text:
1=======================
2A Linux CD-ROM standard
3=======================
4
5:Author: David van Leeuwen <david@ElseWare.cistron.nl>
6:Date: 12 March 1999
7:Updated by: Erik Andersen (andersee@debian.org)
8:Updated by: Jens Axboe (axboe@image.dk)
9
10
11Introduction
12============
13
14Linux is probably the Unix-like operating system that supports
15the widest variety of hardware devices. The reasons for this are
16presumably
17
18- The large list of hardware devices available for the many platforms
19  that Linux now supports (i.e., i386-PCs, Sparc Suns, etc.)
20- The open design of the operating system, such that anybody can write a
21  driver for Linux.
22- There is plenty of source code around as examples of how to write a driver.
23
24The openness of Linux, and the many different types of available
25hardware has allowed Linux to support many different hardware devices.
26Unfortunately, the very openness that has allowed Linux to support
27all these different devices has also allowed the behavior of each
28device driver to differ significantly from one device to another.
29This divergence of behavior has been very significant for CD-ROM
30devices; the way a particular drive reacts to a standard *ioctl()*
31call varies greatly from one device driver to another. To avoid making
32their drivers totally inconsistent, the writers of Linux CD-ROM
33drivers generally created new device drivers by understanding, copying,
34and then changing an existing one. Unfortunately, this practice did not
35maintain uniform behavior across all the Linux CD-ROM drivers.
36
37This document describes an effort to establish Uniform behavior across
38all the different CD-ROM device drivers for Linux. This document also
39defines the various *ioctl()'s*, and how the low-level CD-ROM device
40drivers should implement them. Currently (as of the Linux 2.1.\ *x*
41development kernels) several low-level CD-ROM device drivers, including
42both IDE/ATAPI and SCSI, now use this Uniform interface.
43
44When the CD-ROM was developed, the interface between the CD-ROM drive
45and the computer was not specified in the standards. As a result, many
46different CD-ROM interfaces were developed. Some of them had their
47own proprietary design (Sony, Mitsumi, Panasonic, Philips), other
48manufacturers adopted an existing electrical interface and changed
49the functionality (CreativeLabs/SoundBlaster, Teac, Funai) or simply
50adapted their drives to one or more of the already existing electrical
51interfaces (Aztech, Sanyo, Funai, Vertos, Longshine, Optics Storage and
52most of the NoName manufacturers). In cases where a new drive really
53brought its own interface or used its own command set and flow control
54scheme, either a separate driver had to be written, or an existing
55driver had to be enhanced. History has delivered us CD-ROM support for
56many of these different interfaces. Nowadays, almost all new CD-ROM
57drives are either IDE/ATAPI or SCSI, and it is very unlikely that any
58manufacturer will create a new interface. Even finding drives for the
59old proprietary interfaces is getting difficult.
60
61When (in the 1.3.70's) I looked at the existing software interface,
62which was expressed through cdrom.h, it appeared to be a rather wild
63set of commands and data formats [#f1]_. It seemed that many
65capabilities of a particular drive, in an *ad hoc* manner. More
66importantly, it appeared that the behavior of the standard commands
67was different for most of the different drivers: e. g., some drivers
68close the tray if an *open()* call occurs when the tray is open, while
69others do not. Some drivers lock the door upon opening the device, to
70prevent an incoherent file system, but others don't, to allow software
71ejection. Undoubtedly, the capabilities of the different drives vary,
72but even when two drives have the same capability their drivers'
73behavior was usually different.
74
75.. [#f1]
76   I cannot recollect what kernel version I looked at, then,
77   presumably 1.2.13 and 1.3.34 --- the latest kernel that I was
78   indirectly involved in.
79
80I decided to start a discussion on how to make all the Linux CD-ROM
81drivers behave more uniformly. I began by contacting the developers of
82the many CD-ROM drivers found in the Linux kernel. Their reactions
83encouraged me to write the Uniform CD-ROM Driver which this document is
84intended to describe. The implementation of the Uniform CD-ROM Driver is
85in the file cdrom.c. This driver is intended to be an additional software
86layer that sits on top of the low-level device drivers for each CD-ROM drive.
87By adding this additional layer, it is possible to have all the different
88CD-ROM devices behave **exactly** the same (insofar as the underlying
89hardware will allow).
90
91The goal of the Uniform CD-ROM Driver is **not** to alienate driver developers
92whohave not yet taken steps to support this effort. The goal of Uniform CD-ROM
93Driver is simply to give people writing application programs for CD-ROM drives
94**one** Linux CD-ROM interface with consistent behavior for all
95CD-ROM devices. In addition, this also provides a consistent interface
96between the low-level device driver code and the Linux kernel. Care
97is taken that 100% compatibility exists with the data structures and
98programmer's interface defined in cdrom.h. This guide was written to
99help CD-ROM driver developers adapt their code to use the Uniform CD-ROM
100Driver code defined in cdrom.c.
101
102Personally, I think that the most important hardware interfaces are
103the IDE/ATAPI drives and, of course, the SCSI drives, but as prices
104of hardware drop continuously, it is also likely that people may have
105more than one CD-ROM drive, possibly of mixed types. It is important
106that these drives behave in the same way. In December 1994, one of the
107cheapest CD-ROM drives was a Philips cm206, a double-speed proprietary
108drive. In the months that I was busy writing a Linux driver for it,
109proprietary drives became obsolete and IDE/ATAPI drives became the
110standard. At the time of the last update to this document (November
1111997) it is becoming difficult to even **find** anything less than a
11216 speed CD-ROM drive, and 24 speed drives are common.
113
114.. _cdrom_api:
115
116Standardizing through another software level
117============================================
118
119At the time this document was conceived, all drivers directly
120implemented the CD-ROM *ioctl()* calls through their own routines. This
121led to the danger of different drivers forgetting to do important things
122like checking that the user was giving the driver valid data. More
123importantly, this led to the divergence of behavior, which has already
124been discussed.
125
126For this reason, the Uniform CD-ROM Driver was created to enforce consistent
127CD-ROM drive behavior, and to provide a common set of services to the various
128low-level CD-ROM device drivers. The Uniform CD-ROM Driver now provides another
129software-level, that separates the *ioctl()* and *open()* implementation
130from the actual hardware implementation. Note that this effort has
131made few changes which will affect a user's application programs. The
132greatest change involved moving the contents of the various low-level
133CD-ROM drivers\' header files to the kernel's cdrom directory. This was
134done to help ensure that the user is only presented with only one cdrom
135interface, the interface defined in cdrom.h.
136
137CD-ROM drives are specific enough (i. e., different from other
138block-devices such as floppy or hard disc drives), to define a set
139of common **CD-ROM device operations**, *<cdrom-device>_dops*.
140These operations are different from the classical block-device file
141operations, *<block-device>_fops*.
142
143The routines for the Uniform CD-ROM Driver interface level are implemented
144in the file cdrom.c. In this file, the Uniform CD-ROM Driver interfaces
145with the kernel as a block device by registering the following general
146*struct file_operations*::
147
148        struct file_operations cdrom_fops = {
149                NULL,                   /* lseek */
151                block _write,           /* write--general block-dev write */
153                NULL,                   /* select */
154                cdrom_ioctl,            /* ioctl */
155                NULL,                   /* mmap */
156                cdrom_open,             /* open */
157                cdrom_release,          /* release */
158                NULL,                   /* fsync */
159                NULL,                   /* fasync */
160                NULL                    /* revalidate */
161        };
162
163Every active CD-ROM device shares this *struct*. The routines
164declared above are all implemented in cdrom.c, since this file is the
165place where the behavior of all CD-ROM-devices is defined and
166standardized. The actual interface to the various types of CD-ROM
167hardware is still performed by various low-level CD-ROM-device
168drivers. These routines simply implement certain **capabilities**
169that are common to all CD-ROM (and really, all removable-media
170devices).
171
172Registration of a low-level CD-ROM device driver is now done through
173the general routines in cdrom.c, not through the Virtual File System
174(VFS) any more. The interface implemented in cdrom.c is carried out
175through two general structures that contain information about the
176capabilities of the driver, and the specific drives on which the
177driver operates. The structures are:
178
179cdrom_device_ops
180  This structure contains information about the low-level driver for a
181  CD-ROM device. This structure is conceptually connected to the major
182  number of the device (although some drivers may have different
183  major numbers, as is the case for the IDE driver).
184
185cdrom_device_info
186  This structure contains information about a particular CD-ROM drive,
187  such as its device name, speed, etc. This structure is conceptually
188  connected to the minor number of the device.
189
190Registering a particular CD-ROM drive with the Uniform CD-ROM Driver
191is done by the low-level device driver though a call to::
192
193        register_cdrom(struct cdrom_device_info * <device>_info)
194
195The device information structure, *<device>_info*, contains all the
196information needed for the kernel to interface with the low-level
197CD-ROM device driver. One of the most important entries in this
198structure is a pointer to the *cdrom_device_ops* structure of the
199low-level driver.
200
201The device operations structure, *cdrom_device_ops*, contains a list
202of pointers to the functions which are implemented in the low-level
203device driver. When cdrom.c accesses a CD-ROM device, it does it
204through the functions in this structure. It is impossible to know all
205the capabilities of future CD-ROM drives, so it is expected that this
206list may need to be expanded from time to time as new technologies are
207developed. For example, CD-R and CD-R/W drives are beginning to become
208popular, and support will soon need to be added for them. For now, the
209current *struct* is::
210
211        struct cdrom_device_ops {
212                int (*open)(struct cdrom_device_info *, int)
213                void (*release)(struct cdrom_device_info *);
214                int (*drive_status)(struct cdrom_device_info *, int);
215                unsigned int (*check_events)(struct cdrom_device_info *,
216                                             unsigned int, int);
217                int (*media_changed)(struct cdrom_device_info *, int);
218                int (*tray_move)(struct cdrom_device_info *, int);
219                int (*lock_door)(struct cdrom_device_info *, int);
220                int (*select_speed)(struct cdrom_device_info *, int);
221                int (*select_disc)(struct cdrom_device_info *, int);
222                int (*get_last_session) (struct cdrom_device_info *,
223                                         struct cdrom_multisession *);
224                int (*get_mcn)(struct cdrom_device_info *, struct cdrom_mcn *);
225                int (*reset)(struct cdrom_device_info *);
226                int (*audio_ioctl)(struct cdrom_device_info *,
227                                   unsigned int, void *);
228                const int capability;           /* capability flags */
229                int (*generic_packet)(struct cdrom_device_info *,
230                                      struct packet_command *);
231        };
232
233When a low-level device driver implements one of these capabilities,
234it should add a function pointer to this *struct*. When a particular
235function is not implemented, however, this *struct* should contain a
236NULL instead. The *capability* flags specify the capabilities of the
237CD-ROM hardware and/or low-level CD-ROM driver when a CD-ROM drive
238is registered with the Uniform CD-ROM Driver.
239
240Note that most functions have fewer parameters than their
241*blkdev_fops* counterparts. This is because very little of the
242information in the structures *inode* and *file* is used. For most
243drivers, the main parameter is the *struct* *cdrom_device_info*, from
244which the major and minor number can be extracted. (Most low-level
245CD-ROM drivers don't even look at the major and minor number though,
246since many of them only support one device.) This will be available
247through *dev* in *cdrom_device_info* described below.
248
249The drive-specific, minor-like information that is registered with
250cdrom.c, currently contains the following fields::
251
252  struct cdrom_device_info {
253        const struct cdrom_device_ops * ops;    /* device operations for this major */
255        struct gendisk * disk;                  /* matching block layer disk */
256        void *  handle;                         /* driver-dependent data */
257
259        int speed;                              /* maximum speed for reading data */
260        int capacity;                           /* number of discs in a jukebox */
261
262        unsigned int options:30;                /* options flags */
263        unsigned mc_flags:2;                    /*  media-change buffer flags */
264        unsigned int vfs_events;                /*  cached events for vfs path */
265        unsigned int ioctl_events;              /*  cached events for ioctl path */
266        int use_count;                          /*  number of times device is opened */
267        char name[20];                          /*  name of the device type */
268
269        __u8 sanyo_slot : 2;                    /*  Sanyo 3-CD changer support */
270        __u8 keeplocked : 1;                    /*  CDROM_LOCKDOOR status */
271        __u8 reserved : 5;                      /*  not used yet */
272        int cdda_method;                        /*  see CDDA_* flags */
273        __u8 last_sense;                        /*  saves last sense key */
274        __u8 media_written;                     /*  dirty flag, DVD+RW bookkeeping */
275        unsigned short mmc3_profile;            /*  current MMC3 profile */
276        int for_data;                           /*  unknown:TBD */
277        int (*exit)(struct cdrom_device_info *);/*  unknown:TBD */
278        int mrw_mode_page;                      /*  which MRW mode page is in use */
279  };
280
281Using this *struct*, a linked list of the registered minor devices is
282built, using the *next* field. The device number, the device operations
283struct and specifications of properties of the drive are stored in this
284structure.
285
286The *mask* flags can be used to mask out some of the capabilities listed
287in *ops->capability*, if a specific drive doesn't support a feature
288of the driver. The value *speed* specifies the maximum head-rate of the
289drive, measured in units of normal audio speed (176kB/sec raw data or
290150kB/sec file system data). The parameters are declared *const*
291because they describe properties of the drive, which don't change after
292registration.
293
294A few registers contain variables local to the CD-ROM drive. The
295flags *options* are used to specify how the general CD-ROM routines
296should behave. These various flags registers should provide enough
297flexibility to adapt to the different users' wishes (and **not** the
298arbitrary wishes of the author of the low-level device driver, as is
299the case in the old scheme). The register *mc_flags* is used to buffer
300the information from *media_changed()* to two separate queues. Other
301data that is specific to a minor drive, can be accessed through *handle*,
302which can point to a data structure specific to the low-level driver.
303The fields *use_count*, *next*, *options* and *mc_flags* need not be
304initialized.
305
306The intermediate software layer that cdrom.c forms will perform some
307additional bookkeeping. The use count of the device (the number of
308processes that have the device opened) is registered in *use_count*. The
309function *cdrom_ioctl()* will verify the appropriate user-memory regions
310for read and write, and in case a location on the CD is transferred,
311it will sanitize the format by making requests to the low-level
312drivers in a standard format, and translating all formats between the
313user-software and low level drivers. This relieves much of the drivers'
314memory checking and format checking and translation. Also, the necessary
315structures will be declared on the program stack.
316
317The implementation of the functions should be as defined in the
318following sections. Two functions **must** be implemented, namely
319*open()* and *release()*. Other functions may be omitted, their
320corresponding capability flags will be cleared upon registration.
321Generally, a function returns zero on success and negative on error. A
322function call should return only after the command has completed, but of
323course waiting for the device should not use processor time.
324
325::
326
327        int open(struct cdrom_device_info *cdi, int purpose)
328
329*Open()* should try to open the device for a specific *purpose*, which
330can be either:
331
332- Open for reading data, as done by mount() (2), or the
333  user commands dd or cat.
334- Open for *ioctl* commands, as done by audio-CD playing programs.
335
336Notice that any strategic code (closing tray upon *open()*, etc.) is
337done by the calling routine in cdrom.c, so the low-level routine
338should only be concerned with proper initialization, such as spinning
339up the disc, etc.
340
341::
342
343        void release(struct cdrom_device_info *cdi)
344
345Device-specific actions should be taken such as spinning down the device.
346However, strategic actions such as ejection of the tray, or unlocking
347the door, should be left over to the general routine *cdrom_release()*.
348This is the only function returning type *void*.
349
350.. _cdrom_drive_status:
351
352::
353
354        int drive_status(struct cdrom_device_info *cdi, int slot_nr)
355
356The function *drive_status*, if implemented, should provide
357information on the status of the drive (not the status of the disc,
358which may or may not be in the drive). If the drive is not a changer,
359*slot_nr* should be ignored. In cdrom.h the possibilities are listed::
360
361
362        CDS_NO_INFO             /* no information available */
363        CDS_NO_DISC             /* no disc is inserted, tray is closed */
364        CDS_TRAY_OPEN           /* tray is opened */
365        CDS_DRIVE_NOT_READY     /* something is wrong, tray is moving? */
366        CDS_DISC_OK             /* a disc is loaded and everything is fine */
367
368::
369
370        int tray_move(struct cdrom_device_info *cdi, int position)
371
372This function, if implemented, should control the tray movement. (No
373other function should control this.) The parameter *position* controls
374the desired direction of movement:
375
376- 0 Close tray
377- 1 Open tray
378
379This function returns 0 upon success, and a non-zero value upon
380error. Note that if the tray is already in the desired position, no
381action need be taken, and the return value should be 0.
382
383::
384
385        int lock_door(struct cdrom_device_info *cdi, int lock)
386
387This function (and no other code) controls locking of the door, if the
388drive allows this. The value of *lock* controls the desired locking
389state:
390
391- 0 Unlock door, manual opening is allowed
392- 1 Lock door, tray cannot be ejected manually
393
394This function returns 0 upon success, and a non-zero value upon
395error. Note that if the door is already in the requested state, no
396action need be taken, and the return value should be 0.
397
398::
399
400        int select_speed(struct cdrom_device_info *cdi, int speed)
401
402Some CD-ROM drives are capable of changing their head-speed. There
403are several reasons for changing the speed of a CD-ROM drive. Badly
404pressed CD-ROM s may benefit from less-than-maximum head rate. Modern
405CD-ROM drives can obtain very high head rates (up to *24x* is
406common). It has been reported that these drives can make reading
407errors at these high speeds, reducing the speed can prevent data loss
408in these circumstances. Finally, some of these drives can
409make an annoyingly loud noise, which a lower speed may reduce.
410
411This function specifies the speed at which data is read or audio is
412played back. The value of *speed* specifies the head-speed of the
413drive, measured in units of standard cdrom speed (176kB/sec raw data
414or 150kB/sec file system data). So to request that a CD-ROM drive
415operate at 300kB/sec you would call the CDROM_SELECT_SPEED *ioctl*
416with *speed=2*. The special value 0 means auto-selection, i. e.,
417maximum data-rate or real-time audio rate. If the drive doesn't have
418this auto-selection capability, the decision should be made on the
419current disc loaded and the return value should be positive. A negative
420return value indicates an error.
421
422::
423
424        int select_disc(struct cdrom_device_info *cdi, int number)
425
426If the drive can store multiple discs (a juke-box) this function
427will perform disc selection. It should return the number of the
428selected disc on success, a negative value on error. Currently, only
429the ide-cd driver supports this functionality.
430
431::
432
433        int get_last_session(struct cdrom_device_info *cdi,
434                             struct cdrom_multisession *ms_info)
435
436This function should implement the old corresponding *ioctl()*. For
437device *cdi->dev*, the start of the last session of the current disc
438should be returned in the pointer argument *ms_info*. Note that
439routines in cdrom.c have sanitized this argument: its requested
440format will **always** be of the type *CDROM_LBA* (linear block
441addressing mode), whatever the calling software requested. But
442sanitization goes even further: the low-level implementation may
443return the requested information in *CDROM_MSF* format if it wishes so
444(setting the *ms_info->addr_format* field appropriately, of
445course) and the routines in cdrom.c will make the transformation if
446necessary. The return value is 0 upon success.
447
448::
449
450        int get_mcn(struct cdrom_device_info *cdi,
451                    struct cdrom_mcn *mcn)
452
453Some discs carry a Media Catalog Number (MCN), also called
454Universal Product Code (UPC). This number should reflect the number
455that is generally found in the bar-code on the product. Unfortunately,
456the few discs that carry such a number on the disc don't even use the
457same format. The return argument to this function is a pointer to a
458pre-declared memory region of type *struct cdrom_mcn*. The MCN is
459expected as a 13-character string, terminated by a null-character.
460
461::
462
463        int reset(struct cdrom_device_info *cdi)
464
465This call should perform a hard-reset on the drive (although in
466circumstances that a hard-reset is necessary, a drive may very well not
467listen to commands anymore). Preferably, control is returned to the
468caller only after the drive has finished resetting. If the drive is no
469longer listening, it may be wise for the underlying low-level cdrom
470driver to time out.
471
472::
473
474        int audio_ioctl(struct cdrom_device_info *cdi,
475                        unsigned int cmd, void *arg)
476
477Some of the CD-ROM-\ *ioctl()*\ 's defined in cdrom.h can be
478implemented by the routines described above, and hence the function
479*cdrom_ioctl* will use those. However, most *ioctl()*\ 's deal with
480audio-control. We have decided to leave these to be accessed through a
481single function, repeating the arguments *cmd* and *arg*. Note that
482the latter is of type *void*, rather than *unsigned long int*.
483The routine *cdrom_ioctl()* does do some useful things,
484though. It sanitizes the address format type to *CDROM_MSF* (Minutes,
485Seconds, Frames) for all audio calls. It also verifies the memory
486location of *arg*, and reserves stack-memory for the argument. This
487makes implementation of the *audio_ioctl()* much simpler than in the
488old driver scheme. For example, you may look up the function
489*cm206_audio_ioctl()* cm206.c that should be updated with
490this documentation.
491
492An unimplemented ioctl should return *-ENOSYS*, but a harmless request
493(e. g., *CDROMSTART*) may be ignored by returning 0 (success). Other
494errors should be according to the standards, whatever they are. When
495an error is returned by the low-level driver, the Uniform CD-ROM Driver
496tries whenever possible to return the error code to the calling program.
497(We may decide to sanitize the return value in *cdrom_ioctl()* though, in
498order to guarantee a uniform interface to the audio-player software.)
499
500::
501
502        int dev_ioctl(struct cdrom_device_info *cdi,
503                      unsigned int cmd, unsigned long arg)
504
505Some *ioctl()'s* seem to be specific to certain CD-ROM drives. That is,
506they are introduced to service some capabilities of certain drives. In
507fact, there are 6 different *ioctl()'s* for reading data, either in some
508particular kind of format, or audio data. Not many drives support
509reading audio tracks as data, I believe this is because of protection
510of copyrights of artists. Moreover, I think that if audio-tracks are
511supported, it should be done through the VFS and not via *ioctl()'s*. A
512problem here could be the fact that audio-frames are 2352 bytes long,
513so either the audio-file-system should ask for 75264 bytes at once
514(the least common multiple of 512 and 2352), or the drivers should
515bend their backs to cope with this incoherence (to which I would be
516opposed). Furthermore, it is very difficult for the hardware to find
517the exact frame boundaries, since there are no synchronization headers
518in audio frames. Once these issues are resolved, this code should be
519standardized in cdrom.c.
520
521Because there are so many *ioctl()'s* that seem to be introduced to
522satisfy certain drivers [#f2]_, any non-standard *ioctl()*\ s
523are routed through the call *dev_ioctl()*. In principle, private
524*ioctl()*\ 's should be numbered after the device's major number, and not
525the general CD-ROM *ioctl* number, 0x53. Currently the
526non-supported *ioctl()'s* are:
527
530
531.. [#f2]
532
533   Is there software around that actually uses these? I'd be interested!
534
535.. _cdrom_capabilities:
536
537CD-ROM capabilities
538-------------------
539
540Instead of just implementing some *ioctl* calls, the interface in
541cdrom.c supplies the possibility to indicate the **capabilities**
542of a CD-ROM drive. This can be done by ORing any number of
543capability-constants that are defined in cdrom.h at the registration
544phase. Currently, the capabilities are any of::
545
546        CDC_CLOSE_TRAY          /* can close tray by software control */
547        CDC_OPEN_TRAY           /* can open tray */
548        CDC_LOCK                /* can lock and unlock the door */
549        CDC_SELECT_SPEED        /* can select speed, in units of * sim*150 ,kB/s */
550        CDC_SELECT_DISC         /* drive is juke-box */
551        CDC_MULTI_SESSION       /* can read sessions *> rm1* */
552        CDC_MCN                 /* can read Media Catalog Number */
553        CDC_MEDIA_CHANGED       /* can report if disc has changed */
554        CDC_PLAY_AUDIO          /* can perform audio-functions (play, pause, etc) */
555        CDC_RESET               /* hard reset device */
556        CDC_IOCTLS              /* driver has non-standard ioctls */
557        CDC_DRIVE_STATUS        /* driver implements drive status */
558
559The capability flag is declared *const*, to prevent drivers from
560accidentally tampering with the contents. The capability flags actually
561inform cdrom.c of what the driver can do. If the drive found
562by the driver does not have the capability, is can be masked out by
563the *cdrom_device_info* variable *mask*. For instance, the SCSI CD-ROM
565hence its corresponding flags in *capability* will be set. But a SCSI
566CD-ROM drive might be a caddy system, which can't load the tray, and
567hence for this drive the *cdrom_device_info* struct will have set
569
570In the file cdrom.c you will encounter many constructions of the type::
571
572        if (cdo->capability & ~cdi->mask & CDC _<capability>) ...
573
574There is no *ioctl* to set the mask... The reason is that
575I think it is better to control the **behavior** rather than the
576**capabilities**.
577
578Options
579-------
580
581A final flag register controls the **behavior** of the CD-ROM
582drives, in order to satisfy different users' wishes, hopefully
583independently of the ideas of the respective author who happened to
584have made the drive's support available to the Linux community. The
585current behavior options are::
586
587        CDO_AUTO_CLOSE  /* try to close tray upon device open() */
588        CDO_AUTO_EJECT  /* try to open tray on last device close() */
589        CDO_USE_FFLAGS  /* use file_pointer->f_flags to indicate purpose for open() */
590        CDO_LOCK        /* try to lock door if device is opened */
591        CDO_CHECK_TYPE  /* ensure disc type is data if opened for data */
592
593The initial value of this register is
594CDO_AUTO_CLOSE | CDO_USE_FFLAGS | CDO_LOCK, reflecting my own view on user
595interface and software standards. Before you protest, there are two
596new *ioctl()'s* implemented in cdrom.c, that allow you to control the
597behavior by software. These are::
598
599        CDROM_SET_OPTIONS       /* set options specified in (int)arg */
600        CDROM_CLEAR_OPTIONS     /* clear options specified in (int)arg */
601
602One option needs some more explanation: *CDO_USE_FFLAGS*. In the next
603newsection we explain what the need for this option is.
604
605A software package setcd, available from the Debian distribution
606and sunsite.unc.edu, allows user level control of these flags.
607
608
609The need to know the purpose of opening the CD-ROM device
610=========================================================
611
612Traditionally, Unix devices can be used in two different modes,
613either by reading/writing to the device file, or by issuing
614controlling commands to the device, by the device's *ioctl()*
615call. The problem with CD-ROM drives, is that they can be used for
616two entirely different purposes. One is to mount removable
617file systems, CD-ROM's, the other is to play audio CD's. Audio commands
618are implemented entirely through *ioctl()\'s*, presumably because the
619first implementation (SUN?) has been such. In principle there is
620nothing wrong with this, but a good control of the CD player demands
621that the device can **always** be opened in order to give the
622*ioctl* commands, regardless of the state the drive is in.
623
624On the other hand, when used as a removable-media disc drive (what the
625original purpose of CD-ROM s is) we would like to make sure that the
626disc drive is ready for operation upon opening the device. In the old
627scheme, some CD-ROM drivers don't do any integrity checking, resulting
628in a number of i/o errors reported by the VFS to the kernel when an
629attempt for mounting a CD-ROM on an empty drive occurs. This is not a
630particularly elegant way to find out that there is no CD-ROM inserted;
631it more-or-less looks like the old IBM-PC trying to read an empty floppy
632drive for a couple of seconds, after which the system complains it
633can't read from it. Nowadays we can **sense** the existence of a
634removable medium in a drive, and we believe we should exploit that
635fact. An integrity check on opening of the device, that verifies the
636availability of a CD-ROM and its correct type (data), would be
637desirable.
638
639These two ways of using a CD-ROM drive, principally for data and
640secondarily for playing audio discs, have different demands for the
641behavior of the *open()* call. Audio use simply wants to open the
642device in order to get a file handle which is needed for issuing
643*ioctl* commands, while data use wants to open for correct and
644reliable data transfer. The only way user programs can indicate what
645their *purpose* of opening the device is, is through the *flags*
646parameter (see open(2)). For CD-ROM devices, these flags aren't
647implemented (some drivers implement checking for write-related flags,
648but this is not strictly necessary if the device file has correct
649permission flags). Most option flags simply don't make sense to
650CD-ROM devices: *O_CREAT*, *O_NOCTTY*, *O_TRUNC*, *O_APPEND*, and
651*O_SYNC* have no meaning to a CD-ROM.
652
653We therefore propose to use the flag *O_NONBLOCK* to indicate
654that the device is opened just for issuing *ioctl*
655commands. Strictly, the meaning of *O_NONBLOCK* is that opening and
656subsequent calls to the device don't cause the calling process to
657wait. We could interpret this as don't wait until someone has
658inserted some valid data-CD-ROM. Thus, our proposal of the
659implementation for the *open()* call for CD-ROM s is:
660
661- If no other flags are set than *O_RDONLY*, the device is opened
662  for data transfer, and the return value will be 0 only upon successful
663  initialization of the transfer. The call may even induce some actions
664  on the CD-ROM, such as closing the tray.
665- If the option flag *O_NONBLOCK* is set, opening will always be
666  successful, unless the whole device doesn't exist. The drive will take
667  no actions whatsoever.
668
670-------------------------
671
672You might hesitate to accept this proposal as it comes from the
673Linux community, and not from some standardizing institute. What
674about SUN, SGI, HP and all those other Unix and hardware vendors?
675Well, these companies are in the lucky position that they generally
676control both the hardware and software of their supported products,
677and are large enough to set their own standard. They do not have to
678deal with a dozen or more different, competing hardware
679configurations\ [#f3]_.
680
681.. [#f3]
682
683   Incidentally, I think that SUN's approach to mounting CD-ROM s is very
684   good in origin: under Solaris a volume-daemon automatically mounts a
685   newly inserted CD-ROM under /cdrom/*<volume-name>*.
686
687   In my opinion they should have pushed this
688   further and have **every** CD-ROM on the local area network be
689   mounted at the similar location, i. e., no matter in which particular
690   machine you insert a CD-ROM, it will always appear at the same
691   position in the directory tree, on every system. When I wanted to
692   implement such a user-program for Linux, I came across the
693   differences in behavior of the various drivers, and the need for an
694   *ioctl* informing about media changes.
695
696We believe that using *O_NONBLOCK* to indicate that a device is being opened
697for *ioctl* commands only can be easily introduced in the Linux
698community. All the CD-player authors will have to be informed, we can
699even send in our own patches to the programs. The use of *O_NONBLOCK*
700has most likely no influence on the behavior of the CD-players on
701other operating systems than Linux. Finally, a user can always revert
702to old behavior by a call to
703*ioctl(file_descriptor, CDROM_CLEAR_OPTIONS, CDO_USE_FFLAGS)*.
704
705The preferred strategy of *open()*
706----------------------------------
707
708The routines in cdrom.c are designed in such a way that run-time
709configuration of the behavior of CD-ROM devices (of **any** type)
710can be carried out, by the *CDROM_SET/CLEAR_OPTIONS* *ioctls*. Thus, various
711modes of operation can be set:
712
713CDO_AUTO_CLOSE | CDO_USE_FFLAGS | CDO_LOCK
714   This is the default setting. (With *CDO_CHECK_TYPE* it will be better, in
715   the future.) If the device is not yet opened by any other process, and if
716   the device is being opened for data (*O_NONBLOCK* is not set) and the
717   tray is found to be open, an attempt to close the tray is made. Then,
718   it is verified that a disc is in the drive and, if *CDO_CHECK_TYPE* is
719   set, that it contains tracks of type data mode 1. Only if all tests
720   are passed is the return value zero. The door is locked to prevent file
721   system corruption. If the drive is opened for audio (*O_NONBLOCK* is
722   set), no actions are taken and a value of 0 will be returned.
723
724CDO_AUTO_CLOSE | CDO_AUTO_EJECT | CDO_LOCK
725   This mimics the behavior of the current sbpcd-driver. The option flags are
726   ignored, the tray is closed on the first open, if necessary. Similarly,
727   the tray is opened on the last release, i. e., if a CD-ROM is unmounted,
728   it is automatically ejected, such that the user can replace it.
729
730We hope that these option can convince everybody (both driver
731maintainers and user program developers) to adopt the new CD-ROM
732driver scheme and option flag interpretation.
733
734Description of routines in cdrom.c
735====================================
736
737Only a few routines in cdrom.c are exported to the drivers. In this
738new section we will discuss these, as well as the functions that take
739over the CD-ROM interface to the kernel. The header file belonging
740to cdrom.c is called cdrom.h. Formerly, some of the contents of this
741file were placed in the file ucdrom.h, but this file has now been
742merged back into cdrom.h.
743
744::
745
746        struct file_operations cdrom_fops
747
748The contents of this structure were described in cdrom_api_.
749A pointer to this structure is assigned to the *fops* field
750of the *struct gendisk*.
751
752::
753
754        int register_cdrom(struct cdrom_device_info *cdi)
755
756This function is used in about the same way one registers *cdrom_fops*
757with the kernel, the device operations and information structures,
758as described in cdrom_api_, should be registered with the
759Uniform CD-ROM Driver::
760
761        register_cdrom(&<device>_info);
762
763
764This function returns zero upon success, and non-zero upon
765failure. The structure *<device>_info* should have a pointer to the
766driver's *<device>_dops*, as in::
767
768        struct cdrom_device_info <device>_info = {
769                <device>_dops;
770                ...
771        }
772
773Note that a driver must have one static structure, *<device>_dops*, while
774it may have as many structures *<device>_info* as there are minor devices
775active. *Register_cdrom()* builds a linked list from these.
776
777
778::
779
780        void unregister_cdrom(struct cdrom_device_info *cdi)
781
782Unregistering device *cdi* with minor number *MINOR(cdi->dev)* removes
783the minor device from the list. If it was the last registered minor for
784the low-level driver, this disconnects the registered device-operation
785routines from the CD-ROM interface. This function returns zero upon
786success, and non-zero upon failure.
787
788::
789
790        int cdrom_open(struct inode * ip, struct file * fp)
791
792This function is not called directly by the low-level drivers, it is
793listed in the standard *cdrom_fops*. If the VFS opens a file, this
794function becomes active. A strategy is implemented in this routine,
795taking care of all capabilities and options that are set in the
796*cdrom_device_ops* connected to the device. Then, the program flow is
797transferred to the device_dependent *open()* call.
798
799::
800
801        void cdrom_release(struct inode *ip, struct file *fp)
802
803This function implements the reverse-logic of *cdrom_open()*, and then
804calls the device-dependent *release()* routine. When the use-count has
805reached 0, the allocated buffers are flushed by calls to *sync_dev(dev)*
806and *invalidate_buffers(dev)*.
807
808
809.. _cdrom_ioctl:
810
811::
812
813        int cdrom_ioctl(struct inode *ip, struct file *fp,
814                        unsigned int cmd, unsigned long arg)
815
816This function handles all the standard *ioctl* requests for CD-ROM
817devices in a uniform way. The different calls fall into three
818categories: *ioctl()'s* that can be directly implemented by device
819operations, ones that are routed through the call *audio_ioctl()*, and
820the remaining ones, that are presumable device-dependent. Generally, a
821negative return value indicates an error.
822
823Directly implemented *ioctl()'s*
824--------------------------------
825
826The following old CD-ROM *ioctl()*\ 's are implemented by directly
827calling device-operations in *cdrom_device_ops*, if implemented and
829
830CDROMMULTISESSION
831        Requests the last session on a CD-ROM.
832CDROMEJECT
833        Open tray.
834CDROMCLOSETRAY
835        Close tray.
836CDROMEJECT_SW
837        If *arg\not=0*, set behavior to auto-close (close
838        tray on first open) and auto-eject (eject on last release), otherwise
839        set behavior to non-moving on *open()* and *release()* calls.
840CDROM_GET_MCN
841        Get the Media Catalog Number from a CD.
842
843*Ioctl*s routed through *audio_ioctl()*
844---------------------------------------
845
846The following set of *ioctl()'s* are all implemented through a call to
847the *cdrom_fops* function *audio_ioctl()*. Memory checks and
848allocation are performed in *cdrom_ioctl()*, and also sanitization of
850
851CDROMSUBCHNL
852        Get sub-channel data in argument *arg* of type
853        struct cdrom_subchnl *.
854CDROMREADTOCHDR
856        struct cdrom_tochdr *.
857CDROMREADTOCENTRY
859        of type struct cdrom_tocentry *.
860CDROMPLAYMSF
861        Play audio fragment specified in Minute, Second, Frame format,
862        delimited by *arg* of type struct cdrom_msf *.
863CDROMPLAYTRKIND
864        Play audio fragment in track-index format delimited by *arg*
865        of type struct cdrom_ti *.
866CDROMVOLCTRL
867        Set volume specified by *arg* of type struct cdrom_volctrl *.
868CDROMVOLREAD
869        Read volume into by *arg* of type struct cdrom_volctrl *.
870CDROMSTART
871        Spin up disc.
872CDROMSTOP
873        Stop playback of audio fragment.
874CDROMPAUSE
875        Pause playback of audio fragment.
876CDROMRESUME
877        Resume playing.
878
879New *ioctl()'s* in cdrom.c
880----------------------------
881
882The following *ioctl()'s* have been introduced to allow user programs to
883control the behavior of individual CD-ROM devices. New *ioctl*
884commands can be identified by the underscores in their names.
885
886CDROM_SET_OPTIONS
887        Set options specified by *arg*. Returns the option flag register
888        after modification. Use *arg = \rm0* for reading the current flags.
889CDROM_CLEAR_OPTIONS
890        Clear options specified by *arg*. Returns the option flag register
891        after modification.
892CDROM_SELECT_SPEED
893        Select head-rate speed of disc specified as by *arg* in units
894        of standard cdrom speed (176\,kB/sec raw data or
895        150kB/sec file system data). The value 0 means auto-select,
896        i. e., play audio discs at real time and data discs at maximum speed.
897        The value *arg* is checked against the maximum head rate of the
898        drive found in the *cdrom_dops*.
899CDROM_SELECT_DISC
900        Select disc numbered *arg* from a juke-box.
901
902        First disc is numbered 0. The number *arg* is checked against the
903        maximum number of discs in the juke-box found in the *cdrom_dops*.
904CDROM_MEDIA_CHANGED
905        Returns 1 if a disc has been changed since the last call.
906        For juke-boxes, an extra argument *arg*
907        specifies the slot for which the information is given. The special
908        value *CDSL_CURRENT* requests that information about the currently
909        selected slot be returned.
910CDROM_DRIVE_STATUS
911        Returns the status of the drive by a call to
912        *drive_status()*. Return values are defined in cdrom_drive_status_.
913        Note that this call doesn't return information on the
914        current playing activity of the drive; this can be polled through
915        an *ioctl* call to *CDROMSUBCHNL*. For juke-boxes, an extra argument
916        *arg* specifies the slot for which (possibly limited) information is
917        given. The special value *CDSL_CURRENT* requests that information
918        about the currently selected slot be returned.
919CDROM_DISC_STATUS
920        Returns the type of the disc currently in the drive.
921        It should be viewed as a complement to *CDROM_DRIVE_STATUS*.
922        This *ioctl* can provide *some* information about the current
923        disc that is inserted in the drive. This functionality used to be
924        implemented in the low level drivers, but is now carried out
925        entirely in Uniform CD-ROM Driver.
926
927        The history of development of the CD's use as a carrier medium for
928        various digital information has lead to many different disc types.
929        This *ioctl* is useful only in the case that CDs have \emph {only
930        one} type of data on them. While this is often the case, it is
931        also very common for CDs to have some tracks with data, and some
932        tracks with audio. Because this is an existing interface, rather
933        than fixing this interface by changing the assumptions it was made
934        under, thereby breaking all user applications that use this
935        function, the Uniform CD-ROM Driver implements this *ioctl* as
936        follows: If the CD in question has audio tracks on it, and it has
937        absolutely no CD-I, XA, or data tracks on it, it will be reported
938        as *CDS_AUDIO*. If it has both audio and data tracks, it will
939        return *CDS_MIXED*. If there are no audio tracks on the disc, and
940        if the CD in question has any CD-I tracks on it, it will be
941        reported as *CDS_XA_2_2*. Failing that, if the CD in question
942        has any XA tracks on it, it will be reported as *CDS_XA_2_1*.
943        Finally, if the CD in question has any data tracks on it,
944        it will be reported as a data CD (*CDS_DATA_1*).
945
946        This *ioctl* can return::
947
948                CDS_NO_INFO     /* no information available */
949                CDS_NO_DISC     /* no disc is inserted, or tray is opened */
950                CDS_AUDIO       /* Audio disc (2352 audio bytes/frame) */
951                CDS_DATA_1      /* data disc, mode 1 (2048 user bytes/frame) */
952                CDS_XA_2_1      /* mixed data (XA), mode 2, form 1 (2048 user bytes) */
953                CDS_XA_2_2      /* mixed data (XA), mode 2, form 1 (2324 user bytes) */
954                CDS_MIXED       /* mixed audio/data disc */
955
956        For some information concerning frame layout of the various disc
957        types, see a recent version of cdrom.h.
958
959CDROM_CHANGER_NSLOTS
960        Returns the number of slots in a juke-box.
961CDROMRESET
962        Reset the drive.
963CDROM_GET_CAPABILITY
964        Returns the *capability* flags for the drive. Refer to section
966CDROM_LOCKDOOR
967         Locks the door of the drive. arg == 0 unlocks the door,
968         any other value locks it.
969CDROM_DEBUG
970         Turns on debugging info. Only root is allowed to do this.
971         Same semantics as CDROM_LOCKDOOR.
972
973
974Device dependent *ioctl()'s*
975----------------------------
976
977Finally, all other *ioctl()'s* are passed to the function *dev_ioctl()*,
978if implemented. No memory allocation or verification is carried out.
979
981=========================
982
983- Make a backup of your current driver.
984- Get hold of the files cdrom.c and cdrom.h, they should be in
985  the directory tree that came with this documentation.
986- Make sure you include cdrom.h.
987- Change the 3rd argument of *register_blkdev* from &<your-drive>_fops
988  to &cdrom_fops.
989- Just after that line, add the following to register with the Uniform
990  CD-ROM Driver::
991
992        register_cdrom(&<your-drive>_info);*
993
994  Similarly, add a call to *unregister_cdrom()* at the appropriate place.
995- Copy an example of the device-operations *struct* to your
996  source, e. g., from cm206.c *cm206_dops*, and change all
997  entries to names corresponding to your driver, or names you just
998  happen to like. If your driver doesn't support a certain function,
999  make the entry *NULL*. At the entry *capability* you should list all

1001  has a capability that is not listed, please send me a message.
1002- Copy the *cdrom_device_info* declaration from the same example
1003  driver, and modify the entries according to your needs. If your
1004  driver dynamically determines the capabilities of the hardware, this
1005  structure should also be declared dynamically.
1006- Implement all functions in your <device>_dops structure,
1007  according to prototypes listed in  cdrom.h, and specifications given
1008  in cdrom_api_. Most likely you have already implemented
1009  the code in a large part, and you will almost certainly need to adapt the
1010  prototype and return values.
1011- Rename your <device>_ioctl() function to *audio_ioctl* and
1012  change the prototype a little. Remove entries listed in the first
1013  part in cdrom_ioctl_, if your code was OK, these are
1014  just calls to the routines you adapted in the previous step.
1015- You may remove all remaining memory checking code in the
1016  *audio_ioctl()* function that deals with audio commands (these are
1017  listed in the second part of cdrom_ioctl_. There is no
1018  need for memory allocation either, so most *case*s in the *switch*
1019  statement look similar to::
1020
1022                get_toc_entry\bigl((struct cdrom_tocentry *) arg);
1023
1024- All remaining *ioctl* cases must be moved to a separate
1025  function, *<device>_ioctl*, the device-dependent *ioctl()'s*. Note that
1026  memory checking and allocation must be kept in this code!
1027- Change the prototypes of *<device>_open()* and
1028  *<device>_release()*, and remove any strategic code (i. e., tray
1029  movement, door locking, etc.).
1030- Try to recompile the drivers. We advise you to use modules, both
1031  for cdrom.o and your driver, as debugging is much easier this
1032  way.
1033
1034Thanks
1035======
1036
1037Thanks to all the people involved. First, Erik Andersen, who has
1038taken over the torch in maintaining cdrom.c and integrating much
1039CD-ROM-related code in the 2.1-kernel. Thanks to Scott Snyder and
1040Gerd Knorr, who were the first to implement this interface for SCSI
1041and IDE-CD drivers and added many ideas for extension of the data
1042structures relative to kernel~2.0. Further thanks to Heiko Ei\xC3\x9Ffeldt,
1043Thomas Quinot, Jon Tombs, Ken Pizzini, Eberhard M\xC3\xB6nkeberg and Andrew Kroll,
1044the Linux CD-ROM device driver developers who were kind
1045enough to give suggestions and criticisms during the writing. Finally
1046of course, I want to thank Linus Torvalds for making this possible in
1047the first place.
1048


text:
## anonymous 4 years ago Is this function linear? $\sum_{k=1}^{n}\left(a - a_1\right)\cdot\left(a-a_2\right)\cdots\left(a-a_n\right)$ Where $$a_1,a_2, \ldots, a_n \in \left[0,a\right]$$.

1. anonymous

@mukushla

2. anonymous

$$a$$ is any positive integer.

3. anonymous

I need to determine maximum of the function

4. anonymous

Linear with respect to a

5. anonymous

?

6. anonymous

but I am confused a bit

7. anonymous

This function does not seem to be linear.

8. anonymous

If it's linear it will be maximum at end points right?

9. anonymous

does not. hmm but in the book it says it is and i am confused.

10. anonymous

with respect to a not linear wrt to any of ai linear

11. anonymous

if some function were xy + y + x, would it still be linear? y and x are variables. or (a-y)(a-x).

12. anonymous

i am sorry if i am asking stupid questions, i am just confused

13. anonymous

for function xy + y + x if u fix y(for example) function will be linear wrt x

format: plain


text:
5

# Test the claim that the proportion of people who own cats is significantly different than 10% at the 0.05 significance level. The null and alternative hypothesis wo...

## Question

###### Test the claim that the proportion of people who own cats is significantly different than 10% at the 0.05 significance level. The null and alternative hypothesis would be:Ho:p > 0.1 Ha:p < 0.1 Ho: p > 0.1 Ha: p 0.1 Ho: p = 0.1 Ha:p = 0.1 Ho:p < 0.1 Ha:p > 0.1 Ho: u = 0.1 Ha: / / 0.1 Ho: p < 0.1 Ha: p > 0.1The test is:left-tailedright-tailed two-tailedBased on a sample of 200 people, 1% owned catsThe test statistic is:(Round to 2 decimals)The p-value is:(Round to 2 decimals)B

Test the claim that the proportion of people who own cats is significantly different than 10% at the 0.05 significance level. The null and alternative hypothesis would be: Ho:p > 0.1 Ha:p < 0.1 Ho: p > 0.1 Ha: p 0.1 Ho: p = 0.1 Ha:p = 0.1 Ho:p < 0.1 Ha:p > 0.1 Ho: u = 0.1 Ha: / / 0.1 Ho: p < 0.1 Ha: p > 0.1 The test is: left-tailed right-tailed two-tailed Based on a sample of 200 people, 1% owned cats The test statistic is: (Round to 2 decimals) The p-value is: (Round to 2 decimals) Based on this we: Reject the null hypothesis Do not reject the null hypothesis

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##### $(x, y) \in R$ if $|x-y|=2$
$(x, y) \in R$ if $|x-y|=2$...
##### Describe the steps involved in the mole method.
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##### A manufacturer has modeled its yearly production function $P$ (the monetary value of its entire production in millions of dollars) as a Cobb-Douglas function$$P(L, K)=1.47 L^{0.65} K^{0.35}$$where $L$ is the number of labor hours (in thousands) and $K$ is the invested capital (in millions of dollars). Find $P(120,20)$ and interpret it.
A manufacturer has modeled its yearly production function $P$ (the monetary value of its entire production in millions of dollars) as a Cobb-Douglas function $$P(L, K)=1.47 L^{0.65} K^{0.35}$$ where $L$ is the number of labor hours (in thousands) and $K$ is the invested capital (in millions of dol...
##### I) Find the centre_ radius, and interval of convergence for the given power series_(+96)"" '(13)" unn where Ihe given power series (a) absolutely convergent and(II) Determine all values o (b) condilionally convergent:
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##### Solve the following linear programming problem by maximizing P 2Sx 38y subject t0 the following contraints (graph the contraints and cvaluate the key points of the solution):4006)I0 (symbol is 'greater than or cqual to" in case YOu can"t see the equal (0 pant) 2 [S0 (also "greater than or cqual to')(Xy) point giving max PMax P
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##### Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. $$y=x, y=\sqrt{x} ; \quad \text { about } y=1$$
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text:
# Inverse of Generator of Cyclic Group is Generator

## Theorem

Let $\gen g = G$ be a cyclic group.

Then:

$G = \gen {g^{-1} }$

where $g^{-1}$ denotes the inverse of $g$.

Thus, in general, a generator of a cyclic group is not unique.

## Proof 1

Let $\gen g = G$.

Then from Set of Words Generates Group:

$\map W {\set {g, g^{-1} } } = G$

But:

$\gen {g^{-1} } = \map W {\set {g, g^{-1} } }$

and the result follows.

$\blacksquare$

## Proof 2

Let $C_n = \gen g$ be the cyclic group of order $n$.

By definition, $g^n = e$.

We have that $n - 1$ is coprime to $n$.

So it follows from that Power of Generator of Cyclic Group is Generator iff Power is Coprime with Order that:

$C_n = \gen {g^{n - 1} }$
$g^{n - 1} = g^{-1}$

$\blacksquare$


text:
# Puffin file format

This is a specification for Puffin, a file format designed to store information such as indexes and statistics about data managed in an Iceberg table that cannot be stored directly within the Iceberg manifest. A Puffin file contains arbitrary pieces of information (here called “blobs”), along with metadata necessary to interpret them. The blobs supported by Iceberg are documented at Blob types.

## Format specification

A file conforming to the Puffin file format specification should have the structure as described below.

### Versions

Currently, there is a single version of the Puffin file format, described below.

### File structure

The Puffin file has the following structure

Magic Blob₁ Blob₂ ... Blobₙ Footer

where

• Magic is four bytes 0x50, 0x46, 0x41, 0x31 (short for: Puffin Fratercula arctica, version 1),
• Blobᵢ is i-th blob contained in the file, to be interpreted by application according to the footer,
• Footer is defined below.

Footer has the following structure

Magic FooterPayload FooterPayloadSize Flags Magic

where

• Magic: four bytes, same as at the beginning of the file
• FooterPayload: optionally compressed, UTF-8 encoded JSON payload describing the blobs in the file, with the structure described below
• FooterPayloadSize: a length in bytes of the FooterPayload (after compression, if compressed), stored as 4 byte integer
• Flags: 4 bytes for boolean flags
• byte 0 (first)
• bit 0 (lowest bit): whether FooterPayload is compressed
• all other bits are reserved for future use and should be set to 0 on write
• all other bytes are reserved for future use and should be set to 0 on write

A 4 byte integer is always signed, in a two’s complement representation, stored little-endian.

Footer payload bytes is either uncompressed or LZ4-compressed (as a single LZ4 compression frame with content size present), UTF-8 encoded JSON payload representing a single FileMetadata object.

FileMetadata has the following fields

Field NameField TypeRequiredDescription
propertiesJSON object with string property valuesnostorage for arbitrary meta-information, like writer identification/version. See Common properties for properties that are recommended to be set by a writer.

BlobMetadata has the following fields

Field NameField TypeRequiredDescription
typeJSON stringyesSee Blob types
fieldsJSON list of intsyesList of field IDs the blob was computed for; the order of items is used to compute sketches stored in the blob.
snapshot-idJSON longyesID of the Iceberg table’s snapshot the blob was computed from.
sequence-numberJSON longyesSequence number of the Iceberg table’s snapshot the blob was computed from.
offsetJSON longyesThe offset in the file where the blob contents start
lengthJSON longyesThe length of the blob stored in the file (after compression, if compressed)
compression-codecJSON stringnoSee Compression codecs. If omitted, the data is assumed to be uncompressed.
propertiesJSON object with string property valuesnostorage for arbitrary meta-information about the blob

### Blob types

The blobs can be of a type listed below

#### apache-datasketches-theta-v1 blob type

A serialized form of a “compact” Theta sketch produced by the Apache DataSketches library. The sketch is obtained by constructing Alpha family sketch with default seed, and feeding it with individual distinct values converted to bytes using Iceberg’s single-value serialization.

The blob metadata for this blob may include following properties:

• ndv: estimate of number of distinct values, derived from the sketch.

### Compression codecs

The data can also be uncompressed. If it is compressed the codec should be one of codecs listed below. For maximal interoperability, other codecs are not supported.

Codec nameDescription
lz4Single LZ4 compression frame, with content size present
zstdSingle Zstandard compression frame, with content size present
__

### Common properties

When writing a Puffin file it is recommended to set the following fields in the FileMetadata’s properties field.

• created-by - human-readable identification of the application writing the file, along with its version. Example “Trino version 381”.


text:
# If $f(x)=4x^2+ax+a-3$ is negative for at least one negative $x$ find all possible values of $a$

If $f(x)=4x^2+ax+a-3$ is negative for at least one negative $x$ find all possible values of $a$

I don't know how to find all possible values.

I tried making the lower of the two roots as negative. That does not seem to work out. As $4>0$ the curve will be concave up. There can be numerous cases possible.

Having distinct real roots requires a positive discriminant, i.e.

$$a^2-16(a-3)=(a-4)(a-12)>0,$$ or$$\color{green}{a<4\lor a>12}.$$

Then, having at least one negative root requires that either the sum or the product of the roots be negative, i.e.

$$-a<0\lor a-3<0,$$which is always true and doesn't restrict the solution set !

• You give a very nice argument why one of the roots is negative, +1. – wythagoras Jun 30 '15 at 10:24
• Yep, it avoids computing the roots. Cheers. – Yves Daoust Jun 30 '15 at 10:25
• @YvesDaoust It should be $a<0\lor a-3<0$ – Bhaskar Vashishth Jun 30 '15 at 10:33
• Ooops, $-a<0\lor a-3<0$. Fixed. – Yves Daoust Jun 30 '15 at 10:39

You don't have to solve for the roots (which leads to irrational inequations!) to answer.

Indeed, if $f(x)<0$ for at least one $x$, it has two real roots.The discriminant is $\;\Delta=a^2-16a+48=(a-4)(a-12)$, hence it has two real roots if $a<4$ or if $a>12$.

Suppose $\Delta>0$; we'll determine the sign of the roots: $\;f(0)=a-3$, hence:

• if $a<3$, $0$ separates the roots, i.e. there is one negative and one positive root.
• if $a>3$, $0$ is outside of the interval of the roots, i.e. they're both positive or both negative. In that case, their sign is also the sign of their half-sum, which is $-\dfrac a8$. Thus both are negative.
• if $a=3$, the roots are $0$ and $-\dfrac34$.

In all cases, $f(x)$ has at least one negative root, so that $a$ can take any real value in $\,]-\infty,4[\:\cup\:]12,+\infty[$.

• I think you miscalculated the discriminant... – MathGod Jun 30 '15 at 9:30
• @Ishu: thank you for pointing the error. I've updated my answer. Hope there's no more problems :o) – Bernard Jun 30 '15 at 9:54

Your polynomial is $p(x)=4x^2+ax+a-3$ its discriminant is $\Delta=a^2-16a+48$ and its roots are $\frac{-a\pm \sqrt{\Delta}}{8}$. Now there is a $x$ for which value is negative, and it is a parabola opening upwards, tells us that both roots are real and thus $\Delta>0 \implies a^2-16a+48=(a-4)(a-12)>0$ which happens iff $a\in (-\infty,4)\cup(12,\infty)$.

• Where do you express the condition that there is a negative root ? – Yves Daoust Jun 30 '15 at 10:18
• @YvesDaoust I do not think it is a necessary condition for the question that value is negative for at least one negative $x$, just being negative for any value of $x$, gives the same solution. – Bhaskar Vashishth Jun 30 '15 at 10:23
• Indeed when there are real roots one is always negative, but you must prove it ! – Yves Daoust Jun 30 '15 at 10:28

The function will be positive when $x$ is very large positive or very large negative. This means that there are two solutions, to your equation, of which at least one is negative.

The smallest root is:

$$\frac{-a -\sqrt{a^2-16(a-3)}}{8}$$

So we want to know when $a^2-16(a-3) > 0$, or $a^2-16a+48 > 0$, or $(a-8)^2-16 > 0$, or $(a-8)^2 > 16$, or $$a-8 > 4 \vee a-8 < -4$$

$$a > 12 \vee a < 4$$

When $a$ is positive, both $-a$ and $-a -\sqrt{a^2-16(a-3)}$ are negative, so then the smallest root is negative.

When $a$ is negative, $$\sqrt{a^2-16a+48}=\sqrt{(a-6)^2+12}>\sqrt{(a-6)^2}=|a-6|=6-a$$

With the the second because of the increasing nature of $\sqrt{x}$, the fourth because $a$ and thus $a-6$ negative is. Therefore we have $$\frac{-a -\sqrt{a^2-16(a-3)}}{8}<\frac{-a - (6-a)}{8}=\frac{-6}{8}=-0.75$$

Thus the solution is

$$a > 12 \vee a < 4$$

• The inequalities needn't be strict. – Yves Daoust Jun 30 '15 at 10:19
• @YvesDaoust Yes they do other wise there is exactly one root and is the polynomial only non-positive, it is never negative. – wythagoras Jun 30 '15 at 10:20
• You are right, sorry, I didn't see that the function needed to be strictly negative, just looked at the roots. – Yves Daoust Jun 30 '15 at 10:22

Examining the extrema of $f(x)=4x^2+ax+a-3$ which is a family of upwards open parabolas: $$f'(x) = 8 x + a = 0 \Rightarrow x = -\frac{1}{8}a \\ f''(x) = 8 > 0$$ So we have a local minimum at $x = -(1/8)a$. There the function has the value: $$f(-(1/8)a) = \frac{1}{16}a^2 - \frac{1}{8}a^2 + a - 3 = -\frac{1}{16}a^2 + a - 3$$ The minima regarding different $a$ values lie on the curve $$m(x) = -4x^2-8x -3$$

From $$0 = -\frac{1}{16} a^2 + a - 3 \Rightarrow \\ a^2 = 16 a - 48 \Rightarrow \\ (a - 8)^2 = 64 - 48 = 16 \Rightarrow \\ a = 8 \pm 4$$

we get that the minimum is non-negative for $a \in [4, 12]$ and there we will find no negative value for $f$ at all.

So we have a chance to find negative $f$ values for negative argument only for $a \in I = \mathbb{R} \setminus [4,12]$, where the minimum occurs for a negative function value.

For positive $a\in I$ the minimum is at $x_m = -(1/8) a < 0$, so those $a$ are valid.
For negative $a \in I$ for $x = -1/2 < 0$ we have $f(-1/2) = 1 - a/2 + a - 3 = a/2 - 2 < 0$, so those $a$ are valid as well.

This gives $I = \mathbb{R} \setminus [4,12]$ as set of all feasible $a$.

The graph is a parabola that opens upward. So there are two cases:

1. $f(0) < 0$

2. f has a minimum at $x_0 < 0$ such that $f(x_0) < 0$

In the first case, $f(0) = a -3 < 0 \text{ gives } a<3$

In the second case, $f'(x) = 8x + a$ which has a minimum at $x_0 = -\frac{a}{8}$ where we require $a > 0$. Then $f(x_0) = \frac{a^2}{16} - \frac{a^2}{8} + a - 3 = \frac{-a^2 + 16a - 48}{16} = g(a)$ which is a parabola in a that opens downward, and has roots $a = \frac{-16 \pm \sqrt{256-192}}{-2} = 8 \pm 4$ so $g(a)$ is negative for all $0 < a \le 4$ and $a > 12$.

Putting everything together:

$a \in (-\infty,4) \lor a \in (12,+\infty)$

You can treat this problem with roots indeed, for as soon as you have two distincts roots $x_1$ and $x_2$, all values between the two roots give a negative $f(x)$.

This you can demonstrate easily.

So to have distinct roots, you need to have a strictly positive discriminant, that is

$a^2-4a+12>0$

You also want to have a negative root, that is $-a +\sqrt{a^2-4a+12}<0$

The given expression is a quadratic in $x$ with the leading coefficient $\geq 0$. Since, it is given to be negative for at least one $x$, the conditions are:

1. Discriminant $\geq 0$, that is $a^2-4a+12>0$
2. At least one root is negative. Solve for the roots and make the smaller one less than $0$.


text:
## Calculus 8th Edition

The function is continuous $a = -1.$
By theorem 8 we can write $\lim\limits_{x \to -1} (x + 2x^{3})^{4}$ as $(\lim\limits_{x \to -1} (x + 2x^{3}))^{4}$ This is a polynomial so its limit can be evaluated by substitution. Because it is evaluated by substitution, the value of the function at -1 is equal to the value of the limit at -1 which means that it is continuous.


text:
# Continuous and almost everywhere continuously differentiable with bounded gradient implies Lipschitz?

Let $f:\mathbb{R}^n\to \mathbb{R}$ be continuous and bounded. Let the gradient of $f$ be continuous almost everywhere and let it be uniformly bounded wherever it is defined, i.e. $\|\nabla f(x)\|\le M$ for any $x\in\mathbb{R}^n$, where $M$ is a positive constant. Can we say that $f$ is globally Lipschitz continuous?

If so, where can we find a formal proof?

If not, can we provide a counterexample? (I.e., a function $f:\mathbb{R}^n\to \mathbb{R}$ that is continuous and bounded, has uniformly bounded gradient but is not Lipschitz.)

Remark. This question has been answered (positively) for $n=1$ in If $f$ is continuous and piecewise $C^1$ and $f'$ is bounded a.e., is $f$ Lipschitz?.

Remark. This question has been answered (positively) for $f$ continuously differentiable in A continuously differentiable map is locally Lipschitz. However, from the answer given there, I seem to understand that continuity of the derivatives can be relaxed to boundedness of the derivatives. If this is actually the case, then my question is also answered (positively) in that post. But judging from the following remark, I may be missing something.

Remark This paper http://www.sciencedirect.com/science/article/pii/S0024379512002741 seems to prove that there DO exist counterexamples (i.e. functions that are continuous, have uniformly bounded derivatives but are NOT Lipschitz). However, I must confess I do not understand their proof.

• I am posting this as a separate question than math.stackexchange.com/q/1597154, since I may need to get rid of the compactness of the set where $f$ is defined, and I may want to generalize the result to any $n$. – Antonio Jan 2 '16 at 16:26
• What does "piecewise continuous" mean if $n>1?$ – zhw. Jan 2 '16 at 22:40
• @zhw. Thanks for bringing this up. I am thinking about a function defined on $\mathbb{R}^2$, i.e. on the plane. The plane is partitioned in 3 different regions. Within each of these regions the gradient is continuous, but on the boundaries between different regions, the gradient is not defined. I guess we should say that the gradient is "continuous almost everywhere". Does that help to clarify? – Antonio Jan 3 '16 at 2:07

As stated, the answer to your question is no. The Cantor function is a common counterexample when the derivative is required to exist only almost everywhere. It is continuous, has zero derivative a.e., in particular in an open set of full measure, but it is not Lipschitz continuous, nor absolutely continuous. You can make an analogous example in the cube $\mathbb{R}^n$ considering the Cantor function of the first variable.

If instead you take a function $f\in L^1_{loc}(\mathbb{R}^n)$ and require its gradient to exist in the distributional sense and to be represented by a function in $L^\infty$, then you are by definition stating that $f$ belongs to the Sobolev space $W^{1,\infty}(\mathbb{R}^n)$, which can be proven to coincide with the space of Lipschitz functions, see for example this question. This still holds if you replace $\mathbb{R}^n$ with a convex set $\Omega$ (there are some weaker conditions on $\Omega$ under which it works, but it doesn't work with arbitrary domains). Note that you don't need to assume a priori that $f$ is continuous, nor that the gradient is continuous almost everywhere.

Yes. The given assumptions imply that $f$ is locally absolutely continuous, that is, it is the integral of each of its derivatives (which are defined only at almost all points): $$\tag{*} f(x_1, \ldots x_j+h \ldots x_n) - f(x_1 \ldots x_n) = \int_0^h \frac{\partial f}{\partial x_j}(x_1 \ldots x_j + y \ldots x_n)\ dy, \qquad j=1\ldots n.$$ You can therefore use the mean value theorem as you would normally do for smooth functions.

Proving (*) is a bit annoying for me (I tend to gladly assume that those things are true without bothering to prove them), and is connected to a fine property of functions known as ACL property (see the book on Sobolev spaces by Giovanni Leoni - related).

If you know something about weak or distributional derivatives, then you might be happy to know that they satisfy property (*) (modulo almost-everywhere-related technicalities). Your assumptions give that the almost-everywhere-defined classical derivatives of $f$ are indeed weak derivatives, and so (*) is satisfied. The space of functions that satisfy your assumptions is (contained into) a space usually known as $W^{1, \infty}_{\rm loc}(\mathbb{R}^n)$.

EDIT You are right that my post is way too imprecise. For a rigorous proof using approximation by mollifying, see e.g. Evans's book on PDEs, second edition, Theorem 4 chapter 5 (Characterization of $W^{1,\infty}$ functions), pag. 294.

• In the context I am considering, it is all right to assume that (*) holds. However, the partial derivative $\frac{\partial f}{\partial x_j}$ may not be defined on any of the points on the line from $(x_1,...x_j,...,x_n)$ to $(x_1,...,x_j+h,...,x_n)$. In fact, that line may be one of the boundaries between different continuity regions of the gradient. Do you think that would be a problem? Or can I travel between those two points by integrating the gradient along a different line? – Antonio Jan 2 '16 at 16:54
• This is the almost-everywhere technicality I was talking about. Even if you are so unlucky that you chose a line on which $f$ is singular, your assumptions give that those lines form a null set (that is, a set that is ignored by integration theory). You can therefore act as if your function were smooth. – Giuseppe Negro Jan 2 '16 at 16:58
• NOTE: Anything I have said is hand-waving. That's because of my ignorance. If you want rigorous proofs, then you should look for them on the book by Leoni. – Giuseppe Negro Jan 2 '16 at 16:59
• NOTE 2: Changing the path won't work. You need to integrate along straight paths to prove Lipschitz continuity. – Giuseppe Negro Jan 2 '16 at 17:01
• You are right, there is something I am sweeping under the rug. See my edit for a quick patch. Sorry about the confusion. – Giuseppe Negro Jan 2 '16 at 22:32


text:
This notebook contains an excerpt from the Python Data Science Handbook by Jake VanderPlas; the content is available on GitHub.

The text is released under the CC-BY-NC-ND license, and code is released under the MIT license. If you find this content useful, please consider supporting the work by buying the book!

# Density and Contour Plots¶

Sometimes it is useful to display three-dimensional data in two dimensions using contours or color-coded regions. There are three Matplotlib functions that can be helpful for this task: plt.contour for contour plots, plt.contourf for filled contour plots, and plt.imshow for showing images. This section looks at several examples of using these. We'll start by setting up the notebook for plotting and importing the functions we will use:

In [1]:
%matplotlib inline
import matplotlib.pyplot as plt
plt.style.use('seaborn-white')
import numpy as np

## Visualizing a Three-Dimensional Function¶

We'll start by demonstrating a contour plot using a function $z = f(x, y)$, using the following particular choice for $f$ (we've seen this before in Computation on Arrays: Broadcasting, when we used it as a motivating example for array broadcasting):

In [2]:
def f(x, y):
return np.sin(x) ** 10 + np.cos(10 + y * x) * np.cos(x)

A contour plot can be created with the plt.contour function. It takes three arguments: a grid of x values, a grid of y values, and a grid of z values. The x and y values represent positions on the plot, and the z values will be represented by the contour levels. Perhaps the most straightforward way to prepare such data is to use the np.meshgrid function, which builds two-dimensional grids from one-dimensional arrays:

In [3]:
x = np.linspace(0, 5, 50)
y = np.linspace(0, 5, 40)

X, Y = np.meshgrid(x, y)
Z = f(X, Y)

Now let's look at this with a standard line-only contour plot:

In [4]:
plt.contour(X, Y, Z, colors='black');

Notice that by default when a single color is used, negative values are represented by dashed lines, and positive values by solid lines. Alternatively, the lines can be color-coded by specifying a colormap with the cmap argument. Here, we'll also specify that we want more lines to be drawn—20 equally spaced intervals within the data range:

In [5]:
plt.contour(X, Y, Z, 20, cmap='RdGy');

Here we chose the RdGy (short for Red-Gray) colormap, which is a good choice for centered data. Matplotlib has a wide range of colormaps available, which you can easily browse in IPython by doing a tab completion on the plt.cm module:

plt.cm.<TAB>

Our plot is looking nicer, but the spaces between the lines may be a bit distracting. We can change this by switching to a filled contour plot using the plt.contourf() function (notice the f at the end), which uses largely the same syntax as plt.contour().

Additionally, we'll add a plt.colorbar() command, which automatically creates an additional axis with labeled color information for the plot:

In [6]:
plt.contourf(X, Y, Z, 20, cmap='RdGy')
plt.colorbar();

The colorbar makes it clear that the black regions are "peaks," while the red regions are "valleys."

One potential issue with this plot is that it is a bit "splotchy." That is, the color steps are discrete rather than continuous, which is not always what is desired. This could be remedied by setting the number of contours to a very high number, but this results in a rather inefficient plot: Matplotlib must render a new polygon for each step in the level. A better way to handle this is to use the plt.imshow() function, which interprets a two-dimensional grid of data as an image.

The following code shows this:

In [7]:
plt.imshow(Z, extent=[0, 5, 0, 5], origin='lower',
cmap='RdGy')
plt.colorbar()
plt.axis(aspect='image');

There are a few potential gotchas with imshow(), however:

• plt.imshow() doesn't accept an x and y grid, so you must manually specify the extent [xmin, xmax, ymin, ymax] of the image on the plot.
• plt.imshow() by default follows the standard image array definition where the origin is in the upper left, not in the lower left as in most contour plots. This must be changed when showing gridded data.
• plt.imshow() will automatically adjust the axis aspect ratio to match the input data; this can be changed by setting, for example, plt.axis(aspect='image') to make x and y units match.

Finally, it can sometimes be useful to combine contour plots and image plots. For example, here we'll use a partially transparent background image (with transparency set via the alpha parameter) and overplot contours with labels on the contours themselves (using the plt.clabel() function):

In [8]:
contours = plt.contour(X, Y, Z, 3, colors='black')
plt.clabel(contours, inline=True, fontsize=8)

plt.imshow(Z, extent=[0, 5, 0, 5], origin='lower',
cmap='RdGy', alpha=0.5)
plt.colorbar();

The combination of these three functions—plt.contour, plt.contourf, and plt.imshow—gives nearly limitless possibilities for displaying this sort of three-dimensional data within a two-dimensional plot. For more information on the options available in these functions, refer to their docstrings. If you are interested in three-dimensional visualizations of this type of data, see Three-dimensional Plotting in Matplotlib.


text:
# Dahlquist:Edge Software Protocol

This protocol is for running the Edge software for DNA microarray data anlysis.

## Notes on Getting Edge to Run

• The Edge software would not allow data files to be loaded on Windows, so Kevin wrote to the discussion group for suggestions. Click here to see thread. He was told to do the following fix in a Linux installation:
1. Open the file "knnimpute.r" in a text editor
2. Go to lines 129 and 168, it should read 'PACKAGE="impute"'
3. Change this to 'PACKAGE="knnimpute"'
4. Save the file, then start EDGE and try again.
• This fixed the problem, so we are now running Edge in a Linux environment.

## Directions for Launching Edge at the Keck Lab

• Login with your Keck lab username to mason (the names of the machines are on the lower-left corner of the login screens).
• Right-click on the green tabula rasa.
• Choose Terminal.
• Type:
cd Desktop/edge_1.1.290
R

• At this point, the R prompt shows up. Type:
source("edge.r")
edge()

• The Edge GUI should now appear.

## Analyzing Data with Edge

• Create two tab-delimited text files for "genes" and "covariates".
• Load both into an Edge session.
• Select "Impute Missing Data" from the menu. Calculat Percent Missing Data. The results are:
• Percent of genes missing data:
• Percent of arrays missing data:
• Overall percent of missing data:
• For KNN Parameters, set:
• Percent of missing values to tolerate in a gene: 100 (so all genes included)
• Number of nearest neighbors to use (maximum of 15): 15
• clicked GO to impute missing data.
• Selected "Identify Differentially Expressed Genes"
• Class variable is:
• Covariate giving time points is:
• Covariate corresponding to individuals is:
• Choose spline type, accepted default of Natural Cubic Spline, dimension 4
• Number of null iterations, set to 1000
• Choose a seed for reproducible results, set to 47
• 1000 permutations looks like it will take about 10 minutes.
• Save results as:
• To save the plots, do the following command in the R console window.
savePlot(filename = "PvalHistogram_wt-vs-dCIN5", type = c("png"), device = dev.cur())

• This will save the active plot window under a file name you choose.


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Saturday, September 13, 2014

Bulge Value (code 42) in LWPolyline

Polylines are a pretty handy way to represent and manipulate outlines of objects in CAD software. However, the very robust polyline entity can do so many things that it is somewhat complicated. The light weight polyline (LWPolyline) entity is a trimmed down version of polylines that does the most frequently needed things fairly compactly and straight-fowardly.

LWPolylines are strictly 2D objects. Of course, they can be created in a UCS other than the WCS but they will be planar, and the points stored in them will be 2D points relative to their object coordinate system (OCS—I haven't really worked with OCSs so I'm not much help other than to tell you to learn the Arbitrary Axis Algorithm described in the free DXF Reference from Autodesk, which I've never actually learned properly).

LWPolylines still have the ability to represent arcs. Actually, I don't know how these are represented in Polylines, but in LWPolylines, arcs are fairly simple. All you need is the bulge value (code 42) which is tan(θ/4), where θ is the central angle.
 Fig.1. θ is the central angle.
Which raises an interesting question: why for do we need the tangent of a quarter of the central angle?
 Fig. 2. If the arc above was drawn from A to B, the bulge value would be negative (CW).
One reason is that it is compact. Other than storing points A and B, which you already need, the only additional information to unambiguously denote the arc from A to B is the bulge value. If the radius was stored instead, we would have up to 4 different arcs that could be being referred to. The SVG representation requires you to enter two parameters after the radius that clarify which of the four arcs you want. SVG is trying to be human reader friendly, but in DXF, we're not as worried about that. I find it easier to deal with the bulge as far as computations are concerned.
 Fig. 3. Four possible arcs of a given radius drawn from A to B. Which one do you want?
I imagine the choice of storing the tangent of the angle is computation speed. The choice of a quarter of the angle is probably for two reasons:
1. Tangent has a period of 180°. Now, if you used the tangent of half the angle subtended, this might appear initially to be sufficient, except that you would not be able to use the sign of the value to decide between CW and CCW since you would need the sign to decide between less than or greater than 180° arcs. You would also have an asymptote get in the way of drawing semi-circles. By storing the tangent of a quarter of the arc subtended, we restrict the the domain to angles which yield a positive tangent and so the sign value can be used for something else. It also avoids data representation limitations until you get close to having a complete circle (at which point, the user should probably consider using a circle). I don't know what the solution technically would be for storing a full circle other than a sentinel value or just not letting the user do that—don't know what happens.
2. The diagram above shows some handy interrelationships that allow us to fairly easily calculate the rise (i), the radius, the peak (P), and the center (C). A bit of algebra produces $$r = \frac{u (b^2+1)}{4b}$$ and $$i = \frac{b u}{2}$$, where $$b$$ is the bulge value and $$u$$, $$r$$ and $$i$$ are as in Fig. 2. Calculating $$u$$ unfortunately requires a square root, but no trigonometric function calls. Finding C and P involves using $$r$$ and $$i$$ along with a vector perpendicular to line AB [e.g., $$(x,y) \to (-y,x)$$, but watch which one gets the negative depending on which way you're turning].


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# American Institute of Mathematical Sciences

April  2008, 4(2): 227-246. doi: 10.3934/jimo.2008.4.227

## Finite difference approximation for stochastic optimal stopping problems with delays

 1 Mathematics Division, U. S. Army Research Office, P. O. Box 12211, RTP, NC 27709, United States 2 Department of Mathematics and Center for Research in Scientific Computation, North Carolina State University, Raleigh, NC 27695-8205, United States 3 Department of Mathematics, Towson University, 7800 York Road, Room 316, Towson, MD 21252-0001, United States

Received  March 2007 Revised  September 2007 Published  April 2008

This paper considers the computational issue of the optimal stopping problem for the stochastic functional differential equation treated in [6] The finite difference method developed by Barles and Souganidis [3] is used to obtain a numerical approximation for the viscosity solution of the infinite dimensional Hamilton-Jacobi-Bellman variational inequality (HJBVI) associated with the optimal stopping problem. The convergence results are then established.
Citation: Mou-Hsiung Chang, Tao Pang, Moustapha Pemy. Finite difference approximation for stochastic optimal stopping problems with delays. Journal of Industrial & Management Optimization, 2008, 4 (2) : 227-246. doi: 10.3934/jimo.2008.4.227
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