutthroats that Jack Bauer had been forced to make use of in the devious and treacherous half world of the long war against global terrorism.
Jack said, "Talk fast, Pettibone. Who is Winnetou? Where's Reb Weld? What's the plot against the Round Table?"
Pettibone had recovered from his initial fright. His jawline and chin took on a belligerent set. He said, "I ain't gonna say a goddamned thing and that's the last you're gonna get out of me."
"At least you have the sense not to deny anything. Stay sensible and save yourself a lot of grief."
Pettibone was silent, not even bothering to shake his head. He refused to listen to reason and the clock was running out. Harsh measures were called for. A preliminary roughing up and slapping around failed to make him see the light. More extreme inflictions left him gasping and groaning with pain but unwilling to unburden himself of the relevant facts.
A nasty bit of business forced from him a choking half sob. "Reb'll kill me if I talk!"
That irked Griff. "Listen up, dipshit. Reb's on the run from me and my bro here. You're scared of him? He's scared of us. You're gonna find out why."
Now Pettibone found himself tied to a chair in the garage. His eyes looked like shelled oysters, his glasses had been taken from him earlier at the start of the session.
Fisk's patrol car was parked in the substation parking lot. Rowdy started it up, drove it into the garage, and switched it off. He popped open the hood and got back behind the wheel.
Griff held a pair of battery jumper cables that he'd found in the garage and busied himself under the hood. The jumper cables had spring-hinged, rubber-handled copper pincers at each end. He attached a pair to the twin terminals on the car battery.
He crossed to Pettibone and stood facing him, holding the latter's switchblade. He thumbed the handle stud and the blade came snicking out. It was a long, thin, sharp stiletto. Griff smiled evilly and moved closer to the man tied to the chair. Pettibone's hands were tied with rope behind the back of the chair. He sat rigid, trembling, staring off into the distance.
Griff cut off Pettibone's vest and T-shirt, leaving him bare from the waist up. Pettibone's flesh, rank and unwashed, was the dead-white of creatures that spend their lives in dark caves away from the sun. He was skinny with a prominent collarbone and his rib cage showing so clearly that each separate rib could be counted.
Griff taunted, "What'd you think, I was gonna cut you?" He pressed the handle stud and the blade retracted. He pocketed the weapon. "Maybe later."
He picked up a galvanized metal mop bucket that he'd filled with water and dashed its contents on Pettibone, soaking him above the waist. He grabbed up one of the jumper cables, squeezing the rubber-handled grip. The inside of its saw-toothed jaws were sharp and pointy, the better to clamp down on battery terminals.
Griff said, "We're gonna give your tongue a jump start to set it a-wagging." He fastened the clamp to Pettibone's chest at the right nipple. Pettibone whinnied like a horse breaking a leg. Griff waited until the shrieks died down and said, "Hurts, huh? You wanna talk?"
Pettibone shook his head no. Griff fastened the other jumper cable to Pettibone's chest over his left nipple. Pettibone howled, squirming against the ropes, drumming his booted feet on the garage floor.
Griff surveyed his handiwork with evident satisfaction. "Still won't talk? No? What a dumbass." He shook his head in disbelief. "You called the tune."
He upended the metal mop bucket and placed it over Pettibone's head. Rowdy sat in the driver's seat of the patrol car, resting his elbow on the top of the door and sticking his head out of the window, grinning.
Griff said, "Start 'er up!"
Rowdy switched on the ignition and started the car. The engine noise was loud inside the garage. Live current from the vehicle's nine-volt battery streamed through the jumper cables into Pettibone, the conductivity aided by the water that had doused him.
Pettibone looked like a white marble statue that had gone too long without a cleaning. His back was arched, his flesh rigid. Every muscle, tendon, and sinew stood out in bold relief. He spasmed like an epileptic throwing a fit, his head rattling against the inside of the metal bucket.
Rowdy gunned the motor, sending blue-gray clouds pouring from the exhaust pipe and out the open garage door. Griff studied the face of Jack Bauer, monitoring his reaction. Jack's expression was blandly neutral. He wondered what Griff expected him to do, flinch? He returned the biker's survey with a pleasant smile.
Thirty long seconds passed before Griff made a throat-cutting gesture, signaling Rowdy to switch off the engine. Silence fell like a concrete tomb lid.
Pettibone slumped, sagging against the ropes. He panted for breath between the muffled sobs that came from beneath the bucket. Griff waited a minute until the worst of it had passed before he knocked on the bucket and said, "Ready to start singing yet?"
Pettibone was still holding out. Griff said, "Hit it, Rowdy!"
Rowdy started the car again. Pettibone convulsed as the electricity zapped him, reacting so violently that the bucket was thrown clear from his head to hit the garage ceiling. Griff said, "Wow!"
The shocking ran longer the second time than the first. Griff gave the cutoff signal and Rowdy killed the engine.
Pettibone took longer to recover the second time, too. He shivered, shuddered, sobbed, and shook. He wept and drooled. Griff cupped the other's chin and tilted his head back so he could look him in the face. He said, "How about it?"
Pettibone's gurgled response was hard to make out. Rowdy frowned, said, "What's he saying?"
Jack interpreted. "The same old song: Reb'll kill him if he talks." Griff looked up at Jack. Jack raised his eyebrows as if to say, Is that all you've got?
Griff was really steamed. He told Pettibone, "I'm through playing with you." He unbuckled Pettibone's belt and started opening his pants.
Pettibone vented a fresh round of howls but this time he'd changed his tune. "I'll talk, I'll talk!"
He did.
##
## THE FOLLOWING TAKES PLACE BETWEEN THE HOURS OF 10 P.M. AND 11 P.M. MOUNTAIN DAYLIGHT TIME
Rimrock Road, Colorado
A patrol car bearing the emblem of the state police Mobile Response Team drove south on Rimrock Road. Hardin and Taggart were in it, Taggart driving. Taggart said, "That meeting ran long. I thought it would never end."
Hardin nodded. "Nothing Sheriff Mack likes better than the sound of his own voice."
"'Cept for maybe stuffing his fat face. Too bad that ol' tub of guts won't be at Sky Mount tonight."
Hardin was philosophical. "You can't have everything. Look at the bright side: by tomorrow Mack'll be out on his ass, looking for a new line of work."
Taggart chuckled appreciatively. "I reckon a lot of police bigs'll be finding themselves in that position come sunup."
"But not us, Cole."
"No, sir."
"We're clean as a hound's tooth."
The moon was high, almost directly overhead, a three-quarter bone-white orb that seemed far distant from the mountain landscape. Moonlight reflected off the strand of empty road beyond the reach of the car's headlights; rocky crags and needlelike pinnacles were silhouetted against a purple-black sky speckled with remote points of light that were stars.
Hardin worked the hand mic, radioing the Mountain Lake dispatcher's desk. The only reply was silence. Hardin replaced the mic in its dashboard bracket and settled heavily back in his seat. "Still no answer."
Taggart said, "That don't mean nothing. They might be away from the radio, out back making the handoff, transferring the prisoner to Pettibone. Probably are."
"Kind of late for that. Pettibone should've been and gone by now."
"He might be running late, too."
"Um."
Taggart looked away from the winding ribbon of road unrolling under the headlights to glance at Hardin, the lieutenant's heavy features in profile underlit by the instrument panel's glow. "Ain't worried, are you, Bryce?"
Hardin said, "I just hope Fisk didn't do something stupid."
"Like beating Jack Bauer to death?"
"It could happen."
"Not with Sharon there to ride herd on him."
"You know what that jackass nephew of mine is like, Cole. Once he starts beating on somebody he's hard to stop."
"He likes it too much."
"And he didn't like Bauer, not even a little bit."
Taggart shrugged. "Say he got carried away and Jack is dead. So what?"
Hardin said, "Weld'll be pissed."
"Screw him. As far as I'm concerned he's just another two-bit gun punk and snottier than most."
"On that score, my friend, we're in complete agreement. Unfortunately Mr. Pettibone doesn't share that opinion. He's scared of Weld. He'll be doing plenty of pissing and moaning if Bauer's dead."
"Screw him, too. If he don't like it, tough. Anyhow, we're almost there, so we'll find out what's what soon enough."
The car swung left around a bend and came on a long straightaway. The substation's lights could be seen at the end of it. The car went to it, slowing as it neared its destination.
Taggart said, "Home, sweet home." He turned left on to the drive connecting the road to the parking lot and followed it. Hardin leaned forward in his seat. "I don't see Fisk's car—"
Taggart said, "There it is, in the motor pool."
The car halted outside the garage. Hardin eased up and sat back. He groused, "Damn it, I told that boy not to park there. Honk the horn so he'll come out and move it—"
Headlights came on, filling the car interior with white light. They belonged to the pickup truck that had been standing idling out of sight behind the station. It barreled out with a roar of power and plowed into the patrol car, broadsiding it on the driver's side.
The car wrapped itself around the truck's steel-plated front. Hardin and Taggart received a hell of a jolt, only their seat belts saving them from being tossed around the car's interior. The stunning blow knocked their hats off and left the duo breathless with heads reeling.
Taggart caught a faceful of shattered glass from the driver's side window, which had fragmented upon impact. So had that side's rear window. The windshield frame was bent and the glass frosted. Taggart pawed his face trying to clear his eyes. He shouted, "He crazy—?"
Hardin groaned, his body aching. He felt cut in half from where the seat belt harness had caught him. He raised his hands in a vain attempt to ward off the truck's glaring high beams.
The car shivered as the pickup went into reverse and pulled free of it, backing up to get some running room. The driver stomped the accelerator, and the truck leaped forward to deliver another pulverizing blow to the patrol car.
The second hit transformed the car's shape from a U to a V. The driver's side accordioned. Jagged metal imploded, swatting Taggart. He writhed screaming and thrashing, but the seat belt harness held him in place. The front seat area was diminished by half, pinning its occupants against each other and crowding them against the passenger side. The windshield was gone, the entire sheet of safety glass having popped free of its now warped and distorted frame.
The pickup reversed, shaking itself loose from the car and rolling away from it. Hardin struggled to get free but Taggart's body pinned him against the inside of the door. Taggart wouldn't stop screaming. Hardin pounded him with clublike fists in an attempt to break free or at least silence the screaming, failing at both.
The shrieks dueted with the vroom of the pickup's engine as it made its third and final charge. It hit the car at an angle, shoving it toward the rear of the parking lot. The driver stepped on the gas, pushing the car across the asphalt into a knee-high guardrail.
The car was sandwiched between the rail and the truck. The truck kept pushing. The metal rail bowed outward into empty space, rivets popping. The truck's wheels spun, burning rubber.
There was a giddy sensation of release as the rail gave way. Several feet of ground stood between the edge of the asphalt and eternity. The car slid across them under the truck's relentless pushing and jostling.
The car's passenger side wheels ran out of ground and touched emptiness. There was a bump as the undercarriage hit the edge of the cliff and the car tilted downward. It hung there for a instant before a final nudge from the truck tipped the scales and sent it tumbling off the precipice.
Taggart had stopped screaming but Hardin didn't notice it because he was too busy screaming himself. He screamed all the way down until the car hit a rocky outcropping four or five hundred feet below.
The car bounced off it like a kicked football, sailing into the void for another thousand feet before hitting bottom.
The truck rolled backward away from the edge deeper into the parking lot and halted. Jack Bauer put it into park, unfastened his safety harness, opened the driver's side door, and slid out from behind the steering wheel. He rose, holding on to the side, standing half-in and half-out of the truck cab.
Griff and Rowdy ran out from behind the substation where they'd been hiding and watching. They thought it was a hell of a show and whooped and hollered to show their appreciation.
Jack was oblivious of them, having eyes only for the spot where the car with Hardin and Taggart had gone over the edge. He said, "Adios, amigos. No hard feelings."
He was lying.
##
## THE FOLLOWING TAKES PLACE BETWEEN THE HOURS OF 11 P.M. AND 12 A.M. MOUNTAIN DAYLIGHT TIME
Camp Winnetou, Colorado
Jack Bauer made the mistake of assuming.
Pettibone had told Hardin and Taggart earlier that he'd be taking Jack to Winnetou. Jack had assumed that Winnetou was a code name for someone big, a major player in the Sky Mount strike, possibly even Reb Weld's boss. That Weld had a boss was never in doubt in Jack's mind. The Rebel could never have put an operation like this together in a million years. He lacked the brains, money, and connections. Weld was strictly a hired hand in this deal. Maybe Winnetou was the hidden hand, the shadowy mastermind behind the conspiracy.
Now Jack knew that Winnetou was not a person but a place, a onetime summer camp that had stood shuttered and abandoned for thirty years. It lay in a park just north of Sky Mount, a narrow cleft in the mid-slopes of Thunder Mountain, third and northernmost of the trinity of peaks bordering the estate, Mounts Nagaii and Zebulon being the other two.
A mass of foliage screened the entrance to the dirt road connecting the site with Masterman Way. "Screened" was the word for it because the path wasn't as overgrown as it looked when seen from the paved road. It was camouflaged behind a pair of screens eight feet on a side that were made of chicken wire strung on a wooden framework and hung with bunches of leafy branches to give the impression of a wall of unbroken brush.
They were not unlike the canvas flats that are used in theaters onstage to create the illusion of a scenic background. They were light enough to be handled by one man.
They were rolled back now, pushed out of the way to allow access to the campsite that lay hidden within the woods. It would have been easy to miss the entrance even if they hadn't been there. The trees lining both sides of the path met in an archway above it, their dark boughs interlaced to form a canopy of foliage. The path drove through them like a tunnel, a tunnel whose mouth was sheltered by the real brush hemming it in at both sides.
A pair of metal gateposts each three feet high stood set back a few feet into the passage. The chain that linked them to prevent access now lay flat on the ground. So did the sentry who'd been posted here, whose duty was to keep watch and work the screens and chain to permit the exit and entry of authorized persons.
Pettibone was authorized to engage in such comings and goings. He'd been expected tonight. The pickup truck had halted at the entrance, and the sentry had come out to move aside the camouflaged screens to allow its entry. Griff had been there lurking in the brush, and he crept up behind the sentry and clapped a hand over his mouth to stifle any outcry while he cut his throat. Griff was handy with a knife and he liked the action.
He wiped the blade clean with a handful of leaves and returned the knife to the belt sheath that hung down the side of his left hip. He ran south along the paved road for fifty yards or so before coming to the place where he'd left his bike, a break in the undergrowth where Rowdy sat waiting on his motorcycle.
The bikers kick-started their machines, powerful Harley engines coming alive with a growl of power. Their headlights were dark as they rode to the entrance of the passage.
Jack Bauer sat there behind the wheel of the pickup truck. The slab of steel plate armoring the truck's front was hardly nicked or dented, seemingly impervious to the effects of this day's labors in demolishing three cars: Brad Oliver's vehicle, Jack's CTU Mercedes, and Hardin and Taggart's patrol car. It was a real Deathmobile.
Now Jack was in the driver's seat. He wore his gun holstered under his left arm and a second sidearm in a gun belt holstered on his right hip. The latter was a big .357 that had been provided courtesy of the Mountain Lake substation's armory. A fully-loaded sawed-off twelve-gauge riot shotgun from the same source lay on the passenger seat beside him. It was secured by the seat belt harness. The left side pocket of Jack's coat held spare clips for his pistol, the right held extra shotgun shells.
Griff and Rowdy had also augmented their own firepower with arms and ammo from the armory. Rowdy had a riot shotgun wedged muzzle-down in a hard saddle bucket on the Harley's right side, its butt end nestled against the inside rail of the protective A-bar that was bolted to the top of the back of the seat to serve as a backrest.
Griff sported a pair of .357s. The weapons were in their gun belts, which he wore not at the hips but across his neck and over his shoulders with the holstered pieces nestled butt-out under his arms. The gun belts crossed over his chest and upper back, making a pair of Xs. A bandana was worn knotted across the top of his head to keep his long hair out of his face during the action. The bandana and crossed gun belts heightened his resemblance to an old-time bandito but he rode an iron horse rather than one of the flesh-and-blood variety.
Jack waited for them to join him. He'd warned them of the danger of the green gas and given each of them a slapshot ampoule containing the antidote. The ampoule was in a syrette, a mini-syringe designed for battlefield use. A person exposed to the gas must remove the hypodermic needle's protective plastic cap, jab the spike into the upper thigh, and plunge the thruster home. The needle was tough and able to go straight through pants or other garments when driven into the flesh, a vital time-saving attribute where chemical weapons were involved and every second counted. Jack found himself hoping the bikers wouldn't try it out just to see what kind of a buzz it would give them.
Jack was not alone in the truck cab. Pettibone was there, too. He sat on the floor with his hands cuffed behind him and a second set of cuffs manacling his ankles. He was so skinny that the bracelets easily encircled the bottoms of his pipestem legs. A noose was snugged around his neck, its opposite end tied to the bottom of the passenger side seat. He was left ungagged; from here on in he could make as much noise as he liked. A locker at the substation had yielded a shirt, which now clothed his upper body.
He was silent, perhaps in reaction to the session earlier when he'd spewed a torrent of words, telling about the base camp at Winnetou, its hidden entrance, sentry, and layout. He'd rattled on about Weld, BZ, and the diabolical plan set to be unleashed at zero hour. He now seemed broken but Jack was taking no chances in case Pettibone had misled him or withheld some vital piece of information. Pettibone was going along for the ride and would share in the consequences of any treachery he might have up his sleeve.
The bikers pulled up on either side of the truck cab, Griff on Jack's left and Rowdy on his right. The rumble of their Harleys chorused with the heavy throb of the pickup's powerful motor.
Jack said, "I'll go in first down the middle, you two come in after and take them on the flanks."
Griff looked up at him, impatient. "We know the plan, dude."
Rowdy said, "Let's get it on!"
Jack prodded Pettibone with his boot toe, causing him to look up. Jack said, "Any last-minute information you'd like to volunteer? Because your neck is on the line as much as anyone's."
Pettibone shook his head. Jack pressed, "Nothing you're holding back?"
Pettibone said dully, "You've got it all."
Jack called out through the open windows, "All set?" He looked right, looked left. Rowdy nodded and Griff gave him a thumbs-up.
Jack worked the stick, shifting it into gear. The truck lurched forward. He hit the high beams, filling the leafy archway with bright light. A pair of rutted tracks grooved the ground where other vehicles had been before. The passage followed a long, low incline.
Jack shifted into the next gear, moving the truck along at a moderate pace. It rode fairly smoothly on absorbent shocks and reinforced springs despite the unevenness of the path. He glanced in his rearview mirror where two bright dots appeared as the bikers switched on their machines' headlights.
Branches slapped the truck cab's roof and scratched at its sides as the vehicle lumbered up the passage between the trees. It rolled through the far end into a clearing.
The park wasn't much, just a rocky cleft floored by an acre or so of weedy flats. A handful of cabins were grouped in an inverted U shape at the opposite end of the clearing. A long, low wooden plank building stood at their center. It looked like a shoebox with a roof instead of a lid. Its long side faced the end of the path through the trees.
The cabins were dark, tumbledown shacks, but the long house was in better repair. Its windows were muted squares and oblongs of light. A handful of dark figures milled around in front of it.
Jack pointed the truck at them and headed toward them. He downshifted, slowing as he neared, tapping the horn with his palm heel several times to sound a tinny beep-beep. He stuck his hand out the window and gave what he hoped would be taken for a friendly wave.
The high beams' glare pinned a half-dozen armed men. Jack wondered which of them had taken part in the attack at Silvertop and the cold-blooded execution of the BZ-stricken CTU survivors that had followed it. His own blood was feeling pretty hot at that moment.
Some of the gang had rifles, some had guns, others both. They showed no alarm yet. The truck poked along at a few miles per hour. A couple of men raised hands to shield their eyes against the glare, others turned their heads away from it. One yelled, "Dim those beams!"
Jack switched off his lights. A sudden blackness fell, made heavier by contrast with the harsh glare that had just filled the clearing. Jack speed shifted, stomping the gas while forcing the stick through the different gears as he did so.
The men were backlit by the long house's lights, muted though they were. Jack piloted the car toward them. There were sudden outcries, angry shouts.
Somebody opened fire with an assault rifle. Guns started popping, their muzzle flares spear blades of light. A burst of autofire spanged harmlessly against the truck's steel-plated front.
Jack switched the high beams back on, bathing his opponents in glare. The truck was almost on them. One or two hardy souls stood and fired but the others started breaking for the sides and long house.
A gunman stood in front of the truck shooting at it. A round punched a hole through the windshield, exiting through the roof of the cab. The truck closed with the shooter. He threw up his arms. The truck hit him with a thud and sent him flying.
Jack manhandled the wheel, whipping it to the left. The truck slewed around in a wide curving turn. The men scattered, running in all directions. Jack steered for the nearest, chasing him down. The target ran toward a cabin with the truck at his heels.
He almost made it. The front bumper tagged him and he fell under the wheels. The truck shivered twice as if it had gone over two speed bumps.
Jack swung the wheel around hard left again to keep from hitting the cabin. The right edge of the steel plate struck a corner of it and brushed it aside. The cabin collapsed in a heap of broken logs and the truck kept on going.
The rolling thunder of the truck motor was counterpointed by the angry hornet buzzing of the two Harleys as they entered the scene. One swung left and the other right. Jack couldn't tell which was Griff and which Rowdy.
A racketing fury clouted the driver's side of the truck. Jack didn't like that so well. The machine's flanks were its weak points, the front its strong point. He wheeled it around and drove toward the shooters.
He passed Griff going in the opposite direction chasing a man down. Griff fired a couple of shots across the top of the handlebars at his quarry. Jack flashed past them and missed the outcome of the clash.
A couple of riflemen stood in the space between two cabins on Jack's right, firing at him. He made for them, slugs ricocheting off the truck's steel plate. He threw up a hand to protect his eyes as the windshield disintegrated, spraying him with cubes of broken safety glass. His face and hand were peppered with sharp stinging fragments but not his eyes. He could see fine.
The truck kept going, plowing into the shooters with a one-two combination of thuds, the machine giving a vaguely perceptible shiver as it turned them into broken heaps. The truck rolled up an incline, a tree looming in the lights.
Jack hit the brakes, the truck balking and sliding to a stop with a crunch. The tree broke in mid-trunk where the steel plate had hit it, falling in the opposite direction.
He was after Reb Weld's kill squad, not trees. He threw the gear into reverse, the truck varooming backward and running over the same bodies again. It backed into the clearing, narrowly avoiding hitting Rowdy, who was chasing a man fleeing toward the opening of the passage. Rowdy swerved wide to clear the truck's rear bumper. He shouted something. Jack couldn't make it out but it didn't sound nice.
Griff emerged from behind a cabin near the top of the inverted U. Gunfire zipped around him. It was getting hot so he turned again, weaving between two cabins for cover.
A shotgun boomed. Jack looked in his rearview mirror. Rowdy had halted his cycle to shoot a man. His first shot missed. He stood straddling the bike, shotgun raised to his right shoulder as he swung the muzzle in line with the fugitive and fired again. His quarry went down.
The clearing was empty of fleeing figures but littered with fallen ones. The enemy's firepower was now concentrated in the long house where the remaining shooters were making a stand. They were inside covering behind wooden walls and shooting through windows and the open doorway. Four of them: three at the window and one at the door.
Jack thought that wooden plank walls wouldn't provide much protection against bullets or trucks. He glanced at Pettibone, who huddled cowering in the well under the dashboard on his side of the cab as much as the short rope around his neck allowed. Jack said, "Here we go!"
Pettibone cried, "No, no!"
Jack pointed the truck at the long house and leaned on the horn to get the bikers' attention, filling the clearing with a loud rude braying. He engaged the gear and the truck rolled forward, gathering speed.
The shooters targeted the truck as it closed in on them. Jack hunched down in his seat as low as he could get while still seeing over the top of the dashboard. Line of fire tore up the turf in front of him. The shooters got their range and poured it on into the truck. Bullets spattered the armored front racketing like the proverbial hailstorm on a tin roof.
A row of slugs stitched a cratered line of bullet holes in the cab's rear panel not far from the top of Jack's head.
The long house loomed up, filling Jack's field of vision. He steered toward the window through which a trio of shooters were blasting. The truck's left front tire was shot to pieces, causing it to tilt and veer left.
Jack battled the slide, hauling the steering wheel hard right to compensate for the drift. It took some muscle to keep the machine on course.
The building was fronted by a foot-tall wooden boardwalk. The planks snapped and splintered under the truck's weight, sounding like they were being fed into a wood chipper. They fought the vehicle's progress, trying to slow it down. It bucked and shuddered but continued its advance. The steering wheel fought to break Jack's hold but he clutched it with both hands in a death grip.
Jack stomped the gas pedal to the floor and kept it pinned, goosing a final wild burst of shrieking RPMs out of the engine.
The wall with the window was in his way. The truck punched through it, battering two shooters crouching behind it. They greased the machine's wheels as it thrust into the long house.
The long house had a long hall. Its wooden floor collapsed under the truck. The pickup continued its forward motion, tearing up planks and beams and tossing them to either side. It slid to the middle of the space before jerking to a halt.
Rafters rained down on the truck. The wall behind it had a truck-sized hole in it. Part of it caved, bringing the front half of the roof down with it. The collapse kicked up thick clouds of dry gray dust. Heaps of debris, timbers, and tabletop-sized chunks of plaster came crashing down on the cab's roof and hood.
The middle of the front half of the roof came down but the opposite ends held, the reinforced corner beams staying upright and bearing their load. Jack thought the debris had stopped falling but then another heap came cascading down.
The overhead lights stayed on, so the power source hadn't been cut. That was a break. Jack didn't fancy playing a deadly game of hide-and-go-kill in the dark.
The engine had stalled out. Jack unfastened his seat belt harness and grabbed the riot shotgun by the stock. His left hand gripped the door handle and pulled. The door opened but only a foot or so before stopping, jammed in place by fallen rubble.
He put his shoulder to the door and his weight behind it and tried again, forcing the door open wide enough so he could get out. He put his foot out preparatory to climbing down from the cab. A piece of plaster the size of a card table top fell from above, shattering against the top of the door.
Jack pulled back but the reflex action would have been too late to save him if he'd tried to dismount a heartbeat sooner. He ducked out the open door, dragging the shotgun across the seat with him. The pump-action weapon had a cut-down muzzle and stock, making it the length of a long baton.
An arm stuck out from under the truck. It was still attached to its dead owner. Jack was careful not to step on it, not out of squeamishness but because the footing was uncertain enough without it.
The floor was an obstacle course of holes in the floorboards and piles of rubble. Plaster dust streamed down from above by the handful, powdering him with white particles and flecks. Clouds of the stuff roiled and swirled in the hall like a fog bank rolling in. Dry fog.
Jack moved in a crouch, shotgun leveled. He tried to step carefully but lost his footing on a broken plank and sat down hard. Someone deeper in the hall squirted a burst of autofire in his direction but it missed and passed over his head. He couldn't tell where it had come from, not with the streaming, billowing bank of white particles in midair obscuring his view.
He picked up a piece of plaster and scaled it off to one side. The oldest gag in the book, but it stayed in the book because it worked. The phantom shooter opened fire at the sound of the plaster hitting the floor.
Streaking muzzle flares revealed a ghostly outline of a figure off to Jack's right. Jack loosed a shotgun blast at it, held down the trigger and pumped several more blasts at it. A scream choked off and a body hit the floor, leaving only swirling white dust where the figure had stood.
Jack thought he'd got him but he had to be sure. He advanced slowly, picking his way through the mounds of debris heaped on the floor. He held the shotgun in both hands, leveled at his waist, ready to respond to any sudden threat. The debris lessened as he moved farther away from the front wall. The plaster dust was starting to settle, the white clouds thinning and breaking up.
A dark form lay sprawled on the floor in the area where Jack had fired at his assailant. He approached it cautiously in case the other was shamming, playing possum to take Jack by surprise. He neared the body and saw there was no worry about that. This opponent wasn't coming back for another round, not with the damage the shotgun had done to him.
Footfalls scuttled through disturbed debris behind him. Jack spun, ready to cut loose. A figure jumped up and ran outside through the open doorway. He was out before Jack had a shot at him.
The fugitive ran into a blast of gunfire. A scream sounded, more gunshots, and then the sound of a body hitting the boardwalk.
Jack had no desire to be shot by his allies so he hung back to one side out of the potential line of fire offered by the doorway. "Griff! Rowdy!"
Griff called back, "That you, dude?"
"Yeah!"
"What's happening?"
"It looks clear in here but keep your guard up."
"I always do, man. I'm coming in so don't shoot."
"Don't you shoot, either."
Griff came through the doorway, a gun in each hand. He circled a tangle of broken beams, sidling close to the passenger side of the truck. He stopped short, looking down at something and muttering a stifled exclamation. Jack couldn't see what it was from where he was standing.
Griff pointed his gun downward and fired once. Jack said, "Why'd you shoot?"
Griff said, "A guy was crushed under the rear wheel. He was still alive. It was a mercy killing." He looked up, looked around. "Did we get 'em all?"
Jack said, "The ones in here? I think so, but don't take any chances. There might be one or two that we missed."
Griff scanned the scene, taking in the damages. "You really brought down the house, man."
"I did what I had to do."
"That's cool. I'm into overkill myself."
"Where's Rowdy? Did he make it?"
"Sure. He's indestructible, the big bastard."
Rowdy entered. He gave Jack a dirty look. "You almost ran me over, man!"
Jack said, "Sorry."
"Sorry don't cut it. Next time watch where you're driving."
Griff said, "Lighten up, bro. Save it for Reb."
"Where's Weld? He ain't here. I know, I checked all the bodies while I was making sure they was dead."
Jack said, "Were they?"
Rowdy smiled nastily. "When I left 'em, yeah."
Griff said, "Dude, where's the Rebel? We got us a score to settle with him."
"You and me both," Jack said. "This was his rear guard. They had to be neutralized before we can take him."
Griff smirked. ' "Neutralized.' That's a good one. Maybe you really are a secret agent, the way you talk."
Jack let that one pass. It didn't matter who or what the bikers thought he was. The way to motivate them was to keep their eyes on the prize.
He said, "Reb and his kill squad have already left, gone to Sky Mount on their mission of destruction. That's where we'll find them for the showdown: Sky Mount."
##
## THE FOLLOWING TAKES PLACE BETWEEN THE HOURS OF 12 A.M. AND 1 A.M. MOUNTAIN DAYLIGHT TIME
Beneath Sky Mount, Colorado
Rowdy said, "Look at all the goodies!"
The enemy kill squad had used the long house as their headquarters. The long house had been a mess hall back in the days when Winnetou had been a summer camp. Two-thirds of its space made up the dining area and the remainder the kitchen.
The kitchen had been pressed into service as the killers' command center. Long tables that had once been used for food preparation were now crowded with maps, charts, and diagrams, most of them depicting various aspects of Sky Mount's mansion and grounds. A wall-mounted pantry cabinet held five silenced machine pistols and ammunition for them, the "goodies" to which Rowdy was referring.
Jack Bauer took down one of the weapons from the cabinet pegs that held it and examined it. It was a modern-day Central European–made knockoff of the classic Ingram MAC–10 and MAC–11 submachine guns. SMGs. The lightweight piece was square, boxy, and fitted with a collapsible metal tube stock. With the stock folded down the weapon wasn't much larger or heavier than a conventional semi-automatic pistol. It was chambered for .9mm rounds.
The silencer was the size and shape of the internal cardboard roller inside a wad of paper towels, only rendered in metal.
Jack attached it to the short, snouty gun muzzle. He slapped a magazine clip into the receiver, locked and loaded the piece, and thumbed the selector switch to autofire. He pointed the weapon at a metal bucket on a countertop at the far side of the room and squeezed the trigger, letting off a three-round burst.
The silenced SMG made a quick coughing noise that sounded like the stuttering of a compressed air hose. There was no explosion of gunfire, only a whispered phtt-phtt-phtt!
The metal bucket danced and rattled as it was drilled three times. It bounced off the counter, hit the floor, and rolled.
Griff said, "Nice!"
Jack said, "These could come in handy." He didn't have to tell the bikers twice. They were already helping themselves to the weapons with eager avidity. Jack said, "You know how to use them?"
Griff gave him a disdainful look as if he'd just been insulted. "Are you kidding?"
Rowdy said, "I cut my teeth on these babies."
The bikers stuffed the side pockets of the denim vests bearing the Hellbenders' colors with extra clips for the SMGs. Rowdy said, indignant, "This is probably part of the same load that Reb stole from the club."
Griff said, "We'll return 'em to him with interest—the slugs, anyway. Poetic justice, I call it." He locked and loaded an SMG and pointed it at Pettibone, who stood off to one side trying to make himself inconspicuous. Pettibone's hands were still cuffed behind his back but his feet had been freed. The noose still encircled his neck, its free end of rope hanging down his front.
Pettibone yelped, recoiling. Jack said, "We need him, Griff."
Griff looked disappointed but lowered the weapon. "Well...maybe later."
Jack said, "Why wasn't the rear guard armed with these, Pettibone?"
"They're part of Reb's private stash—for use in the Action only."
That was Pettibone's term for the planned strike against Sky Mount: "the Action." Jack was unsure whether that was the captive's private usage or the group's general label for the strike. Not that it mattered. It meant conspiracy to commit mass murder whatever it was called.
The kitchen was equipped with a rusted metal container that looked like it had once been a cooler or food storage locker. It was eight feet long, four feet high, and three feet deep.
Rowdy test-fired his SMG by shooting at it. The machine pistol went brrrrip! The silencer suppressed not only the reports but the muzzle flare. Jack noted that with approval. It meant that no telltale flashes of light would betray the presence of a shooter. It worked both ways, of course. Reb Weld's hit team was armed with the pieces and would similarly benefit from the suppressors' stealth.
A horizontal line of nickel-sized holes was punched through the container's side. Streams of water leaked through them. Griff fired off a burst, further holing the container. More water came squirting out from the freshly made bullet holes.
Griff cradled the SMG, smiling affectionately down at it. "This is one sweet piece!"
Rowdy said, "Hold your fire bro, I want to see something." He crossed to the container, skirting the growing circle of water puddling on the floor from the bullet holes. He looked inside. "It's filled with ice. Maybe they used it to cool their beer."
Griff said, "Any brewskis in it?"
"Nope."
"Who gives a shit then?"
Jack was mulling over something that had been puzzling him, the presence on the long table of a handful of strips of white cloth. He picked one up. It was four inches high and about eight inches long. Its ends featured vertical strips of some Velcro-like material. They made a circular band when fastened together. He said, "What's this, Pettibone?"
Pettibone said, "I dunno."
"You're alive because you're useful. I'm getting the feeling you're not useful anymore."
"What's the difference? You're just gonna kill me anyway."
"You want it now? Fine. Hey, Griff—"
Pettibone said quickly, "Wait, wait! I just remembered what they're for. They're armbands. All the Action team is wearing them."
"It's a recognition symbol."
"Yeah. The insiders at the estate will be wearing them, too. That way there won't be any screwups if they cross paths during the Action."
Griff sidled over to Pettibone and stuck the muzzle of the silencer-equipped SMG against the underside of the other's chin, causing him to tilt his head back. Griff said, "So you 'just remembered,' huh?"
Pettibone said tightly, "I—I forgot..."
Jack said, "How many insiders are there?"
Pettibone said, "I don't know—" He must have seen something he didn't like in Jack's face because he blurted out, "I don't! That's Reb's business and he don't like nobody sticking their nose into it!"
Griff said, "Oh yeah? We're gonna jump into Reb's business with both feet and see how he likes that."
Pettibone gulped. He had trouble talking with Griff's weapon pressing the underside of his chin but he struggled to get his message across. "Only Reb knows who's on the inside. That's the reason for the white armbands, anybody wearing one is with us and not to be harmed."
Jack said, "You're going along, Pettibone, so whatever happens to us will happen to you first. Keeping that in mind, is there anything else you forgot to tell us?"
"That's all of it, I swear!"
"You'd better be damned sure about that."
"No, that's it, that's all!"
"Okay, you get to breathe awhile longer. Griff..."
Griff shouted, "Bang bang!," laughing as Pettibone flinched. He lowered the weapon to his side but kept giving Pettibone evil looks. Jack picked up one of the white cloth strips, said, "Which arm?"
Pettibone said, "Either one, it don't matter." Jack's hesitation prompted Pettibone to add, "I ain't lying!"
Jack fixed the white cloth around his left biceps, fastening it in place with the adhesive strips. He said to the bikers, "Might as well. We can't afford to overlook anything that gives us a slight edge."
Griff and Rowdy donned the white armbands. Jack put one around Pettibone's upper arm, said, "We don't want his buddies to get the idea that everything's not going according to plan."
He emptied the shotgun shells from his right side jacket pocket, replacing them with extra clips for the SMG. Rowdy said, "You ain't bringing the scattergun?"
Jack said, "I don't know if we want shotguns blasting around explosives and gas grenades. The machine pistol's better suited for the work."
"I'm bringing both. The riot gun'll be handy in case we gotta knock down any doors."
"Use it as a last resort. We're not looking to advertise our presence."
"Better to have it and not need it than need it and not have it."
"I can't argue with that."
Rowdy grabbed some of the shotgun shells and stuffed them into his pants pockets. There were plenty of flashlights around, heavy-duty baton models. Jack helped himself to one. "We'll need these, too."
Rowdy said, "You and Griff take 'em. I'll carry the shotgun in one hand and the machine gun in the other."
Jack and Griff tried out the flashlights to make sure they worked. Griff shone the powerful beam in Pettibone's face, causing the other to squint and turn away. Griff said, "I'm watching you."
Jack said, "Let's go." He, the bikers, and their prisoner crossed to the door. Rowdy said to Griff, "We can come back for the other two machine guns later."
They all exited, the kitchen door putting them on the short, north side of the long house.
The moon was behind a mountain peak and thick darkness lay on the tiny park. Jack and Griff switched on their flashlights. Jack told Pettibone, "Lead on."
Pettibone said, "This way." He rounded the northwest corner to go behind the back of the building, closely followed by the others. A well-worn footpath ran south. The group passed the end of the building and kept on going, crossing a weedy field whose west edge was bordered by thickets of brush. Beyond them a rock wall thrust up for hundreds of feet, part of the mid-slopes of Thunder Mountain.
The path trailed south for a hundred yards leaving the Winnetou campgrounds behind. Ahead rose a rocky promontory several hundred feet high, a spur that thrust out at right angles from Thunder Mountain. The spur was Sky Mount's northern border; on its far side lay the estate. Mansion and grounds stood at the base of the spur's south face.
The spur was a rock curtain cutting off all sight and sound of the estate; not even the glow of the mansion's lights could be seen in the night sky above it. It effectually isolated Sky Mount from Winnetou, ensuring that neither the camp's hooded lights nor the sound of the gun battle earlier could be discerned by those on the estate.
The corner pocket where the spur met Thunder Mountain's east face was blanketed with inky darkness. The path began to rise slightly as it neared the base of the spur. Pettibone stumbled, almost falling. He said, "I can't walk with my hands cuffed behind me."
Griff said, "Yeah? At least you're still upright." He prodded Pettibone with the tip of the machine pistol. "Keep moving if you want to stay that way."
Pettibone lurched forward, staggering onward. A short low rise of loose dirt and rocks now came underfoot, the talus skirting the slope's base. Their goal lay at the top of it. They paused, Jack and Griff's flashlight beams picking out their destination.
A hole gaped in the solid rock wall, a massive door or portal jutting out from it at right angles. Rowdy said, "Damn!"
Griff said, "We know that Mr. Bones here was telling the truth about this one thing, anyhow."
Jack set his SMG and flashlight down on top of a waist-high boulder and took his cell phone out of his pocket. Griff said, "What're you doing?"
Jack said, "Calling for reinforcements, if I can get through."
"You wouldn't be planning on pulling a cross, would you, dude?" Griff's voice took on that too-casual tone that Jack recognized as a harbinger of potential violence.
Jack said, "I'm standing right here, you can listen in. Like I said before, this is no game of cops and robbers, this is national security business. This is the connection that'll get us all off the hook on that Mountain Lake job."
"I was born paranoid, Jack. But you've been righteous so far so I'm willing to play out the string. Go ahead."
Jack saw that Ryan Chappelle had left him another stack of voice mail messages. His mouth took on a wry twist as he ignored them and pressed the speed dial for Orlando Garcia's personal number.
He didn't know whether the cell would work here but he thought it might because he and the Pine Ridge command post were both on the same eastern side of the mountain range, unlike Shadow Valley farther to the west where the intervening peaks had effectively blocked off the signal. His cell had a scrambler to screen against electronic eavesdroppers so the communication would be secure—if it got through.
The call was answered on the third ring. Garcia said without preamble, "I'd just about written you off." The signal was mushy but audible. "Where are you? What's—"
Jack cut across the other's words, "No time for that now. There's a strike planned against Sky Mount and I'm going to stop it if I can. Have a tac squad waiting outside the estate but don't go in before you hear from me. Otherwise you might trigger the massacre we're trying to prevent. Did you get all that?"
"Now hold on a damned minute—"
"Did you get that?"
"Yes."
"Good. If you don't hear from me by two o'clock—that's zero two hundred hours, repeat, zero two hundred hours—I didn't make it and you can start evacuating Sky Mount. You read me?"
"Yes, damn it, but—"
"Zero two hundred hours and then move. Not before."
"Wait—!"
Jack broke the connection and switched off the cell. It was the trust factor all over again. Brad Oliver had been warned in advance that he was going to be apprehended. The word could only have come from someone high up in CTU/L.A. or CTU/DENV. Was it Garcia? Jack didn't know. He wasn't going to tip his hand until he did know one way or the other. He'd told Garcia just enough to salvage the situation at Sky Mount should Jack fail to accomplish his mission but not so much that Garcia could thwart him if he turned out to be working for the other side.
He pocketed the cell and picked up his weapon and flashlight. He said, "Pettibone will go first. If there's any booby traps along the way he'll get it first."
Pettibone said, "There ain't."
"You'll go first anyway."
They went up the slope in single file, Pettibone first, followed by Jack, Griff, and Rowdy. They paused at the lip of the entrance, Jack and Griff shining their flashlights on the portal.
A flat-topped summit, about six feet wide, aproned the hole in the wall. The hole was the mouth of a tunnel that stretched deep into the spur. It had been drilled through rock and lined with poured concrete. Cool, dry air wafted out of it. The night air was already cool but the tunnel was colder.
It was dark, pitch-black, not a glimmer of light showing within. Its rounded archway was seven feet tall and five feet wide. The concrete had been poured so the floor was flat. The door, or hatch, was a massive construction of milled steel three feet thick. A pair of curved hinges of corresponding size secured it to the tunnel. The hinges emerged from slots set in the inside of the frame at the head of the portal. The hatch would fit flush with the frame when it was closed.
Its exterior was faced with rock paneling that blended in with its surroundings. In the years—decades—that it had been in place, it had become overgrown with moss and lichens of the same type as those that clung to the natural rock walls.
The inside center of the hatch featured an oversized spoked metal wheel the size of a big-rig truck's steering wheel. Its hub was mounted on a short, squat column fitted inside a cylindrical shaft set deep in the metal mounting. It could be used to open the hatch or dog it closed. The mechanism could only be operated from the inside.
Rowdy said, "It's like a bank vault door..."
Griff said, "With a fifty-thousand-dollar payoff inside. And it's already open. All we got to do is take Reb's head to collect."
"And not get killed."
"Yeah, that, too."
Pettibone hung back at the threshold. Jack prodded him with the tip of the SMG, said, "Move. And quit dragging your feet. We don't want to be late for the party."
Pettibone started forward, Jack following him inside, the bikers trailing him. The concrete lining was cracked and stained but the excavation was far more stable and solidly built than that of the Silvertop mine. The mine had been damp and clammy but this tunnel was dry and dusty.
Twin flashlight beams reached around Pettibone's skeletal form to probe deep into the tunnel's vitals but not deep enough to reach its end. The floor had a slight incline with a grade of less than five degrees. The group trudged silently upward along it.
The tunnel was stark, functional, utilitarian. The outside world might as well have ceased to exist for all the effect it had here. Its influence was nil. The concrete lining had an absorbent quality that muffled ambient sound. The footfalls of the intruders were whispered rustlings on the treadmill of a seemingly endless trek.
Jack herded Pettibone along, poking him with the SMG on the ever more frequent occasions that he halted his progress. Jack read that as a good sign. It meant they were nearing their goal at the end of the tunnel, a destination Pettibone had no desire to reach.
The tunnel ceased its incline and now began to run level. The flashlight beams all at once encountered something instead of being swallowed up by nothingness. They bounced off a distant shape, an unrecognizable blur.
Jack whispered over his shoulder to the bikers, "Heads up and stand by for action." A lot of ground remained to be covered but the transit was more endurable now that the end was in sight.
Jack tilted his flashlight beam to one side. A fuzzy patch of brightness swam at the end of a long stretch. He touched the tip of the silencer to the back of Pettibone's head where it met the top of his neck, a gentle reminder of the facts of life and death.
Jack didn't know what lay at the end of the tunnel, but whatever it was, he didn't want Pettibone acting up when they got there.
There was no way to hide their approach but he hoped that the opposition would be lulled by a misreading of the situation. They were expecting Pettibone to arrive with a captive Jack Bauer in tow for delivery to Reb Weld. The two of them would surely be escorted by a couple of armed men from the rear guard.
Jack planned to take advantage of that split-second window of opportunity before the foe realized the truth.
The tunnel came to an end, opening on a rectangular chamber twenty feet deep, fifteen feet high, and fifteen feet wide. This end of the shaft was fitted with a massive hatch, twin to the one at the entrance. It opened inward and was swung back all the way on its curved hinges so that its inner face pressed against the chamber wall to the left of the portal.
The first thing Jack saw inside the chamber was a golf cart, the incongruity of its presence here adding an offbeat, surrealistic touch. It was parked at the foot of a loading platform that stood at the chamber's far end.
The platform's top was four feet above the floor. A well in the left corner held a short flight of stone stairs leading up to the platform. A couple of boxes and packing crates stood on top of the platform.
A man sat on top of one of the crates facing the tunnel. A square-edged doorway opened in the wall behind him. An electric lantern was set atop a nearby crate. Its glow seemed as cheery and welcoming as sunshine after the tunnel's dark passage.
The man was stocky and squat with short hair and a goatee. He wore dark clothing and a white band on his upper left arm. He hopped off the crate and stepped forward toward the edge of the platform as the newcomers arrived. He said, "Pettibone, what kept you—Hey!" He turned, grabbing for the SMG on top of the crate.
Jack shoved Pettibone aside and made his play. He went for a head shot in case the other was wearing a bulletproof vest under his shirt.
His weapon made a throat-clearing noise as it squirted three rounds into the sentry's face. He fired it one-handed, holding the flashlight in his other hand. The piece when fired in a short burst had a recoil that was a bit heavier than that of a .45-caliber semi-automatic pistol.
The sentry flopped rearward like he was trying to do a back dive. He bumped into the crate, upsetting but not overturning it. The SMG fell clattering to the platform but didn't go off.
Jack darted across the floor to the stairwell and took the steps two at a time. He crossed the platform to the doorway, covered to one side of it, and peeked around the edge into the space beyond.
The doorway opened into a vast, cavernous area that resembled nothing so much as an underground parking garage and was the size of an airplane hangar. The enclosure was long and low-ceilinged. It was dark except for a line of electric lanterns that had been placed at regular intervals along the center of the floor to the opposite end.
It was empty, unoccupied. Jack Bauer strained his ears listening for the sound of an alarm or hue and cry. None came.
He turned his attention back to the chamber. Griff stood crouched holding his weapon to Pettibone's head. Rowdy climbed the stairs to the top of the platform and joined Jack.
Jack said, low-voiced, "This level is clear—I think." He switched off the flashlight and set it down on a crate. He said, "Cover me."
Rowdy said, "Right." He leaned the shotgun against the wall and held the SMG in both hands.
Jack ducked low and went through the doorway into the sprawling bunkerlike construction that lay on the other side. He dodged to the right, out of the glow of the electric lanterns lining the floor and into the welcoming gloom that hovered on either side of the illuminated central path.
Rowdy covered behind the doorway's edge and stuck the SMG outside, the tip of its silenced snout quivering like a dowsing rod in quest not of water but of human targets.
The floor was carpeted with a layer of dust several inches thick once Jack moved aside from the center space. It smothered the sound of his already light-footed tread. His movements disturbed the dust, each footfall raising puffs of the stuff. It tickled his nose, and he had to fight to keep from sneezing. That would be a hell of a note, to give himself away by sneezing!
Jack advanced, guided by the lamplight glimpsed out of the corner of his eye. The space dwarfed him with its slab-sided monolithic immensity.
His eyes grew accustomed to the dimness, allowing him to make out a line of closed doors in the wall to his right. They were tall, narrow oblongs a shade lighter than the shadows engulfing them. No lights showed behind any of those doors. The wall space between them was lined with stacked cardboard boxes furry with dust.
He went to the opposite end of the bunker. A glow brightened in the right-hand corner as he neared it, an independent light source separate from the lanterns marking out the centerline.
It revealed a stairwell. A flight of stone steps climbed to a landing, then another stairway led to a second landing. The light came from electric lanterns with hooks at the top that were hung from horizontal bars of the metal railing enclosing the open side of the stairs. A closed door was set in a wall at the second, final landing. It was outlined by light coming from the other side of the door.
Jack figured he'd come far enough by himself. No point in having allies if you didn't use them. He turned, heading back toward the antechamber. This time he did it the easy way, following the well-lit center aisle. It was a well-traveled route, judging by the lack of layered dust that pervaded the rest of the bunker.
He halted at the halfway point, motioning for the bikers to join him. They came to him, Griff hustling Pettibone along with them. Jack said, "All clear. A stairway leads to the next level. I didn't want to go past that without Pettibone as a stalking horse."
Griff looked around, his eyes glittering slits, his shoulders hunched as if anticipating a blow. He said feelingly, "What a creep joint!"
Rowdy said, "What is this dump?"
They all spoke in hushed voices. Jack said, "It's a fallout shelter for surviving an atomic war. Must've been built a half century ago."
Rowdy said, "Man, I'd rather get nuked than live in this mausoleum!"
Griff said, "Hey, did you dig that golf cart?"
"Beats walking through the tunnel."
Jack said, "It's a good way to bring supplies in, too. Like bombs and gas grenades."
They went to the far end of the bunker and stood at the bottom of the stairwell. Griff said, "What's on the other side of that door?"
Jack Bauer said, "The showdown."
##
## THE FOLLOWING TAKES PLACE BETWEEN THE HOURS OF 1 A.M. AND 2 A.M. MOUNTAIN DAYLIGHT TIME
Sky Mount, Colorado
Reb Weld did not handle frustration well. He was an action man; he liked to be up and doing. Standing around waiting did not sit well with him. Delay irritated him, especially when it involved matters over which he had no control. He was a control freak, too. He said, "What's the stall, Al?"
Grant Graham started at the sound of Weld's voice. He was standing next to Weld, but he'd been watching Al Baranco rigging the timed detonator to the last set of explosive charges. He was so caught up by Baranco's air of methodical concentration that it came as a jolt when Weld spoke up, shattering his reverie.
Baranco labored on, doing what he'd been doing, giving no sign that he'd heard the Rebel's words. Graham flinched, said out of the side of his mouth, "Take it easy, Reb."
Weld said, "Screw you, Graham." He spoke in a normal conversational tone, contrasting with Graham's husky prison-yard whisper. Graham had taken falls on several felony counts and spent years in Federal penitentiaries.
Talking out of the side of his mouth so only fellow inmates could hear him and prison guards couldn't had become second nature to him. He'd carried the habit with him even though he was outside the walls. Being within arm's length of enough explosives to blow him to atoms as he now was only intensified this habitual trait.
Weld said, "Get with the program and shake a leg, Al."
Graham said, "Reb, please—"
"Shut up."
Al Baranco stopped what he was doing, namely attaching a wire to one of the terminals on the timing device. The device was the size and shape of a paperback book. It had a matte black plastic casing. Its face had a digital display slot screen and a numerical keypad with some additional buttons. Twin terminals protruded from the top of the case above the readout screen, small brightly polished metal knobs with little caps that screwed on or off.
Baranco held a pair of needle-nose pliers that he was using to strip the insulation off the end of one of the wires leading to a packet of blocks of C–4 plastic explosives that had been taped together.
He paused but did not look up. He said, "It's unwise to distract a man when he's setting a bomb timer, Reb."
The mansion at Sky Mount had two underground levels that were officially in use. The one nearer to the surface held storerooms of various types. Food supplies, an extensive wine collection, glassware, table settings, and the like were only a few of the commodities that were resourced there, along with a treasure trove of paintings, statuary, antiques, and other art objects that had been removed from permanent display upstairs and put in storage on Level One.
Level Two, the subcellar, housed the vitals of the mansion, the all-important mechanisms and support systems that kept it going. Here were the banks of fuse boxes, meters, dials, and relays that monitored and controlled the countless miles of electrical wiring that made up the great house's nervous system. Here were the hydraulic pumps and pipes that kept the plumbing running smoothly everywhere from the sinks in the custodians' supply closets to the outdoor Olympic-sized swimming pool—as well as the slightly smaller indoor heated pool.
Here were the boilers and furnaces and fuel tanks that gave the mansion its heat and hot water. This last was the target of Reb Weld and his associates. Sky Mount was equipped with three fuel storage tanks, each the size of a railroad freight car and filled to the brim with heating oil. They stood lined up in a row in a sunken area at one end of the subcellar.
Each tank rested on its own cradle, an intricate webwork of cross-braced metal beams and struts that held them suspended above the floor to allow workmen access to their undersides.
Reb Weld, Graham, and Baranco were grouped alongside the tank in the middle. The other two tanks flanking it had already been rigged with explosive charges. The middle tank had been gimmicked like the other two with blocks of C–4 plastic explosives in the critical junction points that would rip their bellies open and ignite their contents into a colossal firestorm of holocaust proportions.
It was a laborious, time-consuming process. The blocks of plastic explosives and the detonators had previously been stored in the fallout shelter below, an abandoned area whose existence was known only to a few. Reb and his crew had spent the night hand-carrying the blocks up the stairwell and through a secret door near the fuel tank area.
They'd been planted on the undersides of the tanks.
All three sets of charges were separately wired to a single master detonator-timer.
This had been done to save time that would have been eaten up by fixing each load with its own individual timing device and to limit the exposure time of Weld's team in the Level Two area.
Three sets of wires fed into a trunk cable. Baranco was wiring the cable to the twin terminal posts of the master timer. It was set to go off at three o'clock in the morning, when an electric charge would pulse down the trunk cord and along the three separate sets of wires whose detonator tips would simultaneously explode all charges and blow the tanks and the great house above it to kingdom come.
Once Baranco was done wiring and setting the master timer only one final task remained: to switch on the timer on a smaller charge attached to a crate holding the remaining BZ gas grenades. They'd been placed near a ventilator intake grille at the base of a metal conduit duct air shaft. The bomb would be set last but would go off first, at 2:50 A.M., ten minutes before the oil tanks blew.
The green gas would be sucked into the ventilator intake shaft to circulate throughout every room in the mansion, spreading madness and chaos. The pandemonium would have time to reach a crescendo of artificially induced psychotic frenzy among guests and staffers alike before the bombs on the oil tanks blew. The sequence was designed to maximize the body count.
The bombs would ignite a firestorm of volcanic proportions, spewing a flaming geyser upward that would scour the underground levels before rupturing the ground floor and fountaining a white-hot inferno throughout the great house.
The devastation would be awesome, the casualties immense, and the repercussions catastrophic to the nation and the economies of the world.
Reb Weld was looking forward to it. His only regret was that he wouldn't be able to view the spectacle firsthand. Chaos and destruction were his delights. Why? Because that's the way he was made.
It wouldn't be smart to stick around and watch the show, though, much as he'd like to. Or healthy, either. Not with that crazy green gas heralding the apocalyptic hellstorm.
Baranco was the demolitions expert. That action was out of Weld's league. He had to stand by, watch, and wait while Baranco worked his black art.
It was necessary but Weld didn't have to like it. He wasn't called "the Rebel" for nothing. He resented taking a backseat to anybody, especially the so-called experts who knew more about a subject than he did. He'd always been that way, it was his singular defining trait. That and a mean streak as wide as a sixteen-lane superhighway.
He couldn't resist needling Baranco even as the bomb man was engaged in the tricky and delicate work of rigging the last fuel tank bomb. He did it because it was risky and pushed the edge, giving him a fresh jolt of the adrenaline he so inordinately craved. He was an adrenaline addict.
He'd even considered pilfering a BZ grenade from the crated cache to take with him as a souvenir. It'd be a kick to get a taste of the gas itself and see what the head was like. The survival instinct reasserted itself, overpowering that compulsion for crazy kicks. A BZ grenade would be his ticket to the execution chamber should he be caught with it, especially after tonight.
Baranco said, "I'll get on with my work now if that's all right with you, Reb."
Sarcastic bastard! Maybe there'd be a chance to cut him down to size later when his tasks were done. But not now.
Weld said, "Go ahead, nobody's stopping you."
Baranco kept pushing it. He was a needler in his own soft-spoken way, too. "I'd appreciate a little quiet while I'm fixing this last connection. If my hand should slip..."
He didn't have to finish the sentence.
Weld said, "I'll be quiet as a mouse. Not a peep out of me, Al."
"Thank you very much, that will be deeply appreciated."
Baranco set the pliers down in his pocket tool case and began unscrewing the knob at the top of the terminal, preparatory to winding the hooked end of the exposed copper wiring at the base of the post.
And then his head exploded.
Jack Bauer had been standing on the top landing of the fallout shelter stairway with one hand gripping the back of the collar of the shirt that he'd scrounged up at the Mountain Lake substation to replace the garments that Griff had cut off Pettibone. Pettibone had to look normal to deceive his accomplices, or as normal as he could look even when fully clad. Jack's other hand gripped the SMG.
Griff and Rowdy crouched a few steps below him, ready to spring into action. Rowdy had been convinced of the inadvisability of bringing along the riot gun and had stowed it aside at the bottom of the stairwell for retrieval later. The bikers were armed with SMGs and a few handguns and knives were also tucked away on their persons.
Jack said, low-voiced, urgent, "Go!" He prodded Pettibone in the kidneys with the tip of the weapon to reinforce the command.
The door opened outward onto the landing. Pettibone gripped the doorknob, turned it, and pulled it toward him. The door accessed a tiny vestibule with another door at the opposite end. That door opened on to a Level Two walkway overlooking the sunken area where the fuel tanks were located. The sunken area was five feet below floor level with sets of stairs at each corner and bordered by a waist-high rail fence.
The far door was wide open. Jack hustled Pettibone through it, following at his heels. Griff and Rowdy were a pace or two behind.
A man on the walkway was starting toward the vestibule at the same time. He was Loogan, one of Weld's men. He'd been posted to secure the exit and had heard the sound of movement behind the inner door. He was advancing on it as Pettibone, with Jack behind him, came rushing out.
Loogan fired, his silenced SMG sounding a whispered stutter as it loosed a burst into Pettibone's middle. Pettibone fell back, writhing against Jack as the slugs tore into him. His last conscious act in this world was to serve as an unwitting, unwilling human shield for Jack Bauer, catching the full measure of Loogan's triggered slugs.
Jack got his gun hand free and returned fire, stitching Loogan up the middle of the chest. Loogan dropped, his weapon clattering against the walkway. The telltale sound was lost in the vastness of the space, drowned out by the background noise of throbbing pumps and creaking pipes.
Jack eased Pettibone to the floor and stepped around him. Pettibone's popping eyes stared sightless and unblinking as he sprawled inert on the walkway.
Griff and Rowdy came barreling out of the vestibule. Jack held a finger to his lips, signaling for silence. The bikers halted, charged up, kill-ready. The walkway overlooked a long side of the sunken floor, running parallel to the long axis of the nearest tank.
Jack said in stage whisper, "I'll go down here and come on them from under the tank. You go around and pin them from opposite ends." They nodded.
Jack clambered over the guardrail and dropped cat-footed to the sunken floor five feet below. Griff and Rowdy were in motion, scrambling in opposite directions, hustling toward the corners of the tank pit.
Jack slipped across the space to the first tank with shadow stealth. The rigging under the tank was a webwork of black metal beams, braces, and diagonals. The underside of the tank was held suspended in its steel cradle about four feet above the floor. He ducked under the outside rail and began moving forward in a crouch, bent almost double, picking his way through the labyrinth of the undercarriage.
He went as fast as he could but his progress seemed maddeningly slow, with time measured out in every pounding heartbeat. His whole body quivered in anticipation of a shout, a shot, or a scream that would alert his prey to the fact that they were being stalked and hunted.
The steel net let light pass through it screened through a tangle of black beams. He stepped over some, ducked under others. The swelling curve of the cylindrical tank pressed downward as he neared midpoint, forcing him to drop to hands and knees to proceed. He banged his knees, barked his shins, bruised his elbows, and bumped his head in his hurry to gain ground.
The ceiling lifted as the midpoint was left behind and the tank curved upward. The smell of fuel oil was thick in his nostrils and mouth; he could taste it. The layers of cross-bracing between him and his goal thinned, allowing him to see more of the gap between the tank he was under and the one in the middle.
Silhouetted forms flashed ahead and to the right of him. The rise and fall of voices made themselves heard over the pounding of his own pulse that throbbed in his eardrums. Jack forced himself to slow down though it was torturous to fight the overpowering urge to rush into battle. The setup was all-important and he had to be in the optimum position before cutting loose. A misstep could be fatal not only to him but to hundreds of innocent lives.
He hoped with all his being that Rowdy and Griff were taking similar pains. No outcry or outburst had sounded as yet so perhaps they were. Jack couldn't wait for them to shoot a move, though; he had to seize and keep the initiative.
The trio came into his view, grouped in the open area between the tanks and a dozen paces to his left. He crept forward, inching closer.
Reb Weld with his signature platinum-blond crew cut stood a half a head taller than his sideman, a chunky character with a brown rooster-tail haircut. They both stood watching a third man who knelt hunched over on the floor with the master timer in one hand and the exposed copper wiring of the trunk cable bundling all three detonator wires to the different sets of charges in his other hand.
Jack Bauer went down on one knee, resting the silenced muzzle of his SMG in the V made by the intersection of two cross-braces to steady his aim as he lined up the sight posts on the skull of the explosives expert. He set the selector to single shot for better accuracy and rechecked his alignment to make sure he was on target.
Weld was bickering with the demolitions man. The expert was saying sarcastically, "Thank you very much, that will be deeply appreciated."
Jack squeezed the trigger.
A single cough sounded simultaneously with the top of Al Baranco's cranium flying apart. The explosives expert's head was haloed by the corona of pink mist indicative of a brain being blown to pieces.
A perfect head shot, drilling the brain, switching off all neuro-muscular reflexes and reactions, ensuring that Baranco would cease to exist without so much as a twitch.
Things happened fast after that.
Somewhere on one of the walkways there was a startled outcry of pain, a stuttering exchange of gunfire, and a scream.
Al Baranco dropped. Reb Weld and Graham recoiled from the stinging spray of disintegrated bits of bone and brain matter that had spattered them when the slug fragmented Baranco's head above the ears.
Jack flipped the selector to autofire. Graham stood between him and Weld. Jack fired a burst that chopped Graham. Graham threw up his hands over his head and shrieked as the legs were cut out from under him.
Reb Weld was quick! He dove forward away from the gunfire, going into a roll and tumbling out of it before Graham hit the floor. He leaped to his feet and lunged sideways, gaining the cover of one end of the oil tank behind him for protection.
Jack snaked out from under the tank, standing upright in the gap between the tanks. Graham lay rolling around on the floor, beating his hands against the upper thighs of his now useless legs as if they were on fire. Jack stood over him, firing a quick burst into Graham's chest. That switched Graham off but Jack put a few more into his head to be sure.
Reb Weld was racing toward the wall at the end of the sunken floor when Griff popped up at the railing of the walkway above. Both men opened fire at the same time.
Griff missed but Weld didn't, tagging the biker twice. Griff went over backward. Weld tossed back his head and gave a rebel yell of exaltation as he reached the wall and jumped up grabbing for the guardrail.
Jack's burst caught Weld in the back in midair. Weld fell back, crashing to the sunken floor. The SMG fell from his grip and went skittering away from him, out of his reach. Weld rose on his elbows, a bloody smear marking the floor where his back had touched it. He reached for his waistband, clawing at the butt of a gun tucked in the top of his pants.
Jack advanced on Weld, methodically spraying him with autofire. Weld flopped around as the bullets ventilated him.
He weltered in his own gore, tiger-striped with blood. He raised the back of his head off the floor, neck muscles cording and quivering from the strain of trying to see who had done him in.
Griff rose to his knees on the walkway, clutching the rail to keep from falling while he watched Weld's finish.
Jack moved so Weld could see his killer. Weld's eyes widened, then narrowed as he recognized the CTU agent he'd come face-to-face with in the clearing on Pine Ridge before fate had taken a hand by intervening in the form of a charging bear.
He mouthed the word, "You!"
Jack Bauer said, "The Hellbenders send their regards."
He delivered the coup to de grâce to Reb Weld:
A head shot.
Griff had caught two slugs, one in the right side and the other in the left shoulder. He draped his arms over the bottom rail to hold his upper body upright while he stared down at a torn and bloody carcass that used to be Reb Weld.
He held out a red hand to ward off Jack when the latter moved to help him. He managed to choke out a few sentences between gasps and stifled groans.
Griff said, "I'll make it—Rowdy must've ran into trouble—check on him...The big slob was never any good without me."
He added after a pause, "I'll stay here and enjoy the scenery."
Rowdy had run into trouble. He sat on the opposite walkway, back propped upright against the wall. His right arm was at his side, outstretched at the elbow, his hand wrapped around the grip of a still-smoking SMG. His midsection was a red ruin of an anatomy lesson. His left hand held his insides back from tumbling out.
Incredibly he still lived, awareness in his eyes as Jack went to him.
A body lay nearby, twisted in the angular contortions of violent death: the last member of Reb Weld's elite hit team. He lay facedown, reaching for a crate of BZ gas grenades that lay inches short of the fingertips of his clutching hand. A shiny sheet-metal, square-sided length of duct conduit piping was bolted vertically to the wall above the crate, its scooped-mouth bottom covered by a metal grille and hanging two or three feet above the top of the grenade-laden box.
It was the intake port of a ventilator air shaft. Jack Bauer could feel the suction of air currents being drawn upward into its mouth and away through the piping to be carried through the precincts of the mansion above ground.
Jack moved the crate a man's length away from the intake port. A wad of C–4 plastic explosive was wedged into the bottom of the crate, wires trailing up from it and over the side waiting for a timing device to be attached. He gingerly disengaged the tip of the detonator cord from the puttylike mass, disarming it and setting the cord a safe distance away from the crate.
He hunkered down beside Rowdy, leaning forward to catch the big biker's last words. Rowdy said, "Bad luck—he got me before I could get him. Thought I was dead...got a big surprise when he found out I wasn't..."
His tired eyes cut a glance toward the BZ crate. "Guess you found what you wanted, dude..."
He breathed something that Jack was barely able to make out: "Valhalla is calling—"
His last breath.
Jack Bauer defanged the plastic explosives rigged to the fuel tanks, pulling the detonator cords, gathering them up, and depositing them in a safe place on the walkway. He went to see what he could do for Griff. Griff was sitting up, holding a wadded bandana against the wound in his side and using it for a compress.
He said, "Rowdy...?"
Jack said, "He got his man before he died."
Griff nodded. "He went out Hellbender style then."
"That he did. I'll get help. Don't shoot any of my people when they come down here to secure the site."
"I'll try not to."
Jack opened his mouth to say something, but before he could, Griff said, "That's a joke, man. Just pulling your chain...go do what you have to do, I'm okay."
Jack started to walk away. Griff called after him, "Kill 'em all!"
Jack turned, said, "I'll do my best." He moved on.
He went to a stairwell that accessed the next level. He found a morbid surprise at the foot of it. Two bodies lay there where Weld and his crew had put them, undoubtedly intending to come back and carry them topside to the surface and plant them somewhere on the grounds to be found and scapegoated in the aftermath of the horrific hellfire that turned out not to be.
They were the cadavers of Abelson Prewitt and Ingrid Thaler, the grandmaster of the cult of the Zealots and his faithful second-in-command. The bodies were cold to the touch—icy—frozen.
Another piece of the puzzle had fallen into place.
Jack Bauer loaded a fresh clip into his SMG before beginning the long, wearying climb upstairs to the surface.
Don Bass exited a side door in the mansion, hurrying along a flagstone path that curved through silent, nighted gardens toward the guesthouse that served as the command center for the Brand Security Agency cadre overseeing the corps of uniformed and plainclothes guards now on duty on the graveyard shift.
Sleep and the security chief were strangers. Bass's wavy hair stood out in tufts. His eyes were red embers buried deep in hollow, purple-bruised sockets. His movements were stiff-legged, zombielike as he forced himself to scurry at quick time toward the guesthouse turned guardhouse.
A figure stood outlined in the open front doorway waiting for him. It was Larry Noone, Bass's top man and figurative right hand, the man whose urgent phone call summoning his boss had jarred Bass out of fitful light sleep and back into action on the double.
Bass was not so tired, however, that he failed to notice the white armband prominently pinned to the upper arm of Noone's navy blazer. Bass paused at the threshold, clutching the insides of the doorframe with both hands for support while he tried to catch the breath that his hasty arrival had stolen from him.
An expression of concern marked Noone's face. "Are you all right, Chief?"
Bass blustered it out, barking, "Certainly! Just a little winded, that's all. I hustled over here after I got your call. Sounded urgent. What's up?"
Noone said, "Come in and I'll tell you."
Bass marched into the front hall, turning left to follow Noone down a short corridor. The heart of the command center lay on the other side of a closed door at the passage's end, in a room that was an electronic nerve nexus of computerized consoles whose multiscreens imaged real-time feedback from the array of closed-circuit automated TV cameras that kept the mansion and estate under constant surveillance. A graveyard shift of six top operatives would be posted at the monitors, orchestrating the flexible and adaptive Brand Security defense posture.
Bass, frowning, said, "What's with the white brassard, Larry? It's unauthorized as far as I know."
Noone glanced over his shoulder, flashing an enigmatic half smile. "Change of policy."
Bass's frown deepened. He was a stickler for detail. He said, "That's news to me and I set dress code policy."
Noone paused with his hand on the doorknob. "Step right in, Chief, and it will all be explained to you." He opened the door partway, standing aside so Bass could pass him and enter.
Bass was through the doorway and a half-dozen paces inside the command center before the horror of what he saw registered on his benumbed mind.
It was the scene of a massacre. All six board operators, male and female alike, lay strewn about the room in the places where sudden death had found them. Their bullet-riddled bodies bore wounds so numerous that they could only have been inflicted by an automatic weapon. They were torn and tattered. Blood was everywhere. Walls and consoles were cratered with bullet holes.
Don Bass was struck dumb, paralyzed with shock. A timeless interval passed before he drew a shuddering breath. His heart started beating again, hammering with a wild percussive rhythm.
Somehow he managed to turn around and face Larry Noone. Bass was surprised to find that he was not surprised at all to discover his second-in-command pointing a leveled machine pistol at him. There was a certain pride that his deductive and analytical faculties had not deserted him in the fractional span of life left remaining to him to glory in their possession. Noone had to be the killer; his bland demeanor in the face of such carnage proved it.
A distant part of Bass's mind kept on working, noting that the machine pistol was fitted with a suppressor to silence its workings. It would have to be, since it was the weapon that Noone had used to treacherously slay the comrades and coworkers who trusted him without betraying the deed to the numerous guards stationed on the estate.
Don Bass asked only a single question: "Why?"
Noone shrugged, quirking a whimsical smile. There was an oddly elfin aspect to the big man, with his too-large knowing eyes, mouth upturned at the corners, and slightly pointed chin. Don Bass realized that the person he'd worked with, played with, and with whom he'd shared a good part of his adult professional and personal life was a complete stranger to him.
Noone said, "Call it a coup d'état. Change of power. I'm in. You're out. Way out." He held the gun pointed so it would shoot Bass in the belly where it hurt the most.
He said, "Christ! You can't imagine how long I've waited for this day—this night—this moment. I'm enjoying it so much that I hate to see it end."
Noone sighed. "But all good things must come to an end. If it's any consolation to you, Chief—and I'm sure that it's not—you can go to hell knowing that in a very short time you'll have lots of company when Sky Mount and all its lovely creatures go up in flames. I only regret that you won't be here to see it."
He added, "Die hurting, Chief."
Don Bass laughed out loud, a genuine guffaw at the bizarre turns of fate and reversals of fortune that could occur to a man not in a lifetime, but in a handful of seconds. He experienced an explosion of mirth that left him grinning from ear to ear.
Larry Noone arched an eyebrow, surprised by the other's outlandish reaction at the point of death. He wasn't sure what he'd expected from Bass at the finish but it wasn't this. He shrugged it off. "Hysteria. The mind is going. This will be a blessing for you, Don."
Bass said, "Buddy, you're about to find out how right you are."
"Oh really—?"
Larry Noone didn't live long enough to find out the truth of his words. He fell forward facedown to the floor, stone dead. The back of his head had been shot away by the burst of rounds Jack Bauer put into it at point-blank range, disintegrating the rear half of his skull as if it had been scooped out and exhibiting the gooey gray matter that remained.
Jack stood slumped against the doorframe, leaning against it for support. He let his gun hand fall to his side, holding the still smoking SMG that he'd used to liquidate Larry Noone.
Jack said, "Sorry I didn't get here sooner. I saw you running out the side door of the mansion and played a hunch that I'd better follow you and see what's what."
Don Bass said, "Lucky for me that you did."
"Luck is the difference between hanging and not hanging. I know."
"You heard everything?"
"Enough." Jack Bauer glanced at a wall clock. "Five minutes to two. Time enough for you to tell your gate guards to open up and let Garcia's tac squad in."
##
## THE FOLLOWING TAKES PLACE BETWEEN THE HOURS OF 2 A.M. AND 3 A.M. MOUNTAIN DAYLIGHT TIME
Sky Mount, Colorado
Ernie Sandoval said, "You wrecked the Mercedes, you SOB."
Jack Bauer said, "Get Garcia to buy a new one."
"He just might, after this one is all wrapped up."
Don Bass chimed in, "Hell, I'll buy you one."
Sandoval said, "You can't afford it."
"The Masterman Trust can. Let them pick up the tab."
Jack nodded in agreement. "That's the spirit."
The trio were walking briskly side by side down the main corridor of the mansion's east wing. Bass had a set of keys in hand that would open the anteroom doors and the door to Cabot Huntington Wright's inner sanctum, but as it turned out they weren't needed. The anteroom door was unlocked. The room was dark, but light outlined the closed door to Wright's suite of offices.
A platoon of Orlando Garcia's tac squads augmented by an equal number of Inspector Cullen's ATF agents were swarming the estate, securing the grounds, mansion, and all-important subsurface levels where the BZ grenades and plastic explosives lay, defanged for the moment but very much a potential and potent threat until the moment that agents took possession of them, and that moment was right now.
Jack Bauer's focus lay elsewhere, on the dozen quick paces it took him, Bass, and Sandoval to cross the anteroom to Wright's private door. His hand was on the knob, and to his surprise it turned freely and he opened the door and stormed in, the other two at his heels.
Jack said, "You left your door unlocked, Mr. Wright. Careless of you."
Cabot Huntington Wright was at the opposite end of the room, standing behind his desk, stuffing folders of documents into a briefcase that stood open on his desktop. He froze at the trio's entrance, lifting his gaze from what he was doing to the intruders who'd had the audacity to invade his domain.
He looked away first, oddly abashed to be taken in such a manner. His hands were hidden behind the lid of his attaché case, which stood upright.
Jack's hand flashed inside his jacket, coming into view with a pistol that he held pointed at Wright. Wright raised his arms in the classic hands-up position, obscuring but not hiding the white armband circling his dark-suited left arm.
Bass said, "The white brassard! That clinches it."
Jack circled around the desk, still covering Wright. Wright's hands were empty of everything but foldered documents but Jack was taking no chances. He said, "It's already clinched. It was clinched when Chappelle notified Garcia that he'd found the leaker—and the person to whom he'd leaked."
Sandoval had given Jack a quick update on the way to Wright's office. Ryan Chappelle had discovered that a member of his CTU/L.A. staff had passed the word about Brad Oliver's imminent arrest. A survey of regional division headquarters' phone logs had unearthed the culprit, one of Chappelle's top aides. The leaker had confessed when confronted but claimed he had no other motive than to curry favor with the ultra-rich and powerful Cabot Huntington Wright by giving him a friendly heads-up to prepare him for the embarrassment and disruption that would result when Wright's confidential assistant Brad Oliver was arrested by CTU agents for violating the national security.
The leaker's true motive would eventually come to light in the exhaustive investigation to which he'd be subjected. What was key was the identity not of leaker but of leakee. Chappelle tried to notify Jack Bauer to alert him to the identification but he'd been unable to reach him while Jack was otherwise engaged.
Chappelle had finally swallowed his pride and relayed the information directly to Garcia, enduring the humiliation of having to admit to a longtime rival that one of Chappelle's own was the guilty party. The facts were too vital to withhold, and Chappelle put the potentially career-damaging revelations in Garcia's hands, oblivious of how the hierarchs on the seventh floor at Langley might put a black mark in Chappelle's record book because of the dereliction of a trusted aide.
Chappelle was a patriot, and Jack had never doubted that ultimately he would do the right thing and disseminate the information where it would do the most good. But timing is everything, and Jack was heartened that Chappelle had acted sooner rather than later—for later might have been too late.
Jack Bauer now had the guilty party in hand and there was a standard operating procedure for the way things are done no matter how big the culprit is. Jack set the process in motion.
He said, "Please stay where you are, Mr. Wright, and keep your hands up. You're about to undergo what's sure to be a novel experience in your life: being searched for a weapon."
Wright affected a wry smile. He'd never quite lost his composure from the moment the trio barged in to confront him, but he had lost some of his color, the skin blanching and paling under his deep tan. Now the pallor was starting to fade and the color was returning to his cheeks.
Jack gave him a pat-down frisk, feeling around him for a concealed weapon. Jack was taking nothing for granted; for all he knew Wright might have a weapon on his person. It was that kind of a case.
Sandoval searched Wright's briefcase while Jack searched Wright. Wright said, "Don't you want to search, too, Don?"
Bass shook his head. "I'm private, I don't have jurisdiction. They do. You belong to the United States government now."
Jack said, "If not for a little bit of luck it might have been the other way around." He finished his search, said, "He's clean."
Sandoval said, "Nothing in his briefcase but documents."
"I'm sure the analysts will be interested in them."
Wright said, "I'm sure. May I put my hands down now, gentlemen? I confess that the posture is becoming something of a strain."
Jack said, "Go ahead. You can sit down, too—on the other side of the desk. I don't know what kind of gimmicks you might have built into it but I don't intend to find out the hard way."
Wright smiled with seeming affability. "My, my. Paranoia must be the prime attribute of a government snoop and spy." He went around to the front of the desk and made a show of seating himself comfortably in one of the plush visitors' armchairs.
He tapped a forefinger against the side of his forehead. "My weapons are all in here."
Jack said, "Your checkbook is your weapon, and with it you damned near took over the U.S."
"By the way, am I under arrest? And if so, what are the charges?"
"Yes—Cabot Huntington Wright, you're under arrest for conspiracy to commit treason and terroristic acts against the United States of America."
Marion Clary entered at that moment. Her hair was in disarray, she was without makeup, and her attire showed signs of having been thrown on at a moment's notice. Her demeanor varied between confusion and great distress. "Mr.—Mr. Wright? What's going on here?"
Wright rose when she entered, favoring her with a courtly little bow. "Ah Marion, right on time as always. In case you hadn't heard, I'm being arrested for crimes against the state."
Her dominant motif turned to one of outrage. White circles showed around her eyes, and her face suddenly looked strained and haggard. "Is this some kind of a grotesque joke?"
Wright said, "Grotesque it may be, but it's no joke, I fear."
Marion Clary swayed, looking as if she might faint. Don Bass rushed to her to steady her, said, "Marion, please sit down."
She turned on him, tearing her arm free of his supportive grasp. "Keep your filthy hands off of me!"
He said, "Please sit down."
She stared at him, rigid with indignation. Wright indicated the armchair beside his, said, "Marion, yes, please do."
She staggered like a sleepwalker to the chair and plopped down in it. Sandoval crossed to the office door, closed and locked it. He said, "We don't need any more interruptions."
Wright said, "Now that I've been arrested, will you read me my rights and allow me to speak to my lawyer? One of my many lawyers?"
Jack smiled tightly. "Nice try, Mr. Wright, but in cases involving acts of terrorism the normal rules are suspended and don't apply."
Wright's smile could have passed for one of genuine pleasure. "Ahhhh...so that's how you work it."
"That's how the system works. But never mind about that. Let's talk about how you worked it."
"You have the floor, Agent Bauer. I'm all ears."
Jack moved around to the front of the desk, resting his hip on the corner of it. He began, "I suppose in the long run it'll all come down to the question of sanity. Speaking for myself and not as a mental health professional, I believe that you are sane."
Wright looked more pleased than ever. "Thank you, sir!"
"You're an amoral sociopath but that doesn't read as insanity in my book."
Wright's mouth downturned in a little moue of displeasure. "Now now, no name calling. Surely we don't have to descend to that."
"Call it what you like. You're not the first person to see something he wants and do whatever it takes to get it no matter who gets hurt or what the consequences. You just do it on a more grandiose scale. Otherwise you're no different from the thief who knocks an old lady on the head for her social security money."
Wright nodded, putting his hands together and making a steeple out of them. "I see. At least I've graduated from amoral sociopath to mugger. That's progress, I suppose. And what is the object of my heart's desire?"
Jack took the question seriously. "The United States of America. For starters. Beyond that, who knows? Tomorrow—the world?"
"You're telling it. Please continue."
"My pleasure. In the last twenty-four hours I've had a crash course in the theory and practice of Cabot Huntington Wright as applied to the deadly arts of conspiracy, subornment, corruption, violence, terrorism, and mass murder. You might say I've had a total immersion in the dark side of Wright, the side nobody is supposed to see."
"I daresay that qualifies you as an expert, Agent Bauer."
"I daresay," Jack said dryly. "Let's get back to basics. Crime is a matter of means, motive, and opportunity. Start with motive first. You saw a way to make yourself master of the United States. By that I mean you hatched out a scheme to destabilize the economy, bring it to its knees, and take over the nation's leading corporations at fire-sale prices."
Wright nodded encouragingly, a schoolmaster listening to a prize pupil recite his lessons. "And how was I going to achieve that ambitious goal?"
"The old-fashioned way: murder. Murder and money. It all stemmed from your unique position as chairman of the board of the Masterman Trust. That and your role as director of the yearly Round Tables, a gathering of the richest and most powerful of the land under one roof. Your roof."
"Ah yes, the illustrious Round Table. My arrest will come as a great shock to them, all those dynasts and heirs and movers and shakers who've known me as a trusted friend and confidant over these too many years."
Jack quirked a smile. "They'll get over it, especially once they learn what you had planned for them—death by hallucinogenic gas and inferno."
Marion Clary leaned forward in her seat, her hands balled into fists that perched on her upper thighs. "You're insane, positively stark staring mad!"
Jack let it pass, speaking directly to Wright. "I once read that the emperor Caligula expressed the wish that all Rome had but a single head that he might strike it off with one blow."
Wright said definitively, "Caligula was a piker. Strictly small potatoes."
Jack took note of that remark. Perhaps the smooth facade was starting to crack and the real Cabot Huntington Wright emerge. "You went Caligula one better. You gathered up the people who collectively own a majority share of the real wealth in this country—stocks, bonds, real estate, the corporations that keep the wheels turning—and planned to murder them all in their beds and loot their assets at the same time.
"Stealing what isn't yours is the motive. The means were two-fold, financial and homicidal. The financial aspect is your territory, and I'll outline it quickly for the record. You've been betting a hundred million dollars on the swift, sudden downfall of the national economy. You're the spider at the center of a global web of misdirection and deceit. Using an arsenal of financial gimmickry such as dummy and shell corporations, third-party transactions, and the like, you've been short selling an astronomical amount of stock. I'm no financial wizard but I know what that means. My boss Ryan Chappelle is a wizard with the numbers and he explained it to me.
"You bet a fortune that the bottom will fall out of the U.S. economy. If you win, your short selling of stocks will reap you many fortunes. The economy is already so shaky, all it needs is one good push to send the house of cards tumbling down. You decided to supply that push."
Wright harrumphed. "In all fairness, you'll have to admit that the economy is doing an outstanding job of bringing itself down."
"Yeah, but you wanted to take a chainsaw to it. Mass murder is the push. That's where the Round Table comes in. All the heads and majority stockholders of the biggest corporations gathered in one place. Their sudden, violent deaths would deal the economy a body blow, triggering a financial panic that would make the Crash of 1929 look like a one-day selloff in the market. Stocks would plunge to a fraction of their worth. Universal bankruptcy, mass insolvency. And there you'd be with a mountain of money reaped from your short-selling gamble that the economy would suffer such a catastrophic loss—no gamble, but a sure thing.
"With all that cash in hand you could acquire a controlling interest in every corporation in every sector of the economy worth owning: utilities, insurance, energy, health care providers, software, manufacturing, you name it. The whole enchilada. And you'd have it all. Overnight you'd become the uncrowned king of the United States—king by fact if not by law or title. Master of a financial empire that no king, emperor, or mogul even dreamed of."
Wright was unflappable. "A not unworthy ambition, if I say so myself. In all due modesty."
Jack challenged, "Why be modest? Caligula was a piker—so you said. He would have given his eyeteeth to have an axe like the one you created, designed to lop off the heads of all those friends and confidants who've trusted you over the years."
"Do tell."
"An axe made not of finely honed steel sharpened to a razor's edge but of people. Bad people. As choice a crew of thieves, sadists, and killers as ever labored for the hidden puppet master pulling their strings. People like Brad Oliver, who handled some of the financial aspects of your dirty work."
Wright pulled a long face and looked sad. "Ah yes, poor Brad. Such a tragic death, so untimely a loss to one of the brightest rising stars in the fiscal galaxy."
Jack snorted his derision. "I bet. What happened to Brad? Did he get greedy seeing all those vast sums he was in the process of making for you and decide to feather his own nest? Your super-scheme for shorting was slick and stealthy but his pint-sized version to invest a few million of his own on the coming apocalypse was rushed and clumsy. His junior league manipulations showed up on Chappelle's radar screens because that's just what Ryan was looking for, smelly investments made in a hurry on the basis of foreknowledge of imminent catastrophe. Once Chappelle gets a whiff of something like that, he keeps digging into the numbers until he finds out the real score. Oliver's heavy-handed shorting is what put CTU on to the plot against Sky Mount in the first place."
Wright couldn't have been cooler. "Brad's one overriding fault, and I say to you what I would not hesitate to declare under oath in any court in the land, his great sin was avarice. Greed, pure and simple. He overreached himself and paid the price."
Jack countered, "Thanks to you he did. When you found out that his arrest was imminent you greased the skids out from under him, virtually literally. You tipped him off that we were coming to apprehend him, knowing that he would do what he did: take it on the run. Only before you went to him you made sure that one of your hatchetmen had arranged for Mr. Pettibone and his Deathmobile to be outside the gates waiting for him. When Brad tried to make his getaway, Pettibone ran him off the road on a thousand-foot drop to his death. Exit Brad."
Wright made a face. He was really enjoying himself now. "Dear me! Did I do all that? I'm afraid you'll have some difficulty proving that in court, Agent Bauer."
"Don't be so sure. Look at Marion Clary. She looks like she might be remembering something she'd seen but thought nothing of at the time. Like you having a private little chat in your office here with Brad right before he went out and got smeared all over the eastern slope of Mount Zebulon?"
It was a shot in the dark, but Jack figured it was worth a try. The first mention of Oliver's name had triggered a fidgety restlessness in Marion Clary, an agitation that increased as Jack explicated the mode and manner of Oliver's death.
Cabot Huntington Wright condescended to glance at the receptionist. What he saw there compelled him to take a long second look. She openly fretted, chewing her lower lip, her expression stricken, wounded.
He said, "Marion, dear, surely you don't give any credence to this preposterous twaddle?"
She held herself so tightly that it looked like her neck cords would break. Her eyes were open, staring into space. She shook her head with short, tense movements. She said, "Believe it? Of course not! But—but you did call Brad into your office yesterday afternoon to speak with him, and when it was over he had the most dreadful look on his face and he rushed off like a crazy man and drove to his death—"
"Pure coincidence. Brad had a guilty conscience because he feared his financial chicanery was about to come to light. He ran away and had the misfortune to suffer a terrible fatal accident due to his own carelessness and innate dishonesty."
She turned hurt eyes on him. "Cabot, how can you speak so cruelly about poor Brad, who never deliberately hurt anyone in his life?"
"Honesty compels me to speak the truth." Wright tsk-tsked. "I can't believe that you'd be so credulous as to listen to the ravings of this prosecutorial young man. I'm disappointed in you Marion, very disappointed."
She wasn't listening to him. Jack wondered if she was listening to an inner voice instead. He decided to press on. "That brings us to the other edge of your double-headed axe, Mr. Wright. Oliver was the financial edge. The homicidal edge was Larry Noone.
"Noone was a driving wheel in your murder train. He was perfectly placed to do so. As a high-ranking executive of the Brand Agency, Noone had access to his own private intelligence network, one rivaling any in the public sector and less hampered by red tape. By delving into the Brand computerized files he could learn with a keystroke who was dirty, who could be corrupted and who couldn't. Bribe takers, thieves, prostitutes, deviants, strong-arm goons, and contract killers, all listed there in the files. All he had to do was call them up and dangle the baited hook of Cabot Wright's money in front of them.
"It was Noone who found and recruited Reb Weld, using him to assemble a small army of hired killers. Noone who had all the inside information on security arrangements for the Round Table, allowing Weld and friends to circumvent them. Noone who murdered the board operators in the Brand command center tonight to allow Weld and his killer elite to plant bombs and poison gas in the basement on Level Two to blow up the fuel tanks to create a raging inferno to kill hundreds of innocent men, women, and children and burn Sky Mount down to the ground!"
Wright said acidly, "Of course it's in your interest to blacken the character of poor Larry Noone, considering that you're the one who killed him. That, my dear young sir, is not a tower built on groundless speculation and absurd hypothesis but a fact!"
Marion Clary recoiled as though she'd been struck. She said in a whisper that trembled on the edge of a shriek, "Larry Noone is dead, too? My God, no!"
She covered her ears with her hands to keep from hearing any more. Cabot Wright sat back in his chair, favoring Jack with a richly supercilious smile.
Don Bass went to Marion Clary. His expression was compassionate as he gently but firmly took hold of her thin wrists and eased her hands away from her ears. He said, "Marion, you must listen to me. I've never lied to you and I'm not about to start now. As the Lord is my witness, less than an hour ago Larry Noone held me at gunpoint and was about to kill me. This man Jack Bauer saved my life, and that's the honest truth."
Ernie Sandoval had sat silent for a long while taking it all in. He now spoke up. "It's a time for truth, Marion. You can't stick your head in the sand and hope it goes away. Tell her, Jack. Tell her what Cabot Huntington Wright was going to do to destroy her beloved Sky Mount!"
That caught her attention. Her head jerked slightly to one side and her eyes took on a glazed expression. "Cabot Wright...destroy Sky Mount?"
Jack picked up the ball. He addressed his words to Wright, aware that Marion Clary was following them with a dreadful avidity. She could be a key witness in any future trial of Wright; her testimony could be invaluable if she could be convinced to give it freely.
Jack said, "That brings us to the third leg of our murder triangle. Remember, means, motive, and opportunity. The means was money and the people it could buy, whether it was Brad Oliver and his financial sleight-of-hand or Larry Noone and his handpicked assortment of killers.
"The motive was money, too, money and power, with one nightmarish catastrophe that would make Cabot Huntington Wright richer and more powerful than any other man in the history of the world.
"That brings us to opportunity. Like so much else in this case, opportunity wears more than one face. I've already mentioned the opportunity of having the movers and shakers of the national economy conveniently gathered together under one roof to make a big, fat target. But there's another face to that opportunity, one that is and could only be known to a select handful of persons, and you, Mr. Wright, are the most select of that select few."
"You flatter me, Agent Bauer."
"No I don't, not really. I'm just telling the plain truth the way the facts add up. The fact is that there is one secret that you are in a prime possession to know. It's the old story of the Trojan horse: the enemy was already in the citadel, hidden where no one would ever suspect them. With the Greeks and the Trojans it's a wooden horse. With you and Larry Noone's murder squad, it's a fallout shelter built long ago beneath Sky Mount that the world has forgotten but the few remember.
"A fallout shelter built at the height of the nuclear jitters of the Cold War era. A bunkerlike fortress that accesses Level Two through secret doors and hidden passages. A shelter with an escape route in case Sky Mount should be bombed flat and the shelter inhabitants unable to dig themselves out from under a mountain of rubble. So the builder created himself an escape route, drilling a tunnel through and out of a rock spur of Thunder Mountain into a little high mountain valley named Winnetou.
"The escape route, like the shelter itself, was a closely held secret. The creator didn't want the public to know about it. In case of a threatened atomic attack he'd be besieged by hordes of neighbors and strangers all wanting to escape annihilation by holing up in the shelter, too. That wouldn't do, so the shelter was kept secret and the escape route was hidden to look like part of the mountain so no outsiders would ever dream of its existence!
"What happened then? I'm guessing here, but we'll find out the facts soon enough. The builder died, the shelter entrances and exits were sealed and forgotten, and the few others who knew the secret mostly died out. But who would be better placed to know the secret or rediscover it than the Lord High Executor of the Masterman Trust, the master of Sky Mount itself, you, Cabot Huntington Wright!"
Jack Bauer waited for Wright to respond but it was Marion Clary who reacted first. She stood up suddenly, the light of a massive revelation seizing her with an irresistible force.
She blurted out, "It's true! There is an abandoned fallout shelter hidden under Sky Mount! It was built in the nineteen-fifties by F. X. Masterman, the last surviving heir to descend directly from old H. H. Masterman, founder of the family fortune. Francis Xavier Masterman was an eccentric with an obsession about surviving an atomic war. He spent a fortune building his shelter and escape routes. After he died the family wanted nothing more to do with F.X. and his sensational bad publicity so they capped the tunnels, sealed the hatches, pretended it wasn't there, and forgot about it.
"I know about it because I'm the archivist and knowing the history of Sky Mount is my life's work. I know it, yes—but how do you?"
Jack said seriously, "I know it, Ms. Clary, because I've been there. Just tonight I took the grand tour of it to keep a gang of murder-happy psychos from using it to blow up the fuel tanks and turn the mansion into an infernal holocaust! Where did I learn of it? From a sadistic killer named Pettibone who killed that nice young man Brad Oliver and who knows how many others.
"The big question is, who did he learn it from? From his boss, an even worse killer, who learned it from Larry Noone, who learned it from Cabot Huntington Wright! Unless you told Noone—"
"No," she said firmly, shaking her head. "I've always respected the family's wishes for privacy and kept the truth about the shelter a private matter and never spoken of it to any outsiders."
She was holding her body so tight that instead of turning her head she turned her entire body so she could look down at Cabot Wright and stare him in the eye. She went on, "I've never spoken of it to outsiders, but I have gone into detail about it on more than one occasion with my employer, Mr. Wright!"
Wright literally tried to wave it away, dismissing it with a flicking gesture of his hand. "Marion, you're becoming seriously overwrought. I begin to fear for your state of mind."
"You—you would have helped to destroy Sky Mount? All those innocent human lives? All those priceless art treasures?"
"You're being ridiculous, dear. Sit down and take a pill to relax before you give yourself a nervous breakdown."
Cabot Huntington Wright was beginning to show the first signs of agitation. He was restless, unable to sit still. He kept crossing and uncrossing his legs and squirming around in his seat as if unable to get comfortable.
Marion Clary ignored his advice. She did not sit down or take a pill. She stood her place, staring accusingly down at Wright.
Wright turned to the others as if unable to face her stare. "Do you see what you've done, gentlemen—and I use the term loosely—with your monstrous fabrication of lies and half truths, slurs and innuendos? You've driven this poor, simple soul nearly half mad with hysteria!"
Jack Bauer said softly, "Maybe she's starting to realize the truth of what you've done, Mr. Wright. The lies and scheming, the conniving at murder, and more: wholesale mass murder!"
Wright affected an air of extreme nonchalance bordering on indifference. He studied his carefully manicured fingernails, flicked an imaginary spot of dust from his lapel. But he was watching Jack out of the corners of his eyes.
Jack ignored Wright's smooth front and kept hammering his points home. He said, "Speaking of opportunity, that brings up one last important element in your master plan. It was a lucky fluke but you saw it lying there and picked it up for your own use. I'm referring to the presence of Abelson Prewitt and his inner circle of Zealots at the compound at Red Notch. Every conspiracy needs a fall guy, a patsy who can be blamed for the crime, and Prewitt was ripe for the taking. It's the time-honored ploy known as 'Pay the Law.' Give the authorities a ready-made scapegoat for the crime and the manhunt ends. Otherwise they'll keep on looking and possibly even stumble across the real culprits.
"Prewitt was your scapegoat. He was a crackpot cultist who hated the Round Table and all that it represents. There was no real history of violence in his background, but that was no problem. A lot of these cults go on their own way for years before reaching the breaking point and lashing out with overt acts. Prewitt's crank economic theories and overheated rhetoric made him perfect for framing.
"The plan was to lay the blame for the Sky Mount terror strike on Prewitt and his cadre. To carry that out they first had to be disposed of. The Mountain Lake MRT unit did the advance work. They'd all been suborned into working for the plot, bought and paid for. I'm guessing that Larry Noone handled that part of the operation. I wondered how the activity at Winnetou could have gone unreported until I found out a little while ago from Agent Sandoval that Hardin's MRT had the responsibility of patrolling that area and consistently gave it a clean bill of health.
"Red Notch was hit early Thursday morning. The MRT did the advance work of neutralizing O'Hara and Dean, the ATF agents monitoring the compound from the outside. Hardin and Taggart got the drop on the unsuspecting agents and put them out of the way. That left a clear field for Reb Weld's kill squad. They blitzed the compound with BZ gas grenades, the potent hallucinogenic gas incapacitating the cultists. A hermit who witnessed the assault said that it was carried out by 'hog-faced demons.' Hog-faced demons—that's what the killers in their gas masks looked like to him. He wasn't so far off the mark at that.
"The round-up of the Zealots didn't come off without a hitch. There was violence, some blood was spilled. Those bloodstains held the telltale chemical markers allowing for the identification of BZ as the chemical weapon agent. The cultists were herded onto their own bus and driven out to Silvertop in Shadow Valley to be disposed of. All but two of them were slaughtered and dumped down the air shaft of an abandoned mine, their bodies covered with dirt to make sure that they wouldn't be found too soon.
"Prewitt and his top lieutenant, Ingrid Thaler, were killed, too, but their bodies weren't dumped with the others. They were taken to the base camp at Winnetou and kept on ice for future use. I came across the ice chest that had been used to keep the corpses in cold storage at the camp but didn't know what it was for until later tonight. It wasn't until I found Prewitt and Thaler's bodies in Level Two that the significance of the ice chest became clear to me. The cadavers had been frozen to disguise the true time of their death. They were going to be planted outside the mansion to be found in the aftermath of the destruction. By then the heat of the firestorm would have warmed them up.
"Prewitt and Thaler being found dead at the scene of the crime would have clinched the case for the Zealots' responsibility for the terror strike. It would have been open and shut as far as the authorities and the public were concerned and no one need look any further into the matter. A crazy cult would have been blamed for the head shot that took down the national economy.
"Only it didn't work out that way. The strike was thwarted and now we know the real mastermind behind the plot was you, Cabot Huntington Wright!"
Wright was squirming. He'd been unable to sit still as Jack Bauer drove home his summation. He now made a visible effort to regain his self-possession.
Wright tried another tack. "Theories are all very well, Agent Bauer—in theory. But proving them in open court is another matter. Let us suppose for the sake of argument that your allegations against me are true? How do you propose to prove them?
"Brad Oliver is dead. Larry Noone is dead. Going by your theory that I used them as middlemen and cat's-paws to, how did you say it, get my dirty work done, how can you prove it? They can't testify, and by your own logic they're the only ones who could have tied me to these sordid murders and theft of Army secret weapons and legions of hired killers and whatnot. Proof. That's what you've got to have and that's what you lack."
Jack pounced on something Wright had let slip. "I never said anything about Army secret weapons and neither did anyone else. That statement was volunteered by you and I call on everybody else in this room to witness that you said it."
Wright stifled a yawn. "Did I say it? I suppose I did. A man in my position, who meets so many four-star generals, politicians, financiers, defense contractors—yes, and intelligence professionals, Agent Bauer—is liable to hear all sorts of things."
He leaned forward, poking his finger in the air to make his point. "You might say that it was, er, just something in the air. Something in the air that made me chatter idly of Army secret weapons. I might even have heard a rumor about a cache of experimental drug gas that was inexplicably lost, stolen, or strayed somehow. These things get around. Perhaps I discussed the subject during an idle moment with Larry Noone.
"What of it? You can make nothing of that, not in any court in the land. I'm no illegal alien whose head you can throw a black hood over and transport to Guantanamo Bay to torture into signing a confession. I'm an American citizen—an extremely wealthy one, need I add? You'll find it very difficult to tie a tin can to this old dog's tail, Agent Bauer."
Jack said grimly, "You underestimate the effectiveness of the American government, Mr. Wright. An army of civil servants thinking about you every waking hour of the day and minutely examining the fine details of your public and private lives can unearth a hell of a lot of data about secrets, lies—and truth."
Wright laughed. "And you underestimate the power of my attorneys, Agent Bauer. My army of high-powered, high-priced attorneys and their inexhaustible battery of legal tactics and tricks."
He rose. "No, you've got nothing on me—nothing you can prove. Go ahead with this farce of an arrest if you like. My lawyers will have me free and on the street before dawn. Not the street, actually, that's just a manner of speaking. Shall we say, out of jail and in the penthouse?"
The unexpected happened, the unpredictable variable that no one can factor into his calculations. Jack Bauer never saw it coming and he wondered later, even if he had, would he have tried to stop it? He thought not.
But he didn't see it coming and neither did anybody else in the room, least of all Cabot Huntington Wright.
Marion Clary without warning seized up a letter opener from the top of Cabot Wright's desk and with a shrill, wordless screech buried it deep in his chest.
She cried, "It's true! True! True!" She stabbed Wright once with each cry, driving the daggerlike letter opener into his chest and neck.
Ernie Sandoval and Don Bass rushed forward to grab her and wrestle the weapon away from her even as Wright folded at the knees, sagging to the floor and sprawling across it, spilling his lifeblood into the deep pile carpet that greedily absorbed it just as it had tenderly cushioned his fall.
Red blood—not blue.
The best physicians were called, of course, but it was too late. One of the thrusts had pierced his heart, and he died within minutes of the assault.
Marion Clary, dazed by her own violence, kept repeating, "I had to do it, he had to be stopped...I had to do it. No one else could stop him, the monster...I had to do it. He had to be stopped."
She was still repeating it when a pair of white-coated orderlies came to take her away. They were very gentle with her. She would need gentleness.
Don Bass said, "I promise on my honor I will move heaven and earth to see that she never spends a day of her life in jail."
Jack Bauer said, "She won't—I promise you that."
Ernie Sandoval said, "If we had any guts we'd pin a medal on her."
Jack shook his head in wonderment. "The ways of fate are strange. Did you see what she stabbed him with?"
Don Bass said, "The letter opener?"
Jack said, "The antique letter opener that once belonged to Marshal Fouché, Napoleon's spy chief."
He stroked his chin thoughtfully. "The old king of spymasters reached out from beyond the grave on one final mission of righteous retribution!"
Sandoval said, "Amen to that, brother."
Don Bass shook his head. "You fellows are kidding yourselves. It's just coincidence, that's all. Sheer blind luck. She wanted to strike out and picked up the first thing that came to hand."
Jack Bauer laughed, a little self-consciously. "I know what I think. You can think what you want, Don. You can do that. It's still a free country.
"At least for today it is," he added. "That'll hold us till tomorrow."
## About the Author
DAVID JACOBS is the author of over three dozen works of fiction and nonfiction, including most recently, the nonfiction titles The Mafia's Greatest Hits and Snakes on a Plane: The Complete Quote Book. Other nonfiction titles include Notes from The Barn, the companion volume to The Shield TV series; Best of Court TV Volumes I-IV; and The Complete Idiot's Guide to the FBI. His fiction includes the military thriller Return of the Dog Team and the western Big Iron. David is a longtime member of the Mystery Writers of America.
Visit www.AuthorTracker.com for exclusive information on your favorite HarperCollins author.
## [24 DECLASSIFIED Books
From Harper](24declassifiedheadshot_con01.html#adc_01)
HEAD SHOT
TRINITY
COLLATERAL DAMAGE
STORM FORCE
CHAOS THEORY
VANISHING POINT
CAT'S CLAW
TROJAN HORSE
VETO POWER
OPERATION HELL GATE
## Copyright
This book is a work of fiction. The characters, incidents, and dialogue are drawn from the author's imagination and are not to be construed as real. Any resemblance to actual events or persons, living or dead, is entirely coincidental.
24 DECLASSIFIED: HEAD SHOT. Copyright © 2009 by Twentieth Century Fox Film Corporation. All rights reserved under International and Pan-American Copyright Conventions. By payment of the required fees, you have been granted the non-exclusive, non-transferable right to access and read the text of this e-book on-screen. No part of this text may be reproduced, transmitted, down-loaded, decompiled, reverse engineered, or stored in or introduced into any information storage and retrieval system, in any form or by any means, whether electronic or mechanical, now known or hereinafter invented, without the express written permission of HarperCollins e-books.
Adobe Digital Edition April 2009 ISBN 978-0-06-191182-8
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Merkel sets out pragmatic vision for Europe
German chancellor insists that the EU must become deeper before it becomes larger and focus on political needs, rather than aims.
By Simon Taylor
German Chancellor Angela Merkel today set out a vision of the EU that included a stout defence of Germany's opposition to enlargement of the EU without ratification of the Lisbon treaty. She also expressed opposition to greater co-ordination of economic policies through summit meetings of eurozone countries.
Merkel gave her speech at Berlin's Humboldt University, a venue chosen by many German politicians for keynote speeches about the EU. Merkel's speech was, though, markedly different from some others given at the Humboldt. In May 2000, for example, Germany's then foreign minister, Joschka Fischer, called for the creation of a European federation "with a European Parliament and European government with full legislative and executive powers".
By contrast, Merkel made clear she has little time for debates on, as she described it, the "end state" of the EU. "In my view, they tend to make the EU's citizens less, rather than more, trustful about today's EU," she said. "Instead of talking about long-term aims we should look at the next steps that are politically necessary."
One of the next steps, she indicated, must be ratification of the Treaty of Lisbon, which would, if adopted, reform the EU's institutional structure. In particular, its reforms would strengthen the presidency of the European Council, which brings together EU leaders, and streamline its decision-making process.
Merkel insisted that the EU cannot take in several new members without the institutional improvements that Lisbon provides. "The deepening of the European Union of 27 members has priority over rapid enlargement," she said.
She added later, though, that she does not exclude further enlargement. "I'm not in a position to do so, nor would I want to. But further enlargements will not happen at the speed some people assume," she said. Deepening did not mean that, as some Eurosceptics warned, that the EU could become a superstate. "The EU has developed a modern method of deepening. It is not a state and it is not meant to become one", she said.
The principal obstacle to ratification at present is the position of Irish voters, who rejected the treaty last year but will be asked to vote again this autumn. "I trust the Irish will come out in favour of Europe," Merkel said, coupling her comment with a call for other countries that have yet to do so to ratify the treaty soon.
Poland's President Lech Kaczyński and Czech President Václav Klaus have yet to sign the treaty, although the treaty has passed through the Polish and Czech parliaments. Germany's supreme court has yet to rule on the treaty's constitutionality.
German interests, European interests
The precedence that Germany gives to a deeper EU over a larger one is one of the four principles that currently underlie Germany's pursuit of the "ideal" of European unification, she said. Two others include an awareness of the uniqueness of the EU's institutions and of the driving role played by values.
The primary principle, though, is "standing up for German interests in Europe and keeping the good of the EU as whole in mind", she said. "These are two sides of the same coin," she said in a message aimed at critics who claim Germany has suddenly started a selfish pursuit of its own interests.
Addressing domestic critics, Merkel rejected the caricature of Germany as the "paymaster" of the EU as "misleading". Germany's membership of the EU costs every German citizen €263 per year, she acknowledged, but she stressed that Germany "reaps above-average gains from the EU internal market". The country also gets money back, she said, pointing out that over the past six years the new federal states have received €29 billion from the EU's structural funds.
She couched that defence in a broader rejection of a book-keeping approach to the benefits of EU membership. In a speech that highlighted the significance of the twentieth anniversary of the collapse of communism and Germany's "privilege" in being able to help shape the EU from within after the reunification of Germany, Merkel insisted that "any evaluation of the European Union that only takes into finances into account is a totally inadequate reflection of the significance of European reunification for our country".
Merkel also rejected suggestions that Germany is a simply a "broker" of other countries' interests in the EU. That wrongly implied, she said, that Germany "has no or few interests of its own in Europe to stand up for".
"But we don't stand up for our interests with a bulldoggish insistence," she said, in an implicit reference to the way the UK defends its core concerns.
Instead, she said, "Germany's diversified economy, which includes traditional industries, high-technology and agriculture, often quite unavoidably makes us an understanding intermediary for various economic interests".
A Europe undivided economically
Turning to the EU's economy, she distanced herself from calls by France's President Nicolas Sarkozy for EU states to act more closely together, for example, holding more meetings of leaders of the 16 EU countries that use the euro. "I will not stand for divisions within Europe," the German chancellor said. "That is why I resist demands for greater co-ordination of economic policy within the eurozone which are often not thought through to their logical conclusions."
"The [European] Council of the 27 finance ministers must therefore play the key role in co-ordinating economic policy," she said.
She made clear, though, that her emphasis on unity did not mean she sided with attempts by central and eastern European governments to make it easier to join the single currency. "The criteria for membership of the eurozone must not be watered down," she said.
She also mounted a robust defence of the current division of powers between the EU and national governments, particularly when it comes to responding to the economic and financial crisis.
"The powers regarding budget tax and social policy are held by member states for good reason. Measures to boost the economy including bank rescue package are national tasks. The EU forms the necessary regulatory framework within which the member states decide on their options for action," she said.
No more Iraqs
Earlier in the speech she sought to play down differences with France, saying: "I enjoy working closely with President Nicolas Sarkozy. Joint Franco-German initiatives are still the best way of guaranteeing that Europe will agree in making progress."
Merkel insisted that the EU must show a united front on foreign policy, saying that "a situation like that which emerged with respect to the Iraq war in 2003 must never be repeated".
"We must also make sure no cracks appear in the front we present to the world," she stressed. "Europe must never split itself or let itself be divided."
Merkel said that the challenges facing the EU are changing, listing security policy, climate change and defending the social market economy. She said that the EU needs to adjust to these new challenges. "The Union must be able to evolve. This does not mean that the EU must always be given new powers and we don't need new legislation, agencies or funding to solve every problem."
"It's about showing the political will to act in a united and co-ordinated way," she said.
More from ... Simon Taylor
Polish envoy's bid for top EU post sparks Brussels backlash
Officials warn department dealing with bloc's neighbors could become illiberal fiefdom.
EU budget offers debt loophole to battle climate change
Policymakers are trying to turn the economy green while keeping EU finances in check.
Europeans confused about ECB's role, survey finds
A poll shows respondents often had a far more wide-ranging view of the ECB's mandate than is actually the case.
A wonk's guide to the French EU presidency policy agenda
Emmanuel Macron is angling to push ahead new rules on tech companies, change global tax rates and rethink the EU's role in the world.
By Lili Bayer and Zosia Wanat
By Bjarke Smith-Meyer
By Maïa de La Baume, Giorgio Leali, Paola Tamma, Samuel Stolton, Simon Van Dorpe and Zia Weise
Economic governance Enlargement EU reform EU treaties
France Germany Iraq Poland United States | {
"redpajama_set_name": "RedPajamaCommonCrawl"
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\section{Introduction}
Hilbert schemes of points have many applications in different areas of mathematics, for example Haiman proved the $n!$ conjecture using their geometry (see~\cite{Haiman}). Fogarty~\cite{Fogarty} considered these schemes when the ambient variety was a smooth surface and showed that then the Hilbert scheme of points is smooth as well. If we start from a three or more dimensional scheme this is not the case and many questions arise, such as when the Hilbert scheme of points is reduced, Cohen-Maculay, etc. Recently, Jelisiejew closed a long-standing open question in this area and showed that the Hilbert scheme of points on an affine space is not reduced~\cite{Hilbpath}; this problem was open for more than fifty years. Still the question about the smallest counter-example to reducedness remains. Methods of Jelisiejew were partially implicit and the non-reducedness witness was represented by a scheme of degree $5082$ on $\Affine^{14}$. In practise, people analyse the Hilbert schemes of $d$ points, where $d$ is much smaller, usually $d \leq 20$; see for example~\cite{example1} or~\cite{example2}. The question remains whether non-reducedness is present in this range.
The aim of this paper is to provide simpler example of a non-reduced point on the Hilbert scheme with both the dimension of an affine space and the number of points much smaller than in~\cite{Hilbpath}. More precisely we show that the Hilbert scheme of $d$ points on $\Affine^n$ is non-reduced as soon as $d \geq 13, n \geq 6$, which immediately follows from non-reducedness of $\mathrm{Hilb}^{13}_{\Affine^6/\kk}$. In particular we give:
\begin{example}[Corollary~\ref{cor:important}]\label{ex:first}
Let $I$ be the ideal generated by polynomials $f$ in variables $\frac{\partial}{\partial x_1}, \dots, \frac{\partial}{\partial x_6}$ such that $f$ applied to $F = x_1 x_2 x_4 - x_1 x_5^2 + x_2 x_3^2 + x_3 x_5 x_6 + x_4 x_6^2$ is zero or $f$ is of degree greater or equal three. Then $[I] \in \mathrm{Hilb}^{13}_{\Affine^6/\kk}$ is a non-reduced point.
\end{example}
In order to prove non-reducedness of points similar to the above we use the $\G_m$-action on ${\mathcal H} := \mathrm{Hilb}^{d}_{\Affine^n/\kk}$ coming from the action of $\G_m$ on $\Affine^n$ by multiplication. We define the barycenter scheme ${\mathcal B} \subset {\mathcal H}$ which consists of points represented by tuples of points in $\Affine^n$ such that their weighted average is $0$. Next we note that ${\mathcal H} \simeq \Affine^n \times {\mathcal B}$. Moreover, the $\G_m$-action on ${\mathcal H}$ restricts to ${\mathcal B}$. The point from Example~\ref{ex:first} is an example of a hedgehog point:
\begin{definition}\label{def:hedgehog}
A $\G_m$-invariant point $[I] \in {\mathcal B}$ is a \emph{hedgehog point} if it satisfies the following conditions:
\begin{itemize}
\item $(T_{[I]} {\mathcal B})_{>0} = 0$,
\item $T_{[I]} {\mathcal B} \neq (T_{[I]} {\mathcal B})_0$,
\item there is no closed point $[I'] \neq [I]$ such that $\lim_{t \to 0} t \cdot [I'] = [I]$.
\end{itemize}
\end{definition}
The last dot can be stated in other terms. Let $V$ be the subscheme of ${\mathcal B}$ consisting of points for which their $\G_m$-limit at zero is $[I]$. We call it the \emph{negative spike at $[I]$}. The last condition above is equivalent to $0$-dimensionality of $V$ or to the fact that $V$ is topologically a point. This justifies the name "hedgehog point", as ${\mathcal B}$ looks like ${\mathcal B}^{\G_m}$ with infinitesimal spikes, near $[I]$. A more precise definition of the negative spike and being a hedgehog point is given in Definition~\ref{def:hedgehog:precise}.
We prove that:
\begin{theorem}[Hedgehog point theorem~\ref{thm:Hedgehog_point_theorem:precise}]\label{thm:Hedgehog_point_theorem}
A hedgehog point $[I] \in {\mathcal B}$ is non-reduced. Hence $(0, [I]) \in \Affine^n \times {\mathcal B} \simeq {\mathcal H}$ is non-reduced as well.
\end{theorem}
The geometric idea behind the proof is as follows:
\begin{enumerate}
\item By a generalisation of \BBname{} decomposition for the $\G_m$-action on the Barycenter scheme we get that there is an open neighbourhood of $[I]$ such that the barycenter looks like an affine $\Affine^1$-cone over $\G_m$-fixed points. Here the fact that $(T_{[I]} {\mathcal B})_{>0} = 0$ is crucial.
\item The fiber over $[I]$ of this cone is $V$ and it is $0$-dimensional as $[I]$ is a hedgehog point. Assuming (for a contradiction) reducedness of $[I]$ in the barycenter scheme, we get that the aforementioned affine cone is trivial i.e. the barycenter scheme is isomorphic to the fix-points subscheme near $[I]$. This follows from semi-continuity of dimension of fibers for projective morphisms.
\item The tangent space to the fixed-point subscheme of ${\mathcal B}$ is $(T_{[I]} {\mathcal B})_0$ so it is different than $T_{[I]} {\mathcal B}$, because $[I]$ is a hedgehog point. Hence, we get a contradiction with $[I]$ being reduced.
\end{enumerate}
The key tool is the generalisation of \BBname{} decomposition from~\cite{JS19}. It reduces the whole problem to only a few facts about tangents and the fiber $V$.
To construct hedgehog points we start with a cubic in six variables $F \in k[x_1, \dots, x_6]$ and proceed as in Example~\ref{ex:first}. We produce an ideal $I$ which is defined as $\Fperp + S_{\geq 3}$ where $\Fperp \subset S = k[\frac{\partial}{\partial x_1}, \dots, \frac{\partial}{\partial x_6}]$ is the set of differential polynomials, such that when applied to $F$ they give $0$. The cubic $F$ is \emph{general enough} if it lies in the open (and non-empty) set of cubics defined precisely in Definition~\ref{def:nice}. We prove that:
\begin{theorem}[Corollary~\ref{cor:Iishedgehog}]\label{thm:cubics_give_hedgehog}
For general enough cubic $F$ the given point $[I] \in \mathrm{Hilb}^{13}_{\Affine^6/\kk}$ lies in the barycenter subscheme and is a hedgehog point. In particular it is non-reduced.
\end{theorem}
Non-emptiness of the set of general enough cubics is witnessed by $F$ from Example~\ref{ex:first}. Due to the paper~\cite{Shafarevich} algebras with their Hilbert function $(1, 6, 6)$ form an irreducible component of the barycenter scheme. The point $[I]$ described above lies only in that component, what follows from the fact that ${\mathcal B}$ and ${\mathcal B}^{\G_m}$ are homeomorphic near $[I]$ (by point (2) in the above sketch of the proof of Theorem~\ref{thm:Hedgehog_point_theorem:precise}, but without assuming reducedness) and from the fact that ${\mathcal B}^{\G_m}$ consists of algebras of type $(1,6,6)$ near $[I]$. Moreover, a general point of the component containing $[I]$ is smooth, so non-reducedness here is of the "embedded component" type.
After proving theorems~\eqref{thm:Hedgehog_point_theorem} and~\eqref{thm:cubics_give_hedgehog} we determine the negative spike $V$ over $[I]$ coming from a general enough cubic $F$. We prove:
\begin{theorem}[Theorem~\ref{thm:completeV}]
If $I = \Fperp + S_{\geq 3}$ for a general enough cubic $F$, then the negative spike at $[I]$ is isomorphic to $\Spec(S/\Fperp)$.
\end{theorem}
As a subscheme of the Hilbert scheme $V$ is induced by a deformation. We describe a related deformation and call it the \emph{fractal family}. We chose the name fractal because the fiber over the unique closed point and the base space of this deformation are almost the same, so it is like a deformation over itself. This exhibits an interesting phenomenon: the Hilbert scheme which parametrises finite schemes, naturally contains some of them (or very close to such), as closed subschemes. Overall, the isomorphism between $\Spec(S/\Fperp)$ and $V$ comes from the composition of the morphism to ${\mathcal H}$ defined by the fractal family and the projection ${\mathcal H} \simeq \Affine^n \times {\mathcal B} \to {\mathcal B}$ from the Hilbert scheme to its barycenter subscheme.
We briefly discuss the contents of the paper. In Section~\ref{sec:prelims} we introduce Hilbert schemes of points, describe a few groups acting on Hilbert schemes of points on affine spaces and define \BBname{} decomposition. Next we introduce obstruction theories, describe the barycenter scheme and present a simple form of Maculay's duality. In Section~\ref{sec:main} we start with the notion of being general enough for cubics in six variables and the details of the procedure of making a point $[I]$ in the $\mathrm{Hilb}^{13}_{\Affine^6/\kk}$ from a general enough cubic $F$. After that we determine the tangent space to $V$ at $[I]$ and calculate primary obstructions for $V$. It allows us to prove the $0$-dimensionality of $V$ and conduct the main argument of the paper. At the end we determine $V$ completely by constructing the fractal family.
Some interesting
questions remain: for example what is the minimal $n$ such that $\mathrm{Hilb}_{\Affine^n/\kk}$ is
non-reduced or whether the Gorenstein locus of the Hilbert scheme of points is
non-reduced.
\section{Preliminaries}\label{sec:prelims}
We assume familiarity with schemes and their functorial description (e.g. as in~\cite[Part 1, Chapter 2]{FGI+05}). We work in an algebraically closed field $\kk$ of characteristic $0$, although the characteristic and algebraic closedness play a significant role only at a few places.
We will be working a lot with group-scheme actions. A good reference to this topic is again~\cite[Part 1, Chapter 2]{FGI+05}.
Now we list some of the notations appearing later in the paper. The reason for two notations - $S$ and $P$ - denoting the polynomial algebra will become clear after Definition~\ref{def:duality}.
\begin{itemize}
\item $\kk$ denotes the algebraically closed field of characteristic $0$.
\item $X$ denotes scheme locally of finite type over $\kk$.
\item For $x \in X(k)$ the tangent space to $X$ at $x$ will be denoted $T_{x} X$. Moreover, $T_{x}^{\vee} X$ will denote the cotangent space to $X$ at $x$.
\item $T$ denotes a test scheme for a functor.
\item $S$ denotes a polynomial algebra $\kk[\alpha_1, \dots, \alpha_n]$ where $n$ is usually clear from the context.
\item $P$ denotes a polynomial algebra $\kk[x_1, \dots, x_n]$ isomorphic to the one above, but with different names for variables.
\item If $A$ is a $\kk$-algebra, then $S_A$ denotes $S \otimes_{\kk} A = A[\alpha_1, \dots, \alpha_n]$.
\item If $n$ is fixed, we will write $\xx$ or $\aalpha$ for respectively $x_1, \dots, x_n$ or $\alpha_1, \dots, \alpha_n$.
\item $k[\eps]$ denotes the ring $\frac{k[\eps]}{(\eps)^2}$, so whenever something is denoted by $\eps$ we have $\eps^2 = 0$.
\item $\Affine^n$/$\Pn$, $\G_m$ denote respectively the affine/projective space of dimension $n$ and the algebraic torus $\Spec(\kk[t^{\pm 1}])$.
\item If $\G_m$ acts on a vector space $V$, then the induced grading on $V$ is given by $V_n := \{ v \in V : t \in \G_m(k) \textnormal{ acts by multiplying by } t^{-n} \}$.
\item If $B$ is a non-negatively graded $k$-algebra then its \emph{Hilbert function} is a function $\mathbb{N} \to \mathbb{N}$ defined by $n \mapsto \dim_k(B_n)$.
\item For a linear space $W$ let $\Sym_2(W)$ be a subspace of $W \otimes W$ consisting of symmetric tensors. The canonical symmetric bilinear $k$-vector space homomorphism from $W \times W$ to $\Sym_2(W)$ is defined by the formula:
\begin{equation}
(w, v) \mapsto \frac{1}{2}(w \otimes v + v \otimes w).
\end{equation}
This yields a natural duality between $\Sym_2(W)$ and $\Sym_2(W^{\vee})$ induced by the natural duality between $W \otimes W$ and $W^{\vee} \otimes W^{\vee}$. More precisely we define a map:
\begin{equation}~\label{eq:dualityw}
\cdot : \Sym_2(W^{\vee}) \times \Sym_2(W) \to k
\end{equation}
by the formula $(\eta \mu) \cdot (w v) = (\frac{1}{2}(\eta \otimes \mu + \mu \otimes \eta)) \cdot (\frac{1}{2}(w \otimes v + v \otimes w)) := \frac{1}{4}(\eta(w) \mu(v) + \eta(v) \mu(w) + \mu(w) \eta(v) + \mu(v) \eta(w)) = \frac{1}{2}(\eta(w) \mu(v) + \eta(v) \mu(w))$.
This gives an identification of $\Sym_2(W)^{\vee}$ with $\Sym_2(W^{\vee})$.
\item $\delta_{ij}$ denotes the Kronecker delta.
\end{itemize}
\subsection{Hilbert schemes}
%
We recall the definition of the Hilbert functor of points on a scheme.
\begin{definition}\label{def:Hilbert:first}
Let $X$ be a scheme over $\kk$. Then the functor $\mathrm{Hilb}^ d_{X/\kk} : (\mathit{Sch}/\kk)^{op} \longrightarrow \mathbf{Set}$ given by:
\begin{equation*}
\mathrm{Hilb}^ d_{X/\kk}(T) = \left\{ \begin{matrix} Z \subset X_ T\text{ closed subscheme such that }
\\ Z \to T\text{ is finite locally free of degree }d
\end{matrix} \right\}
\end{equation*}
is called a Hilbert functor of $d$ points on $X$. For the definition of finite locally free of degree $d$ see~\cite[Tag~02K9]{stacks_project}. Alternatively one could write finite of degree $d$ and flat.
\end{definition}
\begin{example}\label{ex:functorH}
If $X = \Affine^n$ then for a $\kk$-algebra $A$ we have:
\begin{equation*}
\mathrm{Hilb}^ d_{\Affine^n/\kk}(\Spec(A)) = \left\{ \begin{matrix} I \subset S_A \text{ ideal such that }
\\ S_A/I \text{ is a locally free $A$-module of rank }d
\end{matrix} \right\}
\end{equation*}
In this section we fix $n$ and $d$ and write ${\mathcal H}$ for both this functor and the scheme representing it. If we want to emphasize the number of points $d$ we write ${\mathcal H}^d$. Also, we will denote by $[I]$ an $A$-point corresponding to an ideal $I \subset S_A$ such that $S_A/I$ is locally free of rank $d$.
\end{example}
In the whole paper we will be only concerned with the Hilbert schemes of points on affine spaces. To study local properties of these functors/schemes the following description of the tangent space will be crucial:
\begin{theorem}\label{thm:tangenttohilb}
Fix a $k$-point $[I] \in {\mathcal H}$. Then the tangent space $T_{[I]} {\mathcal H}$ is isomorphic to the $k$-vector space $\Hom_S(I, S/I)$.
\end{theorem}
The full proof of this theorem can be found in~\cite[Corollary 6.4.10]{FGI+05}. We will use this identification a few times, so we recall how to obtain a homomorphism in $\Hom_S(I, S/I)$ from an element of $T_{[I]} {\mathcal H}$.
First, using the fact that $T_{[I]} {\mathcal H} = \Mor \left( \Spec(k[\eps]), {\mathcal H} \ | \ \{\textnormal{pt.}\} \mapsto [I] \right) $ we see that an element of $T_{[I]} {\mathcal H}$ is given by a diagram:
\[\begin{tikzcd}
0 & {I'} & {S[\eps]} & {\frac{S[\eps]}{I'}} & 0 \\
0 & I & {S} & {\frac{S}{I}} & 0
\arrow[from=1-2, to=1-3]
\arrow[from=1-3, to=1-4]
\arrow[from=1-4, to=1-5]
\arrow[from=1-1, to=1-2]
\arrow[from=2-1, to=2-2]
\arrow[from=2-2, to=2-3]
\arrow[from=2-3, to=2-4]
\arrow[from=2-4, to=2-5]
\arrow[from=1-4, to=2-4]
\arrow[from=1-3, to=2-3]
\arrow[from=1-2, to=2-2]
\end{tikzcd}\]
where we use a convention that $S[\eps] := S_{k[\eps]}$. The corresponding element of $\Hom_S(I, S/I)$ is given by $\delta$ on the diagram below:
\[\begin{tikzcd}
&&& 0 \\
&&& {\frac{S}{I}} \\
0 & {I'} & {S[\eps]} & {\frac{S[\eps]}{I'}} & 0 \\
0 & I & {S} & {\frac{S}{I}} & 0 \\
&&& 0
\arrow[from=3-2, to=3-3]
\arrow["\beta", from=3-3, to=3-4]
\arrow[from=3-4, to=3-5]
\arrow[from=3-1, to=3-2]
\arrow[from=4-1, to=4-2]
\arrow["\alpha", from=4-2, to=4-3]
\arrow[from=4-3, to=4-4]
\arrow[from=4-4, to=4-5]
\arrow["\gamma"', from=3-4, to=4-4]
\arrow[from=3-3, to=4-3]
\arrow[from=3-2, to=4-2]
\arrow[from=2-4, to=3-4]
\arrow[from=1-4, to=2-4]
\arrow[from=4-4, to=5-4]
\arrow["s", bend left, from=4-3, to=3-3]
\arrow["\delta", bend left, crossing over, dashed, from=4-2, to=2-4]
\end{tikzcd}\]
Here $s$ is just a natural embedding of $S$ into $S[\eps]$. The homomorphism $\beta \circ s \circ \alpha$ composed with $\gamma$ is $0$ (by diagram chasing) thus we get the existence of $\delta$.
\begin{remark}\label{rem:tangentinverse}
On the other hand, if we are given $\delta \in \Hom_S(I, S/I)$, then the corresponding ideal $I' \subset S[\eps]$ is given by:
\begin{equation}
I' = ( f - \eps \cdot \delta(f) : f \in I ).
\end{equation}
A precise explanation can be found in~\cite{FGI+05}.
\end{remark}
When dealing with locally free/flat modules over local rings as $k[\eps]$ the following algebraic lemma is useful:
\begin{lemma}\label{fact:basislift}
Suppose that $(B, \mm)$ is a local $k$-algebra with residue field $k$ and $C$ is a free $B$-module of rank $d$. Assume that $(c_1, \dots, c_d)$ is a tuple of elements of $C$ such that the tuple $(c_1/\mm, \dots, c_d/\mm)$ is a basis of $C/\mm C$. Then $(c_1, \dots, c_d)$ is a basis of $C$.
\end{lemma}
\begin{proof}
\begin{comment}
Define a $B$-module homomorphism $B^{\oplus d} \to C$ sending the $i$'th basis element to $c_i$. Let $L$ be the cokernel of this map. Tensor with $k = B/\mm$ is right exact and $(c_1/\mm, \dots, c_d/\mm)$ is a basis of $C/\mm C$ so we get that $L/\mm L = 0$. Hence $L = \mm L$ and by Nakayama's lemma $L=0$. Let $K$ be the kernel of $B^{\oplus d} \to C$. Consider the diagram:
\[\begin{tikzcd}
0 & K & {B^{\oplus d}} & C & 0 \\
{0} & {K/\mm K} & {(B/\mm)^{\oplus d}} & {C/\mm C} & 0
\arrow[from=1-1, to=1-2]
\arrow[from=1-2, to=1-3]
\arrow[from=1-3, to=1-4]
\arrow[from=1-4, to=1-5]
\arrow[from=2-1, to=2-2]
\arrow[from=2-2, to=2-3]
\arrow[from=2-3, to=2-4]
\arrow[from=2-4, to=2-5]
\arrow[from=1-2, to=2-2]
\arrow[from=1-3, to=2-3]
\arrow[from=1-4, to=2-4]
\end{tikzcd}\]
The bottom line is left exact, as the upper short exact sequence splits, because $C$ is a free $B$-module. By assumption $(B/\mm)^{\oplus d} \to C/\mm C$ is an isomorphism, because of the construction of $B^{\oplus d} \to C$. Hence, $K/ \mm K = 0$, so $K = \mm K$ and again by Nakayama's lemma we get that $K=0$. Thus the map $B^{\oplus d} \to C$ is an isomorphism, which means that $(c_1, \dots, c_d)$ is indeed a basis of $C$.
\end{comment}
Omitted.
\end{proof}
\subsection{Groups acting on ${\mathcal H}$}
Generally, if an algebraic commutative group $G$ acts on a scheme $X$, then there is an induced action of $G$ on $\mathrm{Hilb}^ d_{X/\kk}$. Indeed, take $T$ - a test scheme. Then the $G$-action $G \times X \to X$ yields (by the base change to $T$) a morphism $G_T \times X_T \to X_T$. If we take $t \in G(T) = \Mor(T, G) = \Mor_T(T, G_T)$, then we get the diagram:
\[\begin{tikzcd}
{G_T \times X_T} & {X_T} \\
{X_T}
\arrow[from=1-1, to=1-2]
\arrow["{(t, id)}", from=2-1, to=1-1]
\arrow["{t \cdot}"', from=2-1, to=1-2]
\end{tikzcd}\]
where the vertical map $X_T \to G_T \times X_T$ is the identity on the second coordinate and factors through the $T$-point $t$ on the first one. Now if $Z \subset X_T$ represents a $T$-point of $\mathrm{Hilb}^ d_{X/\kk}$, then the action of $t$ on $Z$ is defined by the image of map $(t\cdot) : X_T \to X_T$. This map is an isomorphism (because $G$ is an algebraic group), so $t \cdot Z := (t \cdot)(Z) \subset X_T \to T$ is again locally finite of degree $d$.
Below we provide two examples of lifting actions from $\Affine^n$ to ${\mathcal H}$.
\begin{example}\label{ex:torusaction}
Consider the $\G_m$-action on $\Affine^n$ given by the standard grading $\deg(\alpha_i)=1$. Let us describe the induced morphism $\G_m \times {\mathcal H} \to {\mathcal H}$ constructed above on its $T$-points for $T = \Spec(A)$ where $A$ is a $k$-algebra. We are interested in the map:
\begin{equation*}
\G_m(\Spec(A)) \times {\mathcal H}(\Spec(A)) \to {\mathcal H}(\Spec(A))
\end{equation*}
Take $a \in \G_m(\Spec(A)) = A^*$ (invertible elements). As above, $a$ yields an isomorphism:
\[\begin{tikzcd}
{\Affine_A^n} & {\Affine_A^n}
\arrow["{a \cdot}"', from=1-1, to=1-2]
\end{tikzcd}\]
which on the level of algebras comes from the $A$-algebra homomorphism $(a \cdot)^{\#}:S_A = A[\aalpha] \to A[\aalpha]$ sending $\alpha_i$ to $a \alpha_i$. Now, a point $[Z] \in {\mathcal H}(\Spec(A))$ is represented by an ideal $I \subset S_A$ and the new point is given by the ideal $a \cdot I := \ker(q \circ (a \cdot)^{\#})$ where $q: S_A \to \frac{S_A}{I}$ is the quotient map. The justification of this formula follows from the diagrams below (the left hand side is the definition of the action and the right hand side is the corresponding diagram on the level of algebras):
\[\begin{tikzcd}
{\Affine_A^n} & {\Affine_A^n} & {S_A} & {S_A} \\
Z & {a \cdot Z} & {\frac{S_A}{I}} & {\frac{S_A}{a \cdot I}}
\arrow["{a \cdot}"', from=1-1, to=1-2]
\arrow[hook', from=2-1, to=1-1]
\arrow[hook', from=2-2, to=1-2]
\arrow[from=2-1, to=2-2]
\arrow["{(a \cdot)^{\#}}", from=1-4, to=1-3]
\arrow["q", two heads, from=1-3, to=2-3]
\arrow[two heads, from=1-4, to=2-4]
\arrow[from=2-4, to=2-3]
\end{tikzcd}\]
\end{example}
Using the above description of $\G_m$-action on ${\mathcal H}$ one can prove the following result:
\begin{proposition}\label{prop:torusequivariance}
Suppose that $A$ is a ${\mathbb Z}$-graded $k$-algebra and consider a morphism $\phi : \Spec(A) \to {\mathcal H}$ given by a subscheme $Z \subset \Affine_A^n$ cut be the ideal $I \subset S_A$. Consider a ${\mathbb Z}$-grading on $S_A = S \otimes_k A$ given by $(S_A)_l = \bigoplus_{i+j = l} S_i \otimes_k A_j$. If $I$ is homogeneous with respect to this grading, then $\phi : \Spec(A) \to {\mathcal H}$ is $\G_m$-equivariant.
\end{proposition}
\begin{proof}
\begin{comment}
We need to check that the following diagram commutes:
\[\begin{tikzcd}
{\Spec(A)} & {\mathcal H} \\
{\G_m \times \Spec(A)} & {\G_m \times {\mathcal H}}
\arrow["\mu"', from=2-1, to=1-1]
\arrow["\phi"', from=1-1, to=1-2]
\arrow["\nu", from=2-2, to=1-2]
\arrow["{id \times \phi}", from=2-1, to=2-2]
\end{tikzcd}\]
Let $\mu' = \phi \circ \mu$ and $\nu' = \nu \circ (id \times \phi)$. These both are morphisms from $\G_m \times \Spec(A) = \Spec(A[t^{\pm 1}])$ to ${\mathcal H}$ so they are given by some ideals in $S_{A[t^{\pm 1}]}$. It is enough to check that these ideals are the same. Assume for simplicity that $I$ is generated by a single homogeneous $f = \sum_{N} a_{N} \aalpha^{N} \in S_A$ so that $\deg(a_{N}) + |{N}| = l$ for all $N$ (we use the multi-index notation) and some $l \in {\mathbb Z}$. The proof is the same without this assumption, however, one needs to have in mind one more index.
We start with the calculation of the ideal corresponding to the map $\mu'$. Since $\mu' = \phi \circ \mu$ the subscheme $Z'$ such that $\mu'$ is given by $[Z'] \in {\mathcal H}(\G_m \times \Spec(A))$ is a pullback of $Z \subset \Affine_A^n$ by the map $(id \times \mu) : \Affine^n \times (\G_m \times \Spec(A)) \to \Affine^n \times \Spec(A)$. By writing $\G_m \times \Spec(A) = \Spec(A[t^{\pm 1}])$ and looking at algebras we get:
\[\begin{tikzcd}
{\Affine_{A[t^{\pm 1}]}^n} & {\Affine_A^n} & {S_{A[t^{\pm 1}]}} & {S_A} \\
{Z'} & Z & {\frac{S_{A[t^{\pm 1}]}}{I'}} & {\frac{S_A}{I}}
\arrow[from=1-1, to=1-2]
\arrow[hook', from=2-1, to=1-1]
\arrow[hook', from=2-2, to=1-2]
\arrow[from=2-1, to=2-2]
\arrow[from=1-4, to=1-3]
\arrow[two heads, from=1-3, to=2-3]
\arrow[two heads, from=1-4, to=2-4]
\arrow[from=2-4, to=2-3]
\arrow["\lrcorner"{anchor=center, pos=0.125, rotate=90}, draw=none, from=2-1, to=1-2]
\arrow["\lrcorner"{anchor=center, pos=0.125, rotate=90}, draw=none, from=2-3, to=1-4]
\end{tikzcd}\]
Since $Z'$ is a pullback we have $\frac{S_{A[t^{\pm 1}]}}{I'} = S_{A[t^{\pm 1}]} \otimes_{S_A} \frac{S_A}{I}$. By right exactness of the tensor product the ideal $I'$ is generated by the image of the map \[(id \otimes i):S_{A[t^{\pm 1}]} \otimes_{S_A} I \to S_{A[t^{\pm 1}]} \otimes_{S_A} S_A = S_{A[t^{\pm 1}]}\]
where $i:I \to S_A$ is the inclusion. We calculate the image of $1 \otimes f$ via $(id \otimes i)$:
\begin{equation}
1 \otimes f = 1 \otimes (\sum_{N} a_{N} \aalpha^{N}) = \sum_{N} \aalpha^N \otimes a_{N} = \sum_{N} \aalpha^N \cdot a_{N} \cdot t^{\deg(a_{N})} \otimes 1 \mapsto \sum_{N} \aalpha^N \cdot a_{N} \cdot t^{\deg(a_{N})} \in S_{A[t^{\pm 1}]}
\end{equation}
Thus $I' = (f^*)$ for $f^* = \sum_{N} a_{N} t^{\deg(a_N)} \cdot \aalpha^{N}$.
Now we calculate the ideal corresponding to the map $\nu'$. Again, by the fact that $\G_m \times \Spec(A) = \Spec(A[t^{\pm 1}])$ we can see $id \times \phi$ from the diagram at the beginning of this proof as an element of $\G_m (\Spec(A[t^{\pm 1}])) \times {\mathcal H}(\Spec(A[t^{\pm 1}]))$. By the definition of $id \times \phi$ this element is the pair $(t, [Z''])$, where $Z''$ is a pullback of $Z$ as on the diagram:
\[\begin{tikzcd}
{\Affine_{A[t^{\pm 1}]}^n} & {\Affine_A^n} \\
{Z''} & Z
\arrow[hook', from=2-2, to=1-2]
\arrow["\pi"', from=1-1, to=1-2]
\arrow[hook', from=2-1, to=1-1]
\arrow[from=2-1, to=2-2]
\arrow["\lrcorner"{anchor=center, pos=0.125, rotate=90}, draw=none, from=2-1, to=1-2]
\end{tikzcd}\]
Here $\pi$ comes from the projection $\G_m \times \Spec(A) \to \Spec(A)$. Hence the ideal of $Z''$ in $S_{A[t^{\pm 1}]}$ is generated by $I$ so we will also denote it by $I$.
By Example~\ref{ex:torusaction} the image of the pair $(t, [Z''])$ by $\nu$ is given by the subscheme $t \cdot Z'' \subset \Affine_{A[t^{\pm 1}]}^n$. The ideal $I''$ of $t \cdot Z''$ is given by the kernel of the composition:
\[\begin{tikzcd}
{S_{A[t^{\pm 1}]}} & {S_{A[t^{\pm 1}]}} & {\frac{S_{A[t^{\pm 1}]}}{I}}
\arrow["{(t \cdot)^{\#}}", from=1-1, to=1-2]
\arrow[from=1-2, to=1-3]
\end{tikzcd}\]
where $(t \cdot)^{\#} : S_{A[t^{\pm 1}]} \to S_{A[t^{\pm 1}]}$ is a $A[t^{\pm 1}]$-algebras homomorphism sending $\alpha_i$ to $t\alpha_i$. Thus $I''$ is generated by $f(t^{-1} \cdot \aalpha) = \sum_{N} a_{N} \aalpha^{N} \cdot t^{-|N|}$.
The element $t \in A[t^{\pm 1}]$ is invertible so we have:
\begin{equation}
f^* = \sum_{N} a_{N} t^{\deg(a_N)} \cdot \aalpha^{N} = t^l \cdot \sum_{N} a_{N} t^{\deg(a_N) - l} \cdot \aalpha^{N} = t^l \cdot \sum_{N} a_{N} t^{-|N|} \cdot \aalpha^{N} = t^l \cdot f(t^{-1} \cdot \aalpha).
\end{equation}
Hence, $I' = (f^*) = (f(t^{-1} \cdot \aalpha)) = I''$ and we are done.
\end{comment}
Left to the reader.
\end{proof}
\begin{example}\label{ex:additiveaction}
Now we analyse the induced $\Affine^n = \G_a^n$-action on ${\mathcal H}$ coming from the action on $\Affine^n$ by translations. Again, for a $k$-algebra $A$ we get a map:
\begin{equation*}
\Affine^n (\Spec(A)) \times {\mathcal H}(\Spec(A)) \to {\mathcal H}(\Spec(A))
\end{equation*}
but now it will be defined by $((a_1, \dots, a_n), I) \mapsto \ker(q \circ (\Bar{a}+)^{\#})$ where $(\Bar{a}+)^{\#}:S_A \to S_A$ is a $A$-homomorphism sending $\alpha_i$ to $\alpha_i + a_i$.
\end{example}
From Example~\ref{ex:torusaction} we can see that a $k$-point $[I] \in {\mathcal H}$ is $\G_m$-invariant if and only if $I$ is homogeneous with respect to the standard grading on $S$. Thus, if $I$ is homogeneous, then $\G_m$ acts on its tangent space $T_{[I]} {\mathcal H}$. We describe this action in the theorem below:
\begin{theorem}\label{thm:gradingonH}
Let $[I]$ be a $\G_m$-invariant $k$-point of ${\mathcal H}$ (so $I \subset S$ is homogeneous). Then the $\G_m$-action on the tangent space $T_{[I]} {\mathcal H} = \Hom_S(I, S/I)$ induces a grading $\Hom_S(I, S/I) = \bigoplus_{k \in {\mathbb Z}} \Hom_S(I, S/I)_k$, where:
\begin{equation}
\Hom_S(I, S/I)_k = \{ \delta \in \Hom_S(I, S/I) : \delta(I_n) \subset (S/I)_{n+k} \textnormal{ for all integers $n$} \}.
\end{equation}
\end{theorem}
\begin{proof}
\begin{comment}
Take a point $a \in \G_m(k) = k^*$. The $\G_m$-action on ${\mathcal H}$ yields a morphism:
\[\begin{tikzcd}
{\mathcal H} & {\mathcal H}
\arrow["{a \cdot}"', from=1-1, to=1-2]
\end{tikzcd}\]
Now, if $\delta \in T_{[I]} {\mathcal H} = \Mor \left( \Spec(k[\eps]), {\mathcal H} | \{pt.\} \mapsto [I] \right)$, then the morphism $a \cdot \delta$ is given by:
\[\begin{tikzcd}
{\mathcal H} & {\mathcal H} \\
{\Spec(k[\eps])}
\arrow["{a \cdot}"', from=1-1, to=1-2]
\arrow["\delta"', from=2-1, to=1-1]
\arrow["{a \cdot \delta}"', from=2-1, to=1-2]
\end{tikzcd}\]
Take an ideal $I' \subset S[\eps]$ yielding $\delta \in \Hom_S(I, S/I)$ (as in the description after Theorem~\ref{thm:tangenttohilb}). By Example~\ref{ex:torusaction} the ideal $a \cdot I'$ corresponding to $\delta'$ is given by the formula $a \cdot I' := \ker(S[\eps] \to S[\eps] \to \frac{S[\eps]}{I'})$ where the first map is over $k[\eps]$ and takes $\alpha_i$ to $a \alpha_i$. We check what is the formula defining a morphism $\delta' \in \Hom_S(I, S/I)$ constructed from $a \cdot I'$ as in the discussion after Theorem~\ref{thm:tangenttohilb}. Consider the following diagram:
\[\begin{tikzcd}
& {a \cdot I'} && 0 \\
&& {k[\eps][\Bar{\alpha}]} & {\frac{k[\Bar{\alpha}]}{I}} \\
0 & {I'} & {k[\eps][\Bar{\alpha}]} & {\frac{k[\eps][\Bar{\alpha}]}{I'} \simeq \frac{k[\eps][\Bar{\alpha}]}{a \cdot I'}} & 0 \\
0 & I & {k[\Bar{\alpha}]} & {\frac{k[\Bar{\alpha}]}{I}} & 0 \\
&&& 0
\arrow[from=3-2, to=3-3]
\arrow["\beta", from=3-3, to=3-4]
\arrow[from=3-4, to=3-5]
\arrow[from=3-1, to=3-2]
\arrow[from=4-1, to=4-2]
\arrow["\alpha", from=4-2, to=4-3]
\arrow[from=4-3, to=4-4]
\arrow[from=4-4, to=4-5]
\arrow["\gamma"', from=3-4, to=4-4]
\arrow[from=3-3, to=4-3]
\arrow[from=3-2, to=4-2]
\arrow[from=2-4, to=3-4]
\arrow[from=1-4, to=2-4]
\arrow[from=4-4, to=5-4]
\arrow["{\alpha_i \mapsto a \alpha_i}", from=2-3, to=3-3]
\arrow["\beta'", from=2-3, to=3-4]
\arrow[from=1-2, to=2-3]
\arrow["s"{pos=0.7}, bend left=60, crossing over, from=4-3, to=2-3]
\end{tikzcd}\]
Fix a homogeneous $f \in I$. We will calculate $\delta'(f)$. First, $\alpha(f) = f \in k[\Bar{\alpha}]$. Next apply $s$ to $f \in k[\eps][\Bar{\alpha}]$. In order to get to $\frac{k[\eps][\Bar{\alpha}]}{I'} \simeq \frac{k[\eps][\Bar{\alpha}]}{a \cdot I'}$ we apply the map $\alpha_i \mapsto a \alpha_i$ and get $a^{ \deg f} \cdot f(\Bar{\alpha}) \in k[\eps][\Bar{\alpha}]$. Next we take a quotient by $I'$ and we get $a^{\deg f} \cdot f / I'$ which, by the definition of $\delta$ is $a^{\deg f} \cdot \eps \cdot \delta(f)(\Bar{\alpha})$. The last step is to apply the isomorphism $\frac{k[\eps][\Bar{\alpha}]}{I'} \simeq \frac{k[\eps][\Bar{\alpha}]}{a \cdot I'}$. It takes $\alpha_i$ to $a^{-1} \alpha_i$, so its image of $a^{\deg f} \cdot \eps \cdot \delta(f)(\Bar{\alpha})$ is $a^{\deg f - \deg (\delta(f))} \cdot \eps \cdot \delta(f)(\Bar{\alpha})$. In the end we get a formula:
\begin{equation}
\delta' (f) = (a \cdot \delta)(f) = a^{\deg f - \deg (\delta(f))} \cdot \delta (f).
\end{equation}
Hence $\delta \in \Hom_S(I, S/I)_k$ if and only if $a$ acts on $\delta$ by multiplying by $a^{-k}$ if and only if $\delta(I_n) \subset (S/I)_{n+k}$ for all integers $n$. This finishes the proof.
\end{comment}
Omitted.
\end{proof}
There is no analog of this theorem for the $\Affine^n$ additive action on ${\mathcal H}$ (as in Example~\ref{ex:additiveaction}), because there is no $\Affine^n$-invariant point in ${\mathcal H}$. However, one can still ask what happens locally or how does the orbit map (of a point) look like. The following theorem answers the question on the level of tangent spaces:
\begin{theorem}\label{thm:TNTdescribtion}
Let $[I]$ be a $k$-point of ${\mathcal H}$ and let $\Affine^n \to {\mathcal H}$ be its orbit map under the additive action of $\Affine^n$. Then the induced tangent map $\eta : T_{ \{0\} } \Affine^n \to T_{[I]} {\mathcal H}$ sends the $i$'th basis vector $\partial_i$ to $\eta(\partial_i) \in T_{[I]} {\mathcal H} = \Hom_S(I, S/I)$ given by the formula:
\begin{equation}
\eta(\partial_i) (f) = \frac{\partial f}{\partial \alpha_i}.
\end{equation}
Thus, by abuse of notation we will write $\partial_i$ for $\eta(\partial_i)$.
\end{theorem}
\begin{proof}
\begin{comment}
Consider the $i$'th basis vector $\partial_i \in T_{ \{0\} } \Affine^6$. It corresponds to a morphism $\Spec(k[\eps]) \to \Affine^6$ onto $i$-th coordinate and composing with $[I]$'s orbit $\Affine^6 \to {\mathcal H}$ we get $\Spec(k[\eps]) \to {\mathcal H}$ which corresponds to the ideal $I'$. From Example~\ref{ex:additiveaction} (applied to $A = k[\eps]$) it follows that $I'\subset S[\eps]$ is given by the kernel of the composition:
\[\begin{tikzcd}
{S[\eps]} && {S[\eps]} & {\frac{S[\eps]}{I}}
\arrow[from=1-3, to=1-4]
\arrow["{x_j \mapsto x_j + \delta_{ij} \cdot \eps}", from=1-1, to=1-3]
\end{tikzcd}\]
so $I' = \{ h(\alpha_1, \dots, \alpha_6) \in S[\eps] \ | \ h(\dots, \alpha_i + \eps, \dots) \in I \cdot S[\eps] \}$.
To calculate the induced element of $\Hom_{S}(I, S/I)$ we need to look at the diagram:
\[\begin{tikzcd}
&&& 0 \\
&&& {\frac{S}{I}} \\
0 & {I'} & {S[\eps]} & {\frac{S[\eps]}{I'}} & 0 \\
0 & I & S & {\frac{S}{I}} & 0 \\
&&& 0
\arrow[from=3-2, to=3-3]
\arrow["", from=3-3, to=3-4]
\arrow[from=3-4, to=3-5]
\arrow[from=3-1, to=3-2]
\arrow[from=4-1, to=4-2]
\arrow["", from=4-2, to=4-3]
\arrow[from=4-3, to=4-4]
\arrow[from=4-4, to=4-5]
\arrow[""', from=3-4, to=4-4]
\arrow[from=3-3, to=4-3]
\arrow[from=3-2, to=4-2]
\arrow[from=2-4, to=3-4]
\arrow[from=1-4, to=2-4]
\arrow[from=4-4, to=5-4]
\arrow["", bend left, crossing over, from=4-2, to=2-4]
\arrow["", bend left, from=4-3, to=3-3]
\end{tikzcd}\]
For $f(\alpha_1, \dots, \alpha_6) \in I$ we know that $f(\dots, \alpha_i - \eps, \dots) \in I'$. By the Taylor's formula:
\begin{equation}
f(\alpha_1, \dots, \alpha_6) - f(\dots, \alpha_i - \eps, \dots) = \eps \frac{\partial f}{\partial \alpha_i}
\end{equation}
because $\eps^2 = 0$. Thus the image of $\partial_i$ in $\Hom_S(I, S/I)$ is defined by a formula
\begin{equation}
\partial_i(f) = \frac{\partial f}{\partial \alpha_i} \ (mod \ I)
\end{equation}
which finishes the proof.
\end{comment}
We leave it as an exercise for the reader.
\end{proof}
Suppose additionally that $[I]$ is a $\G_m$-invariant point of ${\mathcal H}$. From the description of the grading on $T_{[I]} {\mathcal H}$ in Theorem~\ref{thm:gradingonH} we see that The image of $\eta$ above lies in $\Hom_S(I, S/I)_{-1} \subset \Hom_S(I, S/I)_{< 0}$. Thus it is natural to consider the definition:
\begin{definition}\label{def:TNT}
A $\G_m$-invariant $k$-point $[I] \in {\mathcal H}$ has \emph{trivial negative tangents} (abbreviated TNT), if the map $\eta : \lin_k(\partial_1, \dots, \partial_n) \to \Hom_S(I, S/I)_{< 0}$ from Theorem~\ref{thm:TNTdescribtion} is surjective.
\end{definition}
This just means that the only negative tangents at $[I]$ come from $\Affine^n$ additive action and are given by derivatives.
This definition is taken from~\cite{elemcomp}. Note that because the image of $\eta$ is contained in $\Hom_S(I, S/I)_{-1}$ and $\partial_1, \dots, \partial_n \in \Hom_S(I, S/I)_{-1}$ are linearly independent it follows that $[I]$ as in the definition has TNT if and only if $\dim_k \Hom_S(I, S/I)_{<0} = n$.
\subsection{\BBname{} decomposition}
%
The torus action can be useful to study geometry of Hilbert schemes. To exploit it we will need the theory of \BBname{} decompositions which originates from the papers~\cite{ABB1} and~\cite{ABB2}. Later in the paper, we will work with the \BBname{} decomposition induced by the $\G_m$-action from Example~\ref{ex:torusaction}.
In this subsection we present necessary tools from the paper~\cite{JS19} which generalises \BBname's result. In the paper the authors work with linearly reductive groups, so the class that contains $\G_m$, but we will present the results from there only for the $\G_m$-case.
\begin{definition}\label{def:xplusfunctorial}
Let $X$ be a $\G_m$-scheme. Consider a functor $\Xplus : (\mathit{Sch}/\kk)^{op} \longrightarrow \mathbf{Set}$:
\begin{equation}\label{eq:definitionFunctorial:new}
\Xplus(T) = \left\{ \symbf \colon \mathbb{A}^1 \times T \to \varX\ |\
\symbf \mbox{ is $\Gmult$-equivariant} \right\},
\end{equation}
where $\G_m$ acts on $\Affine^1 \times T$ by the natural action of $\G_m$ on $\Affine^1$ (i.e. by multiplication). It is called the \emph{\BBname \ decomposition} of $X$. Moreover, \BBname{} decomposition is functorial, which means that if $X, Y$ are $\G_m$-schemes and $g : X \to Y$ is $\G_m$-equivariant, then there is $g^+ : X^+ \to Y^+$ such that $g^+(T) : X^+(T) \to Y^+(T)$ is given by $g^+(T)(f : \Affine^1 \times T \to X) = (g \circ f : \Affine^1 \times T \to X) \in Y^+(T)$ for any test scheme $T$.
\end{definition}
From the \BBname \ decomposition of a scheme $X$ we get three natural transformations:
\begin{itemize}
\item $\Xplus(T) \to X(T)$ given by restricting $\symbf$ to $1 \times T$. This is called \emph{forgetting about the limit} and will be denoted by $i_X$.
\item $\Xplus(T) \to X^{\G_m}(T)$ given by restricting $\symbf$ to $0 \times T$. This is called \emph{restricting to the limit} and will be denoted by $\pi_X$.
\item $X^{\G_m}(T) \to \Xplus(T)$ given by $(\symbf : T \to X^{\G_m}) \mapsto ((j \circ \symbf \circ pr_1): T \times \Affine^1 \to X)$, where $j : X^{\G_m} \to X$ is the closed embedding. This morphism is a section of $\pi_X$ and will be denoted by $s_X$.
\end{itemize}
Under this notation the following holds:
\begin{theorem}[Existence of \BBname{} decompositions~{\cite[Theorem~1.1]{JS19}} ]\label{ref:introRepresentability:thm}
Let $\varX$ be a $\G_m$-scheme locally of finite type over
$\kk$. Then the functor $\Xplus$ defined
by~\eqref{eq:definitionFunctorial:new} is represented by a scheme locally
of finite type over $\kk$, denoted also by $\Xplus$.
Moreover, the scheme $\Xplus$ has a natural $\Affine^1$-action. The map
$\iinftyX\colon \Xplus\to \varX^{\G_m}$ is \emph{affine} of finite type and
equivariant.
\end{theorem}
Moreover, if $X$ is separated, then $X^+$ can be seen as a subset of $X$, because we have the following fact:
\begin{fact}~\cite[lemma 5.8]{JS19}\label{fact:mono}
For separated $X$ the morphism $i_X : X^+ \to X$ is a monomorphism of functors.
\end{fact}
Hence, for any scheme $T$ we will treat $X^+(T)$ as a subset of $X(T)$, as long as $X$ is separated. In particular, for a $k$-point $x \in X^{\G_m}$, identifying $x$ with $s(x) \in X^+$ and with $x \in X$ we can treat the tangent space to $X^+$ at $x$ as a subset of $T_{x} X$: \[T_{x} X^+ = X^+( \Spec(k[\eps]) )_{\{\textnormal{pt.}\} \mapsto x} \subset X( \Spec(k[\eps]) )_{\{\textnormal{pt.}\} \mapsto x} = T_{x} X.\]
This fact also justifies why $X^+$ is called a decomposition of $X$. Geometrically speaking, $k$-points of $X^+$ are these points $p$ of $X$ for which the $\G_m$-orbit of $p$ can be extended to a map from $\Affine^1$ to $X$. The image of $0$ of this extended map is $\G_m$-invariant and it is equal $\pi_X(p)$. One can think of $\pi_X(p)$ as the limit $\lim_{t \to 0} t \cdot p$ where dot represents the action of $\G_m$ on $X$. For a proper scheme such limits always exist. A good example to bear in mind is the \BBname \ decomposition of $\G_m$-action on $\Pone$ described below:
\begin{example}
Consider the case when $X = \Pone$ and $\G_m$ fixes $\infty$ and acts on $\Pone \setminus \{ \infty \} = \Affine^1$ by multiplication. Then $\Xplus = \Affine^1 \sqcup \{ \infty \}$ and $X^{\G_m} = \{ 0, \infty \}$. The morphism $\pi_X$ takes $\Affine^1$ to $0$ and $\infty$ to $\infty$. On the other hand, $i_X$ embeds $\Affine^1$ onto $\Pone \setminus\{ \infty \}$ and sends $\infty$ to $\infty \in \Pone$.
In fact for any smooth $X$ the map $\pi_X$ is locally an affine bundle and this was proved by \BBname{} in~\cite{ABB1}, \cite{ABB2}.
\end{example}
As can be seen in the above example, near $0$ the morphism $i_X$ is even an isomorphism. Thus one can study the local geometry of $\Pone$ near $0$ by looking at $(\Pone)^+$. This is not a coincidence as seen in the following proposition which will be crucial for our purposes:
\begin{proposition}\label{ref:comparison:intro:prop}~\cite[Proposition 1.6]{JS19}
Let $\varX$ be separated and locally of finite type.
Let $x\in \varX^{\G_m}$ be a fixed $\kk$-point such that the
cotangent space $T_{x}^{\vee} \varX$ has no negative degrees (with respect to the grading induced by the $\G_m$-action on $\varX$). Then the map $\ioneX\colon \Xplus \to \varX$ is an open embedding near $x' = \isectionX(x)\in \Xplus$.
More precisely, there exists an affine open
$\Affine^1$-stable neighbourhood
$U$ of $x'$ such that $(\ioneX)_{|U}\colon U\to \varX$ is an open
embedding. In particular, $x\in \varX$ has an affine open $\G_m$-stable
neighbourhood.
Conversely, if a fixed point $x$ has an affine open $\Affine^1$-stable neighbourhood
in $\varX$, then $T_{x}^{\vee} \varX$ has no negative degrees.
\end{proposition}
\begin{remark}
The condition of $T_{x}^{\vee} \varX$ having no negative degrees is equivalent to $T_{x} \varX$ having no positive degrees, because the $\G_m$-grading on the dual space is reversed.
\end{remark}
It is worth to mention here, that the idea of using \BBname \ decomposition in order to study Hilbert schemes is not new. However, before the tools only applied to smooth varieties, so these applications were concerning mostly Hilbert schemes of points on smooth surfaces, which by~\cite{Fogarty} are smooth. An example of such an application is presented in the paper~\cite{ES87} by Ellingsrud and Strømme where they calculate the homology groups of the complex manifolds corresponding to Hilbert schemes of points on the projective plane over complex numbers.
%
\subsection{Obstruction theories}
%
We will need obstruction theories so we briefly introduce them here. We follow presentation from~\cite{JS21}. For a more detailed explanation see~\cite{DefHar}.
We will start with some intuition. Let $x \in X$ be a $\kk$-point in a scheme $X$ locally of finite type over $\kk$. Then $x$ is an image of a morphism $\Spec(\kk) \to X$ and one could ask how does this morphism extends to spectras of Artinian rings other then $\kk$ itself. It is natural then to consider the functor $D_{X, x} : \textit{Art}_{\kk} \to \mathbf{Set}$ from the category $\textit{Art}_{\kk}$ of local Artinian $\kk$-algebras with the residue field $\kk$ given by:
\begin{equation}\label{ref:def:eq}
D_{X,x}(R) = \{ \symba: \Spec(R) \to X \textnormal{ | $\symba(\mm_R)=x$}\}.
\end{equation}
It is called the \emph{deformation functor} of $X$ at $x$ and it is pro-representable by the completion of the local ring $\hat{{\mathcal O}}_{\varX, x}$ (i.e. $D_{X,x}(R) \simeq \Hom(\hat{{\mathcal O}}_{\varX, x},R)$).
Lets reformulate the question about extensions of $\Spec(\kk) \to X$ in a purely algebraic way: if $(A, \mm)$ is a Noetherian complete local $\kk$-algebra (e.g. $\hat{{\mathcal O}}_{\varX, x}$), when can we lift a morphism $\symba_0 : A \to B_0$ to a morphism $\symba: A \to B$ for a local surjection of Artinian rings $B \to B_0$ as on the diagram below?
\[\begin{tikzcd}
B & {B_0} \\
A
\arrow[from=1-1, to=1-2]
\arrow["{\symba_0}"', from=2-1, to=1-2]
\arrow["\symba", dashed, from=2-1, to=1-1]
\end{tikzcd}\]
For technical reasons we assume that $\mm_B \cdot \ker(B \to B_0) =0$ - this is called a \emph{small extension}. Every local surjection of Artinian rings is a composition of small extensions so by restricting to such morphisms we don't lose any information.
Let $D=\Hom(A, -)$. By the Cohen structure theorem~\cite{Cohen} $A$ can be presented as a quotient of $\hat{\symbp} := \kk [\![\symbv_1, \ldots , \symbv_d]\!]$ for $d = \dim(A)$ of the form $A = \hat{\symbp} / \symbJo$ where $\symbJo \subset \mm_{\hat{\symbp}}^2$. We can always lift the morphism $\kk [\![\symbv_1, \ldots , \symbv_d]\!] \to B_0$ to $B$ (not necessarily uniquely), so we obtain a diagram:
\[\begin{tikzcd}
0 & \symbJo & {\hat{\symbp}} & A & 0 \\
0 & K & B & {B_0} & 0
\arrow[from=2-3, to=2-4]
\arrow[from=2-2, to=2-3]
\arrow[from=2-1, to=2-2]
\arrow[from=2-4, to=2-5]
\arrow[from=1-3, to=1-4]
\arrow[from=1-4, to=1-5]
\arrow[from=1-2, to=1-3]
\arrow[from=1-1, to=1-2]
\arrow["{\symbb|_{\symbJo}}", from=1-2, to=2-2]
\arrow["\symbb", from=1-3, to=2-3]
\arrow["{\symba_0}", from=1-4, to=2-4]
\end{tikzcd}\]
The restriction $\symbb |_{\symbJo}$ doesn't depend on the choice of $\symbb$ and $\mm_B \cdot K = 0$, so we get a homomorphism $\symbJo/\mm \symbJo \to K$. In other words we get an element of $(\symbJo/\mm \symbJo)^{\vee} \otimes_{\kk} K$ which vanishes if and only if the extension $A \to B$ exists. If we denote $Ob = (\symbJo/\mm \symbJo)^{\vee}$ and $D(C) = \Hom(A,C)$ then we get a functorial (under morphisms of small extensions) sequence:
\begin{equation}\label{ref:smallext:eq}
D(B) \to D(B_0) \to Ob \otimes_{\kk} K
\end{equation}
It is \emph{exact} which in this case means that $\symba_0 \in D(B_0)$ goes to $0$ if and only if there exists $\symba \in D(B)$ which restricts to $\symba_0$. In general obstruction theories provide such exact sequences for every small extension.
\begin{definition}
Let $D$ be a functor $\textit{Art}_{\kk} \to \mathbf{Set}$ such that $D(\kk) = \{ \textnormal{pt.} \}$ and let $T_D := D(\kk[\varepsilon])$. An obstruction theory for $D$ is a finitely dimensional vector space $Ob_D$ over $\kk$ such that for all small extensions $0 \to K \to B \to B_0 \to 0$ there is given a map $ob: D(B_0) \to Ob_D \otimes_{\kk} K$ with the following properties:
\begin{enumerate}
\item for all $\symba_0 \in D(B_0)$, $ob(\symba_0)=0$ if and only if $\symba_0$ extends to $\symba \in D(B)$,
\item if $\symba_0 \in D(B_0)$, then the set of extensions of $\symba_0$ to $\symba \in D(B)$ is affine over $T_D \otimes_{\kk} K$ (we will not use this condition so we do not explain it).
\end{enumerate}
We also require that $ob$ is functorial with respect to morphisms of small extensions which means that if we have a commutative diagram where horizontal lines are small extensions:
\[\begin{tikzcd}
0 & K & B & {B_0} & 0 \\
0 & {K'} & {B'} & {B_0'} & 0
\arrow[from=1-1, to=1-2]
\arrow[from=1-2, to=1-3]
\arrow[from=1-3, to=1-4]
\arrow[from=1-4, to=1-5]
\arrow[from=2-1, to=2-2]
\arrow[from=2-2, to=2-3]
\arrow[from=2-3, to=2-4]
\arrow[from=2-4, to=2-5]
\arrow["p|_K", from=1-2, to=2-2]
\arrow["p", from=1-3, to=2-3]
\arrow["p_0", from=1-4, to=2-4]
\end{tikzcd}\]
then we get the following commutative diagram:
\[\begin{tikzcd}
{D(B)} & {D(B_0)} & {Ob \otimes_{k} K} \\
{D(B')} & {D(B_0')} & {Ob \otimes_{k} K'}
\arrow[from=1-1, to=1-2]
\arrow["ob", from=1-2, to=1-3]
\arrow["D(p)", from=1-1, to=2-1]
\arrow[from=2-1, to=2-2]
\arrow["id \otimes p|_K", from=1-3, to=2-3]
\arrow["ob", from=2-2, to=2-3]
\arrow["D(p_0)", from=1-2, to=2-2]
\end{tikzcd}\]
\end{definition}
\begin{example}\label{ref:primary:ex}
Now we will discuss \emph{primary obstructions}. Let $D \simeq \Hom(A, -)$, be pro-representable by $(A, \mm)$ where $A = \hat{\symbp} / \symbJo$ for some $\symbJo \subset \mm_{\hat{\symbp}}^2$. Then $A/\mm^2 \simeq \hat{\symbp}/\mm_{\hat{\symbp}}^2$ and we get a homomorphism $\eta : A \to A/\mm^2 \simeq \hat{\symbp}/\mm_{\hat{\symbp}}^2$. We can try to prolong it to a morphism $\mu : A \to \hat{\symbp}/\mm_{\hat{\symbp}}^3$. Then assuming that $D$ has an obstruction theory $Ob$ we get a sequence as in~\eqref{ref:smallext:eq} with $B_0 = \hat{\symbp}/\mm_{\hat{\symbp}}^2$, $B = \hat{\symbp}/\mm_{\hat{\symbp}}^3$ and $K = \mm_{\hat{\symbp}}^2/\mm_{\hat{\symbp}}^3$. Hence, an extension $\mu$ exists if and only if $ob(\eta) \in Ob \otimes_{\kk} K$ is $0$. However, in this case an element of $Ob \otimes_{\kk} K$ can be interpreted as a homomorphism $ob_0:K^{\vee} \to Ob$. Consider the identifications:
\begin{equation}\label{eq:KandSym}
K^{\vee} = (\frac{\mm_{\hat{\symbp}}^2}{\mm_{\hat{\symbp}}^3})^{\vee} \simeq \Sym_2(\frac{\mm_{\hat{\symbp}}}{\mm_{\hat{\symbp}}^2})^{\vee} \simeq \Sym_2((\frac{\mm_{\hat{\symbp}}}{\mm_{\hat{\symbp}}^2})^{\vee}) = \Sym_2(T_{\mm}A)
\end{equation}
where the middle isomorphism comes from duality defined by equation~\eqref{eq:dualityw}.
By identifying $K^{\vee}$ with $\Sym_2(T_{\mm}A)$ we can see $ob_0$ as a map $\Sym_2(T_{\mm}A) \to Ob$ or a bilinear symmetric map $T_{\mm}A \times T_{\mm}A \to Ob$.
Now suppose that $(A', \mm')$ is another Noetherian complete local $k$-algebra and we have a surjection $\zeta : A \to A'$. Assume also that $A' = \hat{\symbT}/\symbJ$ for $\hat{\symbT} := k[\![ \sgamma_1, \dots, \sgamma_m]\!]$, $\mm_{\hat{\symbT}} := (\Bar{\sgamma})$, and $\symbJ \subset \mm_{\hat{\symbT}}^2$. We will describe how to use primary obstructions to get some information about $\symbJ$.
First note that we can present $A$ as a quotient of $k[\![ \Bar{\sgamma}, \Bar{\sbeta} ]\!]$ by an ideal contained in $(\Bar{\sgamma}, \Bar{\sbeta})^2$ so that the following diagram commutes:
\[\begin{tikzcd}
A & {A'} \\
{k[\![ \Bar{\sgamma}, \Bar{\sbeta} ]\!]} & {k[\![ \Bar{\sgamma} ]\!]}
\arrow[from=2-2, to=1-2]
\arrow[from=2-1, to=1-1]
\arrow["\zeta", from=1-1, to=1-2]
\arrow["p"', from=2-1, to=2-2]
\end{tikzcd}\]
where $p$ is defined by fixing $\Bar{\sgamma}$ and taking $\Bar{\sbeta}$ to $0$.
After dividing by squares of maximal ideals we get:
\[\begin{tikzcd}
{A/\mm^2} & {A'/(\mm')^2} \\
{\frac{k[\![ \Bar{\sgamma}, \Bar{\sbeta} ]\!]}{(\Bar{\sgamma}, \Bar{\sbeta})^2}} & {\frac{k[\![ \Bar{\sgamma} ]\!]}{(\Bar{\sgamma})^2}}
\arrow["\simeq", from=2-2, to=1-2]
\arrow["\simeq"', from=2-1, to=1-1]
\arrow[from=1-1, to=1-2]
\arrow["p"', from=2-1, to=2-2]
\end{tikzcd}\]
Next we invert isomorphisms to get:
\[\begin{tikzcd}
{A/\mm^2} & {A'/(\mm')^2} \\
{\frac{k[\![ \Bar{\sgamma}, \Bar{\sbeta} ]\!]}{(\Bar{\sgamma}, \Bar{\sbeta})^2}} & {\frac{k[\![ \Bar{\sgamma} ]\!]}{(\Bar{\sgamma})^2}}
\arrow[from=1-2, to=2-2]
\arrow["\eta'", from=1-1, to=2-1]
\arrow[from=1-1, to=1-2]
\arrow["p"', from=2-1, to=2-2]
\arrow["\mu'"{description}, from=1-1, to=2-2]
\end{tikzcd}\]
where $\mu'$ is just the composition. We define maps $\eta: A \to \frac{k[\![ \Bar{\sgamma}, \Bar{\sbeta} ]\!]}{(\Bar{\sgamma}, \Bar{\sbeta})^2}$ and $\mu : A \to \frac{k[\![ \Bar{\sgamma} ]\!]}{(\Bar{\sgamma})^2}$ as compositions of the quotient map $A \to A/ \mm^2$ with $\eta'$ and $\mu'$ respectively. Consider now the diagram:
\[\begin{tikzcd}
0 & {{K :=\frac{(\Bar{\sgamma}, \Bar{\sbeta})^2}{(\Bar{\sgamma}, \Bar{\sbeta})^3}}} & {{\frac{k[\![\Bar{\sgamma}, \Bar{\sbeta}]\!]}{(\Bar{\sgamma}, \Bar{\sbeta})^3}}} & {{\frac{k[\![\Bar{\sgamma}, \Bar{\sbeta}]\!]}{(\Bar{\sgamma}, \Bar{\sbeta})^2}}} & 0 \\
0 & {{K' = \frac{(\Bar{\sgamma})^2}{(\Bar{\sgamma})^3}}} & {{\frac{k[\![\Bar{\sgamma}]\!]}{(\Bar{\sgamma})^3}}} & {{\frac{k[\![\Bar{\sgamma}]\!]}{(\Bar{\sgamma})^2}}} & 0
\arrow[from=1-1, to=1-2]
\arrow[from=1-2, to=1-3]
\arrow[ from=1-3, to=1-4]
\arrow[from=1-4, to=1-5]
\arrow[from=2-1, to=2-2]
\arrow[from=2-2, to=2-3]
\arrow[from=2-3, to=2-4]
\arrow[from=2-4, to=2-5]
\arrow["p", from=1-4, to=2-4]
\arrow[from=1-3, to=2-3]
\arrow["r", from=1-2, to=2-2]
\end{tikzcd}\]
We can apply the functor $D$ and get:
\[\begin{tikzcd}
{D({\frac{k[\![\Bar{\sgamma}, \Bar{\sbeta}]\!]}{(\Bar{\sgamma}, \Bar{\sbeta})^3}})} & {D({\frac{k[\![\Bar{\sgamma}, \Bar{\sbeta}]\!]}{(\Bar{\sgamma}, \Bar{\sbeta})^2}})} & {Ob \otimes_k K} \\
{D({\frac{k[\![\Bar{\sgamma}]\!]}{(\Bar{\sgamma})^3}})} & {D({\frac{k[\![\Bar{\sgamma}]\!]}{(\Bar{\sgamma})^2}})} & {Ob \otimes_k K'}
\arrow[from=1-1, to=1-2]
\arrow["{id \otimes r}", from=1-3, to=2-3]
\arrow["ob", from=1-2, to=1-3]
\arrow[from=1-1, to=2-1]
\arrow["{D(p)}"', from=1-2, to=2-2]
\arrow[from=2-1, to=2-2]
\arrow["ob'", from=2-2, to=2-3]
\end{tikzcd}\]
By the fact that $D(p)(\eta) = p \circ \eta = \mu$ we get that $(id \otimes r)(ob(\eta)) = ob'(\mu)$. This yields the following commutative diagram:
\[\begin{tikzcd}
{K^{\vee}} & Ob \\
{(K')^{\vee}}
\arrow["{ob(\eta)}", from=1-1, to=1-2]
\arrow["{r^{\vee}}", from=2-1, to=1-1]
\arrow["{ob'(\mu)}"', from=2-1, to=1-2]
\end{tikzcd}\]
By identifying $K^{\vee}$ with $\Sym_2(T_{\mm}A)$ and $(K')^{\vee}$ with $\Sym_2(T_{\mm'}A')$ as in equation~\eqref{eq:KandSym} we get the diagram:
\[\begin{tikzcd}
{\Sym_2(T_{\mm}A)} & Ob \\
{\Sym_2(T_{\mm'}A')}
\arrow["{ob(\eta)}", from=1-1, to=1-2]
\arrow["{ob'(\mu)}"', from=2-1, to=1-2]
\arrow["{r^{\vee}}", from=2-1, to=1-1]
\end{tikzcd}\]
The map $ob(\eta)$ is the primary obstruction map $ob_0$ by its definition.
Hence, $ob'(\mu)$ is the restriction of primary obstructions to $\Sym_2(T_{\mm'}A')$. This allows us to give a lower bound for the second degree part of $\symbJ$ if we can calculate the restriction of primary obstruction map $ob_0$ to $\Sym_2(T_{\mm'}A')$:
\begin{proposition}\label{prop:primaryobs_tool}
Consider the map $ob'(\mu)^{\vee} : Ob^{\vee} \to K$. Then:
\begin{equation}
\im (ob'(\mu)^{\vee}) \subset \frac{(\sgamma_1, \dots, \sgamma_m)^3 + J}{(\sgamma_1, \dots, \sgamma_m)^3}.
\end{equation}
\end{proposition}
\begin{proof}
Consider the commutative diagram:
\[\begin{tikzcd}
A & {A'} \\
{\frac{k[\![\sgamma_1, \dots, \sgamma_m]\!]}{(\sgamma_1, \dots, \sgamma_m)^3 + J}} & {\frac{k[\![\sgamma_1, \dots, \sgamma_m]\!]}{(\sgamma_1, \dots, \sgamma_m)^2}}
\arrow["\zeta", from=1-1, to=1-2]
\arrow[from=1-2, to=2-1]
\arrow[from=1-2, to=2-2]
\arrow[from=2-1, to=2-2]
\arrow["\nu"', from=1-1, to=2-1]
\arrow["\mu"{description}, crossing over, from=1-1, to=2-2]
\end{tikzcd}\]
where maps from $A' = k[\![\sgamma_1, \dots, \sgamma_m]\!]$ to ${\frac{k[\![\sgamma_1, \dots, \sgamma_m]\!]}{(\sgamma_1, \dots, \sgamma_m)^3 + J}}$ and ${\frac{k[\![\sgamma_1, \dots, \sgamma_m]\!]}{(\sgamma_1, \dots, \sgamma_m)^2}}$ are quotients, and $\nu, \mu$ are appropriate compositions with $\zeta$. Now look at the map of extensions:
\[\begin{tikzcd}
0 & {K' = \frac{(\sgamma_1, \dots, \sgamma_m)^2}{(\sgamma_1, \dots, \sgamma_m)^3}} & {B :=\frac{k[\![\sgamma_1, \dots, \sgamma_m]\!]}{(\sgamma_1, \dots, \sgamma_m)^3}} & {B_0 := \frac{k[\![\sgamma_1, \dots, \sgamma_m]\!]}{(\sgamma_1, \dots, \sgamma_m)^2}} & 0 \\
0 & {K'' := \frac{(\sgamma_1, \dots, \sgamma_m)^2}{(\sgamma_1, \dots, \sgamma_m)^3 + J}} & {B' := \frac{k[\![\sgamma_1, \dots, \sgamma_m]\!]}{(\sgamma_1, \dots, \sgamma_m)^3 + J}} & {B_0' :=\frac{k[\![\sgamma_1, \dots, \sgamma_m]\!]}{(\sgamma_1, \dots, \sgamma_m)^2} } & 0
\arrow[from=2-2, to=2-3]
\arrow[from=2-3, to=2-4]
\arrow[from=2-1, to=2-2]
\arrow[from=2-4, to=2-5]
\arrow[from=1-2, to=1-3]
\arrow[from=1-1, to=1-2]
\arrow[from=1-4, to=1-5]
\arrow[from=1-3, to=2-3]
\arrow["{=}", from=1-4, to=2-4]
\arrow["u", from=1-2, to=2-2]
\arrow[from=1-3, to=1-4]
\end{tikzcd}\]
By applying $D$, we get the following:
\[\begin{tikzcd}
{D(B)} & {D(B_0)} & {Ob \otimes_k K'} && \mu & {ob'(\mu)} \\
{D(B')} & {D(B_0')} & {Ob \otimes_k K''} & {\nu} & \mu & 0
\arrow[from=1-1, to=1-2]
\arrow[from=1-2, to=1-3]
\arrow["{id \otimes u}", from=1-3, to=2-3]
\arrow["{=}", from=1-2, to=2-2]
\arrow[from=1-1, to=2-1]
\arrow[from=2-1, to=2-2]
\arrow[from=2-2, to=2-3]
\arrow[maps to, from=2-4, to=2-5]
\arrow[maps to, from=1-5, to=2-5]
\arrow[maps to, from=2-5, to=2-6]
\arrow[maps to, from=1-5, to=1-6]
\arrow[maps to, from=1-6, to=2-6]
\end{tikzcd}\]
The fact that $(id \otimes u)(ob'(\mu)) = 0$ translates into:
\[\begin{tikzcd}
{(K')^{\vee}} & Ob \\
{(K'')^{\vee}}
\arrow["{ob'(\mu)}", from=1-1, to=1-2]
\arrow["{u^{\vee}}", from=2-1, to=1-1]
\arrow["0"', from=2-1, to=1-2]
\end{tikzcd}\]
So $ob'(\mu) \circ u^{\vee} = 0$, which means that $\im (ob'(\mu)^{\vee}) \subset \ker u = \frac{(\sgamma_1, \dots, \sgamma_m)^3 + J}{(\sgamma_1, \dots, \sgamma_m)^3}$.
\end{proof}
The following lemma will be useful for calculating primary obstructions. In its statement we use the fact that in the category of rings over $k$ product of $\frac{k[\eps]}{(\eps)^2}$ and $\frac{k[\eps']}{(\eps')^2}$ is $\frac{k[\eps, \eps']}{(\eps, \eps')^2}$.
\begin{lemma}\label{fact:primary_vanish}
Let $\delta: A \to \frac{k[\eps]}{(\eps)^2}$, $\delta' : A \to \frac{k[\eps']}{(\eps')^2}$ and denote by also $\delta, \delta'$ the corresponding elements in $T_{\mm} A$. Suppose that the induced homomorphism $(\delta \times \delta') : A \to \frac{k[\eps, \eps']}{(\eps, \eps')^2}$ can be extended to a homomorphism to $\frac{k[\eps, \eps']}{(\eps^2, \eps'^2)}$:
\[\begin{tikzcd}
A & {\frac{k[\eps, \eps']}{(\eps, \eps')^2}} \\
& {\frac{k[\eps, \eps']}{(\eps^2, \eps'^2)}}
\arrow[from=2-2, to=1-2]
\arrow["\delta \times \delta'", from=1-1, to=1-2]
\arrow["\gamma", dashed, from=1-1, to=2-2]
\end{tikzcd}\]
Then the primary obstruction map $ob_0$ vanishes on $\delta \cdot \delta' \in \Sym_2(T_{\mm} A)$.
\end{lemma}
\begin{proof}
Consider a morphism of small extensions:
\[\begin{tikzcd}
0 & {\frac{(\eps, \eps')^2}{(\eps, \eps')^3}} & {\frac{k[\eps, \eps']}{(\eps, \eps')^3}} & {\frac{k[\eps, \eps']}{(\eps, \eps')^2}} & 0 \\
0 & {\frac{(\eps, \eps')^2}{(\eps^2, \eps'^2)}} & {\frac{k[\eps, \eps']}{(\eps^2, \eps'^2)}} & {\frac{k[\eps, \eps']}{(\eps, \eps')^2}} & 0
\arrow[from=1-2, to=1-3]
\arrow[from=1-3, to=1-4]
\arrow[from=1-4, to=1-5]
\arrow[from=1-1, to=1-2]
\arrow[from=2-1, to=2-2]
\arrow[from=2-2, to=2-3]
\arrow[from=2-3, to=2-4]
\arrow["{=}"', from=1-4, to=2-4]
\arrow[from=2-4, to=2-5]
\arrow[from=1-3, to=2-3]
\arrow["s", from=1-2, to=2-2]
\end{tikzcd}\]
By applying $D = \Hom(A, -)$ on it we get:
\[\begin{tikzcd}
{D(\frac{k[\eps, \eps']}{(\eps, \eps')^3})} & {D(\frac{k[\eps, \eps']}{(\eps, \eps')^2})} & {Ob \otimes_k \frac{(\eps, \eps')^2}{(\eps, \eps')^3}} && {\delta \times \delta'} & {ob(\delta \times \delta')} \\
{D(\frac{k[\eps, \eps']}{(\eps^2, \eps'^2)})} & {D(\frac{k[\eps, \eps']}{(\eps, \eps')^2})} & {Ob \otimes_k \frac{(\eps, \eps')^2}{(\eps^2, \eps'^2)}} & \gamma & {\delta \times \delta'} & 0
\arrow[from=1-1, to=1-2]
\arrow[from=2-1, to=2-2]
\arrow["{=}"', from=1-2, to=2-2]
\arrow[from=1-1, to=2-1]
\arrow[from=2-2, to=2-3]
\arrow["ob", from=1-2, to=1-3]
\arrow["{id \otimes s}"', from=1-3, to=2-3]
\arrow[maps to, from=2-4, to=2-5]
\arrow[maps to, from=1-5, to=2-5]
\arrow[maps to, from=2-5, to=2-6]
\arrow[maps to, from=1-5, to=1-6]
\arrow[maps to, from=1-6, to=2-6]
\end{tikzcd}\]
The fact that $(id \otimes s)(ob(\delta \times \delta')) = 0$ translates to:
\[\begin{tikzcd}
{(\frac{(\eps, \eps')^2}{(\eps, \eps')^3})^{\vee}} & Ob \\
{(\frac{(\eps, \eps')^2}{(\eps^2, \eps'^2)})^{\vee}}
\arrow["{ob(\delta \times \delta')}", from=1-1, to=1-2]
\arrow["{s^{\vee}}", from=2-1, to=1-1]
\arrow["0"', from=2-1, to=1-2]
\end{tikzcd}\]
By the discussion before Proposition~\ref{prop:primaryobs_tool} for $A' = \frac{k[\eps, \eps']}{(\eps, \eps')^2}$ we get that $ob(\delta \times \delta')$ is the restriction of the primary obstruction map $ob_0 : \Sym_2(T_{\mm} A) \to Ob$ to the image of $\Sym_2(T_{(\eps, \eps')} \frac{k[\eps, \eps']}{(\eps, \eps')^2})$ by the tangent map $d(\delta \times \delta')$. However, by definition of $\delta \times \delta'$ we have: \begin{equation}\label{eq:obs_eq_1}
d(\delta \times \delta')(\eps^{\vee}) = \delta
\end{equation}
and
\begin{equation}\label{eq:obs_eq_2}
d(\delta \times \delta')(\eps'^{\vee}) = \delta'
\end{equation}
The map $s^{\vee}$ on the above diagram is the embedding of $k (\eps \cdot \eps')^{\vee}$ into $\Sym_2(k \eps^{\vee} \oplus k \eps'^{\vee})$. Thus, using the identifications~\eqref{eq:obs_eq_1} and~\eqref{eq:obs_eq_2} we get that $ob_0(\delta \cdot \delta')=0$. This finishes the proof.
\end{proof}
\end{example}
\begin{example}~\label{ex:hilbert}
Fix a $\kk$-point $x \in {\mathcal H}$. Then $x$ is given by an ideal $I \subset S$ such that $S/I$ of degree $d$. Consider the functor $D_{{\mathcal H}, x}$ pro-representable by $\hat{{\mathcal O}}_{{\mathcal H}, x}$. Its tangent space is $\Hom_{S}(I, S/I)$ by Theorem~\ref{thm:tangenttohilb}. It is also known that $D_{{\mathcal H}, x}$ has an obstruction theory with the obstruction space $Ob = \Ext^1_{S}(I, S/I)$. For a proof see~\cite[Corollary 6.4.10]{FGI+05}. Thus in this case the primary obstruction map is a function $ob_0 : \Sym_2(\Hom_{S}(I, S/I)) \to \Ext^1_{S}(I, S/I)$.
\end{example}
We will need the method of calculating primary obstructions for the functor in Example~\ref{ex:hilbert}. It comes from the paper~\cite{JS21}.
Consider an ideal $I \subset S$ and its free resolution as an $S$-module $(F_i)_{i \geq 1}$. Then the $S$-module $S/I$ has a free resolution which starts with the quotient map $q:F_0 = S \to S/I$ and later is continued by the resolution $(F_i)_{i \geq 1}$ using the composition $F_1 \to I \subset S = F_0$. For a given $\symbc \in \Hom_S(I, S/I)$, we can lift it to $S$-module homomorphisms as on the diagram below (not necessarily uniquely):
\[\begin{tikzcd}~\label{diagram:hilb_obtruction}
& 0 & I & {F_1} & {F_2} & {F_3} & \dots \\
0 & {S/I} & {F_0} & {F_1} & {F_2} & \dots
\arrow[from=1-3, to=1-2]
\arrow["{d_0}"', from=1-4, to=1-3]
\arrow["{d_1}"', from=1-5, to=1-4]
\arrow["{d_2}"', from=1-6, to=1-5]
\arrow[from=2-6, to=2-5]
\arrow["{d_1}"', from=2-5, to=2-4]
\arrow["{d_0}"', from=2-4, to=2-3]
\arrow["q"', from=2-3, to=2-2]
\arrow[from=2-2, to=2-1]
\arrow[from=1-7, to=1-6]
\arrow["\symbc"'{pos=0.7}, from=1-3, to=2-2]
\arrow["{s_1(\symbc)}"'{pos=0.8}, from=1-4, to=2-3]
\arrow["{s_2(\symbc)}"'{pos=0.8}, from=1-5, to=2-4]
\arrow["{s_3(\symbc)}"'{pos=0.8}, from=1-6, to=2-5]
\end{tikzcd}\]
Under this notation the following theorem holds:
\begin{theorem}~\cite[Theorem 4.18]{JS21}~\label{thm:hilb_obstruction}
Consider a point represented by an ideal $I \subset S$ on the Hilbert scheme ${\mathcal H}$ and the associated obstruction theory with obstruction group $Ob = \Ext^1_{S}(I, S/I)$. Then for $\symbc_1, \symbc_2 \in \Hom_{S}(I, S/I)$ the primary obstruction map sends the class of $\symbc_1 \cdot \symbc_2 \in \Sym_2 (\Hom_{S}(I, S/I))$ to the class of:
\begin{equation}~\label{eq:hilb_obstruction}
q \circ (s_1(\symbc_1) \circ s_2(\symbc_2) + s_1(\symbc_2) \circ s_2(\symbc_1))
\end{equation}
in $\Ext^1_{S}(I, S/I)$.
\end{theorem}
Note that $q \circ s_1(\symbc_1) \circ s_2(\symbc_2) \circ d_2 = \symbc_1 \circ d_0 \circ d_1 \circ d_3(\symbc_2) = \symbc_1 \circ 0 \circ d_3(\symbc_2) = 0$, so~\eqref{eq:hilb_obstruction} indeed yields an element of $\Ext^1_{S}(I, S/I)$.
%
\subsection{Barycenter}
%
It will be convenient for us not to work with the Hilbert scheme ${\mathcal H}$ itself, but with its closed subset consisting of tuples which are centered around $0$. In order to construct it we need to know that the support map which takes a point of the Hilbert scheme and associates with it its support (in the symmetric product of the ambient variety) is a morphism of schemes. This is summarised in the following theorem from~\cite[Theorem 7.1.14]{FGI+05}:
\begin{theorem}\label{thm:HC}
There is a morphism of schemes (called the Hilbert-Chow morphism):
\begin{equation}
\theta : {\mathcal H} \to (\Affine^n)^d \goodquotient \mathbb{S}_d
\end{equation}
which on the level of sets is given by: $[Z] \mapsto \sum_{p} \deg({\mathcal O}_{Z,p}) [p]$, where $Z$ is a subscheme of $\Affine^n$ as in Definition~\ref{def:Hilbert:first}.
\end{theorem}
Here the quotient $(\Affine^n)^d \goodquotient \mathbb{S}_d$ is by the definition the spectrum of the algebra of global sections of $(\Affine^n)^d$ invariant under the action of the symmetric group $\mathbb{S}_d$ acting by permuting points. It is a categorical quotient because $(\Affine^n)^d$ is an affine scheme.
Now we will define the barycenter scheme. The morphism $(\Affine^n)^d \to \Affine^n$ given by taking the average of $d$ vectors is $\mathbb{S}_d$-equivariant, so it descends to a morphism $Av:(\Affine^n)^d \goodquotient \mathbb{S}_d \to \Affine^n$. By taking composition with the Hilbert-Chow morphism we get the barycenter morphism
\begin{equation}
Bar: {\mathcal H} \to \Affine^n.
\end{equation}
\begin{definition}\label{def:Bar}
The barycenter scheme is ${\mathcal B} := Bar^{-1}(\{ 0 \})$. If we want to emphasize $d$ we write ${\mathcal B}^d$.
\end{definition}
To study the barycenter scheme we give a more explicit definition of the Hilbert-Chow morphism, or more precisely a formula for the composition $\Spec(A) \to {\mathcal H} \to (\Affine^n)^d \goodquotient \mathbb{S}_d$ for a ring $A$ such that $\Spec(A) \to {\mathcal H}$ is given by a free family $R$ over $A$. For $r \in R$ let $Nm_{R/A}(r)$ and $Tr_{R/A}(r)$ be respectively the determinant and the trace of the endomorphism of $R$ (as an $A$-module) given by multiplying by $r$. Then the following theorem from the notes~\cite[Theorem 2.16, Corollary 2.18]{notesbertin} holds:
\begin{theorem}~\label{thm:HCaffine}
Suppose a morphism $\Spec(A) \to {\mathcal H}$ is given by a free $A$ - algebra $R$. Then the composition:
\[\begin{tikzcd}
{\Spec(A)} & {{\mathcal H}} & {(\Affine^n)^d \goodquotient \mathbb{S}_d}
\arrow[from=1-1, to=1-2]
\arrow[from=1-2, to=1-3]
\end{tikzcd}\]
comes from the composite homomorphism:
\[\begin{tikzcd}
{(k[\aalpha] \otimes_k \dots \otimes_k k[\aalpha])^{\mathbb{S}_d}} & {(R \otimes_A \dots \otimes_A R)^{\mathbb{S}_d}} & A
\arrow[from=1-1, to=1-2]
\arrow["LNm", from=1-2, to=1-3]
\end{tikzcd}\]
where the first map is the homomorphism $k[\aalpha] \to k[\aalpha] \otimes_k A \to R$ tensored $d$ times, and the second map $LNm$ is the unique $A$-algebra homomorphism that satisfies the equation:
\begin{equation}
LNm(r \otimes \dots \otimes r) = Nm_{R/A}(r).
\end{equation}
Moreover, if we consider the composition of the above morphism $\Spec(A) \to (\Affine^n)^d \goodquotient \mathbb{S}_d$ with the average morphism $Av : (\Affine^n)^d \goodquotient \mathbb{S}_d \to \Affine^n$ we get a morhpsism $\Spec(A) \to \Affine^n = \Spec(k[\aalpha])$ which on the level of algebras is given by the formula:
\begin{equation}
\alpha_j \mapsto \frac{1}{d} Tr_{R/A}(\alpha_j)
\end{equation}
where on the right hand side we identify $\alpha_j$ with its image in $R$.
\end{theorem}
Every point of the Hilbert scheme can be uniquely shifted so that it is centered at $0$, so there is an isomorphism ${\mathcal H} \simeq \Affine^n \times {\mathcal B}$ as one can prove using for example Theorem~\ref{thm:HCaffine}.
\begin{proposition}~\label{prop:hilbandbar}
The barycenter morphism $Bar :{\mathcal H} \to \Affine^n$ is equivariant under the action of the $n$-dimensional additive group $\G_a^n$. Hence ${\mathcal H} \simeq \Affine^n \times {\mathcal B}$.
\end{proposition}
\begin{proof}
We omit the proof of the $\Affine^n$-equivariance of $Bar$.
\begin{comment}
First we prove that the barycenter morphism $Bar :{\mathcal H} \to \Affine^n$ is $\Affine^n$-equivariant. It means that the following diagram commutes:
\[\begin{tikzcd}
{\Affine^n \times {\mathcal H}} & {\Affine^n \times \Affine^n} \\
{\mathcal H} & {\Affine^n}
\arrow["\mu", from=1-1, to=2-1]
\arrow["{+}", from=1-2, to=2-2]
\arrow["{id \times Bar}", from=1-1, to=1-2]
\arrow["Bar", from=2-1, to=2-2]
\end{tikzcd}\]
Here $\mu$ is the $\Affine^n$-action on ${\mathcal H}$. It is enough to check that for all $k$-algebras $A$ the following diagram commutes:
\[\begin{tikzcd}
{\Affine^n(\Spec(A)) \times {\mathcal H}(\Spec(A))} & {\Affine^n(\Spec(A)) \times \Affine^n(\Spec(A))} \\
{{\mathcal H}(\Spec(A))} & {\Affine^n(\Spec(A))}
\arrow["\mu", from=1-1, to=2-1]
\arrow["{+}", from=1-2, to=2-2]
\arrow["{id \times Bar}", from=1-1, to=1-2]
\arrow["Bar", from=2-1, to=2-2]
\end{tikzcd}\]
Fix a pair $(\bar{a}, [Z]) \in \Affine^n(\Spec(A)) \times {\mathcal H}(\Spec(A))$ where $\bar{a} = (a_1, \dots, a_n) \in A^n$. By passing to localisation, we can assume that $[Z] \in {\mathcal H}(\Spec(A))$ is given by a free $A$-module $R$ of degree $d$. By Theorem~\ref{thm:HCaffine} we calculate the images of $(\bar{a}, [Z])$ by both ways and get:
\[\begin{tikzcd}
{(\bar{a}, [Z])} & {(\bar{a}, (\frac{1}{d} \cdot Tr_{R/A}(\alpha_i))_{i=1}^n)} \\
{[\bar{a}+Z]} & {(\frac{1}{d} \cdot Tr_{R/A}(\alpha_i + a_i))_{i=1}^n} & {(a_i + \frac{1}{d} \cdot Tr_{R/A}(\alpha_i))_{i=1}^n}
\arrow[maps to, from=1-1, to=1-2]
\arrow[maps to, from=1-1, to=2-1]
\arrow[maps to, from=1-2, to=2-3]
\arrow[maps to, from=2-1, to=2-2]
\arrow["{?}"{marking}, draw=none, from=2-2, to=2-3]
\end{tikzcd}\]
The image of $[\bar{a} + Z]$ by $Bar$ is the tuple $(\frac{1}{d} \cdot Tr_{R/A}(\alpha_i + a_i))_{i=1}^n$, because the ideal of $\bar{a} + Z$ in $\Affine_A^n$ is given be the kernel of the composite homomorphism:
\[\begin{tikzcd}
{S_A} & {S_A} & R
\arrow[two heads, from=1-2, to=1-3]
\arrow["{(\bar{a} +)^{\#}}", from=1-1, to=1-2]
\end{tikzcd}\]
as in Example~\ref{ex:additiveaction} and under this composition $\alpha_i$ goes to $\alpha_i + a_i \in R$.
By calculation:
\begin{equation}
\frac{1}{d} \cdot Tr_{R/A}(\alpha_i + a_i) = \frac{1}{d} \cdot (Tr_{R/A}(\alpha_i) + Tr_{R/A}(a_i)) = \frac{1}{d} \cdot (Tr_{R/A}(\alpha_i) + d \cdot a_i) = \frac{1}{d} \cdot Tr_{R/A}(\alpha_i) + a_i
\end{equation}
we get that the question mark in the above diagram can be replaced with equality. This finishes the part proving that $Bar$ is $\G_m$-equivariant.
Now we pass to the construction of an isomorphism ${\mathcal H} \simeq \Affine^n \times {\mathcal B}$.
Consider the following diagram:
\[\begin{tikzcd}
{\Affine^n \times {\mathcal B}} & {\Affine^n \times \{0\}} \\
{\Affine^n \times {\mathcal H}} & {\Affine^n \times \Affine^n} \\
{\mathcal H} & {\Affine^n}
\arrow["{id \times 0}", from=1-1, to=1-2]
\arrow["{id \times Bar}", from=2-1, to=2-2]
\arrow["Bar", from=3-1, to=3-2]
\arrow["{+}", from=2-2, to=3-2]
\arrow["\mu", from=2-1, to=3-1]
\arrow["{id \times j}", from=1-1, to=2-1]
\arrow["{id \times 0}"', hook, from=1-2, to=2-2]
\arrow["\simeq", shift left=5, bend left, from=1-2, to=3-2]
\arrow[shift right=2, bend right, from=1-1, to=3-1]
\end{tikzcd}\]
where $j: {\mathcal B} \to {\mathcal H}$ is the closed embedding. The bottom square is commutative because $Bar$ is $\Affine^n$-equivariant. The upper square is a pullback diagram, because of the definition of ${\mathcal B}$. The bottom square is also a pullback, which follows from the existence of group inverse in $\G_a^n \simeq \Affine^n$. Thus the left hand side map $\Affine^n \times {\mathcal B} \to {\mathcal H}$ is a pullback of an isomorphism, so it is also an isomorphism and we get the conclusion.
\end{comment}
The isomorphism ${\mathcal H} \simeq \Affine^n \times {\mathcal B}$ is given by the left vertical map on the diagram:
\[\begin{tikzcd}
{\Affine^n \times {\mathcal B}} & {\Affine^n \times \{0\}} \\
{\Affine^n \times {\mathcal H}} & {\Affine^n \times \Affine^n} \\
{\mathcal H} & {\Affine^n}
\arrow["{id \times 0}", from=1-1, to=1-2]
\arrow["{id \times Bar}", from=2-1, to=2-2]
\arrow["Bar", from=3-1, to=3-2]
\arrow["{+}", from=2-2, to=3-2]
\arrow["\mu", from=2-1, to=3-1]
\arrow["{id \times j}", from=1-1, to=2-1]
\arrow["{id \times 0}"', hook, from=1-2, to=2-2]
\arrow["\simeq", shift left=5, bend left, from=1-2, to=3-2]
\arrow[shift right=2, bend right, from=1-1, to=3-1]
\end{tikzcd}\]
where $j: {\mathcal B} \to {\mathcal H}$ is the closed embedding. The bottom square is commutative because $Bar$ is $\Affine^n$-equivariant. The upper square is a pullback diagram, because of the definition of ${\mathcal B}$. The bottom square is also a pullback, which follows from the existence of group inverse in $\G_a^n \simeq \Affine^n$. Thus the left hand side map $\Affine^n \times {\mathcal B} \to {\mathcal H}$ is a pullback of an isomorphism, so it is also an isomorphism and we get the conclusion. The details are left to the reader.
\end{proof}
The isomorphism ${\mathcal H} \simeq \Affine^n \times {\mathcal B}$ respects torus action:
\begin{proposition}\label{prop:barisgminv}
The barycenter morphism $Bar : {\mathcal H} \to \Affine^n$ is $\G_m$-equivariant. Moreover, the isomorphism ${\mathcal H} \simeq \Affine^n \times {\mathcal B}$ from Proposition~\ref{prop:hilbandbar} is $\G_m$-equivariant, where the torus action on ${\mathcal B}$ is the one induced from $\G_m$-action on ${\mathcal H}$.
\end{proposition}
\begin{proof}
\begin{comment}
We need to prove commutativity of the diagram:
\[\begin{tikzcd}
{\G_m \times {\mathcal H}} & {\G_m \times \Affine^n} \\
{\mathcal H} & {\Affine^n}
\arrow["\nu", from=1-1, to=2-1]
\arrow["{\cdot}", from=1-2, to=2-2]
\arrow["{id \times Bar}", from=1-1, to=1-2]
\arrow["Bar", from=2-1, to=2-2]
\end{tikzcd}\]
where $\nu$ is the $\G_m$-action on ${\mathcal H}$. Exactly as in the proof of Proposition~\ref{prop:hilbandbar} it is enough that for a $k$-algebra $A$ the following diagram commutes:
\[\begin{tikzcd}
{\G_m(\Spec(A)) \times {\mathcal H}(\Spec(A))} & {\G_m(\Spec(A)) \times \Affine^n(\Spec(A))} \\
{{\mathcal H}(\Spec(A))} & {\Affine^n(\Spec(A))}
\arrow["\nu", from=1-1, to=2-1]
\arrow["\cdot", from=1-2, to=2-2]
\arrow["{id \times Bar}", from=1-1, to=1-2]
\arrow["Bar", from=2-1, to=2-2]
\end{tikzcd}\]
Again, we assume that $\Spec(A) \to {\mathcal H}$ is given by a free $A$-module $R$ of degree $d$ and we fix $a \in \G_m(\Spec(A)) = A^*$. By moving the pair $(a, [Z])$ along two paths on the diagram we get:
\[\begin{tikzcd}
{(a, [Z])} & {(a, (\frac{1}{d} \cdot Tr_{R/A}(\alpha_i))_{i=1}^n)} \\
{[a \cdot Z]} & {(\frac{1}{d} \cdot Tr_{R/A}(a \cdot \alpha_i))_{i=1}^n} & {(a \cdot \frac{1}{d} \cdot Tr_{R/A}(\alpha_i))_{i=1}^n}
\arrow[maps to, from=1-1, to=1-2]
\arrow[maps to, from=1-1, to=2-1]
\arrow[maps to, from=1-2, to=2-3]
\arrow[maps to, from=2-1, to=2-2]
\arrow["{=}"{marking}, draw=none, from=2-2, to=2-3]
\end{tikzcd}\]
where the right bottom equality follows from the fact that $Tr_{R/A}$ is an $A$-module homomorphism. This finishes the proof of $\G_m$-equivariance of $Bar$.
Now we prove the "moreover" part of the statement. Recall that the isomorphism ${\mathcal H} \simeq \Affine^n \times {\mathcal B}$ comes from the diagram:
\[\begin{tikzcd}
{\Affine^n \times {\mathcal B}} & {\Affine^n \times \{0\}} \\
{\Affine^n \times {\mathcal H}} & {\Affine^n \times \Affine^n} \\
{\mathcal H} & {\Affine^n}
\arrow["{id \times 0}", from=1-1, to=1-2]
\arrow["{id \times Bar}", from=2-1, to=2-2]
\arrow["Bar", from=3-1, to=3-2]
\arrow["{+}", from=2-2, to=3-2]
\arrow["\mu", from=2-1, to=3-1]
\arrow["{id \times j}", from=1-1, to=2-1]
\arrow["{id \times 0}"', hook, from=1-2, to=2-2]
\arrow["\simeq", shift left=5, bend left, from=1-2, to=3-2]
\arrow[shift right=2, bend right, from=1-1, to=3-1]
\end{tikzcd}\]
which appears in the proof of Proposition~\ref{prop:hilbandbar}.
Since we have proved that $Bar$ is $\G_m$-equivariant it is immediate that all maps on this diagram beside $\mu$ are $\G_m$-equivariant. The $\G_m$-equivariance of $\mu$ is equivalent to commutativity of:
\[\begin{tikzcd}
{\G_m \times(\Affine^n \times {\mathcal H})} & {\G_m \times {\mathcal H}} \\
{\Affine^n \times {\mathcal H}} & {\mathcal H}
\arrow["{id \times \mu}", from=1-1, to=1-2]
\arrow[from=1-1, to=2-1]
\arrow["\mu"', from=2-1, to=2-2]
\arrow["\nu", from=1-2, to=2-2]
\end{tikzcd}\]
We check this on $\Spec(A)$-point of $\G_m \times(\Affine^n \times {\mathcal H})$ given by the triple $(a, (\bar{b}, [Z])) \in (\G_m \times(\Affine^n \times {\mathcal H}))(\Spec(A))$. We have:
\[\begin{tikzcd}
{(a, (\bar{b}, [Z]))} & {(a, [\bar{b} +Z])} \\
{(a \cdot \bar{b}, [a \cdot Z])} & {[a \cdot \bar{b} + a \cdot Z]} & {[a \cdot (\bar{b} + Z)]}
\arrow[maps to, from=1-1, to=1-2]
\arrow[maps to, from=1-2, to=2-3]
\arrow[maps to, from=1-1, to=2-1]
\arrow[maps to, from=2-1, to=2-2]
\arrow["{=}"{marking}, draw=none, from=2-2, to=2-3]
\end{tikzcd}\]
where the equality follows from the distributivity of multiplication over addition. Thus $\mu$ is indeed $\G_m$-equivariant and so also the isomorphism between $\Affine^n \times {\mathcal B}$ and ${\mathcal H}$ which is the composition of $\G_m$-equivariant morphisms $id \times j$ and $\mu$ is $\G_m$-equivariant.
\end{comment}
One should use Theorem~\ref{thm:HCaffine}. Details are left as an exercise for the reader.
\end{proof}
The $\G_m$-action on ${\mathcal B}$ inherited from ${\mathcal H}$ allows us to perform the \BBname{} decomposition on ${\mathcal B}$ and get the diagram:
\[\begin{tikzcd}
{{\mathcal B}^{+}} & {{\mathcal B}} \\
{{\mathcal B}^{\G_m}}
\arrow["\pi", from=1-1, to=2-1]
\arrow["i"', from=1-1, to=1-2]
\arrow["s", bend left, from=2-1, to=1-1]
\end{tikzcd}\]
For a $\G_m$-invariant point $[I] \in {\mathcal B}$ we identify it with $[I] \in {\mathcal B}^{\G_m}$ and $s([I]) \in {\mathcal B}^+$ in the spirit of Fact~\ref{fact:mono}. Now we can finally give a precise definition of a hedgehog point:
\begin{definition}\label{def:hedgehog:precise}
The \emph{negative spike} at a $\G_m$-invariant point $[I]$ is the fiber $V = \pi^{-1}([I]) \subset {\mathcal B}^+$. The point $[I]$ is said to be a \emph{hedgehog point} if the following conditions are satisfied:
\begin{enumerate}
\item $(T_{[I]} {\mathcal B})_{>0} = 0$,
\item $T_{[I]} {\mathcal B} \neq (T_{[I]} {\mathcal B})_0$,
\item $V$ is $0$-dimensional.
\end{enumerate}
\end{definition}
The equivalence of this formulation and and the one in Definition~\ref{def:hedgehog} follows from the fact that $V$ has an $\Affine^1$ action, so it is $0$-dimensional if and only if it is a point (topologically).
\subsection{Macaulay duality}
%
In order to find hedgehog points on the barycenter scheme we will use the construction of \emph{apolar algebras}. More thorough introduction of this topic can be found in~\cite{Jel17}.
\begin{definition}~\label{def:duality}
For $P = k[x_1, \dots, x_n]$ let $\alpha_1 := \frac{\partial}{\partial x_1}, \dots, \alpha_n := \frac{\partial}{\partial x_n}$. Then the polynomial algebra $S = k[\alpha_1, \dots, \alpha_n]$ generated by commuting derivatives acts on $P$ by taking derivatives. This action is called evaluation and is denoted by:
\begin{equation}
\ \wcirc \ : S \times P \to P.
\end{equation}
For $f \in P$ we also define $ev_f : S \to P$ by the formula $ev_f(h) := h \ \wcirc \ f$.
If $I \subset P$ is a $P$-submodule we define its dual as $I^{\perp} := Ann(I) = \{ h(\alpha_1, \dots, \alpha_n) \in k[\alpha_1, \dots, \alpha_n] : ev_f(h) = 0 \textnormal{ for all } f \in I\}$.
The algebra $Apolar(I) := k[\alpha_1, \dots, \alpha_n]/I^{\perp}$ is called the \emph{apolar algebra} of $I$. When $I$ is generated by a single polynomial $F$ we simply write $Apolar(F)$ for $Apolar(I)$ and $\Fperp$ for $I^{\perp}$.
\end{definition}
\begin{example}
If $I$ in the above definition is generated by $F$, then the ideal $I^{\perp} = \Fperp$ is given by the kernel of the map $ev_F:k[\alpha_1, \dots, \alpha_n] \to k[x_1, \dots, x_n]$. The quotient $Apolar(F) = k[\alpha_1, \dots, \alpha_n]/\Fperp$ is isomorphic (as a linear space over $\kk$) to the image \[ ev_F(k[\alpha_1, \dots, \alpha_n]) = \lin_{\kk}(F, \frac{\partial F}{\partial x_1}, \dots, \frac{\partial F}{\partial x_n}, \frac{\partial^2 F}{\partial x_i \partial x_j}, \dots, 1)\] For example if $F = x^2 + y^2$, then $Apolar(F) \simeq \lin_{\kk}(x^2+y^2, x, y, 1)$ - it is four-dimensional, and so it gives a point in ${\mathcal H}^4$.
Moreover, if $F$ is homogeneous, then so is $\Fperp$ hence $Apolar(F)$ is graded.
\end{example}
Later, in Subsection~\ref{sub:flatness} we will need the relative Macaulay duality, so we introduce it here. We restrict ourselves to the case $n=6$.
\begin{notation}
We define rings of new variables $k[\beta_1, \dots, \beta_6]$ and $k[y_1, \dots, y_6]$ so that $\beta_i := \frac{\partial}{\partial y_i}$. Additionally, we define a new free variable $t$ which will correspond to the torus action.
In this case we get the relative evaluation map:
\begin{equation}
ev : k[t][\alpha_1, \dots, \alpha_6, \beta_1, \dots, \beta_6] \times k[t][x_1, \dots, x_6, y_1, \dots, y_6] \to k[t][x_1, \dots, x_6, y_1, \dots, y_6].
\end{equation}
The action of $k[t, \Bar{\alpha}, \Bar{\beta}]$ on $k[t, \Bar{x}, \Bar{y}]$ will be also denoted by $\wcirc$ so that $ev((a, b)) = a \ \wcirc \ b$. Moreover, if $b \in k[t, \Bar{x}, \Bar{y}]$ then $ev_{b} : k[t, \Bar{\alpha}, \Bar{\beta}] \to k[t, \Bar{x}, \Bar{y}]$ is the map $ev_b(a) := a \ \wcirc \ b$.
Fix a cubic $F$. We write $F_{\Bar{x}} := F(\Bar{x}), F_{\Bar{y}} := F(\Bar{y})$ and $F_{\Bar{x}}^{\perp} \subset k[\alpha_1, \dots, \alpha_6]$, $F_{\Bar{y}}^{\perp} \subset k[\beta_1, \dots, \beta_6]$ for the dual ideals. We will also use polynomials $F_{\Bar{x}} \cdot F_{\Bar{y}}$ or $F(\Bar{x} + \Bar{y}), F(t \cdot \Bar{x} + \Bar{y})$ of mixed variables and in such cases we will usually omit in which ring we are calculating those, unless it is unclear.
\end{notation}
The following polynomial will later yield the fractal family mentioned in the introduction.
\begin{definition}\label{def:gammat}
Suppose $g \in k[\aalpha]$ is such that $g \ \wcirc \ F = 1$ and $(Q_i)_{1 \leq i \leq 6}$ are such that $Q_i \ \wcirc \ F = x_i$.
Let $\Gamma(t) \in k[t, \aalpha, \bbeta]$ be defined by:
\begin{equation}
\Gamma(t) := g(\aalpha) + t \cdot \sum_i \beta_i \cdot Q_i(\aalpha) + t^2 \cdot \sum_i \alpha_i \cdot Q_i(\bbeta) + t^3 \cdot g(\bbeta).
\end{equation}
\end{definition}
\begin{lemma}~\label{fact:Gamma}
The following identity holds:
\begin{equation}
\Gamma(t) \ \wcirc \ (F_{\Bar{x}} \cdot F_{\Bar{y}}) = F(t \cdot \xx + \yy).
\end{equation}
\end{lemma}
\begin{proof}
The idea is to use Taylor's formula. First note that:
\begin{equation}
F(\xx + \yy) = F(\xx) + \sum_i \frac{\partial F}{\partial x_i}(\xx) \cdot y_i + \sum_{i,j} \frac{\partial^2 F}{\partial x_i \partial x_j}(\xx) \cdot y_i y_j + \dots
\end{equation}
and also:
\begin{equation}
F(\xx + \yy) = F(\yy) + \sum_i \frac{\partial F}{\partial y_i}(\yy) \cdot x_i + \sum_{i,j} \frac{\partial^2 F}{\partial y_i \partial y_j}(\yy) \cdot x_i x_j + \dots
\end{equation}
On the other hand, $F$ is a cubic, so $F(\xx + \yy)$ can be think of as a cubic with variable $\xx$ and coefficients in $k[\yy]$. From the second Taylor expansion we see that the part of $x$-degree zero is $F(\yy)$ and the part of $x$-degree one is $\sum_i \frac{\partial F}{\partial y_i}(\yy) \cdot x_i$. Moreover, using the first Taylor expansion we get that the part of $F(\xx + \yy)$ of $x$-degree at least two is $F(\xx) + \sum_i \frac{\partial F}{\partial x_i}(\xx) \cdot y_i$. Thus in fact:
\begin{equation}
F(\xx + \yy) = F(\xx) + \sum_i \frac{\partial F}{\partial x_i}(\xx) \cdot y_i + \sum_i \frac{\partial F}{\partial y_i}(\yy) \cdot x_i + F(\yy)
\end{equation}
so also:
\begin{equation}
F(t \cdot \xx + \yy) = t^3 \cdot F(\xx) + t^2 \cdot \sum_i \frac{\partial F}{\partial x_i}(\xx) \cdot y_i + t \cdot \sum_i \frac{\partial F}{\partial y_i}(\yy) \cdot x_i + F(\yy)
\end{equation}
By the assumptions about $g$'s and $Q_i$'s action on $F$ from Definition~\ref{def:gammat} we get the conclusion.
\end{proof}
The following observation will be useful:
\begin{lemma}\label{fact:fracfamilyform}
Let $(F_{\xx} \cdot F_{\yy})^{\perp} \subset k[t, \aalpha, \bbeta]$ and $\Fperpx \subset k[\aalpha], \Fperpy \subset k[\bbeta]$. Then:
\begin{equation}
(F_{\xx} \cdot F_{\yy})^{\perp} = (\Fperpx, \Fperpy).
\end{equation}
\end{lemma}
\begin{proof}
Consider the composition of the evaluation maps on the diagram:
\[\begin{tikzcd}
{k[t, \aalpha, \bbeta]} & {k[t, \xx, \bbeta]} & {k[t, \xx, \yy]}
\arrow["{ev_{F(\xx)}}", from=1-1, to=1-2]
\arrow["{ev_{F(\yy)}}", from=1-2, to=1-3]
\arrow["{ev_{(F_{\xx} \cdot F_{\yy})}}"', bend right, from=1-1, to=1-3]
\end{tikzcd}\]
Note that $\ker ev_{F(\xx)} = k[t, \bbeta] \cdot \Fperpx, \ker ev_{F(\yy)} = k[t, \xx] \cdot \Fperpy$ and $(ev_{F(\xx)})(k[t, \aalpha] \cdot \Fperpy) = k[t, \xx] \cdot \Fperpy$. Take $f \in (ev_{F(\xx)})^{-1}(k[t, \xx] \cdot \Fperpy)$ i.e. in the kernel of the composition. This means that $ev_{F(\xx)} (f) \in k[t, \xx] \cdot \Fperpy$. Take $h \in k[t, \aalpha] \cdot \Fperpy$ such that $ev_{F(\xx)}(h) = ev_{F(\xx)}(f)$. Then $f-h \in \ker ev_{F(\xx)} = k[t, \bbeta] \cdot \Fperpx$. We present $f$ as:
\begin{equation}
f = (f-h) + h \in k[t, \bbeta] \cdot \Fperpx + k[t, \aalpha] \cdot \Fperpy.
\end{equation}
Thus, we have an inclusion $(ev_{F(\xx)})^{-1}(k[t, \xx] \cdot \Fperpy) \subset k[t, \bbeta] \cdot \Fperpx + k[t, \aalpha] \cdot \Fperpy$, since $f$ was an arbitrary element of the left hand side. The other inclusion is immediate.
On the other hand the composition of $ev_{F(\xx)}$ and $ev_{F(\yy)}$ is $ev_{(F_{\xx} \cdot F_{\yy})}$ so by the definition its kernel is $(F_{\xx} \cdot F_{\yy})^{\perp}$. This finishes the proof.
\end{proof}
Also, note that:
\begin{lemma}~\label{fact:free}
The $k[t, \bbeta]$-algebras $\frac{k[t, \aalpha, \bbeta]}{F(t \cdot \xx + \yy)^{\perp}}$ and $\frac{k[t, \bbeta]}{\Fperpy}$ are isomorphic.
\end{lemma}
\begin{proof}
Consider the natural $k$-algebra homomorphism $\mu : k[t, \bbeta] \to \frac{k[t, \aalpha, \bbeta]}{F(\xx + \yy)^{\perp}}$.
First we show that this map is onto. Note that:
\begin{equation}
(\alpha_i - t \cdot \beta_i) \ \wcirc \ F(t \cdot \xx + \yy) = (\frac{\partial}{\partial x_i} - t \cdot \frac{\partial}{\partial y_i})(F(t \cdot \xx + \yy)) = (\frac{\partial F}{\partial x_i}) (t \cdot \xx + \yy) \cdot t - t \cdot (\frac{\partial F}{\partial x_i}) (t \cdot \xx + \yy) = 0
\end{equation}
so $\alpha_i \equiv t \cdot \beta_i \ (mod \ (t \cdot \xx + \yy)^{\perp})$, and we get surjectivity. We will prove now that the kernel of $\mu$ is $\Fperpy$. Take $f = f(\bbeta, t) \in \ker \mu$. This means that:
\begin{equation}
f(\bbeta, t) \ \wcirc \ F(t \cdot \xx + \yy) = 0.
\end{equation}
Let $f(\bbeta, t) = \sum_i f_i(\bbeta) \cdot t^i$. Then:
\begin{equation}
f(\bbeta, t) \ \wcirc \ F(t \cdot \xx + \yy) = \sum_i (f_i(\bbeta) \ \wcirc \ F(t \cdot \xx + \yy) ) \cdot t^i = \sum_i (f_i(\bbeta) \ \wcirc \ F_{\yy})(t \cdot \xx + \yy) \cdot t^i.
\end{equation}
by the chain rule. Suppose (for a contradiction) that there exist $i$ such that $f_i(\bbeta) \ \wcirc \ F_{\yy} \neq 0$. Take the smallest such $i$. Then the polynomial in variables $\xx, \yy$ near $t^i$ in $f(\bbeta, t) \ \wcirc \ F(t \cdot \xx + \yy)$ has to be $0$, so $(f_i(\bbeta) \ \wcirc \ F_{\yy})(t \cdot \xx + \yy)$ has all coefficients near $x^i y^j$ zero. But this is only possible if $f_i(\bbeta) \ \wcirc \ F_{\yy} = 0$ which gives a contradiction. Thus all functions $f_i$ belong to $\Fperpy$ and so $f \in \Fperpy$ which finishes the proof.
\end{proof}
This lemma yields an immediate corollary:
\begin{corollary}~\label{cor:freeness}
The algebra $\frac{k[t, \aalpha, \bbeta]}{F(t \cdot \xx + \yy)^{\perp}}$ is a free $k[t]$-module, because:
\begin{equation}
\frac{k[t, \aalpha, \bbeta]}{F(t \cdot \xx + \yy)^{\perp}} \simeq \frac{k[t, \bbeta]}{\Fperpy} \simeq k[t] \otimes_k \frac{k[\bbeta]}{\Fperpy}.
\end{equation}
\end{corollary}
\section{The main example}\label{sec:main}
In this section we fix symbols $i, s, \pi$ for the maps:
\[\begin{tikzcd}
{{\mathcal B}^{+}} & {{\mathcal B}} \\
{{\mathcal B}^{\G_m}}
\arrow["\pi", from=1-1, to=2-1]
\arrow["i"', from=1-1, to=1-2]
\arrow["s", bend left, from=2-1, to=1-1]
\end{tikzcd}\]
on the \BBname{} decomposition for the $\G_m$-action on ${\mathcal B}$ induced by the $\G_m$-action on ${\mathcal H}$. This makes sense as ${\mathcal B} \subset {\mathcal H}$ is $\G_m$-invariant by Proposition~\ref{prop:barisgminv}.
We start by proving the Hedgehog point theorem which we recall here:
\begin{theorem}[Hedgehog point theorem]\label{thm:Hedgehog_point_theorem:precise}
A hedgehog point $[I] \in {\mathcal B}$ is non-reduced. Hence $(0, [I]) \in \Affine^n \times {\mathcal B} \simeq {\mathcal H}$ is non-reduced as well.
\end{theorem}
\begin{proof}
Assume on the contrary that ${\mathcal B}$ is reduced at $[I]$.
By Proposition~\ref{ref:comparison:intro:prop} the map $i$ is an isomorphism on some open neighbourhood of $[I]$, because $(T_{[I]} {\mathcal B})_{>0} = 0$ as $[I]$ is a hedgehog point. Hence, ${\mathcal B}^+$ is also reduced at $[I]$.
From affineness of $\pi$, see Theorem~\ref{ref:introRepresentability:thm}, we get that ${\mathcal B}^+$ corresponds to some sheaf ${\mathcal M}$ of ${\mathcal O}_{{\mathcal B}^{\G_m}}$-algebras and $\pi$ is the natural morphism from the relative $\Spec$ of ${\mathcal M}$ to ${\mathcal B}^{\G_m}$. Moreover, the sheaf ${\mathcal M}$ has an $\mathbb{N}$-grading (as an ${\mathcal O}_{{\mathcal B}^{\G_m}}$-module) coming from $\Affine^1$-action. Thus, with a convention that $\dim \emptyset = -1$, for a point $p \in {\mathcal B}^{\G_m}$ we get:
\begin{equation}
\dim \pi^{-1}(p) = \dim \Spec_{{\mathcal O}_{{\mathcal B}^{\G_m}}}({\mathcal M})|_{ \{ p \} } = \dim \Proj_{{\mathcal O}_{{\mathcal B}^{\G_m}}}({\mathcal M})|_{ \{ p \} } + 1
\end{equation}
Thanks to the third condition of being a hedgehog point we know that the fiber of $\pi$ over $[I]$ is $0$-dimensional, so $\dim \Proj_{{\mathcal O}_{{\mathcal B}^{\G_m}}}({\mathcal M})|_{ \{ [I] \} } = -1$. By upper semi-continuity of fibers of projective morphisms (e.g.~\cite[Theorem 12.8]{Hartshorne}) there is an open neighbourhood $U$ of $[I]$ inside ${\mathcal B}^{\G_m}$ such that for every $p \in U$ we have $\dim \Proj_{{\mathcal O}_{{\mathcal B}^{\G_m}}}({\mathcal M})|_{ \{ p \} } = -1$, so $\dim \pi^{-1}(p) = 0$. These fibers all have a unique fixed point of $\Affine^1$-action so they are all topologically points. Thus we get a morphism:
\[\begin{tikzcd}
{\pi^{-1}(U)} \\
U
\arrow["\pi|_{\pi^{-1}(U)}", from=1-1, to=2-1]
\end{tikzcd}\]
which is affine and on the level of sets is a bijection. Note that the section $s|_U$ is the inverse to $\pi|_{\pi^{-1}(U)}$ on the level of topological spaces, so $\pi|_{\pi^{-1}(U)}$ is in fact a homeomorphism. Consider the diagram:
\[\begin{tikzcd}
U && {\pi^{-1}(U)} \\
& U
\arrow["{=}"', from=1-1, to=2-2]
\arrow["s|_U", from=1-1, to=1-3]
\arrow["\pi|_{\pi^{-1}(U)}", from=1-3, to=2-2]
\end{tikzcd}\]
The morphism $\pi$ is affine, so separated. Thus, by Cancellation theorem~\cite[10.1.19]{Vakil} we get that $s|_U$ is a closed embedding. But it also a homeomorphism, so it is in fact an isomorphism of schemes. Hence, $T_{[I]} {\mathcal B}^{\G_m} = T_{[I]} U' \simeq T_{[I]} \pi^{-1}(U') = T_{[I]} {\mathcal B}^+ = T_{[I]} {\mathcal B}$. This gives us a contradiction, because $T_{[I]} {\mathcal B}^{\G_m} = (T_{[I]} {\mathcal B})_0$ and $[I]$ is a hedgehog point so $T_{[I]} {\mathcal B} \neq (T_{[I]} {\mathcal B})_0$.
\end{proof}
Now we concentrate on the search for hedgehog points. We fix $n=6$ and $d=13$. Therefore ${\mathcal H} = \mathrm{Hilb}^{13}_{\Affine^6/\kk}$ and ${\mathcal B}$ is the corresponding barycenter scheme so ${\mathcal H} \simeq \Affine^6 \times {\mathcal B}$. Moreover, ${\mathcal H}^{14}$ denotes the Hilbert scheme of $14$ points on $\Affine^6$. We start with the definition of being general enough:
\begin{definition}~\label{def:nice}
Let $F$ be a cubic in six variables $x_1, \dots, x_6$. We call $F$ \emph{general enough} if the following conditions are satisfied:
\begin{enumerate}
%
\item $(F, \frac{\partial F}{\partial x_1}, \dots, \frac{\partial F}{\partial x_6}, x_1, \dots, x_6, 1)$ is a basis of $\im(ev_F)$ so that $\Fperp$ gives a point $[\Fperp] \in {\mathcal H}^{14}$.
%
\item The minimal graded free resolution of $\Fperp$ starts with:
\[\begin{tikzcd}
0 & \Fperp & {S^{\oplus 15}(-2)} & {S^{\oplus 35}(-3)} & \dots
\arrow[from=1-5, to=1-4]
\arrow[from=1-4, to=1-3]
\arrow[from=1-3, to=1-2]
\arrow[from=1-2, to=1-1]
\end{tikzcd}\]
%
\item $[\Fperp] \in {\mathcal H}^{14}$ has TNT (which is equivalent to $\dim (T_{[\Fperp]} {\mathcal H}^{14})_{<0} = 6$, see Definition~\ref{def:TNT}).
%
\end{enumerate}
\end{definition}
Let us elaborate about this definition for a while. First, condition (1) assures that $[\Fperp]$ gives us a point in ${\mathcal H}^{14}$. Secondly, the condition about the minimal graded free resolution encodes two properties: it implies that (a) $\Fperp$ is generated by its second degree $(\Fperp)_2$ (which is $15$-dimensional, because $\dim_k (\Fperp)_2 = \dim_k S_2 - \dim_k (S/\Fperp)_2 = 21 - 6 = 15$), and (b) the kernel of the surjection $\Fperp \leftarrow S^{\oplus 15}(-2)$ is generated by the third degree, so just by the relations between generators of $\Fperp$ multiplied by linear forms. More thorough analysis of this resolution will appear in Subsection~\ref{sub:tangent}. The TNT condition assures the tame behaviour of tangent space to $[\Fperp]$.
Notice that all of these conditions are open (on the moduli space of cubics). Indeed, condition (1) is open, because it can be translated into a statement about the rank of some minors of $ev_F:S \to P$. The openness of condition (2) follows from semi-continuity of Betti numbers (for the definition of Betti numbers see~\cite[Chapter 1B]{Eis1}). At last, the openness of condition (3) is implied by the semi-continuity of dimension of a fixed degree to a point in $({\mathcal H}^{14})^{\G_m}$. It is because if we restrict the tangent sheaf to $({\mathcal H}^{14})^{\G_m}$ it decomposes as a sum of sheaves of integer degrees and the result follows from half-continuity of dimension of a coherent sheaf on a scheme. Moreover, the set of general enough cubics is not only open, but also non-empty, as the below example shows:
\begin{example}\label{ex:nonempty}
Let $F = x_1 x_2 x_4 - x_1 x_5^2 + x_2 x_3^2 + x_3 x_5 x_6 + x_4 x_6^2$ as in Example~\ref{ex:first}. One can check by hand, or using a computer, that this $F$ satisfies conditions (1) and (2) in Definition~\ref{def:nice}. The fact that it satisfies the condition (3) can be deduced from~\cite[lemma 3.6]{Jel18}.
\end{example}
Now we describe a procedure of making a point $[I] \in {\mathcal H}$ from a general enough cubic $F$. By the definition of being general enough the $\kk$-linear space $\lin_{\kk}(F, \frac{\partial F}{\partial x_1}, \dots, \frac{\partial F}{\partial x_6}, \frac{\partial^2 F}{\partial x_i \partial x_j}, \dots, 1)$ is $14$-dimensional and is generated by a basis $(F, \frac{\partial F}{\partial x_1}, \dots, \frac{\partial F}{\partial x_6}, x_1, \dots, x_6, 1)$. Then it follows that the algebra $S/\Fperp$ has a natural grading with the Hilbert function $(1,6,6,1)$. Hence the third degree of $S/\Fperp$ is generated by a single element $g$ and we can assume that $ev_F(g) = 1$. We fix a representative of the class of $g$ in $S$ and denote it also by $g$.
Let $I = (\Fperp, g) = \Fperp + S_{\geq 3} \subset S$. Then $S/I \simeq Apolar(F)/(g)$ is $13$-dimensonal, so it represents a point $[I] \in {\mathcal H}$. Note that the ideal $I$ is homogeneous and the only maximal ideal which contains it is the ideal of $0 \in \Affine^6$. Thus, $[I]$ is in fact an element of ${\mathcal B}^{\G_m} \subset {\mathcal B} \subset {\mathcal H}$ by the description of the Hilbert-Chow morphism in Theorem~\ref{thm:HC} and Definition~\ref{def:Bar}.
In this section we fix a general enough cubic $F$ and $g$ such that $ev_F(g)=1$. Also, $[I]$ denotes a point of ${\mathcal H}$ constructed from $F$ by taking $I = (\Fperp, g)$.
For our applications, the analysis of the $\pi$-fiber over $[I]$ will be important, so we fix a notation $V := \pi^{-1}([I])$. Recall that by Theorem~\ref{ref:introRepresentability:thm}, the fiber $V$ has a natural $\Affine^1$-action and it is affine as the inverse image of an affine subscheme of ${\mathcal B}^{\G_m}$.
The structure of this section is as follows. First we study the tangent space $T_{[I]} {\mathcal H}$ and check that it satisfies assumptions of Proposition~\ref{ref:comparison:intro:prop}. Then we take a closer look at the resolution of the ideal $I$ and using primary obstructions we prove that $V$ is $0$-dimensional. At the end we study a deformation of $S/I$ coming from the fractal family to determine the structure of $V$ completely.
\subsection{The analysis of the point $[I]$}~\label{sub:tangent}
We start with the following proposition:
\begin{proposition}\label{prop:nopositiveH}
The induced action of $\G_m$ on $T_{[I]} {\mathcal H}$ has no positive degrees.
\end{proposition}
\begin{proof}
Suppose $k \geq 1$ and take $\delta \in \Hom_S(I, S/I)_k$. Then $\delta(I_2) \subset (S/I)_{2+k} \subset (S/I)_{\geq 3} = 0$ because $S/I$ has its Hilbert function $(1, 6, 6)$. Similarly, $\delta(I_3) = \delta(I_4) = \dots = 0$. Moreover $I$ is generated by its second and third degree (more precisely by $(\Fperp)_2$ and $g$, because $F$ is general enough) so $\delta(I) = 0$. Thus we have proved that $\Hom_S(I, S/I)_k = 0$ for $k \geq 1$.
\end{proof}
\begin{corollary}~\label{cor:nopositivedeg}
The induced action of $\G_m$ on $T_{[I]} {\mathcal B}$ has no positive degrees.
\end{corollary}
\begin{proof}
This follows from the Proposition~\ref{prop:nopositiveH}, because $T_{[I]} {\mathcal B}$ is a linear subspace of $T_{[I]} {\mathcal H}$.
\end{proof}
\begin{corollary}~\label{cor:bplusisb}
The map $i: {\mathcal B}^+ \to {\mathcal B}$ is an open embedding near $[I]$ i.e. there exist an open $\G_m$-invariant neighbourhood $U \subset {\mathcal B}^+$ of $s([I])$ such that $i_{|U} : U \to {\mathcal B}$ is an open immersion. In particular, we identify $T_{[I]} {\mathcal B}^+ = T_{[I]} {\mathcal B}$.
\end{corollary}
\begin{proof}
This follows from Corollary~\ref{cor:nopositivedeg} and Proposition~\ref{ref:comparison:intro:prop}.
\end{proof}
To study a neighbourhood of $[I] \in {\mathcal H}$ and make use of obstruction theory on ${\mathcal H}$ we need to analyse resolutions of $\Fperp$ and $I$. First, we take the minimal graded free resolution of the $S$-module $\Fperp$ which is of the form below, because $F$ is general enough (Definition~\ref{def:nice}):
\[\begin{tikzcd}
0 & \Fperp & {S^{\oplus 15}(-2)} & {S^{\oplus 35}(-3)} & \dots
\arrow["d_1"', from=1-4, to=1-3]
\arrow["d_0"', from=1-3, to=1-2]
\arrow[from=1-2, to=1-1]
\arrow[from=1-5, to=1-4]
\end{tikzcd}\]
Here $d_0$ sends the natural basis $(E_i)_{1 \leq i \leq 15}$ of $S^{\oplus 15}(-2)$ to fifteen generators of $\Fperp$. The ideal $\Fperp$ is generated by $(\Fperp)_2$ (by $F$ being general enough) which is $15$-dimensional as $\dim_k (\Fperp)_2 = \dim_k S_2 - \dim_k (S/\Fperp)_2 = 21 - 6 = 15$. The number $35$ appears, because if we look at the third degree of $0 \leftarrow \Fperp \leftarrow {S^{\oplus 15}(-2)}$ then we get $0 \leftarrow (\Fperp)_3 \leftarrow S_1 \otimes_{\kk} \bigoplus_{i=1}^{15} \kk E_i$ so the kernel of the map $d_0$ is at least $6 \cdot 15 - \dim_k (\Fperp)_3 = 90 - (\dim_k S_3 - \dim_k (S/\Fperp)_3) = 90 - ({8\choose 3} - 1) = 35$-dimensional.
Next we readjust this resolution so that it works for $I$:
\[\begin{tikzcd}
0 & I & {S^{\oplus 15}(-2) \oplus S(-3)} & {S^{\oplus 35}(-3) \oplus S^{\oplus 6}(-4)} & \dots
\arrow["{d_1}"', from=1-4, to=1-3]
\arrow["{d_0}"', from=1-3, to=1-2]
\arrow[from=1-2, to=1-1]
\arrow["{d_2}"', from=1-5, to=1-4]
\end{tikzcd}\]
Here for $G$ a generator of $S(-3)$, we define $d_0(G) = g$, and for $(H_k)_{1 \leq k \leq 6}$ - generators of $S^{\oplus 6}(-4)$, we put $d_1(H_k) = \sum_{i = 1}^{15} b_{ki} \cdot E_i + \alpha_k \cdot G$ for quadrics $b_{ki}$'s such that $\alpha_k \cdot g + \sum_{i = 1}^{15} b_{ki} \cdot d_0(E_i) = 0$. The existence of such quadrics follows from the fact that $(\Fperp)_4 = S_4$. The ideal $I$ is generated by $\Fperp$ and $g$, so $d_0$ is surjective. Moreover, for $f \in \ker d_0$ there exists $h$ in $S^{\oplus 6}(-4)$ such that $f - d_1(h) \in \ker d_0 \cap S^{\oplus 15}(-2)$. But $\ker d_0 \cap S^{\oplus 15}(-2) = d_1(S^{\oplus 35}(-3))$, because we have started from a resolution of $\Fperp$.
Summarising, we got a resolution of $I$. Using it, we can produce new negative tangents to $[I] \in {\mathcal H}$. Take $Q \in (S/I)_2$. Let $\symbtantwo_Q : S^{\oplus 15}(-2) \oplus S(-3) \to S/I$ be an $S$-module homomorphism defined by $\symbtantwo_Q(E_i) = 0$ and $\symbtantwo_Q(G)=Q$. Consider the following diagram:
\[\begin{tikzcd}
0 & I & {S^{\oplus 15}(-2) \oplus S(-3)} & {S^{\oplus 35}(-3) \oplus S^{\oplus 6}(-4)} \\
&& {S/I}
\arrow["{d_1}"', from=1-4, to=1-3]
\arrow["{d_0}"', from=1-3, to=1-2]
\arrow[from=1-2, to=1-1]
\arrow["{\symbtantwo_Q}"', from=1-3, to=2-3]
\arrow["0", from=1-4, to=2-3]
\arrow["{\symbtan_Q}"', dashed, from=1-2, to=2-3]
\end{tikzcd}\]
Then $\symbtantwo_Q \circ d_1 = 0$, because $\symbtantwo_Q(d_1(S^{\oplus 35}(-3))) \subset \symbtantwo_Q(S^{\oplus 15}(-2)) = 0$ and $\symbtantwo_Q(d_1(H_k)) = \symbtantwo_Q(\sum_{i = 1}^{15} b_{ki} \cdot E_i + \alpha_k \cdot G) = \alpha_k \cdot Q = 0$ as $(S/I)_3 = 0$. Hence, by the exactness of the sequence, we get that $\symbtantwo_Q$ factors and gives us an $S$-module homomorphism which we will call $\symbtan_Q$.
\begin{definition}~\label{def:subspacew}
Let $X := \lin_{\kk}(\symbtan_Q : Q \in (S/I)_2) < \Hom_{S}(I, S/I)$ for $\symbtan_Q$ defined as above, so:
\begin{equation}~\label{eq:defofphi}
\begin{matrix}
\symbtan_Q : I \to S/I \\
(\Fperp)_2 \mapsto 0 \\
g \mapsto Q
\end{matrix}
\end{equation}
Fix $Q_1, \dots, Q_6 \in S$ such that $ev_F(Q_i) = x_i$. Such elements exist because $F$ is general enough. Their classes in $S/I$, which we will denote also by $Q_1, \dots, Q_6$, form a basis of $(S/I)_2$. It follows that $\symbtan_{Q_1}, \dots \symbtan_{Q_6}$ is a basis of $X$. We will also write $\symbtan_i$ for $\symbtan_{Q_i}$.
\end{definition}
\begin{example}~\label{ex:tangentsw}
We will determine the element $[I'] \in \Mor \left( \Spec( k[\eps] ), {\mathcal H} \ | \ \{\textnormal{pt.}\} \mapsto [I] \right) = T_{[I]} {\mathcal H}$ which corresponds to $\symbtan_Q \in W$.
The point $[I']$ is given by an ideal $I' \subset S[\eps]$. By Remark~\ref{rem:tangentinverse} we have:
\[I' = (f - \symbtan_Q (f) \cdot \eps : f \in I).\]
But since $I$ is generated by $\Fperp$ and $g$ we get:
\[ (f - \symbtan_Q (f) \cdot \eps : f \in I) = \Fperp \cdot S[\eps] + (g - \eps Q) \cdot S[\eps].\]
So $I' = \Fperp \cdot S[\eps] + (g - \eps Q) \cdot S[\eps]$.
The $k$-algebra $k[\eps]$ is local, so $\frac{S[\eps]}{I'}$ as a flat, finitely generated $k[\eps]$-module is free. Consider elements $(1, \alpha_1, \dots, \alpha_6, Q_1(\aalpha), \dots, Q_6(\aalpha))$. Their classes mod $(\eps)$ form a basis of $\frac{S}{I}$, so by Lemma~\ref{fact:basislift} we get that $(1, \alpha_1, \dots, \alpha_6, Q_1(\aalpha), \dots, Q_6(\aalpha))$ is a basis of the $k[\eps]$-module $\frac{S[\eps]}{I'}$.
\end{example}
\begin{remark}\label{rem:identification_evaluation}
The evaluation map allows us to define an isomorphism:
\begin{equation}\label{eq:iso_evaluation}
\lambda : X \to k[x_1, \dots, x_6]_1
\end{equation}
by sending $\symbtan_Q$ to $ev_F(Q) \in k[x_1, \dots, x_6]_1$. This identification is natural and under it tangents $\symbtan_{1}, \dots \symbtan_{6}$ correspond to $x_1, \dots, x_6$.
\end{remark}
The map $\lambda$ from Remark~\ref{rem:identification_evaluation} allows us to naturally shift subspace $X$:
\begin{definition}\label{def:subspacey}
Let $\nabla : k[x_1, \dots, x_6]_1 \to k[\partial_1, \dots, \partial_6]_1$ be an isomorphism sending $x_i$ to $\partial_i$. We define:
\begin{equation}
Y := \lin_k(\symbtan_Q - \frac{1}{13} \nabla(\lambda(\symbtan_Q)) : Q \in (S/I)_2).
\end{equation}
In other words $Y = \lin_k(\symbtan_i - \frac{1}{13} \partial_i : i=1, \dots, 6)$. We will write $\symboly_i$ for $\symbtan_i - \frac{1}{13} \partial_i$ so that $Y = \lin_k(\symboly_i : i=1, \dots, 6)$.
\end{definition}
Now we are ready to prove the following geometric characterization of the subspace $Y$:
\begin{proposition}~\label{prop:subspacew}
The subspace $Y$ is the tangent space to the fiber of $\pi : {\mathcal B}^+ \to {\mathcal B}^{\G_m}$ over $[I] \in {\mathcal B}^{\G_m}$. In other words $Y = T_{[I]} V$ under the chain of inclusions:
\begin{equation}
T_{[I]} V < T_{[I]} {\mathcal B}^+ = T_{[I]} {\mathcal B} < T_{[I]} {\mathcal H} = \Hom_{S}(I, S/I).
\end{equation}
\end{proposition}
\begin{proof}
From Proposition~\ref{prop:hilbandbar} we know that ${\mathcal H} = {\mathcal B} \times \Affine^6$, so $T_{[I]} {\mathcal H} = T_{[I]} {\mathcal B} \oplus T_{ \{0\} } \Affine^6$ and the additive action of $v \in \Affine^6(\kk)$ on $[I] = ([I], 0) \in {\mathcal B} \times \Affine^6$ is given by $v + ([I], 0) = ([I], v)$. Thus the subspace $T_{ \{0\} } \Affine^6 < T_{[I]} {\mathcal H}$ is the image of the tangent map of the morphism of $[I]$'s $\Affine^6$-orbit in ${\mathcal H}$.
By Theorem~\ref{thm:TNTdescribtion} we know explicit formulas for elements of this subspace, so we get:
\begin{equation}\label{eq:subspacew1}
T_{[I]} {\mathcal H} = T_{[I]} {\mathcal B} \oplus \lin_k(\partial_1, \dots, \partial_6).
\end{equation}
By Corollary~\ref{cor:bplusisb} we have $T_{[I]} {\mathcal B}^+ = T_{[I]} {\mathcal B}$. The morphisms $\pi, s$ giving the diagram:
\[\begin{tikzcd}
{\mathcal B}^+ \\
{{\mathcal B}^{\G_m}}
\arrow["\pi", from=1-1, to=2-1]
\arrow["s", shift left=1, bend left, from=2-1, to=1-1]
\end{tikzcd}\]
are $\G_m$-equivariant and induce a $\G_m$-invariant decomposition $T_{[I]} {\mathcal B}^+ = \ker d\pi \oplus \im ds$ where a $d$ means the induced morphism on tangent spaces at $[I]$. By the functorial description of ${\mathcal B}^{\G_m}$ we have $T_{[I]} {\mathcal B}^{\G_m} \simeq \im ds = (T_{[I]} {\mathcal B})_0$. Thus, by $\G_m$-invariance $\ker d\pi = (T_{[I]} {\mathcal B})_{\neq 0} = (T_{[I]} {\mathcal B})_{< 0}$ where the last equality holds, because $T_{[I]} {\mathcal B}$ doesn't have any positive degrees by Corollary~\ref{cor:nopositivedeg}. By the definition of $V$ we see that $T_{[I]} V = \ker d\pi < T_{[I]} {\mathcal B}$. Hence, overall we get that:
\begin{equation}\label{eq:subspacew2}
T_{[I]} V = (T_{[I]} {\mathcal B})_{< 0} < T_{[I]} {\mathcal H}.
\end{equation}
In order to determine $(T_{[I]} {\mathcal B})_{< 0}$ we calculate $(T_{[I]} {\mathcal H})_{< 0}$. Take an arbitrary $\delta \in \Hom_S(I, S/I)$ homogeneous of negative degree. It induces the maps from the resolution of $I$:
\[\begin{tikzcd}
0 & I & {S^{\oplus 15}(-2) \oplus S(-3)} & {S^{\oplus 35}(-3) \oplus S^{\oplus 6}(-4)} & \dots \\
&& {S/I}
\arrow[from=1-5, to=1-4]
\arrow["{d_1}"', from=1-4, to=1-3]
\arrow["{d_0}"', from=1-3, to=1-2]
\arrow[from=1-2, to=1-1]
\arrow["\delta"', from=1-2, to=2-3]
\arrow["{\Bar{\delta}}"', from=1-3, to=2-3]
\arrow["0", from=1-4, to=2-3]
\end{tikzcd}\]
where $\Bar{\delta}$ is just the composition. By restricting $\delta$ to $\Fperp$ and $\Bar{\delta}$ to $S^{\oplus 15}(-2)$ we get the diagram:
\[\begin{tikzcd}
0 & \Fperp & {S^{\oplus 15}(-2)} & {S^{\oplus 35}(-3)} & \dots \\
&& {S/I} \\
&& {S/\Fperp}
\arrow[from=1-5, to=1-4]
\arrow["{d_1}"', from=1-4, to=1-3]
\arrow["{d_0}"', from=1-3, to=1-2]
\arrow[from=1-2, to=1-1]
\arrow["{\Bar{\delta}}"', from=1-3, to=2-3]
\arrow["0", from=1-4, to=2-3]
\arrow["\mu", from=3-3, to=2-3]
\arrow["\delta"', from=1-2, to=2-3]
\arrow["{\delta'}"', bend right, dashed, from=1-2, to=3-3]
\arrow["0", bend left, dashed, from=1-4, to=3-3]
\end{tikzcd}\]
where $\mu$ is the natural map. The homomorphism $\delta$ is of negative degree so $\Bar{\delta}$ sends generators $E_i$'s of $S^{\oplus 15}(-2)$ to $(S/I)_{\leq 1}$. However, the quotient $\mu$ is an isomorphism in degrees $\leq 2$. So, in fact $\Bar{\delta}$ factor through $S/\Fperp$ and the composition with $d_1$ is still $0$. Thus, there exists the unique $\delta'$ such that the above diagram commutes. But this $\delta'$ belongs to $\Hom_S(\Fperp, S/\Fperp) \simeq T_{ [\Fperp] } {\mathcal H}^{14}$ so it can only be a linear combination of derivatives $\partial_i$ (because $F$ is general enough so in particular $[\Fperp] \in {\mathcal H}^{14}$ has TNT).
Now we will consider two cases:
\begin{enumerate}
\item $\delta$ is of degree $\leq -2$: Then $\delta'$ is also of degree $\leq -2$ so $\delta' = 0$, because partial derivatives are of degree $-1$ and $\delta'$ is their linear combination. Thus, $\delta|_{(\Fperp)_2} = 0$ or equivalently $\bar{\delta}|_{S^{\oplus 15}(-2)} = 0$. Let $\bar{\delta}(G) = \delta(g) =: s \in (S/I)_{\leq 1}$. We know that $\bar{\delta} \circ d_1 = 0$, so:
\begin{equation}
0 = \bar{\delta} \circ d_1 ( H_k ) = \bar{\delta} (\sum_{i = 1}^{15} b_{ki} \cdot E_i + \alpha_k \cdot G) = \bar{\delta} (\alpha_k \cdot G) = \alpha_k \cdot s \in (S/I)_{\leq 2}
\end{equation}
Note that $(S/I)_{\leq 2} \simeq (S/\Fperp)_{\leq 2}$, so an element from $(S/I)_{\leq 2}$ is $0$ if and only if its evaluation on $F$ is $0$. Hence, we get:
\begin{equation}
ev_F( \alpha_k \cdot s) = \frac{\partial}{\partial x_k} ev_F(s) = 0 \textnormal{ for all $k=1, \dots, 6$}
\end{equation}
The polynomial $ev_F(s)$ is of degree $\geq 2$, because $s \in (S/I)_{\leq 1}$. But this means that $ev_F(s) = 0$ because it has all partial derivatives equal to $0$. Thus $s=0$ in $S/I$ and so in this case $\delta = 0$.
\item $\delta$ is of degree $-1$: A homomorphism $\delta' \in \Hom_S(\Fperp, S/\Fperp)$ is a linear combination of partial derivatives in this case, because $[\Fperp] \in {\mathcal H}^{14}$ has TNT. If $\partial := \sum_{i=1}^6 c_i \partial_i$ is defined using the same coefficients, but for derivatives $\partial_i \in \Hom_S(I, S/I)$, then $\delta - \partial$ is $0$ under the restriction to $(\Fperp)_2$. Thus $\delta - \partial$ has the properties that $(\Fperp)_2 \mapsto 0$ and $g \mapsto s$ for some element $s$ of $(S/I)_2$, because both $\delta$ and $\partial$ are of degree $-1$. Hence $\delta - \partial \in X$ and we see that $\Hom_S(I, S/I)_{-1} = X \oplus \lin_{\kk}(\partial_1, \dots, \partial_6)$, as we have presented $\delta$ as a sum of elements from $\lin_{\kk}(\partial_1, \dots, \partial_6)$ and $X$ in a unique way, namely $\delta = \partial + (\delta - \partial)$.
\end{enumerate}
Summarising, we have proved that:
\begin{equation}
(T_{[I]} {\mathcal H})_{< 0} = (T_{[I]} {\mathcal H})_{-1} = X \oplus \lin_{\kk}(\partial_1, \dots, \partial_6).
\end{equation}
By Definition~\ref{def:subspacew} and Definition~\ref{def:subspacey} we get that $X \oplus \lin_{\kk}(\partial_1, \dots, \partial_6) = Y \oplus \lin_{\kk}(\partial_1, \dots, \partial_6)$ so we have:
\begin{equation}\label{eq:subspacew3}
(T_{[I]} {\mathcal H})_{< 0} = (T_{[I]} {\mathcal H})_{-1} = Y \oplus \lin_{\kk}(\partial_1, \dots, \partial_6).
\end{equation}
We will now show that $Y$ lies in $T_{[I]} {\mathcal B}$.
By Example~\ref{ex:tangentsw} we have an explicit description of morphisms $\symbtan_Q$ as elements of ${\mathcal H}( \Spec(k[\eps]) )$. We apply Theorem~\ref{thm:HCaffine}. Fix $\symbtan_i \in X$ and consider the composition starting with the morphism coming from $\symbtan_i$:
\[\begin{tikzcd}
{\Spec(k[\eps])} & {\mathcal H} & {{(\Affine^6)^{13} \goodquotient \mathbb{S}_{13}}} & {\Affine^6}
\arrow[from=1-2, to=1-3]
\arrow[from=1-1, to=1-2]
\arrow[from=1-3, to=1-4]
\end{tikzcd}\]
The composition, on the level of rings, gives a homomorphism $k[\aalpha] \to k[\eps]$ which is the composition of:
\[\begin{tikzcd}
{k[\aalpha]} & {(k[\aalpha] \otimes_k \dots \otimes_k k[\aalpha])^{\mathbb{S}_d}} & {(\frac{S[\eps]}{I'} \otimes_{k[\eps]} \dots \otimes_{k[\eps]} \frac{S[\eps]}{I'})^{\mathbb{S}_d}} & {k[\eps]}
\arrow[from=1-2, to=1-3]
\arrow["LNm", from=1-3, to=1-4]
\arrow[from=1-1, to=1-2]
\end{tikzcd}\]
where $I' = \Fperp \cdot S[\eps] + (g - \eps Q_i)$ is the ideal from Example~\ref{ex:tangentsw}. By Theorem~\ref{thm:HCaffine} this whole composition takes $\alpha_j$ to $\frac{1}{13} Tr_{\frac{S[\eps]}{I'}/k[\eps]}(\alpha_j)$. By Example~\ref{ex:tangentsw} we can take $(1, \alpha_1, \dots, \alpha_6, Q_1, \dots, Q_6)$ as a basis of $\frac{S[\eps]}{I'}$ over $k[\eps]$. Let $A = (a_{lm})$ be the matrix of multiplying by $\alpha_j$ with respect to this basis. We analyse its diagonal coefficients:
\begin{itemize}
\item $\alpha_j \cdot 1 = \alpha_j$ so $a_{11} = 0$.
\item $\alpha_j \cdot \alpha_k$ is a $k$-linear combination of polynomials $Q_i$ modulo $\Fperp$, so for $k=1, \dots 6$ we have $a_{kk} = 0$.
\item For $k \neq j$ the polynomial $\alpha_j \cdot Q_k$ lies is $\Fperp$ because $ev_F(\alpha_j \cdot Q_k) = ev_{x_k}(\alpha_j) = 0$. Hence $\alpha_j \cdot Q_k = 0$ in $\frac{S[\eps]}{I'}$ and here the diagonal coefficients are also zero.
\item For $k=j$ the polynomial $\alpha_j \cdot Q_j$ is equal to $g$ modulo $\Fperp$, because $ev_F(\alpha_j \cdot Q_j) = ev_{x_j}(\alpha_j) = 1 = ev_F(g)$. But $g \equiv \eps Q_i (mod \ I')$ because $I' = \Fperp \cdot S[\eps] + (g - \eps Q_i)$. Thus, the coefficient near $Q_j$ is $\delta_{ij} \cdot \eps$ in this case.
\end{itemize}
Overall, $\frac{1}{13} Tr_{\frac{S[\eps]}{I'}/k[\eps]}(\alpha_j) = \frac{1}{13} \delta_{ij} \cdot \eps$, so the induced tangent vector in $T_{ \{0\} } \Affine^6$ is $\frac{1}{13} \partial_i$. Thus the image of $\symboly_i = \symbtan_i - \frac{1}{13} \partial_i$ under that tangent map of $Bar$ is $0$. This implies that $Y < \ker d Bar = T_{[I]} {\mathcal B}$.
Now we bring everything together. First, the equation~\eqref{eq:subspacew1} implies that:
\begin{equation}
(T_{[I]} {\mathcal H})_{<0} = (T_{[I]} {\mathcal B})_{<0} \oplus \lin_k(\partial_1, \dots, \partial_6).
\end{equation}
Furthermore, the equation~\eqref{eq:subspacew3} gives as:
\begin{equation}
(T_{[I]} {\mathcal H})_{<0} = Y \oplus \lin_k(\partial_1, \dots, \partial_6).
\end{equation}
But $Y < T_{[I]} {\mathcal B}$ so also $Y < (T_{[I]} {\mathcal B})_{<0}$ as elements of $Y$ are of degree $-1$. Hence we get:
\begin{equation}
Y = (T_{[I]} {\mathcal B})_{<0} = T_{[I]} V
\end{equation}
where the last equality is the equation~\eqref{eq:subspacew2}.
\end{proof}
By Example~\ref{ex:hilbert} the functor $D_{{\mathcal H}, [I]}$ pro-representable by $\hat{{\mathcal O}}_{{\mathcal H}, [I]}$ has an obstruction theory with obstruction space $Ob = \Ext^1_{S}(I, S/I)$.
Since we want to determine primary obstructions to $V$, the following proposition will make the calculations easier:
\begin{proposition}\label{prop:partial_vanish}
Let $ob_0 : \Sym_2(T_{[I]} {\mathcal H}) \to \Ext^1_{S}(I, S/I)$ be the primary obstruction map. Take $\partial \in \lin_k(\partial_1, \dots, \partial_6) < T_{[I]} {\mathcal H}$ and $\delta \in T_{[I]} {\mathcal H}$. Then $ob_0(\partial \cdot \delta) = 0$.
\end{proposition}
\begin{proof}
The idea is to use $\Affine^n$-action to extend $\delta$ and then use Lemma~\ref{fact:primary_vanish}.
Suppose that $\delta$ correspond to the morphism $\delta : \Spec(\frac{k[\eps]}{(\eps)^2}) \to {\mathcal H}$. By composing:
\[\begin{tikzcd}
{\Affine^n \times \Spec(\frac{k[\eps]}{(\eps)^2})} & {\Affine^n \times {\mathcal H}} & {\mathcal H}
\arrow["\mu"', from=1-2, to=1-3]
\arrow["{id \times \delta}"', from=1-1, to=1-2]
\end{tikzcd}\]
we get a $\Affine^n$-equivariant morphism which extends $\delta$. The tangent vector $\partial$ is in $\lin_k(\partial_1, \dots, \partial_6)$ so it is induced by the $\Affine^n$-action on ${\mathcal H}$. Take a morphism $\bar{\partial} : \Spec(\frac{k[\eps']}{(\eps')^2}) \to \Affine^n$ such that its composition with the orbit map of $[I]$ is $\partial : \Spec(\frac{k[\eps']}{(\eps')^2}) \to {\mathcal H}$. Consider the composition $\eta$:
\[\begin{tikzcd}
{\Spec(\frac{k[\eps']}{(\eps')^2}) \times \Spec(\frac{k[\eps]}{(\eps)^2})} & {\Affine^n \times \Spec(\frac{k[\eps]}{(\eps)^2})} & {\Affine^n \times {\mathcal H}} & {\mathcal H} \\
{\Spec(\frac{k[\eps,\eps']}{(\eps^2, \eps'^2)})}
\arrow["{id \times \delta}", from=1-2, to=1-3]
\arrow["{\bar{\partial} \times id}", from=1-1, to=1-2]
\arrow["{=}"{marking}, draw=none, from=2-1, to=1-1]
\arrow["\eta", bend right=10, from=2-1, to=1-4]
\arrow["\mu", from=1-3, to=1-4]
\end{tikzcd}\]
By its definition $\eta$ restricted to $\Spec(\frac{k[\eps']}{(\eps')^2})$ is $\partial$ and $\eta$ restricted to $\Spec(\frac{k[\eps]}{(\eps)^2})$ is $\delta$. Thus the following diagram commutes:
\[\begin{tikzcd}
{\Spec(\frac{k[\eps,\eps']}{(\eps^2, \eps'^2)})} & {\mathcal H} \\
{\Spec(\frac{k[\eps']}{(\eps')^2}) \sqcup_{\Spec(k)} \Spec(\frac{k[\eps]}{(\eps)^2})} \\
{\Spec(\frac{k[\eps,\eps']}{(\eps, \eps')^2})}
\arrow["\eta", dashed, from=1-1, to=1-2]
\arrow[hook', from=2-1, to=1-1]
\arrow["{\partial \sqcup \delta}"', from=2-1, to=1-2]
\arrow["{=}"{marking}, draw=none, from=3-1, to=2-1]
\end{tikzcd}\]
On the level of algebras this diagram is dual to the diagram in the statement of Lemma~\ref{fact:primary_vanish}. Thus the existence of $\eta$ implies that $ob_0(\partial \cdot \delta)=0$ by Lemma~\ref{fact:primary_vanish}.
\end{proof}
Now we work on the resolution of the ideal $I$ in order to use Theorem~\ref{thm:hilb_obstruction} to calculate primary obstruction map restricted to $\Sym_2(T_{[I]} V)$.
Fix $Q \in (S/I)_2$. For $\symbtan_Q : I \to S/I$ we will construct a lift of $\symbtan_Q$ to a homogeneous chain complex map as in the diagram above Theorem~\ref{thm:hilb_obstruction}. The liftings are not unique but we choose them so that they are amenable to calculations.
We have:
\[\begin{tikzcd}~\label{diagram:hilb_obtruction}
& 0 & I & {S^{\oplus 15}(-2) \oplus S(-3)} & {S^{\oplus 35}(-3) \oplus S^{\oplus 6}(-4)} & F_3 \\
0 & {S/I} & {S} & {S^{\oplus 15}(-2) \oplus S(-3)} & {S^{\oplus 35}(-3) \oplus S^{\oplus 6}(-4)} & \dots
\arrow[from=1-3, to=1-2]
\arrow["{d_0}"', from=1-4, to=1-3]
\arrow["{d_1}"', from=1-5, to=1-4]
\arrow["{d_2}"', from=1-6, to=1-5]
\arrow["{d_2}"', from=2-6, to=2-5]
\arrow["{d_1}"', from=2-5, to=2-4]
\arrow["{d_0}"', from=2-4, to=2-3]
\arrow["q"', from=2-3, to=2-2]
\arrow[from=2-2, to=2-1]
\arrow["\symbtan_Q"'{pos=0.8}, from=1-3, to=2-2]
\arrow["{s_1(\symbtan_Q)}"'{pos=0.7}, shift right=1, shorten <=3pt, from=1-4, to=2-3]
\arrow["{s_2(\symbtan_Q)}"'{pos=0.9}, from=1-5, to=2-4]
\arrow["{s_3(\symbtan_Q)}"'{pos=0.9}, from=1-6, to=2-5]
\end{tikzcd}\]
The maps here are defined in the following way:
\begin{itemize}
\item We define $s_1(\symbtan_Q) : S^{\oplus 15}(-2) \oplus S(-3) \to S$ which takes $S^{\oplus 15}(-2)$ to $0$ and the generator $G$ of $S(-3)$ to $Q \in S$ (by abuse of notation we denote by $Q$ both an element of $S/I$ and some quadric element in the inverse image $q^{-1}(Q) \subset S$). Then it is obvious that $\symbtan_Q \circ d_0 = q \circ s_1(\symbtan_Q)$.
\item The second map to define is $s_2(\symbtan_Q)$. We declare it to be $0$ on $S^{\oplus 35}(-3)$. Now it is enough to define it on the basis $(H_k)_{1 \leq k \leq 6}$ of $S^{\oplus 6}(-4)$. Recall that $(E_i)_{1 \leq i \leq 15}$ is the natural basis of $S^{\oplus 15}(-2)$. By the definition $s_1(\symbtan_Q)|_{S^{\oplus 15}(-2)} = 0$, so $s_1(\symbtan_Q) \circ d_1 (H_k) = s_1(\symbtan_Q) (\sum_{i} b_{ki} \cdot E_{i} + \alpha_k \cdot G) = s_1(\symbtan_Q) (\alpha_k \cdot G) = \alpha_k \cdot Q \in S$ for $b_{ki}$'s as in the description of $d_1$. Note that $a_k := ev_F(\alpha_k \cdot Q) \in \kk$ because $\alpha_k \cdot Q \in S_3$. Hence $ev_F(\alpha_k \cdot Q - a_k \cdot g) = 0$, so $\alpha_k \cdot Q - a_k \cdot g \in (\Fperp)_3$ and there exist linear forms $a_{ki}$ such that $\alpha_k \cdot Q - a_k \cdot g$ is the image of $\sum_{i} a_{ki} E_{i}$ by $d_0$ (because $\Fperp$ is generated by $(\Fperp)_2$ by the fact that $F$ is general enough). We fix them and denote by $a_{ki} = a_{ki} (Q)$, since they depend on $Q$. Now we define $s_2(\symbtan_Q) (H_k) := \sum_{i} a_{ki} E_{i} + a_k G$. The constant $a_k$ also depends on $Q$, so we denote it $a_k(Q) = ev_F(\alpha_k \cdot Q)$.
\end{itemize}
This resolution gives us a canonical identification $\Ext^1_{S}(I, S/I) = \frac{\ker d_2^{*}}{\im d_1^{*}}$ where \[ d_1^{*} : \Hom_{S}(S^{\oplus 15}(-2) \oplus S(-3), S/I) \to \Hom_{S}(S^{\oplus 35}(-3) \oplus S^{\oplus 6}(-4), S/I) \] is composing with $d_1$ from the right and \[ d_2^{*} : \Hom_{S}(S^{\oplus 35}(-3) \oplus S^{\oplus 6}(-4), S/I) \to \Hom_{S}(F_3, S/I) \] is composing with $d_2$ from the right.
It will turn out that for our application only a part of the obstruction space will be needed. More precisely, the $S$-module $\Hom_{S}(S^{\oplus 35}(-3) \oplus S^{\oplus 6}(-4), S/I)$ splits as \[ \Hom_{S}(S^{\oplus 35}(-3), S/I) \oplus \Hom_{S}(S^{\oplus 6}(-4), S/I). \] Let $i_2 : S^{\oplus 6}(-4) \to S^{\oplus 35}(-3) \oplus S^{\oplus 6}(-4)$ be the natural embedding. It induces \[ i_2^* : \Hom_{S}(S^{\oplus 35}(-3) \oplus S^{\oplus 6}(-4), S/I) \to \Hom_{S}(S^{\oplus 6}(-4), S/I). \] Consider the projection:
\begin{equation}\label{eq:projection}
\Ext^1_{S}(I, S/I) = \frac{\ker d_2^{*}}{\im d_1^{*}} \to \frac{i_2^*(\ker d_2^*)}{i_2^*(\im d_1^{*})}.
\end{equation}
We have the following proposition, which will be crucial for proving $0$-dimensionality of the scheme $V$:
\begin{proposition}~\label{prop:obscalc}
The primary obstruction map $ob_0 : \Sym_2(T_{[I]} {\mathcal H}) \to \Ext^1_{S}(I, S/I)$ restricted to $\Sym_2(T_{[I]} V) < \Sym_2(T_{[I]} {\mathcal H})$ and composed with the projection~\eqref{eq:projection} yields a $\kk$-linear map:
\begin{equation}
\Omega : \Sym_2(T_{[I]} V) \to \frac{i_2^*(\ker d_2^*)}{i_2^*(\im d_1^{*})},
\end{equation}
whose kernel is $\lin_k(\frac{\partial F}{\partial x_1}(\symboly_1, \dots, \symboly_6), \dots, \frac{\partial F}{\partial x_6}(\symboly_1, \dots, \symboly_6)) < \Sym_2(T_{[I]} V)$.
\end{proposition}
\begin{proof}
Fix $D \in \Sym_2(T_{[I]} V)$ and write it in the form $D = \sum_{1 \leq l, m \leq 6} d_{lm} \symboly_l \symboly_m$, $d_{lm} = d_{ml}$ (we can assume symmetry of $d_{lm}$ since the characteristic of the base field is not two). Recall that $\symboly_i = \symbtan_i - \frac{1}{13} \partial_i$. Using this equation we expand $D$ and get:
\begin{equation}
D = \sum_{1 \leq l, m \leq 6} d_{lm} \symboly_l \symboly_m = \sum_{1 \leq l, m \leq 6} d_{lm} \symbtan_l \symbtan_m + \sum_i \partial_i \cdot \delta_i
\end{equation}
for some $\delta_i \in T_{[I]} {\mathcal H}$. By $k$-linearity of $\Omega$ and Proposition~\ref{prop:partial_vanish} we get that:
\begin{equation}\label{eq:omegacalc1}
\Omega(D) = \Omega(\sum_{1 \leq l, m \leq 6} d_{lm} \symbtan_l \symbtan_m).
\end{equation}
We will use the formula~\eqref{eq:hilb_obstruction} from Theorem~\ref{thm:hilb_obstruction} to determine the right hand side of this equation. First we take any $Q, Q' \in (S/I)_2$ and calculate:
\begin{equation}
q \circ s_1(\symbtan_Q) \circ s_2(\symbtan_{Q'}) (H_k) = q \circ s_1(\symbtan_Q) (\sum_{i} a_{ki}(Q') E_{i} + a_k(Q') G) = q (a_k(Q') \cdot Q) = a_k(Q') \cdot Q,
\end{equation}
so:
\begin{equation}
q \circ (s_1(\symbtan_Q) \circ s_2(\symbtan_{Q'}) + s_1(\symbtan_{Q'}) \circ s_2(\symbtan_{Q})) (H_k) = a_k(Q') \cdot Q + a_k(Q) \cdot Q'.
\end{equation}
If $Q = Q_j$, then by the definition $a_k(Q) = a_k(Q_j) = ev_F(\alpha_k \cdot Q_j) = ev_{x_j}(\alpha_k) = \delta_{kj}$. The third equality follows from the fact that $ev_F(Q_j) = x_j$ and the Definition~\ref{def:duality}. Hence for $Q = Q_j, Q' = Q_l$ we have:
\begin{equation}
q \circ (s_1(\symbtan_{Q_j}) \circ s_2(\symbtan_{Q_l}) + s_1(\symbtan_{Q_l}) \circ s_2(\symbtan_{Q_j})) (H_k) = \delta_{kl} \cdot Q_j + \delta_{kj} \cdot Q_l.
\end{equation}
By equation~\eqref{eq:omegacalc1}, $k$-linearity of $\Omega$ and Theorem~\ref{thm:hilb_obstruction} we get that $\Omega (D) = \sum_{l, m} d_{lm} \Omega(\symbtan_l \symbtan_m)$ is a class (modulo $i_2^*(\im d_1^{*})$) of a homomorphism:
\begin{equation}
r : H_k \mapsto \sum_{l, m} d_{lm} (a_k(Q_l) \cdot Q_m + a_k(Q_m) \cdot Q_l),
\end{equation}
but
\begin{gather}
\sum_{l, m} d_{lm} (a_k(Q_l) \cdot Q_m + a_k(Q_m) \cdot Q_l) = \sum_{l, m} d_{lm} (\delta_{kl} \cdot Q_m + \delta_{km} \cdot Q_l) = \\
\sum_m d_{km} Q_m + \sum_l d_{lk} Q_l = 2 \sum_l d_{lk} Q_l,
\end{gather}
so $D \in \ker \Omega$ if and only if a homomorphism $r : H_k \mapsto 2 \sum_l d_{lk} Q_l$ is in $i_2^*(\im d_1^{*})$.
The condition $r \in i_2^*(\im d_1^*)$ is equivalent to the existence of a map $p$ making the below diagram commute (with $r|_{S^{\oplus 35}(-3)} = 0$):
\[\begin{tikzcd}
{S^{\oplus 15}(-2) \oplus S(-3)} & {S^{\oplus 35}(-3) \oplus S^{\oplus 6}(-4)} & {\sum_{i} b_{ki} \cdot E_{i} + \alpha_k \cdot G} & {H_k} \\
{S/I} && {2 \sum_l d_{lk} Q_l}
\arrow["{d_1}"', from=1-2, to=1-1]
\arrow["p", dashed, from=1-1, to=2-1]
\arrow["r", from=1-2, to=2-1]
\arrow[maps to, from=1-4, to=1-3]
\arrow[dashed, maps to, from=1-3, to=2-3]
\arrow[maps to, from=1-4, to=2-3]
\end{tikzcd}\]
In other words:
\begin{equation}
D \in \ker \Omega \iff r \in i_2^*(\im d_1^*) \iff \textnormal{there exists } p \textnormal{ making the above diagram commute.}
\end{equation}
Such a map $p$ is determined by images of $(E_{i})_{1 \leq i \leq 15}$ and $G$. Hence it exists if and only if there exist $(s_{i})_{1 \leq i \leq 15}$ and $s$ in $S/I$ such that $p(E_{i}) = s_{i}$ and $p(G) = s$ makes the above diagram commute. This imposes the following equations on $(s_{i})_{1 \leq i \leq 15}$ and $s$:
\begin{equation}\label{eq:proof_obs_1}
\sum_{i} b_{ki} \cdot s_{i} + \alpha_k \cdot s = 2 \sum_l d_{lk} Q_l \textnormal{ for $k=1, \dots, 6$} \\
\textnormal{ and } p|_{d_1(S^{\oplus 35}(-3))} = 0.
\end{equation}
Note that $2 \sum_l d_{lk} Q_n$ is of degree $2$ in $S/I$, so we can look only at $s_{i}$'s of degree $0$ (because by the definition $b_{ki}$'s are of degree two) and $s$ of degree 1. In this case $p|_{S^{\oplus 15}(-2)}$ factors through $S/\Fperp$.
Now, using the fact that $0 \leftarrow \Fperp \leftarrow S^{\oplus 15}(-2) \leftarrow S^{\oplus 35}(-3)$ is exact (because $F$ is general enough), we see that the condition $p|_{d_1(S^{\oplus 35}(-3))} = 0$ implies that $p|_{S^{\oplus 15}(-2)}$ factors through $\Fperp$. We get the following commutative diagram:
\[\begin{tikzcd}
0 & \Fperp & {S^{\oplus 15}(-2)} & S^{\oplus 35}(-3) \\
&& {S/I} \\
&& {S/\Fperp}
\arrow[from=1-4, to=1-3]
\arrow[from=1-3, to=1-2]
\arrow["p", from=1-3, to=2-3]
\arrow["{r|_{S^{\oplus 35}(-3)} = 0}", from=1-4, to=2-3]
\arrow[from=1-2, to=2-3]
\arrow[from=1-2, to=1-1]
\arrow["{\hat{p}}"', from=1-2, to=3-3]
\arrow[from=3-3, to=2-3]
\end{tikzcd}\]
where $\hat{p}$ is the unique homogeneous factorisation. Again, by the fact that $F$ is general enough we know that such $\hat{p}$ must be $0$. Indeed, that is because $\Hom_S(\Fperp, S/\Fperp)_{-2} = 0$ as $[\Fperp] \in {\mathcal H}^{14}$ has TNT (see Definition~\ref{def:TNT}). Hence, the constants $s_i$ must be all $0$.
This means that the existence of $p$ satisfying our assumptions is equivalent to the existence of a linear polynomial $s$ such that for $k = 1, \dots, 6$:
\begin{equation}\label{eq:proof_obs_2}
\alpha_k \cdot s = 2 \sum_l d_{lk} Q_l
\end{equation}
Note that the equality~\eqref{eq:proof_obs_2} takes place in $(S/I)_2 = (S/ \Fperp)_2$, so we can apply the isomorphism $ev_F : S/\Fperp \to ev_F(S)$ and get:
\begin{equation}\label{eq:proof_obs_3}
ev_F(\alpha_k \cdot s) = 2 \sum_l d_{lk} x_l \textnormal{ for $k=1, \dots, 6$}
\end{equation}
Moreover, if we find $s$ such that the above equation holds, then also equation~\eqref{eq:proof_obs_2} is satisfied by it, as the evaluation function is an isomorphism on the second degree of $S/ \Fperp$.
The equation~\eqref{eq:proof_obs_3} can be rewritten as:
\begin{equation}\label{eq:proof_obs_4}
\frac{\partial}{\partial x_k} (ev_F(s)) = \frac{\partial}{\partial x_k} (\sum_{l,m} d_{lm} x_l x_m) \textnormal{ for $k=1, \dots, 6$}
\end{equation}
and rearranging the terms, we get:
\begin{equation}\label{eq:proof_obs_5}
\frac{\partial}{\partial x_k} (\sum_{l,m} d_{lm} x_l x_m - ev_F(s)) = 0 \textnormal{ for $k=1, \dots, 6$}
\end{equation}
The quadric $\sum_{l,m} d_{lm} x_l x_m - ev_F(s)$ has all partial derivatives $0$ if and only if it is $0$ itself. Thus $p$ exists iff $\sum_{l,m} d_{lm} x_l x_m = ev_F(s)$ for some $s \in S_1$ iff $\sum_{l,m} d_{lm} x_l x_m \in ev_F(S_1) = \lin_k(\frac{\partial F}{\partial x_1}, \dots, \frac{\partial F}{\partial x_6})$.
The existence of $p$ is equivalent to the fact that $D \in \ker \Omega$, so we get that
\begin{equation}\label{eq:proof_obs_6}
\ker \Omega = \lin_k(\frac{\partial F}{\partial x_1}(\symboly_1, \dots, \symboly_6), \dots, \frac{\partial F}{\partial x_6}(\symboly_1, \dots, \symboly_6)),
\end{equation}
which concludes the proof.
\end{proof}
\subsection{Zero-dimensionality of $V$}\label{sub:descriptionofv}
We will use Proposition~\ref{prop:obscalc} to prove that $V$ is $0$-dimensional.
Let us start with some analysis of $V$. Recall that it is defined as $\pi^{-1}([I])$ for $\pi : {\mathcal B}^+ \to {\mathcal B}^{\G_m}$ and that it is affine with $\Affine^1$-action (by multiplication) where the unique fixed point is $[I]$. Since it is affine let $V = \Spec B$. The action of $\Affine^1$ on $\Spec B$ translates to $\mathbb{N}$-grading on $B$ so $B = \bigoplus_{n \geq 0} B_n$ with $B_0 = k$ as we have only one fixed point. Moreover, we know that $T_{[I]}V$ is $6$-dimensional (Proposition~\ref{prop:subspacew} and Definition~\ref{def:subspacew}), so for $\mm := \bigoplus_{n \geq 1} B_n$ which is the fix-point ideal of $B$, we get that $\dim_{\kk} \frac{\mm}{\mm^2}$ is $6$-dimensional. Fix six homogeneous representants of generators of $\frac{\mm}{\mm^2}$ and call them $\symboly_1^{\vee}, \dots, \symboly_6^{\vee} \in \mm$. Assume they are dual to generators $\symboly_1, \dots, \symboly_6$ of the tangent space at $[I]$. Note that (by what is usually called a graded Nakayama lemma):
\begin{gather}
\mm = (\symboly_1^{\vee}, \dots, \symboly_6^{\vee}) + \mm^2 = (\symboly_1^{\vee}, \dots, \symboly_6^{\vee}) + ((\symboly_1^{\vee}, \dots, \symboly_6^{\vee}) + \mm^2) \cdot \mm = \\
(\symboly_1^{\vee}, \dots, \symboly_6^{\vee}) + \mm^3 = \dots = (\symboly_1^{\vee}, \dots, \symboly_6^{\vee}),
\end{gather}
so $\mm = (\symboly_1^{\vee}, \dots, \symboly_6^{\vee})$. Also, all $\symboly_i^{\vee}$'s are homogeneous of degree $1$, because $\symboly_i$'s are homogeneous of degree $-1$. Thus we can define a graded homomorphism $k[\symb_1, \dots, \symb_6] \to B$ sending $\symb_i$ to $\symboly_i^{\vee}$. It is onto by an induction on the degree argument.
If we denote by $J$ the kernel of this homomorphism, then we get an identification of graded rings $B \simeq k[\symb_1, \dots, \symb_6]/J$ with $J$ - homogeneous. Note that also $J \subset (\symb_1, \dots, \symb_6)^2$, because the tangent space at $[I] \simeq 0$ to $B \simeq k[\symb_1, \dots, \symb_6]/J$ is $6$-dimensional. So to prove that $V$ is $0$-dimensional it is enough to prove that $k[\symb_1, \dots, \symb_6]/J$ is $0$-dimensional. We will do so by obtaining a lower bound on the ideal $J$.
First consider the composition of inclusions and morphisms below:
\begin{equation}
V \subset {\mathcal B}^+ \to {\mathcal B} \subset {\mathcal H}
\end{equation}
By looking near $[I]$ we get a homomorphism ${\mathcal O}_{{\mathcal H}, [I]} \to {\mathcal O}_{V, [I]}$. It is surjective as $V \subset {\mathcal B}^+$ is a closed inclusion, ${\mathcal B}^+ \to {\mathcal B}$ is an isomorphism near $[I]$ and ${\mathcal B} \subset {\mathcal H}$ is a closed inclusion.
We complete it and get a map $\zeta : \hat{{\mathcal O}}_{{\mathcal H}, [I]} \to \hat{{\mathcal O}}_{V, [I]}$ which is also surjective.
Using the map $\zeta$ we will prove that:
\begin{proposition}~\label{prop:boundj}
The vector space $(\Fperp)_2(\Bar{\symb}) := \{ g(\symb_1, \dots, \symb_6) \in k[\symb_1, \dots, \symb_6] : g(\alpha_1, \dots, \alpha_6) \in (\Fperp)_2 \}$ is contained in $J_2$.
\end{proposition}
But before giving the proof of this proposition we give an immediate corollary:
\begin{corollary}~\label{cor:zerodim}
$V$ is $0$-dimensional.
\end{corollary}
\begin{proof}
The cubic $F$ is general enough, so $\Fperp$ is generated by its second degree $(\Fperp)_2$. By Proposition~\ref{prop:boundj} $(\Fperp)_2(\Bar{\symb}) \subset J$, so also $\Fperp(\Bar{\symb}) \subset J$. Hence we get a surjection:
\[\begin{tikzcd}
{\frac{k[\Bar{\symb}]}{\Fperp(\Bar{\symb})}} & {\frac{k[\Bar{\symb}]}{J}}
\arrow[two heads, from=1-1, to=1-2]
\end{tikzcd}\]
The left hand side ring is $0$-dimensional (with respect to the Krull dimension), so the other one also is. Since $V \simeq \Spec(\frac{k[\Bar{\symb}]}{J})$ we get the conclusion.
\end{proof}
\begin{proof}[Proof of the Proposition~\ref{prop:boundj}]
Let $Ob = \Ext^1_{S}(I, S/I)$ and $ob_0 : \Sym_2(T_{[I]} {\mathcal H}) \to Ob$ be the primary obstruction map for $\hat{{\mathcal O}}_{{\mathcal H}, [I]}$.
Let $ob_V : \Sym_2(T_{[I]} V) \to Ob$ be the restriction of $ob_0$ to $\Sym_2(T_{[I]} V)$.
By Proposition~\ref{prop:primaryobs_tool} with $\zeta : \hat{{\mathcal O}}_{{\mathcal H}, [I]} \to \hat{{\mathcal O}}_{V, [I]}$ we get: \[ \im ob_V^{\vee} \subset \frac{(\Bar{\symb})^3 + J}{(\Bar{\symb})^3} = \frac{(\Bar{\symb})^3 + J_2}{(\Bar{\symb})^3}\] Hence, it is enough to prove that $(\Fperp)_2(\Bar{\symb}) \subset \im ob_V^{\vee}$.
Note that if we compose $ob_V$ from the right with $Ob \to \frac{i_2^*(\ker d_2^*)}{i_2^*(\im d_1^{*})}$ we get the map $\Omega$ from Proposition~\ref{prop:obscalc}. Thus we end up with the sequence:
\[\begin{tikzcd}
{\ker \Omega} & {\Sym_2(T_{[I]} V)} & Ob & {\frac{i_2^*(\ker d_2^*)}{i_2^*(\im d_1^{*})}}
\arrow["j"', from=1-1, to=1-2]
\arrow["{ob_V}"', from=1-2, to=1-3]
\arrow[from=1-3, to=1-4]
\arrow["\Omega"', bend left, from=1-2, to=1-4]
\end{tikzcd}\]
so after dualizing we get:
\[\begin{tikzcd}
{(\ker \Omega)^{\vee}} & {{\Sym_2(T_{[I]} V)}^{\vee}} & {Ob^{\vee}} & {(\frac{i_2^*(\ker d_2^*)}{i_2^*(\im d_1^{*})})^{\vee}}
\arrow["j^{\vee}", from=1-2, to=1-1]
\arrow["{ob_V^{\vee}}", from=1-3, to=1-2]
\arrow[from=1-4, to=1-3]
\arrow["{\Omega^{\vee}}", bend right, from=1-4, to=1-2]
\end{tikzcd}\]
Hence $\im \Omega^{\vee} \subset \im ob_V^{\vee} \subset J_2$. Note that $j$ is the kernel of $\Omega$, so $\im \Omega^{\vee} = \ker j^{\vee}$.
We will prove that $\ker j^{\vee} = (\Fperp)_2(\Bar{\symb}) \subset \Sym_2(T_{[I]}^{\vee} V)$ under the identification $\Sym_2(T_{[I]}^{\vee} V) \simeq \Sym_2(T_{[I]} V)^{\vee}$ coming from the natural duality $\cdot : \Sym_2(W^{\vee}) \times \Sym_2(W) \to \kk$ for $W = T_{[I]} V$ (see equation~\eqref{eq:dualityw}). However, for the proof it will be easier to start with a different duality. First let us write $\Sym_2(T_{[I]} V) = k[\symboly_1, \dots, \symboly_6]_2$. Consider the evaluation map from Definition~\ref{def:duality} with variables $x_i$ changed to $\symboly_i$:
\begin{equation}
\ \wcirc \ : k[\alpha_1, \dots, \alpha_6] \times k[\symboly_1, \dots, \symboly_6] \to k[\symboly_1, \dots, \symboly_6]
\end{equation}
In other words $\alpha_i$ acts as $\frac{\partial}{\partial \symboly_i}$. By restricting this map to degree two we get a perfect pairing:
\begin{equation}
\ \wcirc \ : k[\alpha_1, \dots, \alpha_6]_2 \times k[\symboly_1, \dots, \symboly_6]_2 \to k
\end{equation}
This gives us an identification $k[\alpha_1, \dots, \alpha_6]_2 \simeq (k[\symboly_1, \dots, \symboly_6]_2)^{\vee} \simeq \Sym_2(T_{[I]} V)^{\vee}$.
We will now calculate $\ker j^{\vee} \subset \Sym_2(T_{[I]} V)^{\vee} \simeq k[\alpha_1, \dots, \alpha_6]_2$. Take $D \in k[\alpha_1, \dots, \alpha_6]_2$. We have:
\begin{equation}
D \in \ker j^{\vee} \textnormal{ if and only if } D|_{\ker \Omega} = 0,
\end{equation}
From Proposition~\ref{prop:obscalc} we know that $\ker \Omega = \lin_k(\frac{\partial F}{\partial x_1}(\symboly_1, \dots, \symboly_6), \dots, \frac{\partial F}{\partial x_6}(\symboly_1, \dots, \symboly_6))$, so:
\begin{equation}
D|_{\ker \Omega} = 0 \iff D \ \wcirc \ (\frac{\partial F}{\partial x_k}(\symboly_1, \dots, \symboly_6))=0 \textnormal{ for all $k=1, \dots, 6$.}
\end{equation}
We transform this condition using the evaluation action:
\[D \ \wcirc \ (\frac{\partial F}{\partial x_k}(\symboly_1, \dots, \symboly_6)) = (D \cdot \alpha_k) \ \wcirc \ (F(\symboly_1, \dots, \symboly_6)) = (\alpha_k \cdot D) \ \wcirc \ (F(\symboly_1, \dots, \symboly_6)) = \frac{\partial}{\partial \symboly_k}(D \ \wcirc \ F).\]
Since $D \ \wcirc \ F$ is a linear form, the vanishing of $\frac{\partial}{\partial \symboly_k}(D \ \wcirc \ F)$ for all $k=1, \dots 6$ is equivalent to $D \ \wcirc \ F = 0$. Thus we get:
\begin{equation}
D \in \ker j^{\vee} \iff D|_{\ker \Omega} = 0 \iff D \ \wcirc \ F = 0.
\end{equation}
Hence $\ker j^{\vee} = (\Fperp)_2 < k[\alpha_1, \dots, \alpha_6]_2$. The only thing left is to check what happens under the isomorphisms:
\begin{equation}
k[\alpha_1, \dots, \alpha_6]_2 \simeq \Sym_2(T_{[I]} V)^{\vee} \simeq \Sym_2(T_{[I]}^{\vee} V) = \Sym_2(k \symb_1 \oplus \dots \oplus k \symb_6).
\end{equation}
Take $\alpha_i \alpha_j \in k[\alpha_1, \dots, \alpha_6]_2$ and $\symboly_i^{\vee} \symboly_j^{\vee} \in \Sym_2(T_{[I]}^{\vee} V)$ where $(\symboly_i^{\vee})_i$ is the dual basis to $(\symboly_i)_i \subset T_{[I]} V$.
Consider the action of these elements on $\Sym_2(T_{[I]} V)$:
\begin{itemize}
\item $\alpha_i \alpha_j \cdot \symboly_i \symboly_j$ is $1$ if $i \neq j$ and $2$ if $i=j$,
\item $\symboly_i^{\vee} \symboly_j^{\vee} \cdot \symboly_i \symboly_j = \frac{1}{2}(\symboly_i^{\vee}(\symboly_i)\symboly_j^{\vee}(\symboly_j) + \symboly_i^{\vee}(\symboly_j)\symboly_j^{\vee}(\symboly_i))$ so it is $\frac{1}{2}$ if $i \neq j$ and $1$ if $i=j$ (in both cases we use Formula~\eqref{eq:dualityw}).
\end{itemize}
It is easy to check that the action on $\symboly_a \symboly_b$ for $\{a, b\} \neq \{i, j\}$ is zero in both cases. Thus under the above isomorphism $\alpha_i \alpha_j \simeq 2 \symboly_i^{\vee} \symboly_j^{\vee}$.
Hence, after two isomorphisms everything is multiplied by $2$ (when considered in bases $(\alpha_i \alpha_j)$ and $(\symboly_i^{\vee} \symboly_j^{\vee})$). Now, by the definition of $\symb_i$'s (they were coming exactly from the dual basis to $(\symboly_i)_i$) and homogeneity of $(\Fperp)_2$, we get the conclusion.
\end{proof}
\begin{corollary}\label{cor:Iishedgehog}
The ideal $I = (\Fperp, g) = \Fperp + S_{\geq 3}$ yields a hedgehog point $[I] \in {\mathcal B}$. In particular $[I]$ is a non-reduced point of ${\mathcal B}$ and ${\mathcal H}$.
\end{corollary}
\begin{proof}
The point $[I]$ lies in ${\mathcal B}$ as it is supported at $0 \in \Affine^6$. It is $\G_m$-invariant, because $I$ is homogeneous.
Now we check three properties from Definition~\ref{def:hedgehog:precise}:
\begin{enumerate}
\item $(T_{[I]} {\mathcal B})_{>0} = 0$ by Corollary~\ref{cor:nopositivedeg},
\item $T_{[I]} {\mathcal B} \neq (T_{[I]} {\mathcal B})_0$ by Proposition~\ref{prop:subspacew} and Definition~\ref{def:subspacew},
\item the negative spike $V$ at $[I]$ is $0$-dimensional by Corollary~\ref{cor:zerodim}.
\end{enumerate}
The non-reducedness follows from Theorem~\ref{thm:Hedgehog_point_theorem:precise}.
\end{proof}
Hence, by Example~\ref{ex:nonempty} we get:
\begin{corollary}\label{cor:important}
The ideal $I = (\Fperp, g) = \Fperp + S_{\geq 3}$ for $F = x_1 x_2 x_4 - x_1 x_5^2 + x_2 x_3^2 + x_3 x_5 x_6 + x_4 x_6^2$ yields a non-reduced point $[I] \in {\mathcal H}$.
\end{corollary}
\subsection{Fractal family and its flatness}\label{sub:flatness}
%
Since we proved that $V$ is $0$-dimensional in Corollary~\ref{cor:zerodim}, we know that the composition of morphisms:
\begin{equation}\label{eq:compo}
V \subset {\mathcal B}^+ \to {\mathcal B} \subset {\mathcal H}
\end{equation}
is a closed embedding. Indeed, that is because $V$ is concentrated on just one point, the first and the last morphisms are closed embeddings, and $i : {\mathcal B}^+ \to {\mathcal B}$ is an isomorphism on some open neighbourhood of $[I] \in {\mathcal B}^+$. The subscheme $V = \Spec(B) \hookrightarrow {\mathcal H}$ yields a deformation of $[I]$ over $B$. In Subsection~\ref{sub:descriptionofv} we presented $B$ as a quotient $\frac{k[\symb_1, \dots, \symb_6]}{J}$, where $J$ was a homogeneous ideal. Moreover, we proved that $\Fperp(\Bar{\symb}) \subset J$ in Proposition~\ref{prop:boundj} and this was the heart of our argument for $0$-dimensionality of $V$ in Corollary~\ref{cor:zerodim}. Thus, a natural suspicion is that in fact $J = \Fperp(\Bar{\symb})$, or more generally $V \simeq \Spec(S/\Fperp)$. If this was true, then there would exist a morphism $\Spec(S/\Fperp) \to {\mathcal H}$ onto $V$. However, such a morphism would induce a deformation of $S/I$ over $S/\Fperp$ - almost over itself! Moreover, since it would factor through ${\mathcal B}^+$ we should have been able to prolong it to an element of ${\mathcal H}(\Affine^1 \times \Spec(S/\Fperp))$. Suspecting the existence of such a structure, we go the other way around: we find a deformation over $\Affine^1 \times \Spec(S/\Fperp)$ which yields a morphism to ${\mathcal H}$ such that after composing it with the projection ${\mathcal H} = \Affine^6 \times {\mathcal B} \to {\mathcal B}$ the induced morphism to ${\mathcal B}^+$ is an isomorphism onto $V$.
%
\begin{definition}
Consider a $\frac{k[t, \bbeta]}{\Fperpy}$ - algebra $M$ of the form:
\[\begin{tikzcd}
{\frac{k[t, \bbeta]}{\Fperpy}} & {M := \frac{k[t, \aalpha, \bbeta]}{(\Fperpx, \Fperpy, \Gamma(t))}}
\arrow[from=1-1, to=1-2]
\end{tikzcd}\]
where we take $\Gamma(t)$ from Definition~\ref{def:gammat} with $g$ and $Q_i$'s as in Subsection~\ref{sub:tangent}.
We call it the \emph{fractal family}. The name refers to the self-similarity of the base space and the special fiber. The rest of this subsection aims to prove flatness of the fractal family.
\end{definition}
%
Before stating the next algebraic preparation step, we recall some notion from commutative algebra.
\begin{definition}\label{def:modulestructure}
If $C$ is a finite dimensional $k[t]$-algebra, then the $k[t]$-module $\Hom_{k[t]}(C, k[t])$ has a $C$-module structure given by $(c \cdot \symbphi)(d) := \symbphi(c \cdot d)$ for $\symbphi \in \Hom_{k[t]}(C, k[t])$ and $c, d \in C$.
Moreover, if $C = \frac{k[t, \aalpha]}{H^{\perp}}$ for some $H \in k[t, \xx]$, then there is a natural $C$-modules homomorphism:
\begin{equation}
\Phi: C \to \Hom_{k[t]}(C, k[t])
\end{equation}
sending $c \in C$ to $\phi_c := \Phi(c)$ given by the formula:
\begin{equation}
\phi_c (d) = (ev_{c \ \wcirc \ H}(d))(\Bar{0}).
\end{equation}
The evaluation at $\xx = \Bar{0}$ at the end makes sense as $(ev_{c \ \wcirc \ H}(d)) \in k[t, \xx]$ and the result lies indeed in $k[t]$.
\end{definition}
\begin{proposition}~\label{prop:selfdual}
Let $H \in k[t, \xx]$ and consider the algebra $C := \frac{k[t, \aalpha]}{H^{\perp}}$. Suppose that $C$ is a free $k[t]$-module and for all $t_0 \in k$ the dimension of the $k$-linear space $\frac{k[\aalpha]}{H(t_0)^{\perp}}$ is constant.
Then $\Phi : C \to \Hom_{k[t]}(C, k[t])$ is a $C$-module isomorphism.
\end{proposition}
Before the proof let us cite a lemma which is a translation of~\cite[Proposition 2.12]{Jel18} to our context:
\begin{lemma}\label{lem:selfdual}
Assume that $k$ is algebraically closed. If $C = \frac{k[t, \aalpha]}{H^{\perp}}$ for $H \in k[t, \xx]$ is such that for all $t_0 \in k$ the dimension of the $k$-linear space $\frac{k[\aalpha]}{H(t_0)^{\perp}}$ is constant, then for all $t_0 \in k$ there is an isomorphism between $\frac{k[t, \aalpha]}{(H^{\perp}, (t-t_0))}$ and $\frac{k[\aalpha]}{H(t_0)^{\perp}}$ induced by the map:
\begin{equation}
k[t, \aalpha] \to \frac{k[\aalpha]}{H(t_0)^{\perp}}
\end{equation}
which maps $f$ to $ev_{H(t_0)}(f)(t_0)$.
\end{lemma}
\begin{proof}[Proof of Proposition~\ref{prop:selfdual}]
First we prove that $\ker \Phi = 0$:
\begin{equation}
\Phi(c) = 0 \iff \im ev_{c \ \wcirc \ H} \subset (\xx) \iff c \ \wcirc \ H = 0 \iff c = 0 \in \frac{k[t, \aalpha]}{H^{\perp}}.
\end{equation}
The middle equivalence holds, because if $c \ \wcirc \ H \neq 0$, then there exists some partial derivative such that after applying it to $c \ \wcirc \ H$ we get a non-zero element in $k[t]$.
Now our goal is to show that $\Phi$ is onto. Suppose for a contradiction that it is not. Let $D := \Hom_{k[t]}(C, k[t])$. Then we get the following exact sequence of $C$-modules:
\[\begin{tikzcd}
0 & C & {D} & E & 0
\arrow[from=1-1, to=1-2]
\arrow[from=1-2, to=1-3]
\arrow[from=1-3, to=1-4]
\arrow[from=1-4, to=1-5]
\end{tikzcd}\]
where $E \neq 0$. This is also a sequence of $k[t]$-modules and since $E \neq 0$ is finitely generated, there exist an ideal $\mm = (t - t_0) \subset k[t]$ (for some $t_0 \in k$) such that the quotient $E/(t- t_0) E$ is non-zero (take $\mm$ such that $E_{\mm} \neq 0$ and use Nakayama's lemma). The ideal $\mm$ is of this form as $k$ is algebraically closed. We tensor everything by $\frac{k[t]}{(t-t_0)}$ and we get:
\[\begin{tikzcd}
& {C/(t-t_0)C} & {D/ (t-t_0)D} & {E/ (t-t_0)E} & 0
\arrow[from=1-2, to=1-3]
\arrow[from=1-3, to=1-4]
\arrow[from=1-4, to=1-5]
\end{tikzcd}\]
To get the final contradiction, we will prove that ${C/(t-t_0)C} \to {D/ (t-t_0)D}$ is an isomorphism. First, by Lemma~\ref{lem:selfdual} we get that $C/(t-t_0)C \simeq \frac{k[\aalpha]}{H(t_0)^{\perp}}$.
On the other hand, $D/ (t-t_0)D \simeq \Hom_{k}(C/(t-t_0)C, k)$. Indeed, if we apply the functor $\Hom_{k[t]}(C, -)$ on a short exact sequence:
\[\begin{tikzcd}
0 & {(t-t_0)} & {k[t]} & {\frac{k[t]}{(t-t_0)}} & 0
\arrow[from=1-1, to=1-2]
\arrow[from=1-2, to=1-3]
\arrow[from=1-3, to=1-4]
\arrow[from=1-4, to=1-5]
\end{tikzcd}\]
we get:
\[\begin{tikzcd}
0 & {\Hom_{k[t]}(C,(t-t_0))} & {\Hom_{k[t]}(C,k[t])} & {\Hom_{k[t]}(C,\frac{k[t]}{(t-t_0)})} & 0
\arrow[from=1-1, to=1-2]
\arrow[from=1-2, to=1-3]
\arrow[from=1-3, to=1-4]
\arrow[from=1-4, to=1-5]
\end{tikzcd}\]
where exactness follows from the fact that $C$ is a free $k[t]$-module. Now, the last sequence identifies with:
\[\begin{tikzcd}
0 & {(t-t_0) \Hom_{k[t]}(C,k[t])} & {\Hom_{k[t]}(C,k[t])} & {\Hom_{k}(C/(t-t_0)C,\frac{k[t]}{(t-t_0)})} & 0
\arrow[from=1-1, to=1-2]
\arrow[from=1-2, to=1-3]
\arrow[from=1-3, to=1-4]
\arrow[from=1-4, to=1-5]
\end{tikzcd}\]
so we get the desired isomorphism $D/ (t-t_0)D \simeq \Hom_{k}(C/(t-t_0)C, k)$. Now, note that by its definition, the homomorphism \[\frac{k[\aalpha]}{H(t_0)^{\perp}} \simeq {C/(t-t_0)C} \to {D/ (t-t_0)D} \simeq \Hom_{k}(C/(t-t_0)C, k) \simeq \Hom_{k}(\frac{k[\aalpha]}{H(t_0)^{\perp}}, k)\]
is given by the same function $\Phi$ but coming from the Maculay's duality without the variable $t$. The same argument as in the beginning of this proof shows that the obtained map
$\frac{k[\aalpha]}{H(t_0)^{\perp}} \to \Hom_{k}(\frac{k[\aalpha]}{H(t_0)^{\perp}}, k)$
is injective. Now, its surjectivity follows from the equality of dimensions over $k$. But this contradicts the fact that we have an exact sequence:
\[\begin{tikzcd}
& {C/(t-t_0)C} & {D/ (t-t_0)D} & {E/ (t-t_0)E} & 0
\arrow[from=1-2, to=1-3]
\arrow[from=1-3, to=1-4]
\arrow[from=1-4, to=1-5]
\end{tikzcd}\]
with $E/ (t-t_0)E \neq 0$. This finishes the proof of surjectivity of $\Phi$.
\end{proof}
We come back to the analysis of the fractal family.
Since $\Gamma(t) \ \wcirc \ (F_{\Bar{x}} \cdot F_{\Bar{y}}) = F(t \cdot \xx + \yy)$ by Lemma~\ref{fact:Gamma}, we have $(F_{\xx} \cdot F_{\yy})^{\perp} \subset F(t \cdot \xx + \yy)^{\perp} \subset k[t, \aalpha, \bbeta]$. We get a short exact sequence:
\[\begin{tikzcd}
0 & L & {\frac{k[t, \aalpha, \bbeta]}{(F_{\xx} \cdot F_{\yy})^{\perp}}} & {\frac{k[t, \aalpha, \bbeta]}{F(t \cdot \xx + \yy)^{\perp}}} & 0
\arrow[from=1-3, to=1-4]
\arrow[from=1-2, to=1-3]
\arrow[from=1-1, to=1-2]
\arrow[from=1-4, to=1-5]
\end{tikzcd}\]
where $L$ is the kernel of the natural quotient.
This sequence splits, as the right hand side module is a free $\frac{k[t, \bbeta]}{\Fperpy}$-module (isomorphic to $\frac{k[t, \bbeta]}{\Fperpy}$ by Lemma~\ref{fact:free}). After applying the functor $\Hom_{k[t]}(-, k[t])$ we obtain a sequence of $k[t]$-modules:
\[\begin{tikzcd}
0 & {\Hom_{k[t]}(L, k[t])} & {\Hom_{k[t]}(\frac{k[t, \aalpha, \bbeta]}{(F_{\xx} \cdot F_{\yy})^{\perp}},k[t])} & {\Hom_{k[t]}(\frac{k[t, \aalpha, \bbeta]}{F(t \cdot \xx + \yy)^{\perp}},k[t])} & 0
\arrow[from=1-4, to=1-3]
\arrow[from=1-3, to=1-2]
\arrow[from=1-2, to=1-1]
\arrow[from=1-5, to=1-4]
\end{tikzcd}\]
However, by Definition~\ref{def:modulestructure} \[{\Hom_{k[t]}(\frac{k[t, \aalpha, \bbeta]}{(F_{\xx} \cdot F_{\yy})^{\perp}},k[t])}\] has a $\frac{k[t, \aalpha, \bbeta]}{(F_{\xx} \cdot F_{\yy})^{\perp}}$-module structure and \[{\Hom_{k[t]}(\frac{k[t, \aalpha, \bbeta]}{F(t \cdot \xx + \yy)^{\perp}},k[t])}\] has a $\frac{k[t, \aalpha, \bbeta]}{F(t \cdot \xx + \yy)^{\perp}}$-module structure. In particular both of these $k[t]$-modules have a structure of $\frac{k[t, \bbeta]}{\Fperpy}$-module. Thus the above sequence can be considered as a sequence of $\frac{k[t, \bbeta]}{\Fperpy}$-modules.
If we fix a splitting on the first sequence:
\begin{equation}
\frac{k[t, \aalpha, \bbeta]}{(F_{\xx} \cdot F_{\yy})^{\perp}} \simeq L \oplus \frac{k[t, \aalpha, \bbeta]}{F(t \cdot \xx + \yy)^{\perp}}
\end{equation}
we get the induced splitting of $\frac{k[t, \bbeta]}{\Fperpy}$-modules:
\begin{equation}\label{eq:blablabla}
\Hom_{k[t]}(\frac{k[t, \aalpha, \bbeta]}{(F_{\xx} \cdot F_{\yy})^{\perp}}, k[t]) \simeq \Hom_{k[t]}(L, k[t]) \oplus \Hom_{k[t]}(\frac{k[t, \aalpha, \bbeta]}{F(t \cdot \xx + \yy)^{\perp}}, k[t])
\end{equation}
Consider the $k$-algebra $\frac{k[t, \aalpha, \bbeta]}{(F_{\xx} \cdot F_{\yy})^{\perp}}$. It is a free $k[t]$-module, because
\[\frac{k[t, \aalpha, \bbeta]}{(F_{\xx} \cdot F_{\yy})^{\perp}} = \frac{k[t, \aalpha, \bbeta]}{(\Fperpx, \Fperpy)} \simeq \frac{k[\aalpha, \bbeta]}{(\Fperpx, \Fperpy)} \otimes_k k[t]\]
where the first equality follows from Lemma~\ref{fact:fracfamilyform}. Moreover, the polynomial $(F_{\xx} \cdot F_{\yy})$ does not depend on $t$, so the assumptions of Proposition~\ref{prop:selfdual} are satisfied. Thus, the left hand side of equation~\eqref{eq:blablabla} is isomorphic to $\frac{k[t, \aalpha, \bbeta]}{(F_{\xx} \cdot F_{\yy})^{\perp}}$ as a $\frac{k[t, \bbeta]}{\Fperpy}$-module, so it is a free $\frac{k[t, \bbeta]}{\Fperpy}$-module. This means that $\Hom_{k[t]}(L, k[t])$ is a projective $\frac{k[t, \bbeta]}{\Fperpy}$-module (as it is a direct factor in a free $\frac{k[t, \bbeta]}{\Fperpy}$-module). We will now prove that the \Mf is flat by showing it is isomorphic to $\Hom_{k[t]}(L, k[t])$ as a $\frac{k[t, \bbeta]}{\Fperpy}$-module:
\begin{proposition}\label{prop:isotoff}
$\Hom_{k[t]}(L, k[t])$ is isomorphic to the \Mf (as a $\frac{k[t, \bbeta]}{\Fperpy}$-module).
\end{proposition}
\begin{proof}
Let $C := \frac{k[t, \aalpha, \bbeta]}{(F_{\xx} \cdot F_{\yy})^{\perp}}$, $C_0 := \frac{k[t, \aalpha, \bbeta]}{F(t \cdot \xx + \yy)^{\perp}}$ and $q: C \to C_0$ be the quotient map. From Proposition~\ref{prop:selfdual} we know that $\Hom_{k[t]}(C, k[t]) \simeq C$, $\Hom_{k[t]}(C_0, k[t]) \simeq C_0$ by homomorphisms as in Definition~\ref{def:modulestructure}. Thus in order to prove the proposition, it is enough to interpret the image of $\Hom_{k[t]}(C_0, k[t])$ inside $C \simeq \Hom_{k[t]}(C, k[t])$ as the ideal $(\Gamma(t))$.
Consider $c_0 \in C_0$ and let $\Bar{c}_0 \in C$ be a lifting of $c_0$. We will calculate the image of $c_0$ coming from the morphism $C_0 \simeq \Hom_{k[t]}(C_0, k[t]) \to \Hom_{k[t]}(C, k[t]) \simeq C$. First, look at the diagram:
\[\begin{tikzcd}
0 & L & C & {C_0} & 0 \\
&&& k[t]
\arrow[from=1-1, to=1-2]
\arrow[from=1-2, to=1-3]
\arrow["q"', from=1-3, to=1-4]
\arrow[from=1-4, to=1-5]
\arrow["{\symbphi_{c_0}}", from=1-4, to=2-4]
\arrow["{\symbphi_{c_0} \circ q}"', from=1-3, to=2-4]
\end{tikzcd}\]
here $\symbphi_{c_0}$ comes from Definition~\ref{def:modulestructure}, so it is equal to $z \circ ev_{c_0 \ \wcirc \ F(t \cdot \xx + \yy)}$ where $z : k[t, \xx, \yy] \to k[t]$ is the map sending $\xx$ and $\yy$ to $0$. We claim that:
\begin{equation}
\symbphi_{c_0} \circ q = \symbphi_{\Bar{c}_0 \cdot \Gamma(t)}
\end{equation}
Indeed, that it because:
\begin{equation}
\symbphi_{c_0} \circ q = z \circ ev_{c_0 \ \wcirc \ F(t \cdot \xx + \yy)} \circ q = z \circ ev_{\Bar{c}_0 \cdot \Gamma(t) \ \wcirc \ (F_{\xx} \cdot F_{\yy})} = \symbphi_{\Bar{c}_0 \cdot \Gamma(t)}
\end{equation}
So the induced homomorphism $C_0 \simeq \Hom_{k[t]}(C_0, k[t]) \to \Hom_{k[t]}(C, k[t]) \simeq C$ sends $c_0 \mapsto \Bar{c}_0 \cdot \Gamma(t)$ (and this doesn't depend on the choice of $\Bar{c}_0$). Hence the cokernel of this homomorphism is isomorphic to $C/(\Gamma(t))$ so to the \Mf by Lemma~\ref{fact:fracfamilyform}.
\end{proof}
This yields the following:
\begin{corollary}
The \Mf is a flat $\frac{k[t, \bbeta]}{\Fperpy}$-module.
\end{corollary}
\begin{proof}
From Proposition~\ref{prop:isotoff} we know that \Mf is isomorphic to $\Hom_{k[t]}(L, k[t])$ which is a projective $\frac{k[t, \bbeta]}{\Fperpy}$-module by equation~\eqref{eq:blablabla} and the fact that $\frac{k[t, \aalpha, \bbeta]}{(F_{\xx} \cdot F_{\yy})^{\perp}}$ satisfies the assumptions of Proposition~\ref{prop:selfdual}. In particular, the \Mf is flat.
\end{proof}
\subsection{Complete description of the fiber $V$}
In this subsection we prove that the morphism induced by the \Mf to ${\mathcal H}$ composed with the projection ${\mathcal H} \to {\mathcal B}$ is in fact an isomorphism onto $V$.
Let ${\mathcal Z}$ be the universal family over ${\mathcal H}$. The \Mf $M$ gives a morphism $v$:
\[\begin{tikzcd}
{\Spec(M)} & {\mathcal Z} \\
{\Affine^1 \times \Spec(\frac{k[\bbeta]}{\Fperpy})} & {\mathcal H}
\arrow[from=1-2, to=2-2]
\arrow[from=1-1, to=2-1]
\arrow[from=1-1, to=1-2]
\arrow["v", from=2-1, to=2-2]
\arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-1, to=2-2]
\end{tikzcd}\]
By Proposition~\ref{prop:torusequivariance} a morphism $\Affine^1 \times \Spec(\frac{k[\bbeta]}{\Fperpy}) \to {\mathcal H}$ is $\G_m$-equivariant if and only if the associated ideal in $S_{\frac{k[t, \bbeta]}{\Fperpy}} = \frac{k[t, \aalpha, \bbeta]}{\Fperpy}$ is homogeneous with respect to variables $\aalpha$ and $t$ all of degree one.
The \Mf is given by the ideal $(\Fperpx, \Gamma(t)) \subset \frac{k[t, \aalpha, \bbeta]}{\Fperpy}$ so it is homogeneous with respect to this grading, see Definition~\ref{def:gammat}.
Let $\bar{v} : \Spec(\frac{k[\bbeta]}{\Fperpy}) \to {\mathcal H}^+$ be the morphism induced from $v:\Affine^1 \times \Spec(\frac{k[\bbeta]}{\Fperpy}) \to {\mathcal H}$, see Definition~\ref{def:xplusfunctorial}. We write $v|_{ \{1\} } : \Spec(\frac{k[\bbeta]}{\Fperpy}) \to {\mathcal H}$ for the restriction of $v$ to $\{1\} \times \Spec(\frac{k[\bbeta]}{\Fperpy})$.
By Proposition~\ref{prop:hilbandbar} we have ${\mathcal H} \simeq \Affine^n \times {\mathcal B}$ and by Proposition~\ref{prop:barisgminv} the projection $p : {\mathcal H} \to {\mathcal B}$ is $\G_m$-equivariant. Hence, the composition $u = p \circ v$:
\[\begin{tikzcd}
{\Affine^1 \times \Spec(\frac{k[\bbeta]}{\Fperpy})} & {\mathcal H} & {\mathcal B}
\arrow["v", from=1-1, to=1-2]
\arrow["p", from=1-2, to=1-3]
\arrow["u", bend right, from=1-1, to=1-3]
\end{tikzcd}\]
is $\G_m$-equivariant. The morphism $u$ induces a morphism to $\bar{u} : \Spec(\frac{k[\bbeta]}{\Fperpy}) \to {\mathcal B}^+$ by Definition~\ref{def:xplusfunctorial}:
\[\begin{tikzcd}
{\Spec(\frac{k[\bbeta]}{\Fperpy})} & {{\mathcal B}^+} & {{\mathcal B}^{\G_m}} \\
{\Affine^1 \times\Spec(\frac{k[\bbeta]}{\Fperpy})} & {\mathcal B}
\arrow["i"', from=1-2, to=2-2]
\arrow["u"', from=2-1, to=2-2]
\arrow["{\bar{u}}", from=1-1, to=1-2]
\arrow["{\{ 1 \} \times id}"', hook, from=1-1, to=2-1]
\arrow["\pi", from=1-2, to=1-3]
\end{tikzcd}\]
In this subsection we prove that $\bar{u}$ is an isomorphism onto $V \subset {\mathcal B}^+$. We star with the following fact:
\begin{lemma}\label{fact:factorisationV}
The composition $\pi \circ \bar{u}$ is a constant morphism onto the point $[I]$.
\end{lemma}
\begin{proof}
Consider the composition $w := \pi_{{\mathcal H}} \circ \bar{v}$ for the restricting to the limit morphism $\pi_{{\mathcal H}}$ for \BBname{} decomposition for the $\G_m$-action on ${\mathcal H}$. Then the following diagram commutes:
\[\begin{tikzcd}
{\Spec(\frac{k[\bbeta]}{\Fperpy})} & {{\mathcal H}^+} & {{\mathcal B}^+} \\
& {{\mathcal H}^{\G_m}} & {{\mathcal B}^{\G_m}}
\arrow["{p^+}", from=1-2, to=1-3]
\arrow["{\bar{v}}", from=1-1, to=1-2]
\arrow["{\pi_{\mathcal H}}", from=1-2, to=2-2]
\arrow["\pi", from=1-3, to=2-3]
\arrow["{p^{\G_m}}"', from=2-2, to=2-3]
\arrow["{\bar{u}}", bend left, from=1-1, to=1-3]
\arrow["w"', from=1-1, to=2-2]
\end{tikzcd}\]
where $p^+$ and $p^{\G_m}$ are morphism induced by $p$ on \BBname{} decomposition functors and fix-point functors respectively. It is enough to prove that $w : \Spec(\frac{k[\bbeta]}{\Fperpy}) \to {\mathcal H}^{\G_m}$ is a constant map onto $[I]$, because $p^{\G_m}([I]) = [I] \in {\mathcal B}^{\G_m}$.
By the functorial description of the map $\pi$ we see that $w = \pi_{{\mathcal H}} \circ \bar{v} = v|_{ \{0\} \times \Spec(\frac{k[\bbeta]}{\Fperpy}) }$. As $v$ is given by the \Mf $M$ its restriction to zero is given by $M/(t)$ so by the algebra:
\[ M_0 := \frac{k[\aalpha, \bbeta]}{(\Fperpx, \Fperpy, \Gamma(0))} \]
but $\Gamma(0) = g(\aalpha)$ (see Definition~\ref{def:gammat}). Hence $M_0$ is just the product family:
\[ M_0 \simeq \frac{k[\aalpha]}{(\Fperpx, g(\aalpha))} \otimes_k \frac{k[\bbeta]}{(\Fperpy)} \]
As $I = (\Fperpx, g(\aalpha))$ this family corresponds to the constant morphism onto $[I]$. This finishes the proof.
\end{proof}
By Lemma~\ref{fact:factorisationV} the morphism $\bar{u}:\Spec(\frac{k[\bbeta]}{\Fperpy}) \to {\mathcal B}^+$ factors through $V \subset {\mathcal B}^+$, because by its definition $V = \pi^{-1}([I])$. We will write $\bar{u}:\Spec(\frac{k[\bbeta]}{\Fperpy}) \to V$ for the factorisation. We would like to prove that $\bar{u}$ is an isomorphism. In order to do so, first we check that it is $\G_m$-equivariant, where the $\G_m$-action on $\Spec(\frac{k[\bbeta]}{\Fperpy})$ comes from the natural grading of the ring $\frac{k[\bbeta]}{\Fperpy}$ where $\deg(\beta_i) = 1$. We start with the following:
\begin{lemma}\label{fact:gmequibaru}
The morphism $\bar{u}:\Spec(\frac{k[\bbeta]}{\Fperpy}) \to V$ is $\G_m$-equivariant with respect to the natural $\G_m$-action on $\Spec(\frac{k[\bbeta]}{\Fperpy})$ and the usual $\G_m$-action on ${\mathcal H}$.
\end{lemma}
\begin{proof}
First we argue that the morphism $v|_{ \{1\} } : \Spec(\frac{k[\bbeta]}{\Fperpy}) \to {\mathcal H}$ is $\G_m$-equivariant.
The \Mf restricted to $t=1$ is given by the ideal $(\Fperpx, \Gamma(1)) \subset S_{\frac{k[\bbeta]}{\Fperpy}}$. The part $\Fperpx$ is homogeneous with respect to the standard grading on $S$ and $\Gamma(1) = g(\aalpha) + \sum_i \beta_i \cdot Q_i(\aalpha) + \sum_i \alpha_i \cdot Q_i(\bbeta) + g(\bbeta)$ is homogeneous with respect to the grading with $\deg(\alpha_i) = \deg(\beta_i)=1$.
Thus, by Proposition~\ref{prop:torusequivariance} we get the $\G_m$-equivariance of $v|_{\{1\}} : \Spec(\frac{k[\bbeta]}{\Fperpy}) \to {\mathcal H}$.
Now we prove that this suffices to $\G_m$-equivariance of $\bar{u}$. Consider the following commutative diagram:
\[\begin{tikzcd}
{\Spec(\frac{k[\bbeta]}{\Fperpy})} & {{\mathcal H}^+} & {{\mathcal B}^+} \\
& {\mathcal H}
\arrow["{p^+}", from=1-2, to=1-3]
\arrow["{\bar{v}}", from=1-1, to=1-2]
\arrow["{i_{\mathcal H}}", from=1-2, to=2-2]
\arrow["{\bar{u}}", bend left, from=1-1, to=1-3]
\arrow["{v|_{ \{1\} }}"', from=1-1, to=2-2]
\end{tikzcd}\]
We know that $v|_{ \{1\} }, i_{{\mathcal H}}, p^+$ are $\G_m$-equivariant. Thus, the $\G_m$-equivariance of $\bar{v}$ will finish the proof. Denote by $\mu : \G_m \times \Spec(\frac{k[\bbeta]}{\Fperpy}) \to \Spec(\frac{k[\bbeta]}{\Fperpy})$ the $\G_m$-action on $\Spec(\frac{k[\bbeta]}{\Fperpy})$ and $\nu, \nu^+$ the $\G_m$-action on ${\mathcal H}, {\mathcal H}^+$ respectively. We look at the following diagram:
\[\begin{tikzcd}
{\G_m \times \Spec(\frac{k[\bbeta]}{\Fperpy})} & {\G_m \times {\mathcal H}^+} & {\G_m \times{\mathcal H}} \\
{\Spec(\frac{k[\bbeta]}{\Fperpy})} & {{\mathcal H}^+} & {\mathcal H}
\arrow["\mu", from=1-1, to=2-1]
\arrow["{id \times \bar{v}}", from=1-1, to=1-2]
\arrow["{\nu^+}"', from=1-2, to=2-2]
\arrow["{\bar{v}}", from=2-1, to=2-2]
\arrow["{id \times i_{{\mathcal H}}}", from=1-2, to=1-3]
\arrow["\nu", from=1-3, to=2-3]
\arrow["{i_{{\mathcal H}}}", from=2-2, to=2-3]
\arrow["{id \times v|_{ \{1\} }}", bend left, from=1-1, to=1-3]
\arrow["{v|_{ \{1\} }}", bend right, from=2-1, to=2-3]
\end{tikzcd}\]
We want to show that $\nu^+ \circ (id \times \bar{v}) = \bar{v} \circ \mu$. However, by Fact~\ref{fact:mono} $i_{{\mathcal H}}$ is a monomorphism of functors, so this is equivalent to the fact that \[i_{{\mathcal H}} \circ \nu^+ \circ (id \times \bar{v}) = i_{{\mathcal H}} \circ \bar{v} \circ \mu\] By the commutativity of the diagram this translates to $\nu \circ (id \times v|_{ \{1\} }) = v|_{ \{1\} } \circ \mu$. But this is the $\G_m$-equivariance of $v|_{ \{1\} }$ so we are done.
\end{proof}
Now we investigate the behaviour of tangents under $v|_{ \{1\} }$:
\begin{proposition}\label{prop:ending}
Let $(\beta_i^{\vee})_{1 \leq i \leq 6} \subset T_{ (\bbeta) } \Spec(\frac{k[\bbeta]}{\Fperpy})$ be the dual basis to $(\beta_1, \dots, \beta_6) \in (\bbeta)/(\bbeta)^2 = T_{ (\bbeta) }^{\vee} \Spec(\frac{k[\bbeta]}{\Fperpy})$.
The tangent map $d v|_{\{1\}} : T_{ (\bbeta) } \Spec(\frac{k[\bbeta]}{\Fperpy}) \to T_{[I]} {\mathcal H}$ sends $\beta_i^{\vee}$ to $- \symbtan_i$ from Definition~\ref{def:subspacew}.
\end{proposition}
\begin{proof}
An element $\beta_i^{\vee} \in T_{ (\bbeta) } \Spec(\frac{k[\bbeta]}{\Fperpy})$ comes from the morphism $\Spec(k[\eps]) \to \Spec(\frac{k[\bbeta]}{\Fperpy})$ defined by taking $\beta_j$ to $0$ for $j \neq i$ and $\beta_i$ to $\eps$. By composing with $v|_{\{1\}}$ we get a morphism $\Spec(k[\eps]) \to {\mathcal H}$ and it is given by the pullback of the family $\Spec(M')$ over $\Spec(\frac{k[\bbeta]}{\Fperpy})$ to $\Spec(k[\eps])$. Recall that $M' = \frac{k[\aalpha, \bbeta]}{(\Fperpx, \Fperpy, \Gamma(1))}$ and $\Gamma(1) = g(\aalpha) + \sum_i \beta_i \cdot Q_i(\aalpha) + \sum_i \alpha_i \cdot Q_i(\bbeta) + g(\bbeta)$. Thus, the pullback of the \Mf through $\Spec(k[\eps]) \to \Spec(\frac{k[\bbeta]}{\Fperpy})$ gives the family:
\[\begin{tikzcd}
{k[\eps]} & {\frac{k[\aalpha, \eps]}{(\Fperpx, g(\aalpha) + \eps \cdot Q_i(\aalpha))}}
\arrow[from=1-1, to=1-2]
\end{tikzcd}\]
By Example~\ref{ex:tangentsw} this family corresponds to the tangent $- \symbtan_i = - \symbtan_{Q_i}$.
\end{proof}
As a corollary, we prove the following:
\begin{theorem}\label{thm:completeV}
The morphism $\bar{u}:\Spec(\frac{k[\bbeta]}{\Fperpy}) \to V$ is an isomorphism. Moreover, the ideal $J$ from the isomorphism $V \simeq \Spec(\frac{k[\Bar{\symb}]}{J})$ described in the beginning of Subsection~\ref{sub:descriptionofv} is equal to:
\[ (\Fperp)(\Bar{\symb}) := \{ g(\symb_1, \dots, \symb_6) \in k[\symb_1, \dots, \symb_6] : g(\alpha_1, \dots, \alpha_6) \in \Fperp \}. \]
\end{theorem}
\begin{proof}
Consider the diagram:
\[\begin{tikzcd}
& {} \\
{\Spec(\frac{k[\bbeta]}{\Fperpy})} & {{\mathcal H}^+} & {{\mathcal B}^+} \\
{\Affine^1 \times \Spec(\frac{k[\bbeta]}{\Fperpy})} & {\mathcal H} & {\mathcal B}
\arrow["v"', from=3-1, to=3-2]
\arrow["{i_{{\mathcal H}}}", from=2-2, to=3-2]
\arrow["{\bar{v}}", from=2-1, to=2-2]
\arrow[hook, from=2-1, to=3-1]
\arrow["p", bend left, from=3-2, to=3-3]
\arrow["i"', from=2-3, to=3-3]
\arrow["{p^+}", from=2-2, to=2-3]
\arrow["u", bend right, from=3-1, to=3-3]
\arrow["{r^+}", bend left, from=2-3, to=2-2]
\arrow["r", from=3-3, to=3-2]
\arrow["{v|_{ \{1\} }}", from=2-1, to=3-2]
\arrow["{\bar{u}}"', bend left, from=2-1, to=2-3]
\end{tikzcd}\]
where $r$ is the embedding of ${\mathcal B}$ in ${\mathcal H}$ and morphisms $p^+, r^+$ come from $p$ and $r$ respectively, by Definition~\ref{def:xplusfunctorial}. We are interested in the morphism
\[ q := (r \circ i \circ \bar{u}) : \Spec(\frac{k[\bbeta]}{\Fperpy}) \to {\mathcal H} \] We calculate:
\begin{equation}
q = r \circ i \circ \bar{u} = r \circ i \circ p^+ \circ \bar{v} = r \circ p \circ i_{{\mathcal H}} \circ \bar{v} = r \circ p \circ v|_{ \{1\} }
\end{equation}
Thus, the tangent map $dq : T_{ (\bbeta) } \Spec(\frac{k[\bbeta]}{\Fperpy}) \to T_{[I]} {\mathcal H}$ is the composition of $dv|_{ \{1\} } : T_{ (\bbeta) } \Spec(\frac{k[\bbeta]}{\Fperpy}) \to T_{[I]} {\mathcal H}$ and the projection $T_{[I]} {\mathcal H} \to T_{[I]} {\mathcal B} < T_{[I]} {\mathcal H}$.
By Proposition~\ref{prop:ending} if $\beta_i^{\vee}$ is as in the statement of this proposition, then $dv|_{ \{1\} } (\beta_i^{\vee}) = - \symbtan_i$. Note that
\begin{equation}
- \symbtan_i = - (\symbtan_i - \frac{1}{13} \partial_i) - \frac{1}{13} \partial_i = - \symboly_i - \frac{1}{13} \partial_i
\end{equation}
by Definition~\ref{def:subspacey}. By Proposition~\ref{prop:subspacew} $\symboly_i \in T_{[I]} V < T_{[I]} {\mathcal B}$, so the projection of $dv|_{ \{1\} } (\beta_i^{\vee})$ onto $T_{[I]} {\mathcal B}$ is $- \symboly_i$, because $T_{[I]} {\mathcal H} = T_{[I]} {\mathcal B} \oplus \lin_k(\partial_1, \dots, \partial_6)$. Hence
\begin{equation}
dq(\beta_i^{\vee}) = - \symboly_i \textnormal{ for $i = 1, \dots, 6$.}
\end{equation}
After identifying $V$ with the image of the closed embedding given by the composition~\eqref{eq:compo} we get a factorisation:
\begin{equation}\label{eq:last}
q : \Spec(\frac{k[\bbeta]}{\Fperpy}) \to V \subset {\mathcal H}
\end{equation}
Recall that at the beginning of Subsection~\ref{sub:descriptionofv} we presented $V$ as $\Spec(\frac{k[\Bar{\symb}]}{J})$ and $(\symb_i)_{1 \leq i \leq 6}$ was a basis of $T_{[I]}^{\vee} V$ dual to $(\symboly_i)_{1 \leq i \leq 6}$. Thus, by equation~\eqref{eq:last}, we get that:
\begin{equation}\label{eq:lastlast}
(dq)^{\vee}(\symb_i) = - \beta_i \textnormal{ for $i = 1, \dots, 6$.}
\end{equation}
Moreover, by Lemma~\ref{fact:gmequibaru} $\bar{u}$ is $\G_m$-equivariant, so $q$ is $\G_m$-equivariant as well.
In this way we see that $q : \Spec(\frac{k[\bbeta]}{\Fperpy}) \to \Spec(\frac{k[\Bar{\symb}]}{J}) \simeq V$ yields a graded homomorphism of algebras:
\[\begin{tikzcd}
{\frac{k[\bbeta]}{\Fperpy}} & {\frac{k[\Bar{\symb}]}{J}}
\arrow["q^{\#}", from=1-2, to=1-1]
\end{tikzcd}\]
and by equation~\eqref{eq:lastlast} it is defined by $q^{\#}(\symb_i) = - \beta_i$ for $i=1, \dots, 6$. Thus if $f(\Bar{\symb}) \in J$ then $f(-\bbeta) \in \Fperpy$ and because $J$ and $\Fperpy$ are homogeneous it follows that $J \subset \Fperp(\Bar{\symb})$. From Proposition~\ref{prop:boundj} we get that $\Fperp(\Bar{\symb}) \subset J$ so these two ideals are in fact equal. Hence, $q = r \circ i \circ \bar{u}$ is an isomorphism. The morphism $i$ is an isomorphism on some open neighbourhood of $[I]$ by Corollary~\ref{cor:bplusisb} and $r$ is a closed embedding. Thus, $\bar{u}$ is an isomorphism onto $V \subset {\mathcal B}^+$.
\end{proof}
The point $[I] \in {\mathcal H}$ corresponds to the subscheme $\Spec(S/(\Fperp,g)) \subset \Affine^6$ and now we know that the fiber $V$ at $[I]$ is isomorphic to $\Spec(\frac{k[\bbeta]}{\Fperpy})$ so it contains a copy of $\Spec(S/(\Fperp,g))$ as a closed subscheme cut out only by $g$. Hence, the negative spike at $[I]$ almost recovers the ideal $I$. A similar phenomenon can be observed in the geometry of fractals. For example, the Mandelbrot set, which parametrizes functions $f(z) = z^2 + c$ such that the sequence $f(0), f(f(0)), \dots$ is bounded, contains Julia sets of some functions $f_0(z) = z^2 + c_0$. In other words, in the case of fractals the geometry of the Mandelbrot set contains subsets describing behaviour of a single function $f$. It would be interesting to see why such phenomena arise and are these two worlds: (1) of moduli spaces of points and (2) of dynamical behaviour of complex valued functions, somehow related.
%
\section*{Acknowledgements}
The author would like to thank Joachim Jelisiejew for his patience and time.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,294 |
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"redpajama_set_name": "RedPajamaC4"
} | 6,779 |
{"url":"https:\/\/meta.mathoverflow.net\/revisions\/1828\/2","text":"Consider a two-dimensional sphere with a Riemannian metric of total area $$4\\pi$$. Does there exist a subset whose area equals $$2\\pi$$ and whose boundary has length no greater than $$2\\pi$$?","date":"2022-05-24 00:34:52","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 3, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6650054454803467, \"perplexity\": 60.45285686116241}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-21\/segments\/1652662562106.58\/warc\/CC-MAIN-20220523224456-20220524014456-00676.warc.gz\"}"} | null | null |
{"url":"https:\/\/phys.libretexts.org\/Courses\/University_of_California_Davis\/UCD%3A_Physics_9A_Lab\/Lab_2%3A_Graphical_Methods_of_Data_Analysis\/2.2%3A_Activities","text":"Skip to main content\n\n# 2.2: Activities\n\n\n## Equipment\n\n\u2022 butcher paper\n\u2022 masking tape\n\u2022 launch ramp\n\u2022 a steel\u00a0marble\n\u2022 carbon paper\n\u2022 meter stick\n\u2022 plumb bob\n\n## The General Idea\n\nConsider the somewhat contrived story...\n\nA bitter controversy has recently broken out between the residents of a new housing development in the foothills and the Town Council over the safety of a number of large rocks perched on the slope of a hill that ends with a cliff, down to a lake, on the opposite side of which is the housing development. At the heart of the dispute is the relationship between the height above the edge of the cliff at which the rocks are balanced ($$y$$) and the distance, horizontally from the base of the cliff, to which the rocks will range ($$R$$) should they roll down the hill and fly off the cliff. Neither party is convinced by fancy physics calculations, so they both hired engineering firms to conduct modeled experiments to come to a conclusion.\n\nFigure 2.1.1 \u2013 Engineers' Experimental Model\n\nOne group of engineers concluded that $$y$$ and $$R$$ are related by:\n\n$y\\propto R$\n\nA second engineering firm concluded that $$y$$ and $$R$$ are related by:\n\n$y\\propto R^2$\n\nFurther calculations reveal that if the trajectories obey the linear relation, then some of the rocks starting higher on the hill will fly over the lake and land within the housing development, but if the quadratic relation holds, then even the highest rocks on the hill will land in the lake, leaving the housing development undamaged.\n\nYour Task: Set up the same experiment performed by the two engineering firms, and use the graphical technique discussed in the Background Material to draw a conclusion about which of the two relationships given above is the correct one.\n\n## Some Things to Think About\n\nAs you launch into this experiment, here are a few basic considerations to help you accomplish your goal:\n\n\u2022 You are trying to decide between two different functions that may be difficult to distinguish from each other unless one is smart about taking data. \u00a0One important bit of advice in this regard is in the final paragraph of the Background Material. \u00a0It mentions the scaling of the graphs, but how will you perform your experiment to also help in this regard?\n\u2022 One of the pieces of equipment is a plumb bob. \u00a0This device is used to connect two points along a vertical line. \u00a0How is this useful for this experiment?\n\u2022 Humans make all sorts of errors when running experiments. \u00a0In this case, one might record the wrong starting height, or accidentally give the marble a small push. \u00a0How will you protect against the effect of such\u00a0errors?\n\u2022 The engineer's model assumes that the rolling marbles leave the cliff horizontally.\n\u2022 All the experimental runs are performed on the same piece of paper. \u00a0How bad would it be\u00a0if the position of this paper\u00a0is accidentally changed between runs?\n\u2022 As is often the case, the keys to a successful experiment are planning (for which there are hints above), and a well-organized, orderly table of data and computed values.\n\n[As promised in the previous lab, from this point on in Physics 9 labs, you will be given only basic guidance (such as the few tips above). Setting up the means by which you take data and writing the report will be up to your group.]\n\n## Graphing\n\nFor this lab and future labs that involve graphs, you are free to do these however you wish \u2013 you can graph the data by hand or use software.\u00a0One tool you might find useful is\u00a0this\u00a0online graphing calculator. If you click the \"+\" button in the upper-left corner and selecting \"table\", you can enter your data into a table and the points will\u00a0\u00a0be plotted. Then in the upper-right corner you can click the \"share your graph\" icon, which will allow you to print it (which you can do on the lab printer) or save as a document. \u00a0For this lab, we are only looking for the data that looks linear; we are not interested in the actual straight line that best fits the data.\n\n## Lab Report\n\nCraft a lab report for these activities and analysis, making sure to include every contributing\u00a0group member's name on the front page. \u00a0You are strongly encouraged\u00a0to refer back to the Read Me as you do this, to make sure that you are not leaving out anything important. \u00a0You should also feel free to get feedback from your lab TA whenever you find that your group requires clarification or is at an impasse.\n\nEvery member of the group must upload a separate digital copy of the report to their lab assignment in Canvas\u00a0prior to leaving the lab classroom. \u00a0These reports are not to be written outside the lab setting.\n\nThis page titled 2.2: Activities is shared under a CC BY-SA 4.0 license and was authored, remixed, and\/or curated by Tom Weideman directly on the LibreTexts platform.\n\n\u2022 Was this article helpful?","date":"2023-03-31 10:01:37","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5014756321907043, \"perplexity\": 950.2144785433445}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-14\/segments\/1679296949598.87\/warc\/CC-MAIN-20230331082653-20230331112653-00041.warc.gz\"}"} | null | null |
{"url":"https:\/\/ccssmathanswers.com\/comparing-unlike-fractions\/","text":"# Comparing Unlike Fractions \u2013 Definition, Facts, Examples | How do you Compare Unlike Fractions?\n\nUnlike Fractions are fractions with different denominators. For example, $$\\frac { 4 }{ 3 }$$, $$\\frac { 2 }{ 5 }$$, $$\\frac { 1 }{ 9 }$$, $$\\frac { 3 }{ 41 }$$ are some of the examples of unlike fractions. Unequal reactions are completely different from like fractions where the denominator is the same for the like fractions.\n\nCompared to like fractions, it is difficult to perform arithmetic operations on unlike fractions. So, we need to follow some methods to resolve unlike fractions while performing arithmetic operations and Comparing Unlike Fractions. Check out the complete methods and processes to compare unlike fractions in this article. Also, find out all the 6th Grade Math concepts on our website for better preparation.\n\nAlso, find:\n\n## How to Compare Unlike Fractions with LCM?\n\nFollow the below process to compare unlike fractions. The step-by-step method is given below with a detailed explanation.\nStep 1: Firstly, convert the Unlike fractions into like fractions.\nStep 2: Take the LCM of given unlike fractions where LCM is the least common multiple.\nStep 3: Now, we can see each fraction that consists of an equivalent fraction with the same denominator having the obtained LCM.\nStep 4: Now, all the denominators appear similar. We need to compare the numerators having large numbers.\nStep 5: Note down the large number and finalize the large and small fractions.\n\n### How to Compare Fractions if Denominators are Different?\n\nCheck out the below problems and know the complete concept of comparing unlike fractions.\n\nQuestion 1:\nWhich is larger $$\\frac { 3 }{ 2 }$$ or $$\\frac { 5 }{ 6 }$$?\n\nSolution:\nGiven fractions are $$\\frac { 3 }{ 2 }$$ and $$\\frac { 5 }{ 6 }$$\n\n\u2022 To compare the given fractions, first, we need to find the equivalent fractions with a common denominator. Then rearrange the fractions with the help of their numerators.\n\u2022 The fractions $$\\frac { 3 }{ 2 }$$ and $$\\frac { 5 }{ 6 }$$ consists the denomiators as 2 and 6.\n\u2022 The LCM or common denominator of 2 and 6 is 6.\n\n\u2022 Now, multiply the fraction $$\\frac { 3 }{ 2 }$$ with 3 on top and bottom. $$\\frac { 9 }{ 6 }$$\n\u2022 Next, multiply the fraction $$\\frac { 5 }{ 6 }$$ with 1 on top and bottom. $$\\frac { 5 }{ 6 }$$\n\u2022 Now, both fractions are converted to like fractions.\n\u2022 Compare the top numbers of the fractions to find out the large fraction. The top numbers of the fractions are 9 and 5.\n\u2022 9 is greater than 5.\n\nTherefore, $$\\frac { 3 }{ 2 }$$ is greater than $$\\frac { 5 }{ 6 }$$.\n\nQuestion 2:\nWhich is larger $$\\frac { 7 }{ 4 }$$ or $$\\frac { 5 }{ 3 }$$?\n\nSolution:\nGiven fractions are $$\\frac { 7 }{ 4 }$$ and $$\\frac { 5 }{ 3 }$$\n\n\u2022 To compare the given fractions, first, we need to find the equivalent fractions with a common denominator. Then rearrange the fractions with the help of their numerators.\n\u2022 The fractions $$\\frac { 7 }{ 4 }$$ and $$\\frac { 5 }{ 3 }$$ consists the denomiators as 4 and 3.\n\u2022 The LCM or common denominator of 4 and 3 is 12.\n\n\u2022 Now, multiply the fraction $$\\frac { 7 }{ 4 }$$ with 3 on top and bottom. $$\\frac { 21 }{ 12 }$$\n\u2022 Next, multiply the fraction $$\\frac { 5 }{ 3 }$$ with 4 on top and bottom. $$\\frac { 20 }{ 12 }$$\n\u2022 Now, both fractions are converted to like fractions.\n\u2022 Compare the top numbers of the fractions to find out the large fraction. The top numbers of the fractions are 21 and 20.\n\u2022 21 is greater than 20.\n\nTherefore, $$\\frac { 7 }{ 4 }$$ is greater than $$\\frac { 5 }{ 3 }$$.\n\nQuestion 3:\nArrange the fractions $$\\frac { 5 }{ 4 }$$, $$\\frac { 5 }{ 3 }$$, $$\\frac { 7 }{ 2 }$$, $$\\frac { 6 }{ 5 }$$ in ascending order.\n\nSolution:\nGiven fractions are $$\\frac { 5 }{ 4 }$$, $$\\frac { 5 }{ 3 }$$, $$\\frac { 7 }{ 2 }$$, and $$\\frac { 6 }{ 5 }$$.\n\n\u2022 To compare the given fractions, first, we need to find the equivalent fractions with a common denominator. Then rearrange the fractions with the help of their numerators.\n\u2022 The fractions $$\\frac { 5 }{ 4 }$$, $$\\frac { 5 }{ 3 }$$, $$\\frac { 7 }{ 2 }$$, and $$\\frac { 6 }{ 5 }$$ consists the denomiators as 2, 3, 4 and 5.\n\u2022 The LCM or common denominator of 2, 3, 4, and 5 is 60.\n\n\u2022 Now, multiply the fraction $$\\frac { 5 }{ 4 }$$ with 15 on top and bottom. $$\\frac { 75 }{ 60 }$$.\n\u2022 Next, multiply the fraction $$\\frac { 5 }{ 3 }$$ with 20 on top and bottom. $$\\frac { 100 }{ 60 }$$.\n\u2022 Then, multiply the fraction $$\\frac { 7 }{ 2 }$$ with 30 on top and bottom. $$\\frac { 210 }{ 60 }$$.\n\u2022 Next, multiply the fraction $$\\frac { 6 }{ 5 }$$ with 12 on top and bottom. $$\\frac { 72 }{ 60 }$$.\n\u2022 Now, both fractions are converted to like fractions.\n\u2022 Compare the top numbers of the fractions to find out the small and large fractions. The top numbers of the fractions are 210, 100, 75, and 72.\n\u2022 72 < 75 < 100 < 210.\n\nTherefore, $$\\frac { 6 }{ 5 }$$, $$\\frac { 5 }{ 4 }$$, $$\\frac { 5 }{ 3 }$$, and $$\\frac { 7 }{ 2 }$$ is the ascending order of given fractions.\n\nQuestion 4:\nArrange the following fractions in descending order $$\\frac { 3 }{ 4 }$$, $$\\frac { 5 }{ 3 }$$, 1, $$\\frac { 2 }{ 3 }$$, and $$\\frac { 6 }{ 4 }$$.\n\nSolution:\nGiven fractions are $$\\frac { 3 }{ 4 }$$, $$\\frac { 5 }{ 3 }$$, 1, $$\\frac { 2 }{ 3 }$$, and $$\\frac { 6 }{ 4 }$$.\n\n\u2022 To compare the given fractions, first, we need to find the equivalent fractions with a common denominator. Then rearrange the fractions with the help of their numerators.\n\u2022 The fractions $$\\frac { 3 }{ 4 }$$, $$\\frac { 5 }{ 3 }$$, 1, $$\\frac { 2 }{ 3 }$$, and $$\\frac { 6 }{ 4 }$$ consists the denomiators in common as 1, 4 and 3.\n\u2022 The LCM or common denominator of 1, 4, and 3 is 12.\n\n\u2022 Now, multiply the fraction $$\\frac { 3 }{ 4 }$$ with 3 on top and bottom. $$\\frac { 9 }{ 12 }$$\n\u2022 Next, multiply the fraction $$\\frac { 5 }{ 3 }$$ with 4 on top and bottom. $$\\frac { 20 }{ 12 }$$\n\u2022 Then, multiply the number 1 with 12 on top and bottom. $$\\frac { 12 }{ 12 }$$.\n\u2022 Also, multiply the fraction $$\\frac { 2 }{ 3 }$$ with 4 on top and bottom. $$\\frac { 8 }{ 12 }$$\n\u2022 Multiply the fraction $$\\frac { 6 }{ 4 }$$ with 3 on top and bottom. $$\\frac { 18 }{ 12 }$$\n\u2022 Now, both fractions are converted to like fractions.\n\u2022 Compare the top numbers of all the fractions to find out the small and large fractions. The top numbers of the fractions are 9, 20, 12, 8, and 18.\n\u2022 20 > 18 > 12 > 9 > 8.\n\nTherefore, $$\\frac { 5 }{ 3 }$$, $$\\frac { 6 }{ 4 }$$, 1, $$\\frac { 3 }{ 4 }$$, and $$\\frac { 2 }{ 3 }$$ is the descending order of given fractions.\n\n### Converting Unlike Fractions to Like Fractions\n\nIt is important to convert the unlike fractions to like fractions to compare them. You can generally follow two methods to compare unlike fractions to like fractions. Check out the below example to know how to convert and compare unlike fractions by converting them to like fractions.\n\n### Comparing Fractions with Unlike Denominators Examples\n\nExample 1.\nCompare $$\\frac { 7 }{ 6 }$$ and $$\\frac { 5 }{ 9 }$$.\n\nSolution: Method 1: Convert the given unlike fractions into like fractions.\nTo convert the unlike fractions into like fractions, find the LCM of the denominators of the given fractions.\nLCM of denominators 6 and 9 is 18.\n\nNow, multiply the fraction $$\\frac { 7 }{ 6 }$$ with 3 on top and bottom. $$\\frac { 21 }{ 18 }$$. Next, multiply the fraction $$\\frac { 5 }{ 9 }$$ with 2 on top and bottom. $$\\frac { 10 }{ 18 }$$.\nNow, both fractions are converted to like fractions.\nAs both the denominators are equal now, you can compare the numerators to find the largest and smallest number.\n$$\\frac { 21 }{ 18 }$$ > $$\\frac { 10 }{ 18 }$$.\nSo, $$\\frac { 7 }{ 6 }$$ > $$\\frac { 5 }{ 9 }$$.\n\nMethod 2: We can also use cross multiplication to convert the unlike fractions to like fractions.\nSo, firstly take the given fractions. The given fractions are $$\\frac { 7 }{ 6 }$$ and $$\\frac { 5 }{ 9 }$$.\nNow, cross multiply the given fractions.\n7 \u00d7 9 = 63; 6 \u00d7 5 = 30.\n63 > 30.\n\nTherefore, $$\\frac { 7 }{ 6 }$$ is greater than $$\\frac { 5 }{ 9 }$$.\n\n### Comparing Unlike Fractions Solved Examples\n\nQuestion 1.\n\nCompare the given fractions by putting the right sign <, > or =.\n(i) $$\\frac { 2 }{ 5 }$$ ___ $$\\frac { 2 }{ 11 }$$\n(ii) $$\\frac { 7 }{ 3 }$$ ___ $$\\frac { 8 }{ 3 }$$\n(iii) $$\\frac { 3 }{ 9 }$$ ___ $$\\frac { 6 }{ 15 }$$\n(iv) $$\\frac { 4 }{ 3 }$$ ___ $$\\frac { 3 }{ 2 }$$\n(v) 1 ___ 1\n(vi) $$\\frac { 10 }{ 5 }$$ ___ $$\\frac { 10 }{ 7 }$$\n(vii) $$\\frac { 5 }{ 3 }$$ ___ $$\\frac { 12 }{ 9 }$$\n(viii) $$\\frac { 20 }{ 15 }$$ ___ $$\\frac { 28 }{ 21 }$$\n(ix) $$\\frac { 4 }{ 7 }$$ ___ $$\\frac { 10 }{ 13 }$$\n\nSolution:\n\n(i) Given fractions are $$\\frac { 2 }{ 5 }$$ and $$\\frac { 2 }{ 11 }$$.\nLCM of 5 and 11 is 55.\nNow, the fraction becomes $$\\frac { 22 }{ 55 }$$ and $$\\frac { 10 }{ 55 }$$.\n22 > 10. So the answer is >.\n\n$$\\frac { 2 }{ 5 }$$ > $$\\frac { 2 }{ 11 }$$\n\n(ii) Given fractions are $$\\frac { 7 }{ 3 }$$ and $$\\frac { 8 }{ 3 }$$.\nBoth the denominators are the same. So, now compare the numerators.\n7 < 3. So the answer is <.\n\n$$\\frac { 7 }{ 3 }$$ < $$\\frac { 8 }{ 3 }$$.\n\n(iii) Given fractions are $$\\frac { 3 }{ 9 }$$ and $$\\frac { 6 }{ 15 }$$.\nLCM of 9 and 15 is 45.\nNow, the fraction becomes $$\\frac { 15 }{ 45 }$$ and $$\\frac { 18 }{ 45 }$$.\n15 < 18. So the answer is <.\n\n$$\\frac { 3 }{ 9 }$$ < $$\\frac { 6 }{ 15 }$$\n\n(iv) Given fractions are $$\\frac { 4 }{ 3 }$$ and $$\\frac { 3 }{ 2 }$$.\nLCM of 3 and 2 is 6.\nNow, the fraction becomes $$\\frac { 8 }{ 6 }$$ and $$\\frac { 9 }{ 6 }$$.\n8 < 9. So the answer is <.\n\n$$\\frac { 4 }{ 3 }$$ < $$\\frac { 3 }{ 2 }$$.\n\n(v) Given numbers are 1 and 1. So the answer is =.\n\n1 = 1.\n\n(vi) Given fractions are $$\\frac { 10 }{ 5 }$$ and $$\\frac { 10 }{ 7 }$$.\nLCM of 5 and 7 is 35.\nNow, the fraction becomes $$\\frac { 70 }{ 35 }$$ and $$\\frac { 50 }{ 35 }$$.\n70 > 50. So the answer is >.\n\n$$\\frac { 10 }{ 5 }$$ > $$\\frac { 10 }{ 7 }$$.\n\n(vii) Given fractions are $$\\frac { 5 }{ 3 }$$ and $$\\frac { 12 }{ 9 }$$.\nLCM of 3 and 9 is 9.\nNow, the fraction becomes $$\\frac { 15 }{ 9 }$$ and $$\\frac { 12 }{ 9 }$$.\n15 > 12. So the answer is >.\n\n$$\\frac { 15 }{ 9 }$$ > $$\\frac { 12 }{ 9 }$$.\n\n(viii) Given fractions are $$\\frac { 20 }{ 15 }$$ and $$\\frac { 28 }{ 21 }$$.\nLCM of 15 and 21 is 105.\nNow, the fraction becomes $$\\frac { 140 }{ 105 }$$ and $$\\frac { 140 }{ 105 }$$.\n140 = 140. So the answer is =.\n\n$$\\frac { 20 }{ 15 }$$ = $$\\frac { 28 }{ 21 }$$.\n\n(ix) Given fractions are $$\\frac { 4 }{ 7 }$$ and $$\\frac { 10 }{ 13 }$$.\nLCM of 7 and 13 is 91.\nNow, the fraction becomes $$\\frac { 52 }{ 91 }$$ and $$\\frac { 70 }{ 91 }$$.\n52 < 70. So the answer is <.\n\n$$\\frac { 4 }{ 7 }$$ < $$\\frac { 10 }{ 13 }$$.\n\n2. Arrange the given fractions in ascending order.\n(i) $$\\frac { 9 }{ 6 }$$, $$\\frac { 9 }{ 8 }$$, $$\\frac { 9 }{ 4 }$$\n(ii) $$\\frac { 1 }{ 8 }$$, $$\\frac { 1 }{ 2 }$$, 1\n(iii) $$\\frac { 3 }{ 2 }$$, $$\\frac { 9 }{ 4 }$$, $$\\frac { 15 }{ 8 }$$\n(iv) $$\\frac { 4 }{ 5 }$$, $$\\frac { 6 }{ 5 }$$, $$\\frac { 6 }{ 4 }$$\n\nSolution: (i) Given fractions are $$\\frac { 9 }{ 6 }$$, $$\\frac { 9 }{ 8 }$$, and $$\\frac { 9 }{ 4 }$$.\nLCM of 6, 8, and 4 is 24.\nNow, the fractions becomes $$\\frac { 36 }{ 24 }$$, $$\\frac { 27 }{ 24 }$$, and $$\\frac { 54 }{ 24 }$$.\n27 < 36 < 54.\n\nTherefore, $$\\frac { 9 }{ 8 }$$, $$\\frac { 9 }{ 6 }$$, and $$\\frac { 9 }{ 4 }$$ is the ascending order of given fractions.\n\n(ii) Given fractions are $$\\frac { 1 }{ 8 }$$, $$\\frac { 1 }{ 2 }$$, and 1.\nLCM of 8, 2, and 1 is 8.\nNow, the fractions becomes $$\\frac { 1 }{ 8 }$$, $$\\frac { 4 }{ 8 }$$, and $$\\frac { 8 }{ 8 }$$.\n1 < 4 < 8.\n\nTherefore, $$\\frac { 1 }{ 8 }$$, $$\\frac { 1 }{ 2 }$$, and 1 is the ascending order of given fractions.\n\n(iii) Given fractions are $$\\frac { 3 }{ 2 }$$, $$\\frac { 9 }{ 4 }$$, and $$\\frac { 15 }{ 8 }$$.\nLCM of 2, 4, and 8 is 8.\nNow, the fractions becomes $$\\frac { 12 }{ 8 }$$, $$\\frac { 18 }{ 8 }$$, and $$\\frac { 15 }{ 8 }$$.\n12 < 15 < 18.\n\nTherefore, $$\\frac { 3 }{ 2 }$$, $$\\frac { 15 }{ 8 }$$, and $$\\frac { 9 }{ 4 }$$ is the ascending order of given fractions.\n\n(iv) Given fractions are $$\\frac { 4 }{ 5 }$$, $$\\frac { 6 }{ 5 }$$, $$\\frac { 6 }{ 4 }$$.\nLCM of 4, and 5 is 20.\nNow, the fractions becomes $$\\frac { 16 }{ 20 }$$, $$\\frac { 24 }{ 20 }$$, and $$\\frac { 30 }{ 20 }$$.\n16 < 24 < 30.\n\nTherefore, $$\\frac { 4 }{ 5 }$$, $$\\frac { 6 }{ 5 }$$, $$\\frac { 6 }{ 4 }$$ is the ascending order of given fractions.\n\n### Word Problems on Comparing Unlike Fractions\n\nQuestion 1.\n\nSam ate $$\\frac { 3 }{ 18 }$$ part of the chocolate and Sophia ate $$\\frac { 5 }{ 9 }$$ part of the chocolate. Who ate the greater part of the chocolate? What fraction of chocolate was finished by the two girls?\n\nSolution:\n\nGiven that Sam ate $$\\frac { 3 }{ 18 }$$ part of the chocolate and Sophia ate $$\\frac { 5 }{ 9 }$$ part of the chocolate.\nFirstly, take the fractions and convert them into like fractions.\nLCM of 9 and 18 is 18.\nSo, $$\\frac { 3 }{ 18 }$$ and $$\\frac { 10 }{ 18 }$$ are the like fractions now.\nCompare the two fractions $$\\frac { 3 }{ 18 }$$ and $$\\frac { 10 }{ 18 }$$.\nAs 10 is greater than 3, $$\\frac { 5 }{ 9 }$$ is greater than $$\\frac { 3 }{ 18 }$$. Sophia ate a greater part of the chocolate.\nNow, add two fractions to find the chocolate finished by the two girls.\n$$\\frac { 3 }{ 18 }$$ + $$\\frac { 5 }{ 9 }$$ = $$\\frac { 3 + 10 }{ 18 }$$ = $$\\frac { 13 }{ 18 }$$.\n\n$$\\frac { 13 }{ 18 }$$ fraction of chocolate was finished by the two girls.\n\nQuestion 2.\n\nRam traveled $$\\frac { 10 }{ 6 }$$ km of distance in a car and alex traveled $$\\frac { 6 }{ 4 }$$ km of distance in a car. Who covered the greater distance?\n\nSolution:\n\nGiven that Ram traveled $$\\frac { 10 }{ 6 }$$ km of distance in a car and alex traveled $$\\frac { 6 }{ 4 }$$ km of distance in a car.\nFirstly, take the fractions and convert them into like fractions.\nLCM of 6 and 4 is 12.\nSo, $$\\frac { 20 }{ 12 }$$ and $$\\frac { 18 }{ 12 }$$ are the like fractions now.\nCompare the two fractions $$\\frac { 20 }{ 12 }$$ and $$\\frac { 18 }{ 12 }$$.\nAs 20 is greater than 18, $$\\frac { 10 }{ 6 }$$ is greater than $$\\frac { 6 }{ 4 }$$.\n\nTherefore, Ram traveled more distance compared to Alex.\n\nQuestion 3.\n\nYash walked for $$\\frac { 62 }{ 4 }$$ km and William walked for $$\\frac { 27 }{ 2 }$$ km during the weekend. Who cycled more and by how much?\n\nSolution:\n\nGiven that Yash walked for $$\\frac { 62 }{ 4 }$$ km and William walked for $$\\frac { 27 }{ 2 }$$ km during the weekend.\nFirstly, take the fractions and convert them into like fractions.\nLCM of 2 and 4 is 4.\nSo, $$\\frac { 62 }{ 4 }$$ and $$\\frac { 54 }{ 4 }$$ are the like fractions now.\nCompare the two fractions $$\\frac { 62 }{ 4 }$$ and $$\\frac { 54 }{ 4 }$$.\nAs 62 is greater than 54, $$\\frac { 62 }{ 4 }$$ is greater than $$\\frac { 27 }{ 2 }$$.\n$$\\frac { 62 }{ 4 }$$ \u2013 $$\\frac { 27 }{ 2 }$$ = 2\n\nTherefore, Yash walked more compared to William. Yash walked 2km more than William.\n\nQuestion 1. What are unlike fractions?\n\nAnswer: Unlike fractions are the fractions with different denominators. So, the fractions with different denominators like 3, 4, and 5 are called unlike fractions.\n\nQuestion 2. What is the unlike fraction example?\n\nAnswer: The example of unlike fraction is $$\\frac { 4 }{ 3 }$$, $$\\frac { 3 }{ 5 }$$, etc.\n\nQuestion 3. What are the methods to compare unlike fractions?\n\nAnswer: You can use two methods to compare unlike fractions. They are\n(i) Converting unlike fractions to like fractions.\n(ii) Cross multiplication of the fractions to compare unlike fractions.\n\nQuestion 4. How do you compare unlike fractions with LCM?\n\nAnswer: We can compare unlike fractions with LCM.\nFirstly, take the LCM of given fraction denominators.\nThen, convert the fraction according to the obtained LCM.\nThen, finally, compare the fraction depending on the largest number by comparing the numerators of the given fractions.\n\nQuestion 5. What is the difference between like fraction and unlike fraction?\n\nAnswer: The main difference between like fraction and unlike fraction is the like fraction consists of the same denominator where the unlike fraction consists of the different denominator.\n\nScroll to Top","date":"2023-01-27 02:23:41","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8374008536338806, \"perplexity\": 967.885385562458}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-06\/segments\/1674764494852.95\/warc\/CC-MAIN-20230127001911-20230127031911-00529.warc.gz\"}"} | null | null |
Successfully prepare students for the international marketplace. International Business illustrates how successful managers must function in a competitive world. Packed with current examples that reflect the vibrancy of the international business field, this student-friendly text offers a managerial approach that keeps an emphasis on skills development, emerging markets and geographical literacy. | {
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Q: JavaScript RegExp: match, but don't capture There are a thousand regular expression questions on SO, so I apologize if this is already covered. I did look first.
Given the following pattern:
(?:/|-)[0-9]{2}$
And the following strings:
str1 = '65/65/65'
str2 = '65/65/6565'
The matches are:
str1 = '/65' // expected '65'
str2 = '' // as I expected
My intention with ?: was to match, but not include a / or -. What is the correct regular expression to meet my expectations?
A: Normally, this would be done with a lookbehind:
/(?<=[-\/])[0-9]{2}$/
Sadly, JavaScript doesn't support those.
Instead, since you know the length of the "extra bit" (ie. one character, either - or /), it should be simple enough to just .substr(1) it.
A: There is no look behind, but there is exclusive matching (?:) as it won't fail. The issue is that it will be included as part of the overall match (index 0 of the result). Therefore the best alternative is to instead capture the portion you do want. In your example this could be done by surrounding the digit matching in parentheses.
var re = /(?:[/-])([0-9]{2})$/,
str1 = '12/34/56',
str2 = '12/34/5678';
var res1 = str1.match(re); // will want to use res1[1], not res1[0]
var res2 = str2.match(re); // null returned
A: As there's no lookbehind available in Javascript, just wrap the desired part into a capturing group:
var str = '65/66/67';
if(res = str.match(/(?:\/|-)([0-9]{2})$/)) {
console.log(res[1]);
}
See fiddle
Note: (?:\/|-) can be replaced with a character class [\/-] like @anubhava commented.
| {
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\section{Motivation and layout}
Continuous-time quantum walks (CTQW) are intensively studied as simplified models for a wide range of applications, spacing from quantum transport \cite{kendon2011perfect,mulken2011continuous,razzoli11,kulvelis15,tama16,tamascelli19,zatelli20}, e.g. excitonic transport in biochemical complexes involved in some instances of bacterial photosynthesis \cite{kempe2003quantum,mohseni2008environment,Chin2010} , to universal quantum computation \cite{childs09comp} or specific quantum algorithms such as the spatial search \cite{childs04,portugal2013quantum,childs2014,chakraborty2016,Chakraborty20,paris21}. They are usually defined \cite{fahrigutman} on the same line of classical random walks (RW) on undirected, simple graphs, by promoting the graph Laplacian $\mathrm{L}$, which is the generator of the time evolution of an unweighted RW, to an Hamiltonian $\mathrm{H}$.
Going from the classical transfer matrix to the quantum unitary evolution operator, together with the shift of focus from probabilities to amplitudes, make CTQW radically different from classical RW, with reliable prospects of achieving a quantum advantage in specific tasks \cite{chil02}. To mention a few, it is known that CTQW exhibit ballistic propagation of probability on lattices, contrary to the diffusive behavior of classical RW, and that they can solve search problems in shorter times.
\par
Fairly recently, however, it has been suggested that constraining the Hamiltonian of the CTQW to be the Laplacian of the graph is an unnecessary restriction, and richer phenomenology can be observed when the off-diagonal matrix elements of $\mathrm{H}$ are allowed to be generic complex phases, in compliance just with hermiticity: the resulting systems are called \emph{chiral} CTQW \cite{cqw1,cqw2}, because they exhibit asymmetry under time-reversal and directional bias in the propagation of probability. Further motivation for this generalization of CTQW has been provided in \cite{frigerio2021generalized}, where the most general correspondence between classical and quantum continuous-time random walks was also derived. Acknowledging this larger space of opportunities offered by quantum walks, which stems from the one-to-many nature of the step from classical to quantum, one is immediately faced with a new challenge: if assuming $\mathrm{H}= \mathrm{L}$ leads to a single choice of CTQW for a given unweighted graph, enlarging the focus to include chiral CTQW offers many free, real parameters that can be adjusted to optimize the quantum advantage in a specific task, at fixed graph topology. \\
\par
In this paper, we show through a variety of relevant examples and an analytic discussion, that a previously introduced quantity \cite{GBP20}, the quantum-classical distance $\mathcal{D}_{QC}$, is a valuable tool to guide this optimization: it correctly captures the distinction between classical and quantum evolutions of random walks on graphs. Quite generally, quantum walks outperform classical random walks because of faster hitting, which is related to quantum transport and targeting, and faster mixing \cite{mooremixing,ahmadimixing}, when the walker spreads out towards a maximally coherent, uniform superposition of all sites.
By maximizing the value of $\mathcal{D}_{QC}$ at short time scales over the free parameter's space, one is quickly directed towards chiral CTQWs with maximal advantage over the corresponding classical RW, either because of quick hitting or mixing \cite{ahmadimixing,mooremixing} depending on the graph's topology.
Indeed, although from their very introduction in \cite{fahrigutman} the quantum advantage of CTQW has been linked with quantum coherence, it is clear that it cannot reduce to this sole quantum phenomenon, since the target state in a transport task on a graph has very little coherence, for example (although it \emph{does} play a role throughout the evolution).
\par
After setting the notation for continuous-time classical and (chiral) quantum walks in Section \ref{sec:II},
we bring attention to the three dynamical quantities that will be employed throughout the article: the quantum-classical distance, the 1-norm of coherence and the Inverse Participation Ratio, which are all defined in Section \ref{sec:III} and accompanied by a comparison of their short-time expansions for generic chiral Hamiltonians. We then argue the effectiveness of $\mathcal{D}_{QC}$ in the identification of "optimal" phase configurations at fixed topology by looking at four very different and emblematic examples. We start with cycle graphs in Section \ref{sec:IV}, which allow for analytical expressions also in the presence of phases and constitute a testbed for our ideas. For odd cycles we show that the quantum-classical distance correctly signals the best phase for quantum transport and, conversely, spots the characteristic suppression of transport in even cycles for a resonant phase value, which is associated with a dip in the value of $\mathcal{D}_{QC}$.
\par
Then in Section \ref{sec:V} we move to complete graphs, having maximal connectivity. After a preliminary exploration based on randomly generated Hamiltonians, we maximize the quantum-classical distance at short times and find a particular set of hermitian Hamiltonians which permit quantum search without oracle in a time which exactly achieves the optimal quantum speed limit, outperforming Grover's algorithm in the constant pre-factor and holding for any size of the complete graph. Interestingly, here $\mathcal{D}_{QC}$ is larger for evolutions which quickly lead to highly \emph{delocalized} states, in stark contrast with its behavior for cycles. This is consistent with the other typical behavior of quantum walks, which is \emph{fast mixing}. Indeed, the same optimal chiral evolution on complete graphs achieves uniform mixing, contrary to the non-chiral evolution, also with a quadratic speedup with respect to the standard protocol that involves hypercube graphs \cite{mooremixing}.
As a third case, in Section \ref{sec:VI} we examine quantum switches, constructed from a triangle graph with independent chains of sites attached to each vertex of the polygon. It is already known that, for resonant value of the sole free phase, this topology allows for directional quantum transport from one arm of the triangle to another, with minimal losses on the excluded, third arm. We show that the quantum-classical distance again spots the best value of the phase. Moreover, since these graphs are non-regular, an ambiguity about the diagonal entries of $\mathrm{H}$ arises, since the non-chiral association $\mathrm{H} = \mathrm{L}$ indirectly introduces a potential field landscape. We argue that setting all the diagonal phases of $\mathrm{H}$ to the same, arbitrary value, instead of the non-uniform connectivities of the different sites, is the unbiased choice and also the most efficient one for directional quantum transport.
\par
Finally, we tackle the cube graph in Section \ref{sec:VII}, on which perfect quantum transport from one vertex to the opposite one, with the Laplacian as generator, is known to happen. The quantum-classical distance here suggests that phases cannot improve this standard evolution. However, minimal values of $\mathcal{D}_{QC}$ are achieved in correspondence with evolutions that suppress transport at all times on half of the vertices of the cube.
\par
\section{Quantum and classical walks on graphs}
\label{sec:II}
Continuous-time quantum walks are traditionally introduced through an analogy with classical random walks on graphs. For the latter, one considers an undirected simple graph $\mathcal{G} =( V , E) $ of $N$ vertices, where $V$ is the set of vertices or \emph{sites}, and $E$ is the set of edges. To $\mathcal{G}$ corresponds an $N \times N$ symmetric matrix, namely the \emph{Laplacian matrix} of the graph, which is defined as $L = D - A$ where $A$ is the adjacency matrix of $\mathcal{G}$, such that $\left[ A \right]_{jk} = 1$ if there is an edge in $E$ connecting sites $j$ and $k$ and $0$ otherwise, including the diagonal elements, whereas $D$ is a diagonal matrix encoding the connectivities of each vertex, i.e. the number of edges departing from it. Since the sum of the rows and of the colums of $\mathrm{L}$ is zero, it generates a semigroup of bistochastic transformations $\mathcal{E}_{t} = e^{-t \mathrm{L} }$ for $t \in \mathds{R}^{+}$. Therefore, acting on a vector $\underline{p}_0 \in \mathds{R}^{N}$ of occupation probabilities for each site at the initial time, one has a continuous-time, autonomous stochastic process on the graph, which we will call a continuous-time (classical) random walk:
\begin{equation}
\label{eqCRWp}
\underline{p} (t) \ \ := \ \ \mathcal{E}_{t} \left[ \underline{p} \right]\ \ := \ \ e^{-t \mathrm{L} } \underline{p}_{0}.
\end{equation}
Fahri and Gutman \cite{fahrigutman} noticed a suggestive similarity between Eq.(\ref{eqCRWp}) and the Schr\"{o}dinger equation for an $N$-level quantum system. Therefore, they proposed to \emph{define} a continuous-time quantum walk on the same graph $\mathcal{G}$ by promoting $\mathrm{L}$ to a quantum Hamiltonian matrix $\mathrm{H} = \mathrm{L}$ acting on an $N$-dimensional complex Hilbert space $\mathcal{H} \sim \mathds{C}^{N}$ which has a preferred basis $\{ \vert j \rangle \}_{j= 1, ... N}$ whose elements describe localized quantum states on the sites of the graph. The quantum evolution of any initial state $\vert \psi_{0} \rangle \in \mathcal{H}$ now comes automatically \footnote{ Here and in the following we fix $\hbar = 1$. The units of energies are reabsorbed in the time scale so as to conveniently keep everything dimensionless.}:
\begin{equation}
\vert \psi(t) \rangle \ = \ U_{t} \vert \psi_{0} \rangle \ = \ e^{-i t \hat{H} } \vert \psi_{0} \rangle.
\end{equation}
Recently it was pointed out that imposing $\mathrm{H} = \mathrm{L}$ is too restrictive on the quantum side, since a generic Hamiltonian compatible with the graph topology does not need to be a real matrix, but just a Hermitian one. We will focus on \emph{unweighted} graphs, such that the non-zero elements of the adjacency matrix $A$ are equal to $1$ for all the links. Therefore, the most general $\mathrm{H}$ has off-diagonal entries:
\begin{equation}
\left[ \mathrm{H} \right]_{kj} \ \ = \ \ e^{ i \varphi_{kj}} \ \ \ \ ( j \neq k )
\end{equation}
if sites $j$ and $k$ are linked in $\mathcal{G}$, and zero otherwise. Here $\varphi_{kj} \in [0, 2 \pi)$ is a phase depending on the link and $\varphi_{kj} = - \varphi_{jk}$ is required by Hermiticity of $\mathrm{H}$. On the other hand, the diagonal elements of $\mathrm{H}$ must be real numbers, but they do not need to satisfy any further constraint. For regular graphs, the simplest and most unbiased choice is to assume $\left[ \mathrm{H} \right]_{jj} = d$ for all $j = 1,...,N$, so that they are all equal and they can be discarded as they generate an overall, unobservable phase shift on the whole Hilbert space. For non-regular graphs, instead, the choice is less obvious and we shall return to this point in a later section (see also \cite{wong2016laplacian}). Continuous-time quantum walks whose Hamiltonian is a generic Hermitian matrix compatible with the graph topology are also called \emph{chiral} CTQW, to distinguish them from the most standard case with $\mathrm{H} = \mathrm{L}$ (or $\mathrm{H} = \mathrm{A}$), which is by far the most studied in the literature as of now.
\newline
In this work, we are interested in optimizing the additional degrees of freedom of a chiral CTQW on a given graph to achieve the maximal advantage over the corresponding, unique classical RW. It is thus convenient to formulate both evolutions in the Hilbert space formalism to facilitate the comparison. To do this, we first remark that quantum coherent states in the site basis do not exist at the classical level, therefore we shall assume a localize state $\vert j \rangle$ with $j = 1,...,N$ as the initial condition for both the quantum and the classical evolution, to make a fair comparison. Then the classical evolution can be defined by the standard embedding of a classical probability onto a Hilbert space with respect to a preferred basis, that is:
\begin{equation}
\label{eq:CLev}
\mathcal{E}_{t} \big[ \vert j \rangle \langle j \vert \big] = \sum_{k=1}^{N} \langle k \vert e^{-t \mathrm{L}} \vert j \rangle \vert k \rangle \langle k \vert = \sum_{k=1}^{N} p_{kj}(t) \vert k \rangle \langle k \vert
\end{equation}
which ensures that the classically evolved state is always incoherent in the localized basis. Also the quantum evolution can be rewritten in a similar way:
\begin{equation}
\label{eq:Qev1}
\mathcal{U}_{t} \big[ \vert j \rangle \langle j \vert \big] \ = \ e^{-i \hat{H} t} \vert j \rangle \langle j \vert e^{ i \hat{H} t } \ = \ \vert \psi_{j} (t) \rangle \langle \psi_{j} (t) \vert
\end{equation}
where:
\begin{align}
\label{eq:Qev2}
\vert \psi_{j}(t) \rangle \ \ & = \ \ \sum_{k=1}^{N} \alpha_{kj}(t) \vert k \rangle \\
\alpha_{kj}(t) \ \ &= \ \ \langle k \vert e^{-i\hat{H} t} \vert j \rangle \ \ = \ \ \left[ e^{-i \mathrm{H} t} \right]_{kj}.
\end{align}
\subsection{Chiral quantum walks and gauge invariance of transition probabilities}
By a diagonal, unitary change of basis, which is a local phase transformation, many of the phases of $\mathrm{H}$ can be set to $0$. This is particularly useful when considering site-to-site transition probabilities, as it is often the case when dealing with quantum walks. Indeed, quantities such as $P_{j \to k} (t) = \vert \langle k \vert e^{- i H t} \vert j \rangle \vert^{2}$ and functions thereof do not depend on phases which can be removed by those local phase transformations. These are effectively \emph{gauge transformations} and, at least for some regular topologies of the underlying graphs, the change in the Hamiltonian's phases can be accounted for by a gauge transformation of the \emph{gauge connection field}. Since to each edge of the graph can correspond a phase in $\mathrm{H}$ and the phase of the localized state on each vertex can be modified independently, the maximal number of phase degrees of freedom that affect the transition probabilities $P_{j \to k}(t)$ is $|E|-N + 1 = 1 - \chi(\mathcal{G}) $, where $|E|$ is the number of edges of the graph, whereas $\chi(\mathcal{G})$ is the Euler characteristic for generic graphs, including non-planar ones.
For planar graphs, this number is precisely the number of loops, while for non planar ones the number of loops is not obvious to define. In particular, phases cannot affect transition probabilities on tree graphs, while cycle graphs all have just a single relevant phase, which can be distributed over each link or concentrated on a single one. However, given a particular choice for the Hermitian matrix $\mathrm{H}$, it is not trivial to determine whether it is equivalent to a real, symmetric Hamiltonian or not. It has been shown \cite{TRBIAM21} that this is the case if and only if the product of phase factors along each directed, simple, closed path on $\mathcal{G}$ is $1$.
\section{Quantum-classical distance}
\label{sec:III}
\label{sec:2}
By combining Eq.(\ref{eq:CLev}) and Eq.(\ref{eq:Qev1},\ref{eq:Qev2}), it is possible to define a time-dependent quantity which compares the quantum evolution to the classical one, when both start in the same, localized initial state $\hat{\boldsymbol{\rho}}^{0}_{j} = \vert j \rangle \langle j \vert$. This quantity is the \emph{quantum-classical distance} \cite{GBP20}:
\begin{equation}
\mathcal{D}_{QC}^{j} (t) \ \ := \ \ 1 - \sum_{k=1}^{N} p_{kj} (t) \vert \langle k \vert \psi_{j} (t) \rangle \vert^{2} .
\end{equation}
It is still true, by the same token as in \cite{GBP20}, that the minimum value of the quantum-classical distances over all possible \emph{classical} initial states can be attained by a single localized state, at any given time. Therefore we can still define a global quantum-classical distance by \footnote{$V$ is the ordered set of vertices of the graph.}:
\begin{equation}
\mathcal{D}_{QC} (t) \ \ := \ \ \max_{j \in V} \left\{ \mathcal{D}_{QC}^{j} (t) \right\}
\end{equation}
which is independent of the initial state, but only depends upon the graph (specified by $\mathrm{L}$) and the choice of $\mathrm{H}$ compatible with the topology.\\
Checking for a few graphs with different topologies and number of vertices, the overall behaviour of this quantity in time seems very similar to the simpler scenario where $\mathrm{L}$ is a Laplacian and the straightforward identification $\mathrm{H} = \mathrm{L}$ can be applied. In particular, the asymptotic value in the long time limit takes the same expression for any connected graph of size $N$ and it is given by:
\begin{equation}
\mathcal{D}_{QC}^{j} (t \gg 1) \simeq 1 - 1/N.
\end{equation}
Because of this saturation effect of $\mathcal{D}_{QC}$ at large times, fluctuations of the quantum-classical distance can be significant only if they happen before the classical evolution gets close to equilibrium. This is a valuable property: indeed, since the quantum evolution is aperiodic on general graphs, wild behaviors can occur at sufficiently long times and it is therefore mandatory, from a physical point of view, to impose a cutoff on the relevant time scale to perform the required task. A natural choice is precisely the time scale of the classical evolution, especially if a quantum advantage is sought.
\par
\subsection{Quantum-classical distance at short times}
To understand the behavior of the quantum-classical distance at short times, instead, we may expand $p_{kj} (t)$ and $ \vert \langle k \vert \psi_{j} (t) \rangle \vert^{2}$ up to second order in $t$ and obtain
\begin{align}
\mathcal{D}_{QC}^{j} (t) =& \, t \left( \left[ \mathrm{H}^{2} \right]_{jj} - [ \mathrm{H}]_{jj}^{2} \right) \notag \\
& - t^2 \Bigg(- \left[\mathrm{H}^{2} \right]_{jj} + [ \mathrm{H}]_{jj}^{2} + \frac12
\left[\mathrm{H}^{2} \right]_{jj}^{2} \notag \\
& \; - \left[\mathrm{H}^{2} \right]_{jj} [\mathrm{H}]_{jj}^{2} + \frac12 \sum_s \left| \mathrm{H}_{js}\right|^{4}
\Bigg) \ + \ O(t^3)
\end{align}
If we focus on unweighted graphs, which means that we assume $\vert \mathrm{H}_{jk} \vert$ to be either $1$ or $0$ depending on whether vertices $j$ and $k$ are connected by an edge or not, then we can considerably simplify the expression above. Indeed, we find that:
\begin{equation}
\left[ \mathrm{H}^{2} \right]_{jj} - [ \mathrm{H}]_{jj}^{2} \ = \ d_{j} \ = \ \mathrm{L}_{jj}
\end{equation}
where $d_{j}$ is the connectivity of vertex $j$ in the graph. Notice also that:
\begin{align}
\frac12
& \left[\mathrm{H}^{2} \right]_{jj}^{2} - \left[ \mathrm{H}^{2} \right]_{jj} [\mathrm{H}]_{jj}^{2} + \frac12 \sum_s \left| \mathrm{H}_{js}\right|^{4} \ = \\
& = \frac12 \left( \left[ \mathrm{H}^{2} \right]_{jj} - [ \mathrm{H}]_{jj}^{2} \right)^{2} + \frac12 \sum_{s \neq j} \left| \mathrm{H}_{js}\right|^{4} \ = \\
& = \frac{d^{2}_{j} + d_{j}}{2}
\end{align}
where in the last step we used the fact that $\sum_{s \neq j} \left| \mathrm{H}_{js}\right|^{4}$ gets a contribution of $1$ for any $s$ connected to $j$ and $0$ otherwise. Therefore, the short-time expansion of the quantum-classical distance to second order is:
\begin{equation}
\label{eq:DQCshortT}
\mathcal{D}_{QC}^{j} (t) \ = \ d_{j} t \ - \ d_{j} (d_{j} - 1) \frac{t^{2}}{2} \ + \ O(t^3).
\end{equation}
It is relevant to stress that the coefficients of this expansion are dictated just by the connectivity of the starting vertex and they are not influenced by any other parameter involved in $\mathrm{H}$: in particular, \emph{neither the on-site average energies nor the phases of the off-diagonal entries of $\mathrm{H}$ affect the short-time behavior of the quantum-classical distance}.
\subsection{1-norm of Coherence and Inverse Participation Ratio}
In order to understand what type of properties of the quantum walk contributes to the quantum-classical distance, and how $\mathcal{D}_{QC}$ relates to some figure of merits that are considered useful for particular tasks, we now seek the short-time expansion of other two quantities, the 1-norm of coherence and the inverse participation ratio (IPR), to be later compared with Eq.(\ref{eq:DQCshortT}).
\par
Let us start with the 1-norm of coherence. With respect to the localized basis, the coherence of the pure state $\vert \psi_{j}(t) \rangle$ given by Eq.(\ref{eq:Qev1}) can be written as \cite{GBP20}:
\begin{equation}
\mathcal{C}_{j} (t) \ \ = \ \ \left( \sum_{k} \vert \alpha_{kj} (t) \vert \right)^{2} - 1
\end{equation}
with $\alpha_{jk}(t)$ defined according to Eq.(\ref{eq:Qev2}). Here and in the following, it will be helpful to start from the fourth-order short time expansion of $\vert \alpha_{kj} (t) \vert^{2}$:
\begin{align}
\label{eq:alfa2smallt}
\vert \alpha_{kj}(t) \vert^{2} &= \ \ \delta_{jk} \ - \ t^{2} \left( \delta_{jk} \left[ \mathrm{H}^{2} \right]_{jj} - \vert \mathrm{H}_{jk} \vert^{2} \right) \ + \\
& \ + \ t^3 \mathrm{Im} \left[ \mathrm{H}_{jk} \left[ \mathrm{H}^{2} \right]_{kj} \right] \ + \notag \\
& + \ t^4 \left( \frac{1}{12} \delta_{jk} \left[ \mathrm{H}^{4} \right]_{jj} - \frac{1}{3} \mathrm{Re} \left[ \mathrm{H}_{jk} \left[ \mathrm{H}^{3} \right]_{kj} \right] + \right. \notag \\
& \ + \ \left. \frac14 \vert \left[ \mathrm{H}^{2} \right]_{jk} \vert^{2} \right) \ + \ O(t^5) . \notag
\end{align}
From this, it follows that:
\begin{align}
& \sum_{k} \vert \alpha_{kj} (t) \vert \ = \ \sqrt{ 1 - d_{j} t^{2} + O(t^4)} \ + \\
& + \ t \sum_{k \neq j} \vert \mathrm{H}_{jk} \vert \sqrt{ 1 - t\mathrm{Im} \left[ \mathrm{H}_{jk} \left[ \mathrm{H}^{2} \right]_{kj} \right] + O(t^2)} \ + \notag \\
& + \ \frac{t^2}{2} \sum_{k \neq j , \mathrm{H}_{jk} = 0} \vert \left[ \mathrm{H}^{2} \right]_{jk} \vert \ + \ O(t^3) \notag
\end{align}
where we splitted the initial sum over $k$ in a first term with $k=j$, a second sum with $k \neq j$ but which gets contributions only for terms having $\mathrm{H}_{jk} \neq 0$, and a final sum that takes into account the last terms with $k \neq j$ and $\mathrm{H}_{jk} = 0$. Expanding the square roots and collecting powers of $t$, one obtains:
\begin{align}
& \sum_{k} \vert \alpha_{kj} (t) \vert \ = \ 1 + t d_{j} - \frac12 t^{2} \Bigg( d_{j} \ + \\
& + \ \sum_{k \neq j}\mathrm{Im} \left[ \mathrm{H}_{jk} \left[ \mathrm{H}^{2} \right]_{kj} \right] + \sum_{\substack{k \neq j \\ \mathrm{H}_{jk} = 0}} \vert \left[ \mathrm{H}^{2} \right]_{jk} \vert \Bigg) + O(t^3). \notag
\end{align}
Now notice that:
\begin{align*} \
\sum_{k \neq j}\mathrm{Im} \left[ \mathrm{H}_{jk} \left[ \mathrm{H}^{2} \right]_{kj} \right] \ &= \ \sum_{k}\mathrm{Im} \left[ \mathrm{H}_{jk} \left[ \mathrm{H}^{2} \right]_{kj} \right] \ = \\
&= \ \mathrm{Im} \left[ \mathrm{H}^{3} \right]_{jj} \ = \ 0 .
\end{align*}
Overall, for coherence we find:
\begin{align}
\mathcal{C}_{j}(t) & \ = \ \ 2 d_{j} t \ \ + \\
& + \ \Big[ d_{j} ( d_{j} - 1) - \sum_{\substack{k \neq j \\ \mathrm{H}_{jk} = 0 }} \vert \left[ \mathrm{H}^{2} \right]_{jk} \vert \Big] t^{2} \ + O(t^3). \notag
\end{align}
One can appreciate the fact that coherence has a very similar structure to quantum-classical distance at short times, but it is affected by other degrees of freedom of $\mathrm{H}$. Indeed, the term $\sum_{\substack{k \neq j \\ \mathrm{H}_{jk} = 0}} \vert \left[ \mathrm{H}^{2} \right]_{jk} \vert$ has a simple graphical interpretation: $ \vert \left[ \mathrm{H}^{2} \right]_{jk} \vert$ for $k \neq j$ and for $\mathrm{H}_{jk} = 0$ is a sum over all possible paths of length $2$ connecting vertex $j$ with a vertex $k$ at distance $2$ from $j$, where each path is weighted by the product of the phases of its two links in the direction from $j$ to $k$. Since this is the modulus of a sum of phases, interference may happen. Therefore $\mathcal{C}_{j}$ can get a phase-dependent second order contribution in $t$ only if the graph contains a 4-cycle passing through $j$ and bearing a non-zero overall phase, so that there will be at least two paths joining the same $k$ with $j$ and weighted by different phase factors. \\
To quantify the spreading in time of the QW from the initial vertex, a useful figure of merit is provided by the {inverse participation ratio} :
\begin{equation}
\mathcal{I}_{j} (t) \ \ = \ \ \sum_{k} \vert \alpha_{kj}(t) \vert^{4}
\end{equation}
which decreases while the QW spreads over the vertices of the graph. The second-order small time expansion of $\mathcal{I}_{j}(t)$ is derived from Eq.(\ref{eq:alfa2smallt}):
\begin{equation}
\mathcal{I}_{j}(t) \ \ = \ \ 1 - 2 d_{j} t^{2} \ + \ O(t^4).
\label{ipr}
\end{equation}
The above expression is clearly related to coherence and $\mathrm{D}^{j}_{QC}$, but unlike for coherence, the IPR is determined just by the connectivity of the starting vertex at least up to second order in $t$.
The conclusion we can draw is that the short-time expansion of the quantum-classical distance correlates with coherence and IPR, but in a way that depends non-trivially upon the topology; indeed, $\mathcal{D}_{QC} (t)$ compares the classical and quantum \emph{probability distributions} at a given time and these can be similar irrespective of the coherences. In the following, we shall argue that this is in fact a merit of $\mathcal{D}_{QC}$.
\section{Cycle graphs}
\label{sec:IV}
The simplest example to investigate the role of phases in the Hamiltonian of CTQW for quantum transport is provided by cycle graphs. As discussed previously, by gauge invariance we can write the generic Hamiltonian $\mathrm{H}$ for a chiral CTQW on a cycle graph as follows:
\begin{equation}
\mathrm{H} \ \ = \ \ \left(
\begin{array}{cccccc} d_{1} & e^{i \theta} & 0 & 0 & \ldots & e^{-i \theta} \\
e^{-i\theta} & d_{2} & e^{i \theta} & 0 & \ldots & 0 \\
0 & e^{-i\theta} & d_{3} & e^{i\theta} & \ldots & 0 \\
\vdots & \vdots & & & \vdots & \vdots \\
e^{i \theta} & 0 & 0 & \ldots & e^{-i\theta} & d_{N} \end{array}
\right)
\end{equation}
where $\theta \in [0, 2 \pi)$ is the only relevant phase, which has been distributed equally over all links for convenience, while $d_{1},...,d_{N} \in \mathds{R}^{+}$. In this work we focus on the role of phases, hence we will take $d_{1} = ... = d_{N} = d$ and further impose $d=0$ without loss of generality.
The eigenvectors of $\mathrm{H}$ will still be Bloch states:
\begin{equation}
\vert \lambda_{j} \rangle \ \ := \ \ \frac{1}{\sqrt{N}} \sum_{k=1}^{N} e^{i\frac{2 \pi j k}{N}} \vert k \rangle
\end{equation}
while the eigenvalues will be shifted according to:
\begin{equation}
\lambda_{j} \ \ := \ \ 2 \cos \left( \theta + \frac{2 \pi j}{N} \right)
\end{equation}
where $j = 1, ..., N$.
Notice that, although for $\theta = 0$ the spectrum is doubly degenerate, for almost every $\theta \neq 0$ the degeneracy is lifted. Moreover, despite $\theta \in [ 0, 2 \pi )$, one can always choose a representative phase in the reduced scheme, that is $\theta \in [ 0, \frac{2 \pi}{N} ) $, much like the first Brillouin zone for crystal momentum, without affecting transition probabilities between sites (see below). When $N$ is large, the effect of $\theta$ on the spectrum is clearly minor. The transition probabilities associated with this chiral quantum walk on a cycle are:
\begin{align}
P_{j \to k}(t) \ &:
= \frac{1}{N^2} \Big\vert \sum_{s=1}^{N} e^{2 i \left[ \frac{ \pi (k-j) s}{N} - t \cos \left( \theta + \frac{2 \pi s}{N} \right) \right] } \Big\vert^{2}. \label{eq:pjkring}
\end{align}
We again stress the fact that, by virtue of gauge invariance, all the quantities that can be computed from these probabilities, assuming a localized initial state as before, would be the same for a more general $\mathrm{H}$ on the ring of the form:
\begin{equation}
\label{eq:Hring}
\mathrm{H} \ \ = \ \ \left(
\begin{array}{cccccc} d& e^{i \phi_{1}} & 0 & 0 & \ldots & e^{-i \phi_{N}} \\
e^{-i\phi_{1}} & d & e^{i \phi_{2}} & 0 & \ldots & 0 \\
0 & e^{-i\phi_{2}} & d & e^{i \phi_{3}} & \ldots & 0 \\
\vdots & \vdots & & & \vdots & \vdots \\
e^{i \phi_{N}} & 0 & 0 & \ldots & e^{-i \phi_{N-1} } & d \end{array}
\right) \notag
\end{equation}
by simply putting $\theta = \frac{1}{N} \sum_{j=1}^{N} \phi_{j}$ in Eq.(\ref{eq:pjkring}).
In the large ring limit, $N \to \infty$, since the single relevant phase $\theta$ can be concentrated as $N \theta$ on any single link at will by leveraging gauge invariance, the transition probabilities will be the same as for the standard continuous-time quantum walk on a ring:
\begin{equation}
\lim_{N \to \infty} P_{j \to k}(t) \ \ = \ \ \vert J_{ \vert k - j \vert} (2t) \vert^{2}
\end{equation}
where $J_{n}(x)$ is the $n$-th Bessel function. The same result can also be derived directly from Eq.(\ref{eq:pjkring}). This also implies that the effect of phases on the quantum evolution starting on any fixed vertex will show up only when the probability will have reached the opposite side of the cycle, so that phase-dependent interference can happen. Since the initial propagation on the cycle is known to be ballistic with velocity $2$, in order to reach the opposite side of the cycle the walker will take a time approximately given by $\tau = N/4$, which therefore is also the time it takes for $\theta \neq 0$ to have some appreciable effect on the on-site probabilities. However, this prediction can be inaccurate for small rings, because of the non-negligible tails of the ballistic wavefronts. \\
To understand the influence of phases on quantum transport on the cycle after this minimal time, let us first consider the standard $\theta = 0$ case. Here, starting with an initially localized state $\vert j \rangle$, all sites that are symmetrical with respect to the initial vertex must have the same occupation probability at all times, since the symmetry is preserved by the non-chiral Hamiltonian. In particular, the probability of finding the walker on each site except the starting one (and the one opposite to it in the case of even cycles) cannot exceed $\frac12$ at any time.
When the phase is added and $N$ is even, for any $\theta$ one can show that this symmetry is preserved. Indeed, even-cycles belong to the family of bipartite graphs, defined as those graphs whose vertices can be divided into two sets such that two vertices in the same set are never connected by an edge. For such graphs, one can apply a gauge transformation that flips the sign of all the basis elements of one set, while leaving the other set untouched, thereby implementing the transformation $\mathrm{H} \to - \mathrm{H}$, equivalent to time reversal, without affecting the transition probabilities. In particular, for even cycles this implies that the probability of landing at site $k$ or at its symmetric counterpart $N - k + 1$ is the same at all times also for $\theta \neq 0$.
Therefore, any chiral quantum walk on an even cycle with $N$ sites starting at site $j$ can never be localized with probability greater than $\frac12$ on any site different from the initial one and its opposite at $k = j + \frac{N}{2}$. As for the opposite site, which can be interesting as a target for transport, the situation is less clear. At fixed time, the optimal phase to maximize the probability $P_{j \to j+N/2}(t)$ can be different from zero. However, it seems that the highest peaks of $P_{j \to j+N/2}(t)$ for a wide range of times are always attained for $\theta = 0$. This can be appreciated in Fig.\ref{fig:1}, where we plotted the transition probabilities between opposite sites vs. time for cycles of $8$ and $10$ sites and for different values of $\theta$. We restricted the plotting region to probability values in $[0.7,1]$ because we considered $0.7$ as a lower bound to the acceptable fidelity of transport. Again, transition probabilities between other pairs of sites are neglected because they are always upper bounded by $\frac12$.
\begin{figure*}[!ht]
\centering
\subfigure{\includegraphics[scale=0.57]{Fig1_ring_n8.pdf}}\hfill
\subfigure{\includegraphics[scale=0.59]{Fig1_ring_n10.pdf}}
\caption{\small Transition probability from initial site $1$ to site $5$ of a 8-cycle (left) and to site $6$ of a 10-cycle (right) as functions of time and for different values of the phases. The non-chiral $\theta = 0$ case is plotted in black. For the 8-cycle the other phases are $\theta = 0.04$ (green), $\theta = 0.13$ (orange) and $\theta = 0.23$ (light blue). For the 10-cycle we chose $\theta = 0.027$ (green), $\theta = 0.13$ (orange) and $\theta = 0.28$ (light blue). Notice that the probability axis plot range is restricted to $[0.7,1]$ because it is the most informative region for quantum transport.}
\label{fig:1}
\end{figure*}
These observations suggest that chiral CTQWs provide no advantage for optimal quantum transport on even cycles over their non-chiral counterparts. Nevertheless, we highlight the fact that $P_{j \to j + N/2}(t)$ can be completely suppressed at all times by choosing $\theta = \pi/N$ (or, equivalently, a phase of $\pi$ on a single, generic link), a phenomenon that, together with the unbroken reflection symmetry for generic $\theta$, has already been attributed to the fact that even cycles are \emph{bipartite} graphs. We notice that, in the context of excitonic transport in biochemical complexes, this sets a possible prediction to be contrasted with observations, since finding a ring-like structure with an even number of units could exclude that phases play a role.\\
When the cycle is odd, the conclusion can change dramatically. Here there is no opposite vertex to the starting one and, by the previous argument, the non-chiral QW ($\theta = 0$) can never localize on a site different from the initial one with probability greater than $\frac12$. However, almost every $\theta \neq 0$ breaks the reflection symmetry with respect to the starting point, opening the possibility for enhanced quantum site-to-site transport. In Fig.\ref{fig:2} we plotted the transition probabilities from site $1$ to various other sites of a 5-cycle and a 7-cycle for values of $\theta = \frac{\pi}{10}, \frac{\pi}{14}$, respectively ((a) and (b)). For any other value of the phase in the relevant range $[0, \frac{2 \pi}{N} ]$, in each case the highest peaks are considerably lowered. It seems that the value $\frac{\pi}{2N}$ (where $N = 5,7$ respectively) is \emph{resonant}, although the pattern is very disordered and small changes in the precise value of the phase induce considerable shifts in the peaks at long times, which is a signature of chaos. This is illustrated by Fig.\ref{fig:2} (c) and (d), where we displayed the same set of transition probabilities for the same cycles, but with a slightly off-resonant value of $\theta$ ($\theta = 0.29$ for the 5-cycle to be compared with the resonant value of $\frac{\pi}{10} \simeq 0.31$ and $\theta = 0.21$ for the 7-cycle to be compared with the resonant value of $\frac{\pi}{14} \simeq 0.22$ ). We stress that all these peaks displayed in the plots for odd cycles are entirely due to the introduction of the phase degree of freedom, since \emph{non-chiral} CTQW on odd cycles starting in a localized state will never be found with probability higher than $\frac12$ on other sites.
\begin{figure*}[!ht]
\centering
\subfigure[a]{\includegraphics[scale=0.59]{Fig2_ring_n5_phiR.pdf}}\hfill
\subfigure[b]{\includegraphics[scale=0.59]{Fig2_ring_n7_phiR.pdf}}
\subfigure[c]{\includegraphics[scale=0.59]{Fig2_ring_n5_ph0-29.pdf}}\hfill
\subfigure[d]{\includegraphics[scale=0.59]{Fig3_ring_n7_phi0-21.pdf}}
\caption{\small Transition probability from initial site $1$ to site $n$ on a 5-cycle ((a) and (c)) and on a 7-cycle ((b) and (d) ) as functions of time, for the resonant phase values ($\frac{\pi}{10}$ for the 5-cycle (a) and $\frac{\pi}{14}$ for the 7-cycle (b) ) and for slightly off-resonant phase values ($0.29$ for the 5-cycle (c) and $0.21$ for the 7-cycle (d) ). Different colors represent different target sites: $n=2$ in light blue, $n=3$ in orange, $n=4$ in green and $n=5$ in red. }
\label{fig:2}
\end{figure*}
We will now see how the quantum-classical distance can capture these results for the prototypical example of cycle graphs and how it can indicate the optimal phase, putting order to the opaque relationship between the values of $\theta$ and the transition probabilities. A reasonably compact expression for the quantum-classical distance at time $t$ as a function of $\theta$ can be derived from Eq.(\ref{eq:pjkring}) and the standard formulas for a continuous-time classical random walk on cycles:
\begin{equation}
\label{eq:DQCring}
\begin{aligned}
& \mathcal{D}_{QC} (t ; \theta ) \ := \ 1 \ - \ \frac{e^{-2t}}{N^2} \times \sum_{k,s=1}^{N} \exp \left[ 2 t \cos \frac{ 2 \pi k}{N} \ + \right. \\
& \left. - \ 4 i t \sin \left( \theta + \frac{ \pi (2s + k)}{N} \right) \sin \frac{ \pi k}{N} \right].
\end{aligned}
\end{equation}
\par
\begin{figure*}[!ht]
\centering
\subfigure{\includegraphics[scale=0.82]{Fig4_dqc_n10.pdf}}\hfill
\subfigure{\includegraphics[scale=0.77]{Fig4_dqc_n7.pdf}}
\caption{\small Gain in quantum-classical distance $ \Delta \mathcal{D}_{QC}(t)$ vs. time for a 10-cycle (left) and a 7-cycle (right), with respect to the quantum-classical distance for the standard evolutions generated by the Laplacian (blue baseline in both plots). For the 10-cycle phases are increasing in the relevant range $[0, \frac{\pi}{10}]$ with a corresponding color hue from blue to green. For the 7-cycle the relevant range of phases is $[0, \frac{\pi}{14}]$ and $\theta$ increases from blue to red hues. Notice that the maximum in time of $\mathcal{D}_{QC}$ is not necessarily related with the time at which transport occurs, because the former quantity is also influenced by the time scale of the classical evolution. }
\label{fig:3}
\end{figure*}
\par
\begin{figure*}[!ht]
\centering
\subfigure{\includegraphics[scale=0.58]{Fig5_n5_COH.pdf}}\hfill
\subfigure{\includegraphics[scale=0.64]{Fig5_n5_IPR.pdf}}
\caption{\small Coherence and IPR vs. time for a 5-cycle and for different values of $\theta \in [0, \frac{\pi}{10})$, corresponding to color shades from blue to orange. The black line represents the resonant phase $\frac{\pi}{10}$. }
\label{fig:4}
\end{figure*}
Since the dynamics is the same independently of the starting vertex, the expression for the quantum-classical distance can be computed for any localized initial state $ \vert j \rangle$ and maximization over $j$ is not necessary. It is a simple task to show that
\[ \mathcal{D}_{QC} \left(t ; \frac{\pi}{N} + \phi \right) = \mathcal{D}_{QC} \left(t; \frac{\pi}{N} - \phi\right)\]
and, when $N$ is odd, also
\[ \mathcal{D}_{QC} \left(t ; \frac{\pi}{2N} + \phi \right) = \mathcal{D}_{QC} \left(t; \frac{\pi}{2N} - \phi\right). \]
Therefore, when $N$ is even $\mathcal{D}_{QC}(t; \theta)$ attains all its possible values, at fixed $t$, for $\theta$ in the range $[0, \frac{\pi}{N}]$ and when $N$ is odd for $\theta$ in the range $[0, \frac{\pi}{2N}]$.
Already for modestly sized polygons ($N \geq 7$), the variations of $\mathcal{D}_{QC}(t)$ with $\theta$ are very small. Indeed one can expand $\mathcal{D}_{QC} (t)$ around $t=0$ and notice that all terms up to order $N-1$ in $t$ when $N$ is even, and up to order $2N-1$ in $t$ when $N$ is odd, are independent of $\theta$, while an oscillatory $\theta$ dependence starts at higher orders. Because of this feeble $\theta$ dependence, we considered the difference between the quantum-classical distance with phase $\theta$ and the same quantity with zero phase, i.e. $ \Delta \mathcal{D}_{QC}(t;\theta) = \mathcal{D}_{QC} (t ; \theta ) - \mathcal{D}_{QC} (t ; 0)$. As can be appreciated in Fig.\ref{fig:3}, this quantity brings a clear order to the irregular behaviour that was seen in the transition probabilities.
This ordering is also insightful when one considers the following: for even cycles, $\mathcal{D}_{QC} (t; \theta) - \mathcal{D}_{QC} (t; 0)$ has a negative, significant dip at approximately the same time for all values of $\theta$, and it is the lowest at $\theta = \frac{\pi}{N}$, the phase which fully suppresses transport to the opposite vertex; conversely, for odd cycles we find that the same quantity \emph{peaks} at approximately the same time, with the highest peak for $\theta = \frac{\pi}{2N}$, again the resonant phase, which optimizes transport for odd cycles. Moreover, the heights of the peaks are monotonically increasing in absolute value with $\theta \in [0, \frac{\pi}{2 N} ) $, while the depths of the dips for even $N$ are monotonically increasing with $\theta \in [0, \frac{\pi}{N} ) $. Therefore we see that finding the global maximum of $\Delta \mathcal{D}_{QC}$, with respect to $\theta$ and $t$, at a maximum (resp. minimum) of $\mathcal{D}_{QC}(t; \theta)$ in time, correctly spots the least classical (resp. the nearest to classical) evolution among all possible chiral CTQW on the same cycle.
Intuitively, we can understand the reason of its effectiveness: $\mathcal{D}_{QC}(t)$ is maximized for the quantum evolution that departs the most from the classical one, the latter being slowly and diffusively
spreading towards the homogeneous distribution. Therefore, a chiral CTQW which evolves to nearly localized states and could be optimal for transport, will also maximizes $\mathcal{D}_{QC}(t)$. To further support this claim, we can look at the phase dependence of the functions $\mathcal{C}(t)$ and $\mathcal{I}(t)$, as in Fig.\ref{fig:4} for a 5-cycle. Again, the resonant phase (black line, $\theta = \frac{\pi}{10}$) is associated with higher localization (lower coherence and higher IPR values) especially at short times, and also the overall trend seen for the quantum-classical distance is respected at least at short times, with values of $\theta$ close to $0$ leading to the opposite behavior. We conclude this Section by noting that some results about so-called \emph{pretty good universal transport} for chiral CTQW on cycles with a prime number of vertices are known in the mathematical literature \cite{mathcycles}, although they have little role in a physical context since no bound on the time needed for transport is considered there.
\vspace{5mm}
\par
\section{Complete graphs}
\label{sec:V}
Having discussed cycle graphs characterized by minimal connectivity, we will now examine the regular graphs with maximal connectivity, i.e. complete graphs. In a complete graph just a minority of phases of $\mathrm{H}$ can be ignored by gauge invariance, and we expect that the role played by the new degrees of freedom will be major for this topology.
At this point it is worth mentioning that for standard quantum walks with real Hamiltonian generators, the dynamics of the walker on a complete graph with localized starting condition is equivalent to the dynamics on a star graph with the same number of vertices, when starting at the core vertex \cite{Xu_2009}. In other words, if the Hamiltonian is real, many links of the complete graph can be eliminated without changing the evolution. This is intuitive by symmetry arguments: since all vertices are connected to the initial one, the amplitudes in all the vertices except the first will be equal at all times and probability will never flow through links that connect these vertices, therefore they are irrelevant for the dynamics. Interestingly, the addition of phases radically changes this conclusion since, as we shall see, search to the quantum speed limit and without an oracle can be achieved on a complete graph with appropriately chosen phases, while this is clearly impossible with the star graph which has no non-trivial phase degrees of freedom, being a tree graph. The upshot is that the generalization to Hermitian Hamiltonians and chiral CTQWs is more powerful than previously imagined, because it explicitly differentiates between graph topologies that, for some initial conditions, would be completely equivalent for evolutions generated by the simple Laplacians.\\
\par
\subsection{Random chiral Hamiltonians for complete graphs}
We initially investigated the effect of randomly added phases on quantum-classical distance, 1-norm of coherence and IPR. Because of the large number of free parameters, we adopted different strategies to better explore the parameters space. In Fig.\ref{fig:5} a comparison between the averaged coherence and IPR for different phase choices on the complete graph with $N=13$ is shown. The blue curves corresponds to the non-chiral choice with $\mathrm{H}= \mathrm{L}$. The orange curves are averages over 400 configurations where a single phase $e^{i \phi}$ is generated randomly and then attached to all the links of the complete graph in the direction $j \to k$ for $k > j$.
We infer that a typical phase different from $\phi = 0$ attached to each link increases the average coherence and decreases the IPR with respect to $\mathrm{H} = \mathrm{L}$.
If, instead, we stochastically distribute two random phases $e^{i \phi_{1} }, e^{i \phi_{2}}$ among the edges in the given direction, there is an even greater increase in the coherence and decrease in the IPR (dark green curves, again resulting from an average of 400 random configurations). Finally, if an independent, randomly generated phase is attached to each link, the resulting average behaviour is described by the black curves. Notice that now the order is irrelevant, since all orders will be explored if the phase of each link is independent and sampled in the full range $[0,2\pi)$. Clearly this rule entails all the previous ones, but the \emph{typical} configuration contributing to the black curve will be one in which there is no correlation between phases on different links and there is essentially full \emph{disorder} in the phase degrees of freedom of the Hamiltonian. Now the IPR stabilizes to the lowest values between the examined ones, while the coherence is maximal with respect to the previous cases. These two quantities therefore hints at a \emph{phase-disorder-induced delocalization} in the evolution of a localized state on a complete graph \footnote{Here by \emph{phase disorder} we mean the randomness in the choice of phases for each link, much like the disorder of the on-site potential in Anderson localization, and \emph{not} the disorder induced by uncertainty on the values of the Hamiltonian's parameters.}. These results were checked also for $N=16$ and $N=17$ and appear robust irrespective of the number of sites.
\begin{figure*}[!ht]
\centering
\subfigure{\includegraphics[scale=0.65]{Fig6_complete_n13_compI.pdf}}\hfill
\subfigure{\includegraphics[scale=0.65]{Fig6_complete_n13_compC.pdf}}\hfill
\caption{\small IPR $\mathcal{I}(t)$ and coherence $\mathcal{C}(t)$ vs. time of a continuous-time quantum walker on a complete graph with $N=13$ sites with localized initial condition. The standard evolution generated by $\mathrm{H} = \mathrm{L}$ is depicted by the blue curves. The orange curves are averages over $400$ Hamiltonians with a single, randomly generated phase attached to all links in the "positive" direction. The dark-green curves resulted from the average of $400$ Hamiltonians with two, independent randomly generated phase randomly attributed to each link in positive direction. Black curves correspond to the random assignment of independent phases to each link, still averaged over $400$ runs. }
\label{fig:5}
\end{figure*}
Let us now look at the difference $\Delta \mathcal{D}_{QC}$ between the quantum-classical distance for these different averaged evolutions and the quantum-classical distance for the reference case $\mathrm{H}= \mathrm{L}$ (Fig.\ref{fig:7}, same color code). We see a glaring relation between this quantity and IPR or coherence: any addition of non-zero phase seems to add to the quantum-classical distance (at least \emph{on average}), and the increase goes along with the higher delocalization, as signaled by large values of the coherence and low values of the IPR.
Mind that, strictly speaking, we are considering the quantum-classical distance \emph{at fixed initial state $\vert j \rangle$}, since the full $\mathcal{D}_{QC}(t)$ would require maximization over all initial states at each time, once $\mathrm{H}$ and its phases have been specified. A crucial point here is the correlation between $\Delta \mathcal{D}_{QC}$ and localization (high values of $\mathcal{I}$ and low values of $\mathcal{C}$), which is opposite with respect to the relation found for cycle graphs.
Here those dynamics leading to less localized states are more quantum, according to the quantum-classical distance. This fact could be partially predicted from the short-time expansion of Eq.(\ref{eq:DQCshortT}) which is not monotonic in the connectivity $d_{j}$, unlike the expansions for $\mathcal{C}(t)$ and $\mathcal{I}(t)$ to the same order in $t$. This is another evidence that $\mathcal{D}_{QC}(t)$ stands out against other dynamical quantities in the successful recognition of optimal chiral quantum evolutions. Of course, the relevant task is graph-dependent, being either an hitting-type or a mixing-type task, but remarkably the quantum-classical distance seems able to optimize the parameters in both scenarios.
\begin{figure}[!ht]
\centering
\includegraphics[scale=0.67]{Fig6_complete_n13_compDQC.pdf}
\caption{\small Difference $\Delta \mathcal{D}_{QC}(t)$ between the quantum-classical distances of the various, averaged chiral evolutions and the non-chiral $\mathrm{H} = \mathrm{L}$ one, for a complete graph with $N=13$ sites and localized initial condition of the CTQW. Color code and meaning as for Fig.\ref{fig:5}. The reference case with $\mathrm{H}= \mathrm{L}$ is the base line in blue. }
\label{fig:7}
\end{figure}
\subsection{Optimization of the quantum-classical distance}
Guided by these observations, the next natural step is to maximize $\mathcal{D}_{QC}(t)$ over all possible phases degrees of freedom in $\mathrm{H}$ for the complete graph.
This has to be performed at a fixed time, that we chose afterwards by identifying the time at which $\mathcal{D}_{QC}$ attains its maximum;
however, it seems that the increasing order of $\mathcal{D}_{QC}$ curves is time independent for complete graphs (as suggested by Fig.\ref{fig:7}) and the result is the same if another time is set for the numerical maximization. Starting with random guesses, the optimization converges to different Hamiltonians each time. However, a common feature of these optimal matrices is easy to spot:
\begin{enumerate}
\item \label{cond:1} The first column of an optimal $\mathrm{H}_{O}$ on the complete graph of $N$ sites is orthogonal, with respect to the Hermitian product on $\mathds{C}^{N}$, to all the rows of $\mathrm{H}$ except the first one (assuming that all the diagonal elements have been fixed to $0$, without loss of generality)
\end{enumerate}
The first column is singled out because of the choice of $\vert 1 \rangle$ as the (arbitrary) initial condition. Let us call $\underline{h}$ the first column of an optimal Hamiltonian $\mathrm{H}_{O}$. Because of the topology and the choice of diagonal elements, its entries will be:
\begin{equation}
h_{1} = 0 \ , \ \ \ \ h_{j} = e^{i \phi_{j}} \ \ \forall j = 2,...,N
\end{equation}
and, since $\mathrm{H}_{O}$ is Hermitian, its first row will be ${ ( \underline{h}^{*} )}^{T}$, so that the scalar product between the first row and the first column is $N-1$. Let us denote by $\underline{e}_{1}$ the localized state $\vert 1 \rangle$ in matrix notation, which is the first vector of the canonical basis. Then we have:
\begin{equation}
\begin{aligned}
\mathrm{H}_{O}^{2n} \cdot \underline{e}_{1} \ & = \ (N-1)^{n} \underline{e}_{1} \\
\mathrm{H}_{O}^{2n+1} \cdot \underline{e}_{1} \ &= \ (N-1)^{n} \ \mathrm{H}_{O} \cdot \underline{e}_{1} \ = \ (N-1)^{n} \underline{h}
\end{aligned}
\end{equation}
The evolution of the initial state localized at site $1$ then follows immediately:
\begin{equation}
\label{eq:evcomplOpt}
\begin{aligned}
& e^{- i t \mathrm{H}_{O}} \cdot \underline{e}_{1} \ = \\
& = \ \cos [\sqrt{N-1} t ] \underline{e}_{1} - \frac{i}{\sqrt{N-1}} \sin [ \sqrt{N-1} t ] \underline{h}
\end{aligned}
\end{equation}
Notice that $\underline{h}$, as a vector of amplitudes, represents a state which is balanced between all sites except the first, to which it is orthogonal. Therefore, the evolution of Eq.(\ref{eq:evcomplOpt}) is a cyclic rotation between the initial state localized at site $1$ and the equally spread state over all other sites except the initial one, necessarily passing through an intermediate flat state:
\begin{equation}
\vert f \rangle \ = \ \frac{1}{\sqrt{N} } \Big( \vert 1 \rangle \ + \ \sum_{j=2}^{N} e^{i \phi_{j}} \vert j \rangle \Big)
\end{equation}
where the amplitude to find the walker in any site of the graph is equal to $N^{-1/2}$. The time needed to reach $\vert f \rangle$ for the first time is given by:
\begin{equation}
\label{eq:tf}
t_{f} \ = \ \frac{1}{\sqrt{N-1}} \arccos \frac{1}{\sqrt{N}}
\end{equation}
while the time needed to reach the state $\vert \underline{h} \rangle = \frac{1}{\sqrt{N-1}} \sum_{j=2}^{N} h_{j} \vert j \rangle $ associated with the vector $\underline{h}$ and orthogonal to the initial state $\vert 1 \rangle$ is:
\begin{equation}
t_{h} \ = \ \frac{ \pi }{2 \sqrt{N-1}}.
\end{equation}
It should be emphasized that $\vert f \rangle$, as for any flat state, has maximal coherence value of $N-1$ and minimal IPR value of $\frac{1}{N}$.
Exploiting the simple structure of these optimal evolutions, an exact expression for their quantum-classical distance can also be derived:
\begin{equation}
\mathcal{D}_{QC}^{O} (t) = 1 - \frac{1 - e^{-N t} }{N} - e^{-N t} \frac{1 + \cos [ 2 \sqrt{N-1} t]}{2}.
\end{equation}
\subsection{Search to the quantum speed limit without an oracle}
Another interesting property of the "optimal" evolution described by Eq.(\ref{eq:evcomplOpt}) can be appreciated if we choose $\vert \underline{h} \rangle \ = \ \frac{i}{\sqrt{N-1}} \sum_{j=1}^{N} \vert j \rangle$, i.e. all the phases of the first column of $\mathrm{H}_{O}$ equal to $i$. This can always be achieved by an appropriate gauge transformation on any Hamiltonian that already fulfills Condition \ref{cond:1}. In this case, $\vert f \rangle = \frac{1}{\sqrt{N}} \sum_{j=1}^{N} \vert j \rangle$ is the flat state with relative phases all equal, therefore the backwards evolution from $\vert f \rangle$ is a \emph{solution to the search problem} with target vertex $1$, starting from the unbiased state and without an oracle (but with biased phases), in a time $t_{f}$, i.e. $ e^{ i \mathrm{H}_{O} t_{f} } \vert f \rangle = \vert 1 \rangle $.
Since coherence and IPR for a localized initial condition are gauge invariant quantities, the blue curves in Fig.\ref{fig:5} imply that it is not possible to reach a flat state from a localized one with any $\mathrm{H}$ which is gauge-equivalent to $\mathrm{L}$. Thus, Condition \ref{cond:1} and also our result for the search problem \emph{require} a \emph{nontrivial configuration} of the phase degrees of freedom. It is interesting to compare our search time $t_{f}$ with Grover's time and with the quantum speed limit for this evolution. Grover's time $t_{g}$ is the time required for the Grover's Hamiltonian $\mathrm{H}_{G} = \mathrm{L} - N \vert 1 \rangle \langle 1 \vert$ to search for the state $\vert 1 \rangle$ when starting from the flat state $\vert f \rangle$. The operator $N \vert 1 \rangle \langle 1 \vert$, called the \emph{oracle}, is needed to break the symmetry between all the vertices of the complete graph and to guide the evolution towards the target vertex. For a complete graph with $N$ vertices, Grover's time is:
\begin{equation}
t_{g} \ = \ \frac{\pi}{2 \sqrt{N} } .
\end{equation}
For any $N>2$ one has $t_f < t_{g} < t_h$, therefore our search Hamiltonian is \emph{faster} than Grover's one, and does not require an oracle.
Of course, in order for this result to hold, a bias has to be present in the \emph{phases} of our optimal $\mathrm{H}$ so that the target vertex can be singled out during the evolution. This bias is in fact embodied by Condition \ref{cond:1}.
It is now meaningful to ask whether this time $t_{f}$ reaches the ultimate time bound allowed by quantum mechanics for the evolution between these initial and final state.
To this end, we briefly recall the notion of quantum speed limits \cite{QSL03a,QSL03,QSL04,QSL17}. Consider two states $\vert a\rangle, \vert b \rangle$ on an Hilbert space $\mathcal{H} \sim \mathds{C}^{N}$ that have the \emph{same} average energy \footnote{This is clearly a necessary condition in order for the unitary evolution generated by $\hat{H}$ to bring $\vert a \rangle$ to $\vert b \rangle$. } with respect to a time-independent Hamiltonian operator $\hat{H}$ on $\mathcal{H}$. Then the \emph{quantum speed limit} $\tau_{QSL}$ is a lower bound on the time needed for the unitary evolution $e^{-i t \hat{H}}$ to rotate $\vert a \rangle$ to $\vert b \rangle$, and it is provided by the following expression:
\begin{equation}
\label{eq:QSL1}
\tau_{QSL} := \max \left\{ \frac{ \arccos \vert \langle b \vert a \rangle \vert}{ \Delta \hat{H}} \ , \ \frac{2 ( \arccos \vert \langle b \vert a \rangle \vert )^{2} }{ \pi ( \langle \hat{H} \rangle - E_{0} ) } \right\}
\end{equation}
where $\Delta\hat{H} = \sqrt{\langle \hat{H}^{2} \rangle - \langle \hat{H} \rangle^{2}}$ is the standard deviation of energy of the states ( $\vert a \rangle$ or $\vert b \rangle$ ), $\langle \hat{H} \rangle$ their average energy and $E_{0}$ is the ground state's energy.
The first quantity inside the $\max$ of the quantum speed limit can be readily computed in our case, for $\vert a \rangle = \vert f \rangle$ and $\vert b \rangle = \vert 1 \rangle$; indeed, the variance $\Delta \hat{H}$ does not depend on the phases and it is equal to $\sqrt{N-1}$ for any Hamiltonian with constant diagonal terms and compatible with the complete graph's topology. Therefore:
\begin{equation}
\label{eq:QSL2} \frac{ \arccos \vert \langle 1 \vert f \rangle \vert }{\Delta \hat{H}} \ = \ \frac{1}{\sqrt{N-1}} \arccos \frac{1}{\sqrt{N}} \ = \ t_{f}
\end{equation}
Since we already know that $t_{f}$ cannot be smaller than $\tau_{QSL}$, the $\max$ in the definition (\ref{eq:QSL1}) of $\tau_{QSL}$ and the result of Eq.(\ref{eq:QSL2}) already imply that $\tau_{QSL} = t_{f}$.
This was verified by computing the second quantity in Eq.(\ref{eq:QSL1}) and checking that it is always smaller than the first for $N>3$ \footnote{The check was performed numerically because $E_{0}$ depends on the phases in a non trivial way, so that the second quantity is harder to compute in general. }. It can be shown that $\tau_{QSL}$ is also the quantum speed limit for Grover's Hamiltonian. Indeed, we highlight that $\Delta \hat{H}$ attains its largest value for the complete graph, among all the topologies that connect $N$ vertices in a simple, connected graph. We conclude that our construction exploits phases to achieve quantum search between $N$ orthogonal states without an oracle and in the least possible time allowed by quantum mechanics, without altering the on-site energies at will.
We remark that the scaling behaviour $O( N^{-1/2})$ of Grover's time is known to be already the best one, and indeed our search time $t_{f}$ follows the same asymptotic scaling for large $N$. The construction that we provided, on the other hand, achieves the goal of optimizing the constant pre-factor, which is sub-optimal for Grover's algorithm. The comparison in log-scale is shown in Fig.\ref{fig:8}.
\begin{figure}[!ht]
\centering
\includegraphics[scale=0.8]{Fig8_QSL.pdf}
\caption{\small Optimal search time exploiting phases $t_{f}$ (black line) and Grover's time $t_{G}$ as functions of the number of sites $N$, with log scale on both axes. $t_{f}$ is always smaller than $t_{G}$, and it is also equal to the quantum speed limit $\tau_{QSL}$ }
\label{fig:8}
\end{figure}
To further illustrate the differences between the evolution induced by Grover's Hamiltonian $\mathrm{H}_{G}$ and the optimal solution $\mathrm{H}_{O}$ that we found, we plotted in Fig.\ref{fig:9} the time behavior of the quantities $\mathcal{C}(t), \mathcal{I}(t)$ and also the gain in quantum-classical distance $\Delta \mathcal{D}_{QC}$ with respect to the reference $\mathrm{H} = \mathrm{L}$, again for a complete graph with $N=13$ sites and starting at vertex $\vert 1 \rangle$. Black curves depict the evolution generated by $\mathrm{H}_{O}$, while Grover's evolution is in light-green. The blue curves correspond to the non-chiral $\mathrm{H} = \mathrm{L}$ choice. The insets show a magnification of the regions where the relevant times happen: the red circle indicates the time $t_{f}$, the red square designates time $t_{G}$ and finally the rotated square indicates $t_{h}$.
\begin{figure*}
\centering
\subfigure{\includegraphics[scale=0.533]{Fig9_n13_search_coh.pdf}}\hfill
\subfigure{\includegraphics[scale=0.533]{Fig9_n13_search_IPR.pdf}}\hfill
\subfigure{\includegraphics[scale=0.553]{Fig9_n13_search_DQC.pdf}}
\caption{\small Comparison for coherence $\mathcal{C}(t)$, IPR $\mathcal{I}(t)$ and gain in quantum-classical distance $\Delta \mathcal{D}_{QC}(t)$ for evolutions generated by $\mathrm{H}_{O}$ (black curves), $\mathrm{H}_{G}$ (light-green curves) and $\mathrm{H}= \mathrm{L}$ (blue curves) for a complete graph of $N=13$ sites and starting with a localized state. The insets in plots of $\mathcal{C}(t)$ and $\mathcal{I}(t)$ show a magnification of the region for $0.3 < t < 0.5$, where times $t_{f}$ (red circle), $t_{G}$ (red square) and $t_{h}$ (rotated red square) are located. }
\label{fig:9}
\end{figure*}
Importantly, the gain in quantum-classical distance with respect to the non-chiral $\mathrm{H} = \mathrm{L}$ choice is also maximal for the evolution generated by $\mathrm{H}_{O}$, even when compared with Grover's evolution, a non obvious fact since the maximization was performed without considering diagonal degrees of freedom. Moreover, the optimal evolution outperforms the best evolutions found in Fig.\ref{fig:5} by randomly generating phases according to IPR and coherence, and correspondingly $\Delta \mathcal{D}_{QC}$ is in fact higher at all times.
It remains to be shown that Condition \ref{cond:1} can indeed be fulfilled by some choice of phases for any $N$, without relying on the optimization of $\mathcal{D}_{QC}$. This is carried out in detail in Appendix \ref{apx:constr}, but it should be clear that Condition \ref{cond:1} arose solely from the maximization of the quantum-classical distance, thereby corroborating the power of this method. \\
To conclude with this class of examples, let us remark that with non-chiral CTQWs on complete graphs with $N$ vertices it is impossible to observe \emph{instantaneous uniform mixing} \cite{ahmadimixing}, where the walker completely delocalizes to a uniform superposition of all basis states at a certain instant during its evolution, except for $N = 2, 3, 4$. Our result shows that, with the generalization to \emph{chiral} quantum walks, but retaining unitarity, instantaneous uniform mixing is achievable on \emph{any} complete graph in a remarkably short time $t_{f}$ given by Eq. \eqref{eq:tf}, which is $O(1/\sqrt{N})$ for large $N$. Importantly, this is a quadratic speedup with respect to the $O(1)$ scaling of mixing on hypercube graphs \cite{mooremixing} \footnote{ In the literature on quantum algorithms, it is common wisdom to rescale the adjacency matrix of regular graphs by their connectivity to compare evolution times. With that convention, the time needed to achieve uniform mixing with the scaled adjacency matrix on hypercube graphs is $O(N)$, while the time to perform the same task on complete graphs with our \emph{chiral} protocol is $O(\sqrt{N}$). } and it also holds for generic number of sites, whereas the hypercube protocol applies only if $N$ is a power of $2$.
\par
\section{Quantum switches}
\label{sec:VI}
Here we consider graphs like the one depicted in Fig.\ref{fig:10}, which are seldom referred to as \emph{quantum switches}. Since the graph is planar and there is a single loop, just one phase will affect transition probabilities between sites, and we attach it on the link which closes the triangle and is opposite to vertex $1$, calling it $e^{i \phi}$ in the direction specified in the figure.
\begin{figure}
\centering
\includegraphics[scale=0.5]{Fig10_qswitch12.pdf}
\caption{\small Graph of the quantum switch with 12 sites. The walker starts at site $1$ (boxed in blue) and the link between site $5$ and site $6$ bears a phase of $e^{i \phi}$ in the Hamiltonian (black arrow). }
\label{fig:10}
\end{figure}
These graphs were considered in \cite{cqw1} as examples of the advantage provided by \emph{chiral} quantum walks over the ones defined by the Laplacian or the adjacency matrix. Indeed, it was shown that a resonant value of $\phi = \frac{\pi}{2}$ suppresses transport from vertex $1$ to vertex $11$, with reference to Fig.\ref{fig:10}, while enhancing transport from vertex $1$ to vertex $12$. Unlike all the other graphs considered so far, the quantum switch is non-regular, meaning that the connectivity is not the same for all sites and the Laplacian or the adjacency matrix of the graph generate different quantum evolutions. Here we first consider the adjacency matrix $\mathrm{A}$ for a $12$-sites switch, and then attach a phase to the link between vertex $5$ and vertex $6$. In Fig.\ref{fig:11} we again plot the difference between $\mathcal{D}_{QC}(t)$ for the Hamiltonian with such a phase and the same quantity without the phase. The value of $\phi$ is increasing between $0$ and $\frac{\pi}{2}$ from light gray to black curves, and we checked that larger values of $\phi$ are redundant.
\begin{figure}
\centering
\includegraphics[scale=0.68]{Fig11_dqc_switch_12.pdf}
\caption{\small Difference between $\mathcal{D}_{QC}$ for the adjacency matrix of the $12$-sites quantum switch with an adjoined phase and the same quantity for the standard adjacency matrix. The value of $\phi$ ranges from $0$ to $\frac{\pi}{2}$ for increasingly darker shades of gray. The black line results from the resonant phase $\frac{\pi}{2}$ and the blue baseline is the non-chiral evolution.}
\label{fig:11}
\end{figure}
The comparison with coherence and IPR shows a similar relation to that seen for cycle graphs: more localized evolutions, leading to higher values of IPR and lower values of coherence, are associated with higher gains in $\Delta \mathcal{D}_{QC}$. The resonant phase $\phi = \frac{\pi}{2}$ is clearly identified as the black curve which maximizes $\Delta \mathcal{D}_{QC}$. It achieves a transport fidelity of $ \sim 0.77$ to the target state $ \vert 12 \rangle$ when starting in $\vert 1 \rangle$ in a time $t \sim 5$. Surprisingly, this transport probability is considerably higher than the value that would be reached in a comparable time range for a simple chain of $8$ sites, i.e. if the third arm of the triangle (composed of vertices $6$, $7$, $9$ and $11$) were not there to begin with, assuming the Laplacian as generator in this latter case.\\
When we take the Laplacian in place of the adjacency matrix as the starting point, instead, the result differs qualitatively. Now for any choice of the free phase, localization on one branch of the switch is never as effective as the previous case. Accordingly, $\mathcal{D}_{QC}$ attains lower values now and its variations with phases are smaller, favoring the choice $\mathrm{H} = \mathrm{A}$ for quantum transport on this topology.
This example also provides further motivation in favor of the use of the adjacency matrix in place of the Laplacian for CTQWs on finite, non-regular graphs. Indeed, while having the connectivities on the diagonal is natural if the differential form of the Laplacian has to be recovered in the continuum limit (without an additional potential field landscape), this request is not so meaningful for small, non-regular graphs that do not embody a discretization of a continuous space in any obvious way.
\par
\section{Cube graph}
\label{sec:VII}
On the cube graph, perfect transport between opposite vertices is possible with a standard CTQW \cite{childs2004spatial}. Moreover, being part of the hypercubes family, the CTQW starting from any localized state will evolve unitarily towards a maximally coherent state, exhibiting instantaneous uniform mixing \cite{mooremixing}. In other words, both goals considered in this work are already achieved by a standard, non-chiral CTQW on the cube graph. Notice also that, once the phases degrees of freedom are taken into account, instantaneous uniform mixing and search without an oracle become equivalent: if the former is possible, starting from a localized state, the final uniform state will in general encode the information about the starting vertex in the relative phases. These can be transferred to the Hamiltonian with a gauge transformation, then by reversing the sign of $\mathrm{H}$ one obtains a chiral CTQW which evolves the uniform state with no relative phases towards a localized one, effectively performing search without an oracle. However, since this is achieved just by a gauge transformation, it will not affect $\mathcal{D}_{QC}$ and it is for all purposes equivalent to the initial non-chiral CTQW. In accordance with these observations, the optimization of the quantum-classical distance over the phases degrees of freedom on the cube graph does not provide better options: the best case for quantum hitting and mixing is already the non-chiral one. On the other hand, by minimizing $\mathcal{D}_{QC}(t)$ at short times, one finds a choice of phases that completely suppresses transport to half of the vertices of the cube, more precisely to those that are not connected to the starting one.
\par
\section{Conclusion}
The opportunities stemming from a generalization of the dynamical generator of continuous-time quantum walks to a generic Hermitian matrix compatible with the graph topology have just begun to be explored and recognized in the quantum information literature. Together with the new phase degrees of freedom that permit this generalization, also comes the issue of finding the best Hamiltonian for a certain quantum task on a given graph topology. After defining continuous-time classical and (chiral) quantum walks on graphs and gauge transformations on the latter, we put at the center three gauge-invariant dynamical quantities that should help the exploration of the effects of the phase degrees of freedom on the evolution of chiral CTQW. The first two, namely the 1-norm of coherence in the on-site basis and the Inverse Participation Ratio, quantify the degree of quantum coherence and of localization of quantum walker's state at any given time, the former being relevant because it is an inherently quantum property, while the latter can spot quantum transport. The third dynamical indicator is the quantum-classical distance, which aims at gauging the difference between a (possibly chiral) quantum evolution and the unique classical one on the same graph. Comparing the short-time expansions of the three indicators, we see that there is a correlation between them, but with no unambiguous common structure. Relying on four significant examples of graphs (some of which are actually infinite graph families) we argue that the quantum-classical distance effectively spots the "optimal" chiral quantum walk from the point of view of different tasks, depending on the topology.
\par
Cycle graph provide the first example and a testbed for the proposal, since some analytical results can be derived in this simple case. The maximization of $\mathcal{D}_{QC}(t)$ with respect to the phase correctly suggests that quantum transport on odd cycles is enhanced for a resonant value of the phase, while confirming that on even cycles the standard choice of the Laplacian is already the best one. For complete graphs, the number of relevant phases is very large, and a preliminary exploration of the parameters space by randomly generating chiral Hamiltonians suggests a clear correlation between greater values of $\mathcal{D}_{QC}$ and \emph{phase-disorder-induced delocalization}, witnessed by low values of IPR and high values of coherence when all phases are random and independent. A systematic maximization of $\mathcal{D}_{QC}(t)$ (at fixed, short times) indeed identifies a chiral quantum evolution on the complete graph which achieves maximal coherence and lowest IPR in very short times. A neat property of these optimal Hamiltonians is recognized, allowing us to show that they can be used for quantum search to the optimal quantum speed limit and without an oracle, outperforming Grover's algorithm in the constant pre-factor. \\
\par The third example is already known from the literature on chiral CTQWs as a \emph{quantum switch}. Again, maximization of $\mathcal{D}_{QC}(t)$ identifies the best phase for directional transport on these topologies. Also, since these graphs are not regular, an ambiguity between the use of the Laplacian or of the adjacency matrix arises, and we argue that the latter is better for directional transport and perhaps more natural.
\par Finally we examined the cube graph, which is known to exhibit perfect quantum transport for $\mathrm{H} = \mathrm{L}$, as part of the family of hypercubes. Numerical simulations suggest that indeed, for this topology, the standard non-chiral case has already the highest value of $\mathcal{D}_{QC}(t)$ against all other possible choices of phases, indicating that faster quantum transport could be impossible here (without ad hoc modifications of on-site energies). However, the \emph{minimal} value of the quantum-classical distance also spots an interesting chiral dynamics, where half of the vertices of the cube, namely those disjoint from the initial one, are never visited by the walker. Together with analogous conclusions for even cycles, this suggests that \emph{minimization} of $\mathcal{D}_{QC}(t)$ can identify \emph{suppression of transport} whenever this can happen. \\
\par
\begin{appendix}
\section{Explicit construction of complete graph chiral Hamiltonians for quantum search to the speed limit}
\label{apx:constr}
First notice that, for any $N \geq 4$, Condition \ref{cond:1} cannot be fulfilled by simply filling the off-diagonal entries of $\mathrm{H}$ with any combination of $\pm 1$ and zeroes on the diagonal (with the constraint of having an Hermitian matrix compatible with the complete graph's topology). One has necessarily to resort to complex numbers. When $N$ is even, a combination of $\pm i$ does the trick. Indeed, by choosing:
\begin{equation}
\begin{aligned}
& \left[ \mathrm{H} \right]_{1j} = i , \ \ \ \ \ \ \left[ \mathrm{H} \right]_{jj} = \left[ \mathrm{H} \right]_{11} = 0 \ \ \ \ \forall j = 2,...,N \\
& \left[ \mathrm{H} \right]_{jk} = (-1)^{j+k} i \ \ \ \ \ \ \forall \ k > j > 1 \\
\end{aligned}
\end{equation}
and with the Hermitian constraint $\left[ H \right]_{jk} = \left[ H \right]^{*}_{kj} \ \forall j,k = 1,..., N $. It is immediate to check that indeed, for $N$ even, the first column of $\mathrm{H}$ is orthogonal to all the rows except the first one, with which it has an inner product of $N-1$. For example, for $N=6$:
\begin{equation}
\mathrm{H} \ = \ \left( \begin{array}{cccccc}
0 & i & i & i & i & i \\
-i & 0 & -i & i & -i & i \\
-i & i & 0 & -i & i & -i \\
-i & -i & i & 0 & -i & i \\
-i & i & -i & i & 0 & -i \\
-i & -i & i & -i & i & 0 \\
\end{array} \right)
\end{equation}
\\
For odd $N$, we found the following construction:
\begin{equation}
\begin{aligned}
& \left[ \mathrm{H} \right]_{1j} = - i , \ \ \ \ \ \ \left[ \mathrm{H} \right]_{jj} = \left[ \mathrm{H} \right]_{11} = 0 \ \ \ \ \forall j = 2,...,N \\
& \left[ \mathrm{H} \right]_{jk} = \exp \left\{ \frac{ 2 \pi i}{N-2} \left[ k - j +\frac{N-3}{2} \right] \right\} , \ \ \ \forall k > j > 1 \\
\end{aligned}
\end{equation}
E.g., for $N=5$:
\begin{equation}
\mathrm{H} \ = \ \left( \begin{array}{cccccc}
0 & i & i & i & i \\
-i & 0 & e^{- \frac{ 2 i \pi}{3}} & 1 & e^{ \frac{2 i \pi}{3}} \\
-i & e^{ \frac{2 i \pi}{3}} & 0 & e^{- 2i \frac{\pi}{3}} & 1 \\
-i & 1 & e^{ \frac{2 i \pi}{3}} & 0 & e^{- \frac{2 i \pi}{3}} \\
-i & e^{- \frac{ 2 i \pi}{3}} & 1 & e^{ \frac{2 i \pi}{3}} & 0 \\
\end{array} \right)
\end{equation}
\end{appendix}
| {
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{"url":"https:\/\/mail.aimsciences.org\/article\/doi\/10.3934\/dcds.2004.10.581","text":"# American Institute of Mathematical Sciences\n\nJuly\u00a0 2004,\u00a010(3):\u00a0581-587. doi:\u00a010.3934\/dcds.2004.10.581\n\n## On the law of logarithm of the recurrence time\n\n 1 Department of Mathematics, Korea Advanced Institute of Science and Technology, Daejeon, 305-701, South Korea 2 School of Mathematics, Korea Institute for Advanced Study, Seoul, 130-722, South Korea\n\nReceived\u00a0 November 2002 Revised\u00a0 May 2003 Published\u00a0 January 2004\n\nLet $T$ be a transformation from $I=[0,1)$ onto itself and let $Q_n(x)$ be the subinterval $[i\/2^n,(i+1)\/2^n)$, $0 \\leq i < 2^n$ containing $x$. Define $K_n (x) =$min{$j\\geq 1:T^j (x)\\in Q_n(x)$} and $K_n(x,y) =$min{$j\\geq 1:T^{j-1} (y) \\in Q_n(x)$}. For various transformations defined on $I$, we show that\n\n$\\lim_{n\\to\\infty}\\frac{\\log K_n(x)}{n}=1 \\quad$and$\\quad \\lim_{n\\to\\infty}\\frac{\\log K_n(x,y)}{n}=1 \\quad$a.e.\n\nCitation: Chihurn Kim, Dong Han Kim. On the law of logarithm of the recurrence time. Discrete and Continuous Dynamical Systems, 2004, 10 (3) : 581-587. doi: 10.3934\/dcds.2004.10.581\n [1] Omer Gursoy, Kamal Adli Mehr, Nail Akar. Steady-state and first passage time distributions for waiting times in the MAP\/M\/s+G queueing model with generally distributed patience times. Journal of Industrial and Management Optimization, 2022, 18 (4) : 2505-2532. doi: 10.3934\/jimo.2021078 [2] Jean-Ren\u00e9 Chazottes, Renaud Leplaideur. Fluctuations of the nth return time for Axiom A diffeomorphisms. Discrete and Continuous Dynamical Systems, 2005, 13 (2) : 399-411. doi: 10.3934\/dcds.2005.13.399 [3] Piotr Oprocha. Chain recurrence in multidimensional time discrete dynamical systems. Discrete and Continuous Dynamical Systems, 2008, 20 (4) : 1039-1056. doi: 10.3934\/dcds.2008.20.1039 [4] Huong Le Thi, St\u00e9phane Junca, Mathias Legrand. First Return Time to the contact hyperplane for $N$-degree-of-freedom vibro-impact systems. Discrete and Continuous Dynamical Systems - B, 2022, 27 (1) : 1-44. doi: 10.3934\/dcdsb.2021031 [5] Hideaki Takagi. Unified and refined analysis of the response time and waiting time in the M\/M\/m FCFS preemptive-resume priority queue. Journal of Industrial and Management Optimization, 2017, 13 (4) : 1945-1973. doi: 10.3934\/jimo.2017026 [6] Yutaka Sakuma, Atsushi Inoie, Ken\u2019ichi Kawanishi, Masakiyo Miyazawa. Tail asymptotics for waiting time distribution of an M\/M\/s queue with general impatient time. Journal of Industrial and Management Optimization, 2011, 7 (3) : 593-606. doi: 10.3934\/jimo.2011.7.593 [7] Daniel Glasscock, Andreas Koutsogiannis, Florian Karl Richter. Multiplicative combinatorial properties of return time sets in minimal dynamical systems. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 5891-5921. doi: 10.3934\/dcds.2019258 [8] Andrey Shishkov. Waiting time of propagation and the backward motion of interfaces in thin-film flow theory. Conference Publications, 2007, 2007 (Special) : 938-945. doi: 10.3934\/proc.2007.2007.938 [9] Zsolt Saffer, Wuyi Yue. A dual tandem queueing system with GI service time at the first queue. Journal of Industrial and Management Optimization, 2014, 10 (1) : 167-192. doi: 10.3934\/jimo.2014.10.167 [10] J\u00e9r\u00f4me Buzzi, V\u00e9ronique Maume-Deschamps. Decay of correlations on towers with non-H\u00f6lder Jacobian and non-exponential return time. Discrete and Continuous Dynamical Systems, 2005, 12 (4) : 639-656. doi: 10.3934\/dcds.2005.12.639 [11] Baoyin Xun, Kam C. Yuen, Kaiyong Wang. The finite-time ruin probability of a risk model with a general counting process and stochastic return. Journal of Industrial and Management Optimization, 2022, 18 (3) : 1541-1556. doi: 10.3934\/jimo.2021032 [12] Y. Charles Li, Hong Yang. On the arrow of time. Discrete and Continuous Dynamical Systems - S, 2014, 7 (6) : 1287-1303. doi: 10.3934\/dcdss.2014.7.1287 [13] Sim\u00e3o P. S. Santos, Nat\u00e1lia Martins, Delfim F. M. Torres. Variational problems of Herglotz type with time delay: DuBois--Reymond condition and Noether's first theorem. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 4593-4610. doi: 10.3934\/dcds.2015.35.4593 [14] Atsushi Kawamoto. H\u00f6lder stability estimate in an inverse source problem for a first and half order time fractional diffusion equation. Inverse Problems and Imaging, 2018, 12 (2) : 315-330. doi: 10.3934\/ipi.2018014 [15] Y. Gong, X. Xiang. A class of optimal control problems of systems governed by the first order linear dynamic equations on time scales. Journal of Industrial and Management Optimization, 2009, 5 (1) : 1-10. doi: 10.3934\/jimo.2009.5.1 [16] Qiuying Li, Lifang Huang, Jianshe Yu. Modulation of first-passage time for bursty gene expression via random signals. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1261-1277. doi: 10.3934\/mbe.2017065 [17] Arthur Bottois, Nicolae C\u00eendea. Controllability of the linear elasticity as a first-order system using a stabilized space-time mixed formulation. Mathematical Control and Related Fields, 2022\u00a0 doi: 10.3934\/mcrf.2022028 [18] Lars Olsen. First return times: multifractal spectra and divergence points. Discrete and Continuous Dynamical Systems, 2004, 10 (3) : 635-656. doi: 10.3934\/dcds.2004.10.635 [19] Massimiliano Tamborrino. Approximation of the first passage time density of a Wiener process to an exponentially decaying boundary by two-piecewise linear threshold. Application to neuronal spiking activity. Mathematical Biosciences & Engineering, 2016, 13 (3) : 613-629. doi: 10.3934\/mbe.2016011 [20] Jiao Song, Jiang-Lun Wu, Fangzhou Huang. First jump time in simulation of sampling trajectories of affine jump-diffusions driven by $\\alpha$-stable white noise. Communications on Pure and Applied Analysis, 2020, 19 (8) : 4127-4142. doi: 10.3934\/cpaa.2020184\n\n2021\u00a0Impact Factor:\u00a01.588\n\n## Metrics\n\n\u2022 HTML views\u00a0(0)\n\u2022 Cited by\u00a0(6)\n\n\u2022 on AIMS","date":"2022-10-05 22:39:22","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.4631155729293823, \"perplexity\": 5287.3680987928665}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-40\/segments\/1664030337668.62\/warc\/CC-MAIN-20221005203530-20221005233530-00026.warc.gz\"}"} | null | null |
As a little girl, I loved going to my grandmother's house during my school holidays. I looked forward to staying with her for long talks, baked cookies and quiet days listening to her stories. I was brokenhearted when she passed away. She left me her house because she knew how happy I was when I stayed there. But very soon after I moved in, I desperately needed Canada geese control in NJ to cut down on the number of geese on my new property.
When I was young, my grandmother loved to point out when the geese were coming back to the city after being away during the winter months. They always flew south, and she would point out when she could hear them flying into the city over the days and weeks. She always treated them respect, and she even threw out piece of stale bread for them to eat. There were not that many in the area back then, so it was not really a problem and did not cause even more of them to come around. I used to love to go outside and help her to feed the geese.
Over time, their breeding got out of control, and there are so many around the place now. It is no fun to be woken up early in the morning by their loud noises. And sometimes they fight with one another, and that is very jarring to me. They leave droppings in my pool, on my deck and on my car. There are just too many of them now. This was upsetting to me because I really enjoyed having them in years past.
After talking to an employee at the company who said that they could remove them, they said that I could try just moving around 75 percent of them only. They expected that it would be years before many more moved in. So, that's what I did, and all is well now. | {
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var async = require('async');
var bower = require('bower');
var colors = require('colors');
var exec = require('child_process').exec;
var fs = require('fs');
var inquirer = require('inquirer');
var isRoot = require('is-root');
var npm = require('npm');
var path = require('path');
var rimraf = require('rimraf');
var which = require('which');
var util = require('../util');
var EventEmitter = require("events").EventEmitter;
var format = require('util').format;
module.exports = function(args, options, callback, ee) {
var projectName, projectFolder, framework, template, messages, directory;
var tasks = [
preflight, prompt, drawImg//, gitClone, folderSetup, npmInstall, bowerInstall
];
// Each function below is executed in order
async.series(tasks, finish);
// 1. Check that the process isn't root, and that Git is installed
function preflight(cb) {
if (isRoot()) {
console.log(util.mascot('face', util.messages.noRoot));
process.exit(1);
}
which('git', function(er) {
if (er) {
console.log(util.messages.gitNotInstalled);
process.exit(69);
}
cb();
});
}
// 2. Find out what the user wants to do
function prompt(cb) {
inquirer.prompt(util.questions(options), function(answers) {
// The variables we need either came from the prompts, or the console arguments
projectName = answers.directory || options.directory;
framework = answers.framework || options.framework;
template = answers.template || options.template || null;
projectFolder = path.join(process.cwd(), projectName);
messages = util.messages(projectName);
cb();
});
}
function drawImg(cb){
var hello = formatHello(messages.helloYeti, framework);
console.log(util.mascot(framework, hello));
}
// 7. Finish the process with a status report
function finish(err, results) {
// Indexes 4 and 5 of results are the npm/Bower statuses
// All the rest should be undefined
console.log(results);
var allGood = results.indexOf(false) === -1;
if (allGood)
console.log(messages.installSuccess);
else
console.log(messages.installFail);
console.log(messages.gitCloneSuccess);
if (results[4])
console.log(messages.npmSuccess);
else
console.log(messages.npmFail);
if (results[5])
console.log(messages.bowerSuccess);
else if (fs.existsSync('bower.json'))
console.log(messages.bowerFail);
if (allGood)
console.log(messages.installSuccessFinal);
else
console.log(messages.installFailFinal);
if (typeof(callback)!=='undefined') callback();
}
}
function formatHello(str, framework) {
framework = framework.charAt(0).toUpperCase() + framework.slice(1)
str = str.join('\n');
str = str.replace('%s', framework);
return str.split('\n');
}
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{"url":"http:\/\/forge.ipsl.jussieu.fr\/nemo\/changeset\/9394","text":"# Changeset 9394\n\nIgnore:\nTimestamp:\n2018-03-13T21:21:44+01:00 (2 years ago)\nMessage:\n\nFix several typos, reverse (biblio then index) and shrink the manual backmatter (columns, font size, separator height) #1793\n\nLocation:\nbranches\/2017\/dev_merge_2017\/DOC\nFiles:\n13 edited\n\nUnmodified\nRemoved\n\n\u2022 ## branches\/2017\/dev_merge_2017\/DOC\/tex_main\/NEMO_manual.sty\n\n\u2022 ## branches\/2017\/dev_merge_2017\/DOC\/tex_main\/NEMO_manual.tex\n\n r9393 %% Bibliography \\cleardoublepage \\phantomsection \\addcontentsline{toc}{chapter}{Bibliography} \\bibliography{..\/tex_main\/NEMO_manual} %% Index \\cleardoublepage \\phantomsection \\addcontentsline{toc}{chapter}{Index} \\printindex %% Bibliography \\addcontentsline{toc}{chapter}{Bibliography} \\bibliography{..\/tex_main\/NEMO_manual} \\end{document}\n\u2022 ## branches\/2017\/dev_merge_2017\/DOC\/tex_sub\/chap_CONFIG.tex\n\n r9393 $z$-coordinates and is forced with tidal lateral boundary conditions using a flather boundary condition from the BDY module. The AMM configuration\u00a0 uses the GLS (key\\_zdfgls) turbulence scheme, the VVL non-linear free surface(key\\_vvl) and time-splitting (key\\_dynspg\\_ts). The AMM configuration\u00a0 uses the GLS (\\key{zdfgls}) turbulence scheme, the VVL non-linear free surface(\\key{vvl}) and time-splitting (\\key{dynspg\\_ts}). In addition to the tidal boundary condition the model may also take\n\u2022 ## branches\/2017\/dev_merge_2017\/DOC\/tex_sub\/chap_DIA.tex\n\n r9393 with the size of the array passed to iomput. The grid\\_ref attribute refers to definitions set in iodef.xml which, in turn, reference grids and axes either defined in the code (iom\\_set\\_domain\\_attr and iom\\_set\\_axis\\_attr in iom.F90) and axes either defined in the code (iom\\_set\\_domain\\_attr and iom\\_set\\_axis\\_attr in \\mdl{iom}) or defined in the domain\\_def.xml file. $e.g.$: \\begin{xmllines} Note, if your array is computed within the surface module each \\np{nn\\_fsbc} time\\_step, add the field definition within the field\\_group defined with the id ''SBC'': $<$field\\_group id=''SBC''...$>$ which has been defined with the correct frequency of operations (iom\\_set\\_field\\_attr in iom.F90) which has been defined with the correct frequency of operations (iom\\_set\\_field\\_attr in \\mdl{iom}) \\item[4.] add your field in one of the output files defined in iodef.xml (again see subsequent sections for syntax and rules) \\subsubsection{Other controls of the XML attributes from NEMO} The values of some attributes are defined by subroutine calls within NEMO (calls to iom\\_set\\_domain\\_attr, iom\\_set\\_axis\\_attr and iom\\_set\\_field\\_attr in iom.F90). Any definition given in the xml file will be overwritten. By convention, these attributes are defined to ''auto'' (for string) or ''0000'' (for integer) in the xml file (but this is not necessary). The values of some attributes are defined by subroutine calls within NEMO (calls to iom\\_set\\_domain\\_attr, iom\\_set\\_axis\\_attr and iom\\_set\\_field\\_attr in \\mdl{iom}). Any definition given in the xml file will be overwritten. By convention, these attributes are defined to ''auto'' (for string) or ''0000'' (for integer) in the xml file (but this is not necessary). Here is the list of these attributes:\\\\\n\u2022 ## branches\/2017\/dev_merge_2017\/DOC\/tex_sub\/chap_DIU.tex\n\n r9393 \\end{itemize} Models are provided for both the warm layer, diurnal\\_bulk.F90, and the cool skin, cool\\_skin.F90.\u00a0 Foundation SST is not considered as it can be obtained Models are provided for both the warm layer, \\mdfl{diurnal_bulk}, and the cool skin, \\mdl{cool_skin}.\u00a0 Foundation SST is not considered as it can be obtained either from the main NEMO model ($i.e.$ from the temperature of the top few model levels) or from some other source.\n\u2022 ## branches\/2017\/dev_merge_2017\/DOC\/tex_sub\/chap_DYN.tex\n\n r9393 For term \\textit{ttt} in the momentum equations, the logical namelist variables are \\textit{ln\\_dynttt\\_xxx}, where \\textit{xxx} is a 3 or 4 letter acronym corresponding to each optional scheme. If a CPP key is used for this term its name is \\textbf{key\\_ttt}. The corresponding If a CPP key is used for this term its name is \\key{ttt}. The corresponding code can be found in the \\textit{dynttt\\_xxx} module in the DYN directory, and it is usually computed in the \\textit{dyn\\_ttt\\_xxx} subroutine.\n\u2022 ## branches\/2017\/dev_merge_2017\/DOC\/tex_sub\/chap_LDF.tex\n\n r9393 The specification of the space variation of the coefficient is made in \\mdl{ldftra} and \\mdl{ldfdyn}, or more precisely in include files \\textit{traldf\\_cNd.h90} and \\textit{dynldf\\_cNd.h90}, with N=1, 2 or 3. \\hf{traldf\\_cNd} and \\hf{dynldf\\_cNd}, with N=1, 2 or 3. The user can modify these include files as he\/she wishes. The way the mixing coefficient are set in the reference version can be briefly described \\subsubsection{Constant mixing coefficients (default option)} When none of the \\textbf{key\\_dynldf\\_...} and \\textbf{key\\_traldf\\_...} keys are When none of the \\key{dynldf\\_...} and \\key{traldf\\_...} keys are defined, a constant value is used over the whole ocean for momentum and tracers, which is specified through the \\np{rn\\_ahm0} and \\np{rn\\_aht0} namelist the eddy induced coefficient has to be defined. Its space variations are controlled by the same CPP variable as for the eddy diffusivity coefficient ($i.e.$ \\textbf{key\\_traldf\\_cNd}). \\key{traldf\\_cNd}). (5) the eddy coefficient associated with a biharmonic operator must be set to a \\emph{negative} value.\n\u2022 ## branches\/2017\/dev_merge_2017\/DOC\/tex_sub\/chap_OBS.tex\n\n r9393 In addition to \\emph{OPA\\_SRC} the offline obs oper requires the inclusion of the \\emph{OOO\\_SRC} directory. \\emph{OOO\\_SRC} contains a replacement \\textbf{nemo.f90} and \\textbf{nemogcm.F90} which overwrites the resultant \\textbf{nemo.exe}. This is the approach taken of the \\emph{OOO\\_SRC} directory. \\emph{OOO\\_SRC} contains a replacement \\mdl{nemo} and \\mdl{nemogcm} which overwrites the resultant \\textbf{nemo.exe}. This is the approach taken by \\emph{SAS\\_SRC} and \\emph{OFF\\_SRC}.\n\u2022 ## branches\/2017\/dev_merge_2017\/DOC\/tex_sub\/chap_SBC.tex\n\n r9393 A generic interface has been introduced to manage the way input data (2D or 3D fields, like surface forcing or ocean T and S) are specify in \\NEMO. This task is archieved by fldread.F90. like surface forcing or ocean T and S) are specify in \\NEMO. This task is archieved by \\mdl{fldread}. The module was design with four main objectives in mind: \\begin{enumerate} \\begin{itemize} \\item \\mdl{nemogcm} : This routine initialises the rest of the model and repeatedly calls the stp time stepping routine (step.F90) \\item \\mdl{nemogcm} : This routine initialises the rest of the model and repeatedly calls the stp time stepping routine (\\mdl{step}) Since the ocean state is not calculated all associated initialisations have been removed. \\item\u00a0 \\mdl{step} : The main time stepping routine now only needs to call the sbc routine (and a few utility functions). and CICE CPP keys \\textbf{ORCA\\_GRID}, \\textbf{CICE\\_IN\\_NEMO} and \\textbf{coupled} should be used (seek advice from UKMO if necessary).\u00a0 Currently the code is only designed to work when using the CORE forcing option for NEMO (with \\textit{calc\\_strair~=~true} and \\textit{calc\\_Tsfc~=~true} in the CICE name-list), or alternatively when NEMO is coupled to the HadGAM3 atmosphere model (with \\textit{calc\\_strair~=~false} and \\textit{calc\\_Tsfc~=~false}). \\textit{calc\\_strair}\\forcode{ = .true.} and \\textit{calc\\_Tsfc}\\forcode{ = .true.} in the CICE name-list), or alternatively when NEMO is coupled to the HadGAM3 atmosphere model (with \\textit{calc\\_strair}\\forcode{ = .false.} and \\textit{calc\\_Tsfc}\\forcode{ = false}). The code is intended to be used with \\np{nn\\_fsbc} set to 1 (although coupling ocean and ice less frequently should work, it is possible the calculation of some of the ocean-ice fluxes needs to be modified slightly - the\n\u2022 ## branches\/2017\/dev_merge_2017\/DOC\/tex_sub\/chap_TRA.tex\n\n r9393 For each equation term\u00a0 \\textit{TTT}, the namelist logicals are \\textit{ln\\_traTTT\\_xxx}, where \\textit{xxx} is a 3 or 4 letter acronym corresponding to each optional scheme. The CPP key (when it exists) is \\textbf{key\\_traTTT}. The equivalent code can be The CPP key (when it exists) is \\key{traTTT}. The equivalent code can be found in the \\textit{traTTT} or \\textit{traTTT\\_xxx} module, in the NEMO\/OPA\/TRA directory. The choice is made in the \\textit{\\ngn{namtra\\_adv}} namelist, by setting to \\forcode{.true.} one of the logicals \\textit{ln\\_traadv\\_xxx}. The corresponding code can be found in the \\textit{traadv\\_xxx.F90} module, The corresponding code can be found in the \\mdl{traadv\\_xxx} module, where \\textit{xxx} is a 3 or 4 letter acronym corresponding to each scheme. By default ($i.e.$ in the reference namelist, \\ngn{namelist\\_ref}), all the logicals \\np{ln\\_zero\\_top\\_layer} specifies that the restoration coefficient should be zero in the surface layer. Finally \\np{ln\\_custom} specifies that the custom module will be called. This module is contained in the file custom.F90 and can be edited by users. For example damping could be applied in a specific region. This module is contained in the file \\mdl{custom} and can be edited by users. For example damping could be applied in a specific region. The restoration coefficient can be set to zero in equatorial regions by specifying a positive value of \\np{nn\\_hdmp}.\n\u2022 ## branches\/2017\/dev_merge_2017\/DOC\/tex_sub\/chap_misc.tex\n\n r9393 duplicated rows and columns due to cyclic or north fold boundary condition as well as overlap MPP areas). The self-compensated summation method should be used in all summation in i- and\/or j-direction. See closea.F90 module for an example. in i- and\/or j-direction. See \\mdl{closea} module for an example. Note also that this implementation may be sensitive to the optimization level.\n\u2022 ## branches\/2017\/dev_merge_2017\/DOC\/tex_sub\/chap_time_domain.tex\n\n r9393 where the subscript $F$ denotes filtered values and $\\gamma$ is the Asselin coefficient. $\\gamma$ is initialized as \\np{rn\\_atfp} (namelist parameter). Its default value is \\np{rn_atfp}\\forcode{ = 10.e-3} (see \\S~\\ref{STP_mLF}), Its default value is \\np{rn\\_atfp}\\forcode{ = 10.e-3} (see \\S~\\ref{STP_mLF}), causing only a weak dissipation of high frequency motions (\\citep{Farge1987}). The addition of a time filter degrades the accuracy of the\nNote: See TracChangeset for help on using the changeset viewer.","date":"2020-08-04 15:03:55","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8533698916435242, \"perplexity\": 9328.183581590982}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-34\/segments\/1596439735867.94\/warc\/CC-MAIN-20200804131928-20200804161928-00107.warc.gz\"}"} | null | null |
TPCN Flack Attack: No Nutritional Content
Tax-funded anti-abortion group uses PR campaign to smear Chronicle, rather than truly responding to our story
By Jordan Smith, Fri., Feb. 23, 2007
The Alternatives to Abortion program diverted $5 million from actual health-care workers such as these at East Austin's Planned Parenthood clinic, to fund nonmedical "crisis pregnancy centers" that directly served only 11 women last year.
On Feb. 13, The Austin Chronicle acquired a seven-page document, produced on Texas Pregnancy Care Network letterhead, recounting a host of complaints about our Jan. 26 news feature, "No Real Alternative," detailing TPCN's administration of the new, state-funded Alternatives to Abortion program. The Feb. 1 memorandum is unsigned, but we've been told that it originated with the office of Sen. Tommy Williams, R-The Woodlands, who authored and carried the budget rider that created the Alternatives to Abortion program back in 2005. More precisely, the document appears to have been produced by the high-powered political consulting firm the Eppstein Group, led by Bryan Eppstein, which has done campaign work for many GOP lawmakers, including Williams. We tried to confirm this provenance from the horses' mouths, but at press time, neither Eppstein nor Williams' aide had returned numerous calls requesting comment on the document – supposedly aimed at "correcting" a story we had published. (Download the whole document here.)
That's not surprising; during the time the story was being reported, TPCN declined to respond to numerous direct inquiries about its government-funded program.
According to the memo, a "tabloid publication" (that's us) had published an article about TPCN that was rife with errors; this memo, its unidentified author writes, is intended to provide "responses to several of the most egregious errors reported in the story" – including that we misled readers about the program's purpose and progress and raised questions about certain TPCN efforts in a way that unfairly implied an underlying ill motive.
The complaints themselves weren't surprising – our assessment of the TPCN program was hardly flattering. Yet none of these alleged errors were ever raised with the Chronicle: Following publication, neither Williams, Eppstein, nor TPCN Executive Director Vincent Friedewald contacted us to complain about errors, to request a retraction, or even to offer a written response for publication. Instead, the memorandum was submitted to the Houston Chronicle, in an effort to have retracted a Feb. 1 Houston Chronicle editorial that cited our story and called into question the process by which funding for the Alternatives to Abortion program was appropriated – through a rider offered by Williams, during a meeting of the Senate Finance Committee in March 2005.
As we've reported over the past year, Williams' rider set aside $5 million (from family-planning funds) over the biennium to administer a program explicitly intended to "promote childbirth" over abortion, primarily through a network of nonprofits, mostly "crisis pregnancy centers" – most run by vociferously anti-abortion groups that offer no medical services whatsoever. Lawmakers later told us they felt misled by Williams' assertion that the rider would not impact funding for traditional providers of preventative medical services for poor women – as in fact it has done.
Traditional family-planning service providers – like Planned Parenthood (the best-known but by no means the only group providing reproductive health care for low-income, underinsured women) – offer a host of medical services (including gynecological exams and cancer and diabetes screenings), as well as counseling on a wide range of topics (including adoption, abortion, parenting, and domestic and sexual violence). By contrast, the providers funded by Alternatives to Abortion provide no medical services at all but must simply guarantee that they will not "promote, refer or perform" abortions.
The rider passed without any real debate and without the consideration of lawmakers outside the Senate Finance Committee. A year later, in March 2006, the Texas Health and Human Services Commission awarded the $2.5-million-per-year contract to the TPCN, a brand-new nonprofit whose directors, according to their résumés, have no experience in nonprofit administration nor in women's health care. Under the contract, TPCN gets a lump sum from the state each month – unlike most other programs funded with federal welfare money, like women's health-care services, which are reimbursed after services are rendered and documented. In January, we reported that under this system, during FY 06, TPCN had invoiced the state for more than $600,000. Although well more than $100,000 supposedly went to pay for "client services," it wasn't until August that the group reported spending any money on services; in that invoice, TPCN reported spending $50.98 on client services. In all, according to TPCN's numbers, in FY 06 the group served just 11 clients statewide – in essence, at that time, the program had cost $58,086 per client.
An unsigned memorandum on TPCN letterhead claims to respond to "several of the most egregious errors" in a recent Austin Chronicle story. In fact, the "errors" are imaginary, and TPCN will not respond to our questions about its programs.
In its Feb. 1 memorandum to the Houston Chronicle, TPCN and its defenders assert that our reporting of the FY 06 numbers – based directly on TPCN and HHSC documents, since TPCN declined to speak with us – was "misleading." How it was misleading isn't clear. The letter says that TPCN was only ready to serve clients during the last three weeks of FY 06, and therefore, apparently, serving 11 clients was a great show of success. While it is true that TPCN didn't ink its contract with HHSC until March, it is also true that TPCN itself designed the rollout of the program and provided HHSC in late February 2006 a revised plan of action, along with estimates for the amount of money needed to get the program up and running and the number of clients that would be served by the end of the fiscal year. In the end, TPCN missed the mark on many of its own self-designed performance measures, including an estimate that total start-up costs would come in at less than $300,000 and that as many as 350 clients would have received services by Aug. 31. (The memo also offers a revised version of cost-per-client at $52.95, but that number is accurate only if what TPCN calls "client visits" were actual counseling sessions, not walk-throughs, and if it's allowed to ignore several hundred thousand dollars in its own "administrative" costs – a real stretch.)
The other objections from TPCN are similarly empty in substance. One is that our story "raised questions about the content and quality of educational materials approved by HHSC and purchased by TPCN for distribution to its Service Providers." This is true: Although we sought to review the materials prior to the story's publication, we were denied access – first by HHSC, which claimed it did not have the materials, and then by TPCN's Friedewald, who sent us an e-mail (which we published in its entirety), informing us that "as a small private business" TPCN didn't have the time or inclination to respond to open-records requests. (Apparently, they're much too busy writing self-justifying memoranda.)
Shortly after our article was published, we were contacted by HHSC and told that the agency had acquired the materials and would be able to make them available for inspection. We reviewed them earlier this month – under the watchful eye of an HHSC representative, who told us she was directed to sit with us during the entire visit.
The TPCN memorandum asserts that the group only buys educational materials that "meet high quality standards"; that cite "legitimate authorities, such as the Centers for Disease Control and Prevention, or … articles published in peer-reviewed medical literature"; and that it buys no "materials that are political, judgmental, reflect activist sentiments or that contain spiritual or religious content." Our review of the TPCN materials contradicts those assertions. We found biased materials – many that appear to spin medical information toward a particular political point of view, such as that found in abstinence-only sex education – and materials that cite dubious sources other than medical and/or health authorities.
For example, some sources of information were commercial interests and not legitimate scientific sources. The pamphlet "Coming Home With Your Baby" refers to a commercial Web site, MothersBliss.com, based in the United Kingdom (which sells items for pregnant women, such as lollipops that are supposed to help with morning sickness) as a source of medical information regarding jaundice in newborns. But the MothersBliss Web site does not cite any legitimate medical authority for any of its information, and a site disclaimer says it is not in the business of "giving medical counseling or other professional services or advice" and is not intended "in any way" to "take the place of a doctor's or a health professional's advice and treatment." (At press time, company representatives had not responded to our request for sourcing information.)
Additionally, many of the educational materials – pamphlets, brochures, books, DVDs, and videotapes – paid for by TPCN with $75,000 in taxpayer money, according to the state contract – are already available for purchase (or, in some cases, for free) from the few legitimate medical authorities they do cite. For example, the pamphlet "Your Baby Learns From Every Experience," which TPCN bought from the self-described pro-life publisher Heritage House 76, cites as references the National Institutes of Health and the D.C.-based nonprofit Zero to Three (whose work is supported, in part, by the federal government and the Boston University Medical Campus), both of which will provide educational materials to nonprofit groups like TPCN without using the Heritage House middleman.
In short, the state of Texas (and Texas taxpayers) have paid TPCN to buy materials that are either religious propaganda or that could be acquired directly elsewhere – often at no cost.
There is at least one thing in the TPCN memo that is absolutely correct. It complains that the Chronicle's "critiques" of the Texas Pregnancy Care Network were based in part on the fact that TPCN doesn't provide any health care or medical services to women. But "The Program," they write, "was never intended to provide medical services."
TPCN Strikes Back!
The Texas Pregnancy Care Network didn't like our story – but couldn't bear to tell us.
The 'Chronicle' Responds to TPCN
The TPCN complaints about Chronicle reporting are much ado about not very much.
More abortion
After a Month of Chaos, Abortion Care Returns to Texas
Clinics, patients, and the law get put through the ringer over care
Mary Tuma, May 1, 2020
Lege Lines: Dem Lawmakers Boycott Anti-Abortion Bill Hearing
Brass ovaries vs. HB 16, and other battle lines drawn at the Capitol
Michael King, March 29, 2019
More by Jordan Smith
'Chrome Underground' Goes Classic Car Hunting
Motoreum's Yusuf & Antonio talk about the biz and their reality TV debut
APD Brass Shifts Up, Down, Across
Musical chairs at Downtown HQ
abortion, Texas Pregnancy Care Network, Tommy Williams, Eppstein Group, Vincent Friedewald, Planned Parenthood, crisis pregnancy centers
Sightlines Discussion with Delita Martin
Big Medium
Our Lady of the Sacred Part: Vulva Pope at The Vortex | {
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The Company has appointed a data protection officer to oversee compliance with this privacy notice. If you have any questions about this privacy notice or about how we handle your personal information, please contact Legal Department, Namesco Limited, Acton House, Perdiswell Park, Worcester, Worcestershire, WR3 7GD.
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If you wish to exercise any of these rights, please contact our data protection officer. We may need to request specific information from you in order to verify your identity and check your right to access the personal information or to exercise any of your other rights. This is a security measure to ensure that your personal information is not disclosed to any person who has no right to receive it.
In the limited circumstances where you have provided your consent to the processing of your personal information for a specific purpose, you have the right to withdraw your consent for that specific processing at any time. This will not, however, affect the lawfulness of processing based on your consent before its withdrawal. If you wish to withdraw your consent, please contact our data protection officer. Once we have received notification that you have withdrawn your consent, we will no longer process your personal information for the purpose you originally agreed to, unless we have another legal basis for processing.
If you have any questions about this privacy notice or how we handle your personal information, please contact our data protection officer as follows: Legal Department, Namesco Limited, Acton House, Perdiswell Park, Worcester, Worcestershire, WR3 7GD. | {
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With a different approach than traditional Dermatologists in Ahwa and in other parts of Gujarat, CureSkin is quick, easy and provides a satisfaction guarantee. This is the reason why it's the most reasonable choice to go for, for any Skin and Hair concerns.
Ahwa is a fantastic town to live. At the same time, it is not the city to find a proper Dermatologist. far-off parts of Gujarat, counting the Dang district, it can be very hard to find a professional Dermatologist who can go ahead and verily help you fix your Skin or Hair concern.
Even if you manage to find a Dermatologist, it takes quite a long time to fix an appointment, post this, you are expected to spend some more time standing in queues and waiting areas just to get a basic diagnosis and you're finally given a list of medicines which you are required to go and buy from pharmacies.
With CureSkin, you don't face any difficulty as the Diagnosis is done by the App with a photo of your issue INSTANTLY, and the regimen is shipped to your residence. Different from a regular Dermatologist, CureSkin's internal panel of Dermatologists and Skin Care experts reliably check up with you all through your regimen to make sure that you're improving, with no additional charges!
The system of getting your Hair and Skin conditions cured through CureSkin is a rather convenient one. You just have to download the app, submit some brief details about your Skin/Hair health, snap a photo of your Skin problem and let the app do its' job!
The CureSkin app will then notify you what your Skin condition is called and will supply you a tailor-made Skin/Hair care regimen that will not only fix your condition, but will go one step beyond and stop it from coming back again. Advice on Lifestyle adjustments, food habits etc will also be given, with regular follow-ups by CureSkin's internal team of qualified Dermatologists and Skin and Hair Care experts. These experts have served hundreds of individuals much like yourself in Ahwa successfully, and help even more every day!
CureSkin has successfully assisted hundreds of users in Ahwa with their Skin and Hair problems. What are you waiting for? Get the CureSkin App right away! | {
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