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This Is Your Captain Speaking
The secret life behind the cockpit door. Rob Graves, an airline Captain, betrays the profession telling all the details of what it's like to be an airline pilot from dodging thunderstorms to life in hotels.
Does Falling Asleep at the Controls Make You a Bad Pilot?
Delta 767 and Greek F-16s in formation.
Not necessarily, though it might very well make you a dead pilot if you were in a single pilot aircraft such as an F-16 or Cessna. But as far as airliners go, if the pilots take a snooze at altitude with the autopilot flying, the airplane stays on course and airspeed. Falling asleep is a physiological incident which can be due to many different reasons, but one thing it is not, is a moral failing. Let me explain.
In a recent event over Greek airspace, a Delta Airlines 767 flying a charter for the US military entered Greek airspace and did not check in with air traffic control. The Greeks then launched two F-16s which intercepted the airliner about 40 minutes after it entered into Greek airspace. Shortly after the intercept, the aircraft reestablished communications with air traffic control and proceded onto its destination of Kuwait.
There have been some unsubstantiated reports in Greek media that the fighter pilots saw the airline pilots unresponsive in their seats. The reports also claim that it was calls from the flight attendants who noticed the intercept which alerted the pilots to the situation.
Delta, for its part, reported that the aircraft couldn't make contact with Greek controllers after their handoff from a previous sector. That's plausible, though losing communications in an extremely busy part of the world for 40 minutes does seem unlikely. Maybe 5 or 10, but 40 is more difficult to swallow.
They Want You to Lose Communication
A little known fact is that many jurisdictions around the world actually love it when an airliner from a wealthy nation flies into their airspace without making contact. They then get to launch an intercept or search and rescue (SAR) forces and then send the bill to whoever has the deepest pockets. That would be Delta and the US government in this case.
Notice that it only took 18 minutes from the time the airliner entered Greek airspace to the launch of the fighters. There are rumors around that Greece itself is in some financial straits. They're smelling a payday. Sure there are legitimate reasons to intercept a comm-out airliner in today's crazy terrorist besotted world, but money also makes a good motivator as well. Win-win I suppose.
At any rate, any pilots worth their salt flying in this part of the world must know that going comm-out while crossing a flight identification region (FIR) boundary while bound for the Middle East will not be good.
So what other reason might there be?
They Were Snoozing (Maybe)
The nature of this job is many days and time zones away from home, back side of the clock flying, lousy diet, and hotel beds which only get the straw changed every other year. I jest about the straw beds, but I often feel like I've slept on one as my back will let me know when trying to roll out of bed.
My point is that getting a reasonable amount of sleep on the road is a serious challenge, and that is if everything goes right. A noisy or inop air conditioner, maids knocking on the door early in the morning after a late night arrival or my personal favorite, hammer drills in a nearby room from construction crews can make a good night of sleep nearly impossible.
Even though most airlines have good fatigue policies which allow pilots to decline a flight with no sanction, there is no guarantee that halfway through a flight which you felt fine to start that you won't simply find it impossible to keep your eyes open. This can be in the middle of the day, perhaps right after lunch while sitting on the sunny side of the jet.
Do you do what you can to prevent this? Sure. Get up and stretch, get a cup of coffee, or take a restroom break. Even after all that you might still be droopy. So it is by far from implausible that this happened to both of the guys or gals up front.
Are They in Trouble?
No, they are not. As I mentioned above, falling asleep is a physiological incident and not a moral failing. There will be no scene from 12 O'clock High with Gregory Peck chewing out a guard who has fallen asleep at his post. What will happen is the crew will fill out various safety and fatigue reports (if that indeed is what happened) and life will go on. And if it was merely a case of lost comm, mostly the same thing will happen.
Their reports will go into a safety database where de-identified safety data will hopefully guide any policy changes which need to be made. Yea, that may be somewhat idealistic, but the point is the crew will live to fly another day.
And as I always brief to my copilots, I never want to wake up and find them sleeping! (Just kidding!)
Posted by The Captain at 9:08 PM 3 comments: Links to this post
Filed under: Part 117, regulation, rest, safety
Egyptair 804
Egyptair A320
Egyptair 804 disappeared over the Mediterranean Sea last week. The aircraft, an Airbus A320, was carrying 56 passengers and 10 crew when it vanished from radar screens at about 2:30 am local time (00:30Z). There were no survivors.
While recovery of the wreckage is underway, the only other notable factor concerning the accident was a flurry of automated maintenance messages received from the aircraft shortly before its disappearance.
The system, known as Aircraft Communications Addressing and Reporting System (ACARS), is used to send data messages about the operational and mechanical status of an airliner to the company for various uses. For instance, when you bring up the arrival status of your flight on your phone, that exact arrival time is derived from data sent over ACARS. The system also sends maintenance status reports to help track the mechanical status of the aircraft and also to alert ground maintenance personnel of impending problems.
This system sent the following messages:
00:26Z 3044 ANTI ICE R WINDOW
00:26Z 561200 R SLIDING WINDOW SENSOR
00:26Z 2600 SMOKE LAVATORY SMOKE
00:27Z 2600 AVIONICS SMOKE
00:28Z 561100 R FIXED WINDOW SENSOR
00:29Z 2200 AUTO FLT FCU 2 FAULT
00:29Z 2700 F/CTL SEC 3 FAULT
no further ACARS messages were received.
These messages indicate that smoke was detected by the lavatory and avionics smoke detectors and that the window heat computer detected an overheat condition. The avionics bay is an area below the cockpit where most of the aircraft's computers and radios are located. What these messages mean when taken together is anyone's guess. They could possibly indicate the presence of an onboard fire or might only indicate multiple erroneous inputs from a failure of the reporting system.
At this point no determination can be made about the nature of the malfunctions or their origins. This will no doubt have to wait until the data and voice recorders are recovered. This recovery effort is underway.
Filed under: accident investigation, avionics, terrorism
How to Dodge a Thunderstorm and Not Spill Your Coffee
They have big teeth too!
Well it's spring. It is a time when, according to Tennyson, a young man's fancy lightly turns to thoughts of love. Being an old grizzled coot firmly in the grip of matrimony for more than a score of years now, my fancy begrudgingly turns instead to the spring return of every pilot's nemesis, the thunderstorm.
I fondly recall a time years ago while sitting in Air Force ground school in Lubbock, Texas. The instructor drew a shape on the board and asked us students to identify the shape. It looked like the state of Texas which some brave soul ventured to point out. The instructor corrected him with a shouted "NO". "That," he exclaimed, "is an airborne emergency!"
His meaning was that flying in the state of Texas in the spring meant doing battle with, or rather attempting to avoid doing battle with the towering giants which would daily sweep across the panhandle. Because even today, with all of our technology and shiny fast moving flying machines, there is still virtually no airplane that can survive a thunderstorm penetration. And I'm including everything from fighters to helicopters to especially airliners in this category.
But wait, you cry! How about those hurricane hunter planes? Aren't hurricanes worse than thunderstorms? Actually no, they're not. At least not in an airplane. A hurricane can certainly muss your hair while airborne, but they are mostly lower altitude phenomena and have limited vertical development. There may be thunderstorms embedded in a hurricane which the hunter aircraft are careful to avoid, but overall, nothing on earth can match the fury of a well developed hammer-headed storm reaching to over 50,000 ft.
Ok, but can't we just fly over them? No, again. Most airliners can only climb up to the low 40s whereas I mentioned above, a large storm can easily reach to 50k. Even should the storm be a "smaller" one and top out in the 30s, the area above the storm can be quite turbulent as the storm grows and should hail get ejected out of the top of one of these babies, that's exactly where you'll find it. So no, going over a developing cumulonimbus is never a good idea. It is always best to avoid a storm laterally.
Wandering into one of these bad boys will easily make the top of your list of worst things to happen in an airplane. The gusts from violent up and down drafts will produce severe turbulence and the volume of rain your engines will swallow can easily put the fire out. An encounter with hail will scratch your shiny paint job just microseconds before it permanently bashes your radome and the leading edges of your wings into a very non aerodynamic shape. And that is if your windshield doesn't shatter.
Thunderstorms are to be avoided at all cost.
Houston to Chicago
Our trip had us returning from an international destination to be followed by the last leg to Chicago. Then we were done. My F/O lived in base so he was just going home while due to the late arrival time, I was going to the crash pad to be followed by an early morning flight home. We parked the jet at the international terminal, cleared customs and went to look for some dinner before meeting another jet in the domestic terminal. My F/O was diverted away from the automated kiosk into the longer line to an agent for some reason so I'd catch up with him later.
My big decision now became where to eat. The oriental place is good but expensive, while the Mexican place had a long line. I opted for Subway. Nothing like living on airport food. The other option is to carry a food bag, which many crew members do, but an international flight means no fresh food can pass. Little wonder that so many airline pilots are fat. (That's my story, and I'm sticking to it!)
I made a quick trip by the pilot lounge and then went to the gate to meet the incoming aircraft. We were supposed to push at 8:00 pm but the board said the flight was expected out at 8:25. It pulled into the gate dutifully at 7:50 meaning that this was planned for a 35 minute turn. We used to do these in as little as 15 or 20 minutes, but the planes are bigger and people are slower and carry more stuff to stow, hence the 35 minute turn.
Weather Reroute
The reason I stopped by the pilot lounge was to get a wifi signal to update the weather app on my iPad. The company spent a ton of money to give us all iPads with a comprehensive weather app. The only problem is that it's useless without a wifi connection, and we're not allowed to use the onboard wifi. This makes the whole thing quite useless for weather. Rumor has it that this will change soon but I'm not holding my breath.
What I did see during my brief lounge visit was that a large system of thunderstorms was camped out just east of the Mississippi river valley, stretching from perhaps eastern Arkansas up to southern Michigan. I guessed that this meant that we were not going to be flying the most direct route. It turns out I was correct.
When I got the flight plan, it showed that we were going to take a very circuitous route to Chicago. We were routed north over Oklahoma City and then to Omaha followed by Fort Dodge, IA at which point we'd make a turn to the east. I don't know if our dispatcher came up with this route by himself or whether it was given to him by air traffic control, but it was what we were given.
What we were also given was a ton of fuel. The normal route should take about two hours, but this roundabout route was going to be 2+45. The 737 burns about five thousand pounds an hour, plus we have to have our required reserves of 45 minutes of extra gas. On top of that we had gas for two alternates listed as Detroit and Columbus, and also extra fuel in case we had to hold. All of that fuel added up to about 27 thousand pounds.
Now it has been said that the only time you've got too much fuel on board is if you are on fire. While true, having too much fuel when you're trying to land on a short runway in bad weather can be a problem as well. More gas means a heavier airplane. And the Chicago weather was forecast to be ok for our arrival, but thunderstorms were threatening. Another limitation is our structural landing weight limit. There is an upper weight limit for landing that must be met. Carrying too much fuel might put us above that limit in which case we'd have to burn it off or divert. 737s cannot dump fuel like many other aircraft.
The dispatcher put a note on the release to pump us up to 29 thousand pounds if other limits were met, but come push time, the extra fuel never showed up. I was fine going without it as we were planned to land with 12 thousand pounds, about double what we normally have on landing.
After a last minute runway change requiring new takeoff data, we blasted off at about 8:45. The sun had set and there was a beautiful blue orange glow to the west. Also visible was a rather large storm several hundred miles distant. It turned out to be over central Texas but we could see it perfectly from Houston. Air traffic control stopped our climb at 37,000 feet, 2000 feet shy of our planned 39,000 ft. I assumed that this was due to other traffic conflicts.
My hat is off to the controllers who must not only keep all the airplanes separate, but who must also thread all their traffic through a constantly shifting map of storms. As an additional challenge, their radar systems do not show weather very well. They rely upon pilots to tell them where they can and cannot go. It's crazy, but I have apps on my phone that have better weather readouts than that.
We watched the lights of the Dallas Metroplex go by followed by Ok City and Kansas City off to our right. Prior to getting to Omaha we were cleared to cut the corner to Fort Dodge. It was at this point that things started getting funky. We were first cleared one arrival over Joliet, and then quickly by another arrival farther to the north. Finally, the controller just gave us a northeasterly heading and a descent.
Storms were approaching Chicago from the west and were closing the normal arrival routes. Controllers have to get very creative when this happens. We were lucky that our arrival was so late as the traffic flows were diminished. Had this happened during peak arrival times, we would likely have held or diverted. The new plan was to take us in north of the city, over the lake and south to Midway. The controller's challenge was to get us across the arrival traffic going into O'Hare. For this he had us expedite our descent to below their arrivals.
I was also thinking about my possible options should Midway get hit by a storm. One of our alternates, Columbus, was already being hit by a storm but Detroit was still good. Since we were only a few miles south of Milwaukee by this point, I checked the weather there. Their visibility was only 1/8th of a mile which made that not a great choice. Behind us, though, was Minnie which had good weather.
Like a quarterback, you're always keeping an eye on your open receivers while you avoid the linebackers.
Smooth as Butter!
We got fussed at several times to expedite our descent below the O'Hare arrival corridor. The problem is that to increase descent, one must increase airspeed by pointing the nose lower. We were also getting the poop knocked out of us by turbulence, and our turbulence airspeed is only 280 kts. So we do the best we can to get down without beating ourselves or our passengers up too much.
Once we were below the bumps and over the lake we broke out of the weather and everything smoothed out. We were given a turn to the south and enjoyed a beautiful view of downtown Chicago from the north which we almost never see. After passing downtown we were given a vector to the downwind to runway 4R at Midway.
Abeam the airport the controller cut us loose and cleared us for the visual. This rarely happens any more and I used the opportunity to hand fly the approach just like we always did in years gone by. Off went the autopilot and autothrottles. My flying skills may get a little rusty when using all the magic day after day, but the rust knocks right off and I rolled out on glidepath right at a thousand feet fully configured.
The aviation gods smiled upon me because I had one of the best touchdowns at Midway that I've had in a long time. The runways are short, so you normally just try to not land long and accept whatever kind of touchdown you can manage.
Well, we taxied clear, contacted ground and the company to let them know we were there, and of course, as day follows night, after a long day and delays, we have to hold out for our gate. Then we get assigned another gate, then back to yet a third gate, and then we wait for marshallers to guide us in. The important point here to remember is that if you shut down an engine, you will then likely be required to turn into your running engine. Think of trying to turn left on ice when only your left rear tire is turning. It doesn't want to go.
Anyway, we get it shut down, I even get a compliment about the landing, and then it's off to the crash pad for about four hours of shuteye before my flight home. Well as they say, it's hard work, but it still beats working for a living.
Posted by The Captain at 12:34 AM 4 comments: Links to this post
Filed under: airlines, war story, weather
How to Fly with a Jerk
Yea, they're out there. The perfectionists, the disgruntled, the narcissists and especially the ones who are going to teach you the "proper" way to fly. Spend enough time in the front of an airliner and sooner or later you will be paired to fly with a jerk. Short of calling in sick, you will have to spend the next several days locked in a closet sized space for hours on end with the south side of a northbound horse.
So how will you cope? Remember that you have an important job to do and to do it well, you have to work with the other pilot. There was a time when the standard method of dealing with a jerk was to simply not deal with him or her. This could've meant each pilot scanning the 10 or 2 o'clock positions permanently while never interacting save for required checklist callouts.
That strategy clearly has drawbacks and can seriously degrade safety. There's a reason that airliners still have two pilots (for now). You're a professional even if the other guy or gal is less so, and the folks in back are counting on both pilots up front to conduct themselves as such. With all that in mind, here's the Captain's guide to surviving a pairing with a jerk! (Me included.)
Don't Keep It to Yourself
I think that most people are generally conflict averse. I know I am. Most of us don't go around looking for a fight or trying to cause trouble, especially on the job. We want to get through our day, finish with zero airspeed at the correct gate and go to the hotel. So when a situation crops up that causes us some consternation, a first reaction may be to just keep it to ourselves. After all we place a high value on good cockpit relations and rationalize that some internal discomfort is not too high a price to maintain a cordial cockpit atmosphere.
Don't fall for this rationalization. It is built on a fallacy. For starters, if the other guy is doing something that bothers you, he may not even realize it. Secondly, your discomfort won't go away. It will only fester and get worse. Trust me on this. And if you keep your feelings to yourself for any length of time, it can get really awkward when you finally do speak up. It is always best to get things off of your chest as early as possible.
Don't Ever Accept Non-Standard Operation
Again, there may have been a time in the distant past where the Captain was omnipotent and not to be questioned no matter what he did, even if it fell outside of standard operating procedures (SOP). Those times are absolutely long gone. You have a legal and moral duty to make sure that the aircraft is always operated by the book. Does this mean that you must become a pecksniff and point out every deviation no matter how minor? No it does not; there is a middle ground between being a bag of sand in the seat and what we sometimes refer to as a "check F/O".
Getting back to the professionalism part, when the aircraft is operated in a non-standard fashion and something bad happens, claiming that the other guy was flying is of course no defense. My advice is to predetermine your own personal boundaries and then stick to them. You'll sleep better at night. If push comes to shove, which it hopefully never will, take the airplane if you must, and after landing grab your gear and get off the airplane. This will of course involve scheduling and chief pilots but your career is worth it.
And I want to add a note here about bringing in third parties to a dispute. I believe in handling conflict personally whenever possible. The outside resources available to you are your union's professional standards people and your chief pilot. The first thing either of these people will ask is whether you discussed the matter with the other person. If you reply in the negative, your credibility drops. Nobody will care about your issue more than you, so start there.
The pro standards folks and (most) chiefs are great, but my thoughts are that if the guy you're flying with is that big of a jerk, the pro standards folks are probably already well acquainted with him. Taking an issue to a chief pilot should be an absolute last resort. You don't want to be known as the guy who runs to management with a personality conflict. Gross or continued non-standard flying? Maybe. Just be ready to defend your actions.
Sometimes You Have to Just Suck it Up
I personally like to let my F/Os fly the airplane any way they'd like as long as it is within SOP, but that's just me. Some cappies like to dictate how the plane should be flown or how a system should be operated. According to most airlines' procedures and FARs, they are within their rights to have it done the way they want it done, again as long as it is within SOP.
As an F/O, you learn to become a chameleon and do it the way the boss likes it to be done. For instance, in the age old argument of whether or not to run the packs on high on a hot day in a 737, everyone has an opinion. But if the guy in the left seat has a preference, his vote is the one that counts. Hopefully he will be willing to listen to your reasons. If not, there's not much that can be done until you upgrade yourself.
My airline has a "bid avoidance" feature that can be used by F/Os to avoid people with whom they really don't want to fly, but it comes with the price of subrogating their own seniority. This feature didn't exist when I was in the right seat so the "suck it up" or "talk it out" methods were all that were available. It did, though, in some measure force people to get along to some degree.
Take a Look in the Mirror
If you find that you seem to end up having more than a few tiffs in the cockpit, then perhaps the problem doesn't lie with the other guy. If it seems to you that everyone you fly with is a pig-headed jerk, then that may become a clue for you. I think this advice is especially useful for captains. F/Os are naturally inclined to become chameleons, so if you find that your cockpit has a tense atmosphere or you spend a lot of time eating alone in the hotel, perhaps some introspection is merited.
Captains, on the other hand, really get very little feedback. Some are conscientious enough to ask how they're doing, but many are not. Most people with a blind spot are unaware that they have one, which is why my above advice to not keep things to yourself may be more useful than first thought.
Do I have war stories about conflict in the cockpit? Yes I do. I think conflict of some sort with another pilot is inevitable given enough time. My personal technique is to try to not bring up topics which might be touchy such as union issues or politics. I'll talk about that stuff all day long if it comes up, but it is much easier to "kick someone's anthill over" when discussing those things as opposed to Da Bears. And I'm not a huge sports fan, but being at least somewhat conversant in sports can be helpful.
Unlike many other workplace environments, a cockpit is an especially bad place for conflict. Both pilots must make an effort to recognize and to quarantine conflict. Getting things on the table early and using techniques such as humor can go a long way to defuse the bad feels. Pilots aren't necessarily known for being great "people people" but getting along up front should be considered as essential a skill as maintaining airspeed on final.
Of course, I always get to fly with my favorite jerk.
Filed under: airlines, pilots, safety
After spending 21 years flying for the US Air Force, I now fly a Boeing 737 for a major American airline.
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Kristen Stewart & Noah Centineo Look Nothing Like Original 'Charlie's Angels' Heartthrobs: See Makeovers
They're Peter Kavinsky and Bella Swan no more! On the set of the new 'Charlie's Angels' movie, Noah Centineo and Kristen Stewart looked gritty and tough, the complete opposite from the roles that made them famous!
The old Kristen Stewart, 28, can't come to the phone right now. Why? Because it looks like this new, hardcore Kristen beat her up and stole her wallet so she'd have money for smokes. The Twilight actress looked completely different from Bella in the first set photos from the upcoming Charlie's Angels shoot. With blonde streaks in her hair and a cigarette behind her ear, Kristen's character looked like she was in need of a cup of coffee and a twelve-hour nap. Noah Centineo, 22, was also a little RATE – "Rough Around The Edges." The To All The Boys I've Loved Before star looked a little shaggy and unshaven, a departure from the role from his star-making Netflix film.
Ella Balinska, 22, was also on set, shooting a scene that had her wield a gun while popping up through a car's sunroof. Currently, Noah, Ella and Kristen's characters are unknown, but from first glance, they look pretty dangerous. It's quite a departure from the days of Twilight and The Fosters for Kristen and Noah. Fans will have to wait until September 2019 release to see them in action, but we can expect for killer pics like this in the meantime.
It also seems that this upcoming movie is going to be a completely different tone from the two Charlie's Angels movies of the early 2000s. Watch. I'm gonna say all this, and then you're gonna be like, 'You guys didn't do that at all," Kristen Stewart told Variety in September. She said that the movie "is still gonna be fun and 'Charlie's Angels'-y, but will also be current and modern."
"God, it's so funny. I know if I say this a certain way, I know that this will be written down. But it's not such a bad thing," she added. "It's kind of like a 'woke' version. … There's a kitsch nature to the last ones that was super fun. But nowadays, if you see a woman in combat, everything should be completely and utterly within their ability." She also said that instead of just three Angels, there will be a whole network of "women across the entire globe are connected and helping each other." That would explain how Elizabeth Banks, Djimon Hounsou and Patrick Stewart are all cast as "Bosley."
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Jobs at Bed Bath & Beyond in Royal Palm Beach, FL
17 job openings within 25 miles
Retail Cashier
Bed Bath & Beyond - Boynton Beach, FL
Our mission at Bed Bath & Beyond is to make it easy for our customers, and our associates, to feel at home. We've made enhancements to our store environment and implemented health and safety best practices to ensure a clean and safe shopping environment for customers and a safe workplace for our associates. As a Cashier, you will be responsible for exceeding our customers' evolving expectatio...
Careers at Bed Bath & Beyond
Jobs in Boynton Beach, FL
Bed Bath & Beyond Jobs in Boynton Beach, FL
Our mission at Bed Bath & Beyond is to make it easy for our customers, and our associates, to feel at home. We've made enhancements to our store environment and implemented health and safety best practices to ensure a clean and safe shopping environment for customers and a safe workplace for our associates. As a Sales Associate, you will be responsible for exceeding our customers' evolvi...
Bed Bath & Beyond - Boca Raton, FL
Jobs in Boca Raton, FL
Bed Bath & Beyond Jobs in Boca Raton, FL
Retail Department Supervisor
Our mission at Bed Bath & Beyond is to make it easy for our customers, and our associates, to feel at home. We've made enhancements to our store environment and implemented health and safety best practices to ensure a clean and safe shopping environment for customers and a safe workplace for our associates. As a Department Supervisor, you will be responsible for exceeding our customers'...
Our mission at Bed Bath & Beyond is to make it easy for our customers, and our associates, to feel at home. We've made enhancements to our store environment and implemented health and safety best practices to ensure a clean and safe shopping environment for customers and a safe workplace for our associates. Our stores are very entrepreneurial, we have a unique approach to customer servi...
Bed Bath & Beyond - Delray Beach, FL
Jobs in Delray Beach, FL
Bed Bath & Beyond Jobs in Delray Beach, FL
Bed Bath & Beyond - Royal Palm Beach, FL
Jobs in Royal Palm Beach, FL
Bed Bath & Beyond Jobs in Royal Palm Beach, FL
Retail Stock Associate
Our mission at Bed Bath & Beyond is to make it easy for our customers, and our associates, to feel at home. We've made enhancements to our store environment and implemented health and safety best practices to ensure a clean and safe shopping environment for customers and a safe workplace for our associates. As a Stock Associate, you will be responsible for exceeding our customers' evolv...
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The third annual "Adopt-A-Chant Campaign," sponsored by the Coastal Carolina Department of Athletics and its Student-Athlete Advisory Council, is underway to help provide 31 children in Horry County a happy holiday season.
This is a substantial increase over last year, when Coastal Carolina Athletics was able to successfully adopt six children.
The 31 children submitted a "wish list" and Coastal Carolina teams, individual student-athletes and members of the athletics administration are gathering items such as toys, clothes and shoes to provide help for those who otherwise could not afford it.
While Coastal Athletics should be able to fulfill the "wish list," donations are welcome to ensure all gifts requested get distributed this season. To make a monetary donation or to find out what items could be purchased off the "wish list," please contact Cari Rosiek at 843-349-4126.
The Coastal Carolina Department of Athletics is also serving the community through several other avenues this holiday season by participating in Toys for Tots, Operation Christmas Child and distributing 1,400 pounds of food for local food banks.


Our law firm specializes in family law, including divorce, custody, parenting time, child support, domestic violence, prenuptial agreements, reconciliation, property settlements, cohabitation agreements, separation agreements, including trial and appellate practice. We also specialize in mediation and collaborative divorce, tailored to our clients' needs.
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We work hard to improve the quality of life for our clients, our community, and our families. We promote the "best interests of the children" and assist parents in preserving and protecting their children's interests.
We help our clients succeed in their lives post divorce.


text:
package knot.core.stream

import java.nio.file.Path

import knot.core.stream.graphs.SourceGraph
import knot.core.stream.graphs.builders.SourceBuilder
import knot.core.stream.ops.SourceOps
import knot.core.stream.sources.{FileSource, IteratorSource, SingleSource}
import knot.core.{Decorations, Done}
import knot.data.ByteNode

import scala.collection.immutable

trait Source[O, M] extends StreamNode[SourceOps[O, M]] {
}

object Source {

  def from[T, M](source: Source[T, M]): SourceGraph[T, M] = new SourceGraph[T, M](
    SourceBuilder(source, Decorations.empty)
  )

  def from[T, M](source: SourceGraph[T, M]): SourceGraph[T, M] = new SourceGraph[T, M](
    source.builder,
    source.decorations
  )

  def fromFile(path: Path, chunkSize: Int, maxChunk: Int): SourceGraph[ByteNode, Done] = from[ByteNode, Done](FileSource(path, chunkSize, maxChunk))

  def fromIterator[T](iterable: immutable.Iterable[T]): SourceGraph[T, Done] = from(IteratorSource(iterable))

  def single[T](element: T): SourceGraph[T, Done] = from(SingleSource(element))

  def concat[T, M, M2](first: Source[T, M], second: Source[T, M2]): SourceGraph[T, M] = from(first).concat[M2](from(second))
}

redpajama_set_name: RedPajamaGithub


text:
The 2013 BRD Brașov Challenger was a professional tennis tournament, played on clay courts, that took place in Brașov, Romania from 2 to 8 September 2013. It was the 18th edition of the BRD Brașov Challenger tournament and part of the 2013 ATP Challenger Tour.

In singles, Austrian player Andreas Haider-Maurer defeated fellow Austrian Gerald Melzer. In doubles, Ukrainian Aleksandr Nedovyesov and Jaroslav Pospíšil of the Czech Republic defeated Romanian players Teodor-Dacian Crăciun and Petru-Alexandru Luncanu.

Singles main draw entrants

Seeds

 1 Rankings are as of August 26, 2013.

Other entrants
The following players received wildcards into the singles main draw:
  Patrick Ciorcilă
  Florin Mergea
  Dragoș Cristian Mîrtea
  Andrei Pătrui

The following players received entry from the qualifying draw:
  Maxim Dubarenco
  Kyle Edmund
  Mathias Bourgue
  Teodor-Dacian Crăciun

The following players received entry as a lucky loser the singles main draw:
  Oliver Golding

Champions

Singles

 Andreas Haider-Maurer def.  Gerald Melzer 6–7(9–11), 6–4, 6–2

Doubles

 Oleksandr Nedovyesov /  Jaroslav Pospíšil def.  Teodor-Dacian Crăciun /  Petru-Alexandru Luncanu 6–3, 6–1

External links

BRD Brasov Challenger
BRD Brașov Challenger
2013 in Romanian tennis
September 2013 sports events in Romania

redpajama_set_name: RedPajamaWikipedia


text:
\section{Introduction}

Let $T>0$ and $\Om\subset\BR^d$, $d=1,2,3,\ldots$, be either a connected bounded domain with a smooth boundary $\pa\Om$ or $\Om=\BR^d$. For $0<\al\le2$, consider an initial(-boundary) value problem for a time-fractional diffusion(-wave) equation
\begin{equation}\label{eq-IBVP-u}
\begin{cases}
(\pa_t^\al+\cL)u(\bm x,t)=g(\bm x-\bm\ga(t)), & {\bm x}\in\Om,\ 0<t<T,\\
\begin{cases}
u(\bm x,0)=0 & \mbox{if }0<\al\le1,\\
u(\bm x,0)=\pa_t u(\bm x,0)=0 & \mbox{if }1<\al\le2,
\end{cases} &{\bm x}\in\Om,\\
u(\bm x,t)=0\quad\mbox{if }\Om\mbox{ is bounded}, & {\bm x}\in\pa\Om,\ 0<t<T.
\end{cases}
\end{equation}
Here, $\cL$ is an elliptic operator with respect to the spatial variable $\bm x\in\BR^d$, and $\pa_t^\al$ denotes the Caputo derivative with respect to the time variable $t\in\BR_+:=(0,\infty)$, which will be precisely defined in Section \ref{sec:2}. The function $g$ is an approximation of Dirac's delta function in $\BR^d$, and $\bm\ga:\BR_+\longrightarrow\BR^d$ describes the orbit of a moving source in $\BR^d$. The governing equation in \eqref{eq-IBVP-u} is called a (time-fractional) diffusion equation when $\al\in(0,1]$, whereas is called a (time-fractional) diffusion-wave equation or a (time-fractional) wave equation when $\al\in(1,2]$. Hence, the system \eqref{eq-IBVP-u} approximates a moving point source problem for the (time-fractional) diffusion(-wave) equation.

In this paper, we are interested in the inverse moving source problem of recovering an unknown orbit function $\bm\ga(t)$ from the solution data detected at a finite number of interior receivers. More precisely, we investigate the following problem.

\begin{problem}[Inverse moving source problem]\label{prob-IMSP}
Let $u$ be the solution to \eqref{eq-IBVP-u} and pick $N$ interior points $\bm x^j\in\Om\ (j=1,\ldots,N)$. Provided that the source profile $g(\bm x)$ is suitably given, determine the source orbit $\bm\ga(t)\ (0\le t\le T)$ by the multiple point observations of $u$ at $\{\bm x^j\}_{j=1}^N\times[0,T]$.
\end{problem}

We remark that the relation between the orbit function and the received dynamical signals is nonlinear, whereas the operator which maps the source profile function $g$ to the forward solution $u$ is linear. Hence, the problem considered in this paper is a nonlinear inverse issue.

We refer to Isakov \cite{Isakov89} for an overview of uniqueness and stability results on inverse source problems. The approaches of applying Carleman estimates and the unique continuation of evolutionary equations have been widely used in the literature and have led to uniqueness and stability results for both inverse coefficient and inverse source problems with the dynamical data over a finite time (see e.g., \cite{ACY-EJAM04,BK1981,CM2006,K1992,JLY2017,Ya1995,Ya1999} as an incomplete list). The concept of increasing stability was explored in \cite{Isakov07} and later investigated further for inverse source problems in \cite{CIL2016}. For stationary (non-moving) sources, the uniqueness in determining source positions with boundary surface data was deduced in \cite{EH}, and those for upper and lower estimates of source positions were derived in \cite{KY01,KY05} in one and  higher dimensions.

To the best of the authors' knowledge, literature on inverse moving source problems arising in fractional diffusion(-wave) equations is rather limited and even remain open. A logarithmic stability and an iterative inversion scheme were considered in \cite{LZ17} for recovering temporal source terms in fractional diffusion equations. Using the moment theory, a uniqueness result to inverse moving source problems in electromagnetism was proved (see \cite{HKLZ}) using boundary surface data. In a series of works \cite{OIO11,N12,O19}, numerical algorithms were examined for reconstructing a moving orbit from boundary data of solutions of the scalar wave equation.

The aim of this paper is to derive stability and uniqueness results with a finite number of interior monitoring points for the fractional model \eqref{eq-IBVP-u} with $\al\in(0,2]$. Our arguments rely on the fixed point theory similarly to \cite{SY11,FK16,WLL16} but are modified to be applicable to the system \eqref{eq-IBVP-u}. For this purpose, we have deduced Duhamel's principle for time-fractional partial differential equations of any order (see Lemma \ref{lem-Duhamel}) and a uniform solution representation for the fractional order $\alpha\in(0,2]$ via the Fourier transform in the whole space (see Lemma \ref{lem-FP-V}). These lemmas have generalized the corresponding well-known results for equations with integral orders in bounded domains, making new contributions to the theory of fractional equations. The interior observation data are here formulated from the theoretical viewpoint, and in a forthcoming paper, on the basis of the current work, we will discuss the inverse problem with more practical data.

The remaining part of this paper is organized as follows. Preliminary knowledge and our main results for fractional equations in bounded and unbounded domains will be stated in Section \ref{sec:2}. Three auxiliary lemmas will be proved in Section \ref{app-proof}. The proofs of our stability result for localized {moving sources} (Theorem \ref{thm-local}) and the uniqueness for non-localized ones (Corollary \ref{cor-global}) will be carried out in Section \ref{sec:4}.

\section{Preliminaries and Main Results}\label{sec:2}

Throughout this paper, we set $\BN:=\{1,2,3,\ldots\}$, and by $C>0$ we denote generic constants which may change from line to line. For $\be\in\BR$, denote the largest integer smaller than or equal to $\be$ by the floor function $\lfloor\be\rfloor$, and the smallest integer larger than or equal to $\be$ by the ceiling function $\lceil\be\rceil$. For $\be\in[0,1]$, define the Riemann-Liouville integral operator $J^\be$ by
\[
J^\be f(t):=\left\{\!
\begin{alignedat}{2}
& f(t), & \quad & \be=0,\\
& \f1{\Ga(\be)}\int_0^t\f{f(s)}{(t-s)^{1-\be}}\,\rd s, & \quad & 0<\be\le1,
\end{alignedat}
\right.\quad f\in C[0,\infty),
\]
where $\Ga(\,\cdot\,)$ is the Gamma function. Then for $\be\in\BR_+$, the Caputo derivative $\pa_t^\be$ and the Riemann-Liouville derivative $D_t^\be$ are defined as
\[
\pa_t^\be:=J^{\lceil\be\rceil-\be}\circ\f{\rd^{\lceil\be\rceil}}{\rd t^{\lceil\be\rceil}},\quad D_t^\be:=\f{\rd^{\lceil\be\rceil}}{\rd t^{\lceil\be\rceil}}\circ J^{\lceil\be\rceil-\be},
\]
where $\circ$ denotes the composition. For the solution representation, we recall the familiar Mittag-Leffler function
\[
E_{\be,\mu}(z):=\sum_{\ell=0}^\infty\f{z^\ell}{\Ga(\be\ell+\mu)},\quad\be\in\BR_+,\ \mu\in\BR,\ z\in\BC,
\]
which satisfies the frequently used estimate (see Podlubny \cite[Theorem 1.5]{P99})
\begin{equation}\label{eq-est-ML}
|E_{\be,\mu}(z)|\le\f C{1+|z|},\quad\be\in(0,2),\ \mu\in\BR_+,\ \f{\pi\be}2<|\mathrm{arg}\,z|\le\pi.
\end{equation}
For later use, we state the following formula concerning the Riemann-Liouville derivative and Mittag-Leffler functions.

\begin{lemma}\label{lem-RL-ML}
For $\be\in\BR_+$ and $\la\in\BR$, we have
\[
D_t^{\lceil\be\rceil-\be}\left(t^{\lceil\be\rceil-1}E_{\be,\lceil\be\rceil}(\la\,t^\be)\right)=t^{\be-1}E_{\be,\be}(\la\,t^\be).
\]
\end{lemma}

Next, we generalize useful Duhamel's principle to time-fractional evolution equations with arbitrary orders $\be\in\BR_+$.

\begin{lemma}[Fractional Duhamel's principle]\label{lem-Duhamel}
Let $\be\in\BR_+$ and $\Om\subset\BR^d$ be a domain. Let $F:\Om\times(0,T)\longrightarrow\BR$ be a smooth function, and $\cP$ be a linear partial differential operator with respect to $\bm x$ defined in $\Om$ whose coefficients are independent of $t$. If a smooth function $u$ satisfies
\begin{equation}\label{eq-Duhamel-u}
\begin{cases}
(\pa_t^\be+\cP)u=F & \mbox{in }\Om\times(0,T),\\
\pa_t^m u=0\ (m=0,\ldots,\lceil\be\rceil-1) & \mbox{in }\Om\times\{0\},
\end{cases}
\end{equation}
then $u$ allows the representation
\begin{equation}\label{eq-Duhamel}
u(\,\cdot\,,t)=\int_0^t D_t^{\lceil\be\rceil-\be}v(\,\cdot\,,t;s)\,\rd s,\quad0<t<T,
\end{equation}
where $v(\,\cdot\,,\,\cdot\,;s)$ is a smooth function satisfying the following homogeneous equation with a parameter $s\in(0,T)$:
\begin{equation}\label{eq-Duhamel-v}
\begin{cases}
(\pa_t^\be+\cP)v=0 & \mbox{in }\Om\times(s,T),\\
\pa_t^m v=0\ (m=0,\ldots,\lceil\be\rceil-2),\ \pa_t^{\lceil\be\rceil-1}v=F(\,\cdot\,,s) & \mbox{in }\Om\times\{s\}.
\end{cases}
\end{equation}
Here we automatically interpret
\begin{align*}
D_t^\be v(\,\cdot\,,t;s) & =\f{\pa_t^{\lceil\be\rceil}}{\Ga(\lceil\be\rceil-\be)}\int_s^t\f{v(\,\cdot\,,r;s)}{(t-r)^{\be-\lfloor\be\rfloor}}\,\rd r,\\
\pa_t^\be v(\,\cdot\,,t;s) & =\f1{\Ga(\lceil\be\rceil-\be)}\int_s^t\f{\pa_r^{\lceil\be\rceil}v(\,\cdot\,,r;s)}{(t-r)^{\be-\lfloor\be\rfloor}}\,\rd r
\end{align*}
for $\be\not\in\BN$, since $v(\,\cdot\,,t;s)$ is only defined for $t<s$.
\end{lemma}

The above lemma generalizes similar results in \cite{LRY16,L17}, where the source term was assumed to take the form of separated variables. For other fractional varieties of Duhamel's principle, we refer to Umarov and Saidamatov \cite{US07}, Zhang and Xu \cite{ZX11}.

In the sequel, all vectors are by default column vectors unless specified otherwise. For instance, we write $\bm x=(x_1,\ldots,x_d)^\T\in\BR^d$ and $\nb f=(\pa_1f,\ldots,\pa_d f)^\T$, where $(\,\cdot\,)^\T$ stands for the transpose and $\pa_k=\f\pa{\pa x_k}$ ($k=1,\ldots,d$). The inner product in $\BR^d$ is denoted by $\bm x\cdot\bm y$, and the Euclidean distance $|\cdot|$ is induced as $|\bm x|=(\bm x\cdot\bm x)^{1/2}$. Given a matrix $\bm\Psi=(\psi_{jk})\in\BR^{d\times d}$, the $\ell^2$ norm $|\,\cdot\,|$ and the Frobenius norm $\|\cdot\|_\F$ of $\bm\Psi$ are defined as
\[
|\bm\Psi|:=\max_{|\bm x|=1}|\bm\Psi\bm x|,\quad\|\bm\Psi\|_\F:=\left(\sum_{j,k=1}^d\psi_{jk}^2\right)^{1/2}.
\]
By the norm equivalence in finite dimensional vector spaces, there exists a constant $C>0$ such that
\begin{equation}\label{eq-norm}
C^{-1}|\bm\Psi|\le\|\bm\Psi\|_\F\le C|\bm\Psi|,\quad\forall\,\bm\Psi\in\BR^{d\times d}.
\end{equation}
The open ball centered at $\bm x\in\BR^d$ with radius $r>0$ is denoted by $B_r(\bm x):=\{\bm y\in\BR^d\mid|\bm y-\bm x|<r\}$, and especially we abbreviate $B_r(\bm0)=B_r$.

For a domain $\Om\subset\BR^d$, denote the usual $L^2$ inner product by $(\,\cdot\,,\,\cdot\,)$, and let $H^p(\Om)$ ($p\in\BR$) be the $L^2$-based Sobolev spaces (see Adams \cite{A75}). If $\Om$ is a connected bounded domain, then the elliptic operator $\cL$ in the initial-boundary value problem \eqref{eq-IBVP-u} is defined as
\[
\cL:H^2(\Om)\cap H_0^1(\Om)\longrightarrow L^2(\Om),\quad f\longmapsto-\mathrm{div}(\bm A\nb f)+c\,f,
\]
where the matrix $\bm A=(a_{jk})_{1\le j,k\le d}\in(C^1(\ov\Om))^{d\times d}$ is symmetric and strictly positive definite uniformly on $\ov\Om$, and $c\in L^\infty(\Om)$ is non-negative. In this case, the operator $\cL$ generates an eigensystem $\{(\la_n,\vp_n)\}_{n\in\BN}$ such that $\cL\vp_n=\la_n\vp_n$ in $\Om$. Moreover, it is well known that $0<\la_1<\la_2\le\cdots$, $\la_n\to\infty$ as $n\to\infty$, and $\{\vp_n\}\subset H^2(\Om)\cap H_0^1(\Om)$ forms a complete orthonormal system of $L^2(\Om)$.

For a square integrable function $f\in L^2(\BR^d)$, denote its Fourier transform by
\[
\wt f(\bm\xi)=(\cF f)(\bm\xi):=\f1{(2\pi)^{d/2}}\int_{\BR^d}f(\bm x)\,\e^{-\ri\,\bm x\cdot\bm\xi}\,\rd\bm x.
\]
For $p\in\BR$, the $H^p(\BR^d)$ norm can be represented by using Fourier transform as
\[
\|f\|_{H^p(\BR^d)}=\left(\int_{\BR^d}(1+|\bm\xi|^2)^p|\wt f(\bm\xi)|^2\,\rd\bm\xi\right)^{1/2},\quad f\in H^p(\BR^d).
\]
If $\Om=\BR^d$, then we define the elliptic operator $\cL$ in the initial value problem \eqref{eq-IBVP-u} as
\[
\cL:H^2(\BR^d)\longrightarrow L^2(\BR^d),\quad f\longmapsto-\mathrm{div}(\bm A\nb f)+\bm b\cdot\nb f+c\,f,
\]
where we assume that the matrix $\bm A=(a_{jk})_{1\le j,k\le d}$, the vector $\bm b=(b_1,\ldots,b_d)^\T$ and the scalar $c$ are constants, and $\bm A$ is strictly positive definite. Especially, only in the case of $\al=2$ we additionally assume $\bm b=\bm0$ and $c\ge0$.

Regarding the initial(-boundary) value problem
\begin{equation}\label{eq-IBVP-V}
\begin{cases}
(\pa_t^\al+\cL)V=0 & \mbox{in }\Om\times\BR_+,\\
\begin{cases}
V=v_0 & \mbox{if }0<\al\le1,\\
V=0,\ \pa_t V=v_1 & \mbox{if }1<\al\le2
\end{cases} & \mbox{in }\Om\times\{0\},\\
V=0\quad\mbox{if }\Om\mbox{ is bounded} & \mbox{on }\pa\Om\times\BR_+,
\end{cases}
\end{equation}
we provide the well-posedness results in the following lemma.

\begin{lemma}\label{lem-FP-V}
Let $\Om\subset\BR^d$ be either a connected bounded domain or $\Om=\BR^d$, and $v_0\in H^p(\Om),v_1\in H^{p-2/\al}(\Om)$ for some fixed $p\in\BR$. Then there exists a unique solution $V\in C([0,T];H^p(\Om))$ to \eqref{eq-IBVP-V}. Moreover, if $\Om$ is bounded, then the solution to \eqref{eq-IBVP-V} takes the form
\begin{equation}\label{eq-sol-IBVP-V}
V(\bm x,t)=\sum_{n=1}^\infty(v_{\lceil\al\rceil-1},\vp_n)\,t^{\lceil\al\rceil-1}E_{\al,\lceil\al\rceil}(-\la_n t^\al)\vp_n(\bm x).
\end{equation}
If $\Om=\BR^d$, then the solution to \eqref{eq-IBVP-V} takes the form
\begin{equation}\label{eq-sol-IVP-V}
V(\bm x,t)=\left\{\!\begin{alignedat}{2}
& \f{t^{\lceil\al\rceil-1}}{(2\pi)^{d/2}}\int_{\BR^d}\wt v_{\lceil\al\rceil-1}(\bm\xi)E_{\al,\lceil\al\rceil}(-S(\bm\xi)t^\al)\,\e^{\ri\,\bm\xi\cdot\bm x}\,\rd\bm\xi, & \quad & 0<\al<2,\\
& \f1{(2\pi)^{d/2}}\int_{\BR^d}\wt v_1(\bm\xi)S(\bm\xi)^{-1/2}\sin(S(\bm\xi)^{1/2}t)\,\e^{\ri\,\bm\xi\cdot\bm x}\,\rd\bm\xi, & \quad & \al=2,
\end{alignedat}\right.
\end{equation}
where $S(\bm\xi):=\bm A\bm\xi\cdot\bm\xi+\ri\,\bm b\cdot\bm\xi+c$.
\end{lemma}

In a bounded domain, the solution representation by the eigensystem is well {known} (see Sakamoto and Yamamoto \cite{SY11}). However, solutions to \eqref{eq-IBVP-V} in the whole space $\BR^d$ seem not well investigated to the best of our knowledge, and we refer to Eidelman and Kochubei \cite{EK04} for the fundamental solution. In such a sense, formula \eqref{eq-sol-IVP-V} in Lemma \ref{lem-FP-V} gives a novel solution representation via the Fourier transform.

Now we are well prepared to discuss Problem \ref{prob-IMSP}. We begin with specifying the choices of the source profile $g(\bm x)$ and the orbit $\bm\ga(t)$. Assume that $g$ is smooth and compactly supported, i.e., $g\in C_0^\infty(\Om)$ and there exists a constant $\de>0$ such that $\supp\,g\subset B_\de$. A typical choice of $g$ can be the following bell-shaped function
\begin{equation}\label{eq-bell}
g(\bm x)=\left\{\!\begin{alignedat}{2}
& C\exp\left(\f1{|\bm x|^2-\de^2}\right), & \quad & |\bm x|<\de,\\
& 0, & \quad & |\bm x|\ge\de.
\end{alignedat}\right.
\end{equation}
For the unknown $\bm\ga$, basically we restrict it in the admissible set
\begin{equation}\label{eq-def-U0}
\cU_0:=\{\bm\ga\in(C^\infty[0,T])^d\mid\bm\ga(0)=\bm0,\ \|\bm\ga'\|_{C[0,T]}\le K,\ \bm\ga(t)+\supp\,g\subset\Om,\ 0\le t\le T\},
\end{equation}
where $K>0$ is a constant. In other words, we restrict our consideration in such orbits that they are smooth and start from the origin with a maximum velocity.

First we investigate a special case of a localized moving source. More precisely, for a sufficiently small $\ve>0$, we further restrict the unknown orbit in
\begin{equation}\label{eq-def-U1}
\cU_1:=\{\bm\ga\in\cU_0\mid\|\bm\ga\|_{C[0,T]}\le\ve\},
\end{equation}
which means $\{\bm\ga(t)\}_{0\le t\le T}\subset B_\ve$ for all $\bm\ga\in\cU_1$.

Since there are $d$ components in the orbit, it is natural to take at least $d$ observation points for the {unique} identification. Within the admissible set $\cU_1$, we pick the minimum necessary $d$ observation points $\bm x^j$ ($j=1,\ldots,d$) and make the following key assumption: there exists a constant $C>0$ depending on $g$, $\{\bm x^j\}_{j=1}^d$ and $\ve$ such that
\begin{equation}\label{eq-assume-obs}
\left|\begin{pmatrix}
\nb g(\bm y^1) & \nb g(\bm y^2) & \cdots & \nb g(\bm y^d)
\end{pmatrix}^{-1}\right|\le C,\quad\forall\,\bm y^j\in\ov{B_\ve(\bm x^j)}\,,\ j=1,\ldots,d.
\end{equation}
In other words, we assume that the matrix $(\nb g(\bm y^1)\ \cdots\ \nb g(\bm y^d))$ is invertible for all $(\bm y^1,\ldots,\bm y^d)\in\prod_{j=1}^d\ov{B_\ve(\bm x^j)}\,$.

\begin{example}
We rephrase assumptions \eqref{eq-def-U1} and \eqref{eq-assume-obs} in the case of $d=1$. For any $\ga\in\cU_1$, we have $\ga(0)=0$ and $|\ga(t)|\le\ve$, $|\ga'(t)|\le K$ for $0\le t\le T$. As for the observation point $x^1$, the assumption \eqref{eq-assume-obs} means
\[
|g'(y)|\ge C^{-1}>0,\quad\forall\,y\in[x^1-\ve,x^1+\ve],
\]
which implies $[x^1-\ve,x^1+\ve]\subset\supp\,g'$.
\end{example}

Now we can state Lipschitz stability and uniqueness results for Problem \ref{prob-IMSP} with the observation data taken at $\{\bm x^j\}_{j=1}^d\times[0,T]$.

\begin{theorem}\label{thm-local}
Fix $\bm\ga_1,\bm\ga_2\in\cU_1$, where $\cU_1$ was defined by \eqref{eq-def-U1}. Denote by $u_1$ and $u_2$ the solutions to \eqref{eq-IBVP-u} with $\bm\ga=\bm\ga_1$ and $\bm\ga=\bm\ga_2$, respectively. If the set of observation points $\{\bm x^j\}_{j=1}^d$ satisfies \eqref{eq-assume-obs}, then there exists a constant $C>0$ depending on $g,$ $\{\bm x^j\}_{j=1}^d$ and $\cU_1$ such that
\[
\|\bm\ga_1-\bm\ga_2\|_{C[0,T]}\le C\sum_{j=1}^d\|\pa_t^\al(u_1-u_2)(\bm x^j,\,\cdot\,)\|_{C[0,T]}.
\]
Especially, $u_1(\bm x^j,\,\cdot\,)=u_2(\bm x^j,\,\cdot\,)\ (j=1,\ldots,d)$ on $[0,T]$ implies $\bm\ga_1=\bm\ga_2$ on $[0,T]$.
\end{theorem}

Our main result Theorem \ref{thm-local} requires condition \eqref{eq-assume-obs} for $\bm x^j$, $j=1,2,\ldots,N$, and especially the number $N$ of the monitoring points $\bm x^j$ should be at least $d$ which is the spatial dimensions. This is reasonable because as unknowns we have to determine $d$ components of $\bm\ga(t)$, and our data are $N$ functions in $t\in[0,T]$.

The key to proving the above theorem is reducing the original problem to a vector-valued Volterra integral equation of the second kind with respect to the difference $\bm\ga_1-\bm\ga_2$. To this end, the representations of solutions to \eqref{eq-IBVP-u} are essential, where Lemmas \ref{lem-Duhamel} and \ref{lem-FP-V} play important roles. Such an argument is also witnessed in \cite{SY11,FK16,WLL16} which also rely on similar non-vanishing assumptions as \eqref{eq-assume-obs}. Nevertheless, due to the nonlinearity of our problem with respect to the orbits, assumption \eqref{eq-assume-obs} looks more complicated than that in \cite{SY11,FK16,WLL16}.

Remarkably, the constant $C$ in the stability estimate of Theorem \ref{thm-local} does not depend on the order $\al\in(0,2]$. Indeed, such a uniform estimate of Lipschitz type is achieved at the cost of accessing the $\al$th order derivative of the observation data. Meanwhile, one can also see from the proof that the ill-posedness resulted from $\al$ is overwhelmed by the key assumption \eqref{eq-assume-obs} along with the admissible set $\cU_1$.

In Theorem \ref{thm-local}, the Lipschitz stability with minimum possible observation points is achieved within the admissible set $\cU_1$ in \eqref{eq-def-U1}, which is rather restrictive. Moreover, since \eqref{eq-assume-obs} implies $\bm x^j\in\supp(\nb g)$ ($j=1,\ldots,d$), the required observation condition seems also strict in practice. On the opposite direction, we can remove the localization assumption $\|\bm\ga\|_{C[0,T]}\le\ve$ in \eqref{eq-def-U1} and obtain a uniqueness result at the cost of very dense observation points.

\begin{corollary}\label{cor-global}
Fix $\bm\ga_1,\bm\ga_2\in\cU_0$, where $\cU_0$ was defined by \eqref{eq-def-U0}. Denote by $u_1$ and $u_2$ the solutions to \eqref{eq-IBVP-u} with $\bm\ga=\bm\ga_1$ and $\bm\ga=\bm\ga_2$, respectively. Assume that there exist a finite set of observation points $X:=\{\bm x^j\}_{j=1}^N$ and a constant $\ve>0$ such that for any $\bm y\in\Om\cap\ov{B_{K T}}\,$, there exist $d$ observation points $\{\bm x^j(\bm y)\}_{j=1}^d\subset X\cap B_\de(\bm y)$ and a constant $C>0$ such that
\begin{equation}\label{eq-assume-obs'}
\left|\begin{pmatrix}
\nb g(\bm z^1) & \cdots & \nb g(\bm z^d)
\end{pmatrix}^{-1}\right|\le C,\quad\forall\,\bm z^j\in\ov{B_\ve(\bm x^j(\bm y)-\bm y)}\,,\ j=1,\ldots,d.
\end{equation}
Then the relation $u_1(\bm x^j,\,\cdot\,)=u_2(\bm x^j,\,\cdot\,)\ (j=1,\ldots,N)$ on $[0,T]$ implies $\bm\ga_1=\bm\ga_2$ on $[0,T]$.
\end{corollary}

As one can imagine, the above corollary follows from the repeated application of Theorem \ref{thm-local}, where the invertibility assumption \eqref{eq-assume-obs'} is a generalization of \eqref{eq-assume-obs}. It suffices to restrict $\bm y\in\Om$ in the ball $\ov{B_{K T}}$ because $\|\bm\ga\|_{C[0,T]}\le K T$ for any $\bm\ga\in\cU_0$ by the definition \eqref{eq-def-U0} of $\cU_0$. Since $\Om\cap\ov{B_{K T}}$ is bounded, the number $N$ of observation points can definitely be finite.

\begin{example}
In the one-dimensional case, if $g$ takes the form of a bell-shaped function \eqref{eq-bell}, then it is readily seen that a choice of $\ve$ and $X$ in Corollary $\ref{cor-global}$ can be
\[
\ve=\f\de9,\quad\bm x^j=\f{(-1)^j\lfloor j/2\rfloor\de}4,\quad N=\lceil4(K T+\de)/\de\rceil.
\]
\end{example}

\section{Proofs of Lemmas \ref{lem-RL-ML}--\ref{lem-FP-V}}\label{app-proof}

\begin{proof}[Proof of Lemma $\ref{lem-RL-ML}$]
By the definitions of Mittag-Leffler functions and the Riemann-Liouville derivative, we direct calculate
\begin{align*}
D_t^{\lceil\be\rceil-\be}\left(t^{\lceil\be\rceil-1}E_{\be,\lceil\be\rceil}(\la\,t^\be)\right) & =\f1{\Ga(\be-\lfloor\be\rfloor)}\sum_{\ell=0}^\infty\f{\la^\ell}{\Ga(\be\ell+\lceil\be\rceil)}\f\rd{\rd t}\int_0^t\f{s^{\be\ell+\lceil\be\rceil-1}}{(t-s)^{\lceil\be\rceil-\be}}\,\rd s\\
& =\f1{\Ga(\be-\lfloor\be\rfloor)}\sum_{\ell=0}^\infty\f{\Ga(\be\ell+\lceil\be\rceil)\Ga(\be-\lfloor\be\rfloor)\la^\ell(t^{\be(\ell+1)})'}{\Ga(\be\ell+\lceil\be\rceil)\Ga(\be\ell+\lceil\be\rceil+\be-\lfloor\be\rfloor)}\\
& =\sum_{\ell=0}^\infty\f{\be(\ell+1)\la^\ell t^{\be(\ell+1)-1}}{\Ga(\be(\ell+1)+1)}=t^{\be-1}\sum_{\ell=0}^\infty\f{(\la\,t^\be)^\ell}{\Ga(\be\ell+\be)}=t^{\be-1}E_{\be,\be}(\la\,t^\be),
\end{align*}
where we have used the formula $\Ga(\be+1)=\be\,\Ga(\be)$.
\end{proof}

\begin{proof}[Proof of Lemma $\ref{lem-Duhamel}$]
The case of $\be\in\BN$ is straightforward and we only give a proof for the case of $\be\not\in\BN$. Actually, it suffices to verify that the function $u$ defined by \eqref{eq-Duhamel} satisfies \eqref{eq-Duhamel-u}. Since $F,u,v$ are assumed to be smooth, we can take any derivatives when needed.

First, it follows from the definition of the Riemann-Liouville derivative that
\begin{equation}\label{eq-Duhamel-pf1}
D_t^{\lceil\be\rceil-\be}F(\,\cdot\,,t)=\f1{\Ga(\be-\lfloor\be\rfloor)}\f\pa{\pa t}\int_0^t\f{F(\,\cdot\,,t-s)}{s^{\lceil\be\rceil-\be}}\,\rd s=\f1{\Ga(\be-\lfloor\be\rfloor)}\left(\f{F(\,\cdot\,,0)}{t^{\lceil\be\rceil-\be}}+\int_0^t\f{\pa_s F(\,\cdot\,,s)}{(t-s)^{\lceil\be\rceil-\be}}\,\rd s\right).
\end{equation}
Next, from \eqref{eq-Duhamel-u} we calculate
\begin{align*}
\pa_t u(\,\cdot\,,t) & =D_t^{\lceil\be\rceil-\be}v(\,\cdot\,,t;t)+\int_0^t\pa_t D_t^{\lceil\be\rceil-\be}v(\,\cdot\,,t;s)\,\rd s\\
& =\left\{\!\begin{alignedat}{2}
& D_t^{1-\be}F(\,\cdot\,,t)+\int_0^t D_t^{1-\be}\pa_t v(\,\cdot\,,t;s)\,\rd s, & \quad & 0<\be<1,\\
& \int_0^t D_t^{\lceil\be\rceil-\be}\pa_t v(\,\cdot\,,t;s)\,\rd s, & \quad & \be>1,
\end{alignedat}\right.
\end{align*}
where we used the initial condition at $t=s$ in \eqref{eq-Duhamel-v}. Inductively, we obtain
\begin{equation}\label{eq-diff-u}
\pa_t^m u(\,\cdot\,,t)=\left\{\!\begin{alignedat}{2}
& \int_0^t D_t^{\lceil\be\rceil-\be}\pa_t^m v(\,\cdot\,,t;s)\,\rd s, & \quad & m<\lceil\be\rceil,\\
& D_t^{\lceil\be\rceil-\be}F(\,\cdot\,,t)+\int_0^t D_t^{\lceil\be\rceil-\be}\pa_t^{\lceil\be\rceil}v(\,\cdot\,,t;s)\,\rd s, & \quad & m=\lceil\be\rceil.
\end{alignedat}\right.
\end{equation}
Since $v$ is sufficiently smooth, for $m<\lceil\be\rceil$ we pass $t\to0$ in \eqref{eq-diff-u} to find
\[
\pa_t^m u(\,\cdot\,,0)=0,\quad m=0,\ldots,\lceil\be\rceil-1,
\]
i.e., the function $u$ satisfies the initial condition in \eqref{eq-Duhamel-u}. Meanwhile, substituting \eqref{eq-diff-u} with $m=\lceil\be\rceil$ into the definition of the Caputo derivative gives
\begin{equation}\label{eq-Duhamel-pf2}
\pa_t^\be u(\,\cdot\,,t)=\f1{\Ga(\lceil\be\rceil-\be)}\int_0^t\f{\pa_s^{\lceil\be\rceil}u(\,\cdot\,,s)}{(t-s)^{\be-\lfloor\be\rfloor}}\,\rd s=:\f{\I_1+\I_2}{\Ga(\lceil\be\rceil-\be)},
\end{equation}
where
\[
\I_1:=\int_0^t\f{D_s^{\lceil\be\rceil-\be}F(\,\cdot\,,s)}{(t-s)^{\be-\lfloor\be\rfloor}}\,\rd s,\quad\I_2:=\int_0^t\f1{(t-s)^{\be-\lfloor\be\rfloor}}\int_0^s D_s^{\lceil\be\rceil-\be}\pa_s^{\lceil\be\rceil}v(\,\cdot\,,s;r)\,\rd r\rd s.
\]
For $\I_1$, the application of \eqref{eq-Duhamel-pf1} yields
\begin{align}
\I_1 & =\f1{\Ga(\be-\lfloor\be\rfloor)}\int_0^t\f1{(t-s)^{\be-\lfloor\be\rfloor}}\left(\f{F(\,\cdot\,,0)}{s^{\lceil\be\rceil-\be}}+\int_0^s\f{\pa_r F(\,\cdot\,,r)}{(s-r)^{\lceil\be\rceil-\be}}\,\rd r\right)\rd s\nonumber\\
& =\f1{\Ga(\be-\lfloor\be\rfloor)}\left(F(\,\cdot\,,0)\int_0^t\f{\rd s}{(t-s)^{\be-\lfloor\be\rfloor}s^{\lceil\be\rceil-\be}}+\int_0^t\pa_r F(\,\cdot\,,r)\int_r^t\f{\rd s}{(t-s)^{\be-\lfloor\be\rfloor}(s-r)^{\lceil\be\rceil-\be}}\,\rd r\right)\nonumber\\
& =\Ga(\lceil\be\rceil-\be)\left(F(\,\cdot\,,0)+\int_0^t\pa_r F(\,\cdot\,,r)\,\rd r\right)=\Ga(\lceil\be\rceil-\be)\,F(\,\cdot\,,t).
\end{align}
For $\I_2$, by suitably exchanging the order of integration, we utilize the definition of Caputo and Riemann-Liouville derivatives to calculate
\begin{align}
\I_2 & =\int_0^t\f1{(t-s)^{\be-\lfloor\be\rfloor}}\int_0^s D_s^{\lceil\be\rceil-\be}\pa_s^{\lceil\be\rceil}v(\,\cdot\,,s;r)\,\rd r\rd s\nonumber\\
& =\f1{\Ga(\be-\lfloor\be\rfloor)}\int_0^t\f1{(t-s)^{\be-\lceil\be\rceil}}\int_0^s\pa_s\int_0^{s-r}\f{\pa_s^{\lceil\be\rceil}v(\,\cdot\,,s-\tau;r)}{\tau^{\lceil\be\rceil-\be}}\,\rd\tau\rd r\rd s\nonumber\\
& =\f1{\Ga(\be-\lfloor\be\rfloor)}\int_0^t\f1{(t-s)^{\be-\lceil\be\rceil}}\int_0^s\f{\pa_r^{\lceil\be\rceil}v(\,\cdot\,,r;r)}{(s-r)^{\lceil\be\rceil-\be}}\,\rd r\rd s\nonumber\\
& \quad\,+\f1{\Ga(\be-\lfloor\be\rfloor)}\int_0^t\f1{(t-s)^{\be-\lceil\be\rceil}}\int_0^s\!\!\int_r^s\f{\pa_\tau^{\lceil\be\rceil+1}v(\,\cdot\,,\tau;r)}{(s-\tau)^{\lceil\be\rceil-\be}}\,\rd\tau\rd r\rd s\nonumber\\
& =\f1{\Ga(\be-\lfloor\be\rfloor)}\int_0^t\pa_r^{\lceil\be\rceil}v(\,\cdot\,,r;r)\int_r^t\f{\rd s}{(t-s)^{\be-\lfloor\be\rfloor}(s-r)^{\lceil\be\rceil-\be}}\,\rd r\nonumber\\
& \quad\,+\f1{\Ga(\be-\lfloor\be\rfloor)}\int_0^t\!\!\int_0^\tau\pa_\tau^{\lceil\be\rceil+1}v(\,\cdot\,,\tau;r)\int_\tau^t\f{\rd s}{(t-s)^{\be-\lfloor\be\rfloor}(s-\tau)^{\lceil\be\rceil-\be}}\,\rd r\rd\tau\nonumber\\
& =\Ga(\lceil\be\rceil-\be)\left(\int_0^t\pa_r^{\lceil\be\rceil}v(\,\cdot\,,r;r)\,\rd r+\int_0^t\!\!\int_0^\tau\pa_\tau^{\lceil\be\rceil+1}v(\,\cdot\,,\tau;r)\,\rd r\rd\tau\right).
\end{align}
On the other hand, since the operator $\cP$ is independent of $t$, we calculate $-\cP u$ as
\begin{align}
-\cP u(\,\cdot\,,t) & =-\int_0^t D_t^{\lceil\be\rceil-\be}\cP v(\,\cdot\,,t;s)\,\rd s=\int_0^t D_t^{\lceil\be\rceil-\be}\pa_t^\be v(\,\cdot\,,t;s)\,\rd s\nonumber\\
& =\int_0^t\f{\pa_t}{\Ga(\be-\lfloor\be\rfloor)}\int_s^t\f1{(t-r)^{\lceil\be\rceil-\be}}\f1{\Ga(\lceil\be\rceil-\be)}\int_s^r\f{\pa_\tau^{\lceil\be\rceil}v(\,\cdot\,,\tau;s)}{(r-\tau)^{\be-\lfloor\be\rfloor}}\,\rd\tau\rd r\rd s\nonumber\\
& =\f1{\Ga(\be-\lfloor\be\rfloor)\Ga(\lceil\be\rceil-\be)}\int_0^t\pa_t\int_0^{t-s}\f1{r^{\lceil\be\rceil-\be}}\int_s^{t-r}\f{\pa_\tau^{\lceil\be\rceil}v(\,\cdot\,,\tau;s)}{(t-r-\tau)^{\be-\lfloor\be\rfloor}}\,\rd\tau\rd r\rd s\nonumber\\
& =\f1{\Ga(\be-\lfloor\be\rfloor)\Ga(\lceil\be\rceil-\be)}\int_0^t\!\!\int_0^{t-s}\f{\pa_t}{r^{\lceil\be\rceil-\be}}\int_s^{t-r}\f{\pa_\tau^{\lceil\be\rceil}v(\,\cdot\,,\tau;s)}{(t-r-\tau)^{\be-\lfloor\be\rfloor}}\,\rd\tau\rd r\rd s\nonumber\\
& =\f1{\Ga(\be-\lfloor\be\rfloor)\Ga(\lceil\be\rceil-\be)}\left\{\int_0^t\!\!\int_0^{t-s}\f{\pa_s^{\lceil\be\rceil}v(\,\cdot\,,s;s)}{r^{\lceil\be\rceil-\be}(t-s-r)^{\be-\lfloor\be\rfloor}}\,\rd r\rd s+\I_3\right\}\nonumber\\
& =\int_0^t\pa_s^{\lceil\be\rceil}v(\,\cdot\,,s;s)\,\rd s+\f{\I_3}{\Ga(\be-\lfloor\be\rfloor)\Ga(\lceil\be\rceil-\be)},
\end{align}
where
\begin{align}
\I_3 & :=\int_0^t\!\!\int_0^{t-s}\f1{r^{\lceil\be\rceil-\be}}\int_0^{t-s-r}\f{\pa_t^{\lceil\be\rceil+1}v(\,\cdot\,,t-r-\tau;s)}{\tau^{\be-\lfloor\be\rfloor}}\,\rd\tau\rd r\rd s\nonumber\\
&=\int_0^t\!\!\int_s^t\f1{(r-s)^{\lceil\be\rceil-\be}}\int_r^t\f{\pa_t^{\lceil\be\rceil+1}v(\,\cdot\,,t-\tau+s;s)}{(\tau-r)^{\be-\lfloor\be\rfloor}}\,\rd\tau\rd r\rd s\nonumber\\
& =\int_0^t\!\!\int_0^\tau\pa_t^{\lceil\be\rceil+1}v(\,\cdot\,,t-\tau+s;s)\int_s^\tau\f{\rd r}{(\tau-r)^{\be-\lfloor\be\rfloor}(r-s)^{\lceil\be\rceil-\be}}\,\rd s\rd\tau\nonumber\\
& =\Ga(\be-\lfloor\be\rfloor)\Ga(\lceil\be\rceil-\be)\int_0^t\!\!\int_0^\tau\pa_t^{\lceil\be\rceil+1}v(\,\cdot\,,t-\tau+s;s)\,\rd s\rd\tau\nonumber\\
& =\Ga(\be-\lfloor\be\rfloor)\Ga(\lceil\be\rceil-\be)\int_0^t\!\!\int_0^{t-s}\pa_\tau^{\lceil\be\rceil+1}v(\,\cdot\,,\tau+s;s)\,\rd\tau\rd s\nonumber\\
& =\Ga(\be-\lfloor\be\rfloor)\Ga(\lceil\be\rceil-\be)\int_0^t\!\!\int_0^\tau\pa_\tau^{\lceil\be\rceil+1}v(\,\cdot\,,\tau;s)\,\rd s\rd\tau.\label{eq-Duhamel-pf3}
\end{align}
The combination of \eqref{eq-Duhamel-pf2}--\eqref{eq-Duhamel-pf3} immediately indicates
\begin{align*}
(\pa_t^\be+\cP)u(\,\cdot\,,t) & =\f{\I_1+\I_2}{\Ga(\lceil\be\rceil-\be)}-\int_0^t\pa_s^{\lceil\be\rceil}v(\,\cdot\,,s;s)\,\rd s-\f{\I_3}{\Ga(\be-\lfloor\be\rfloor)\Ga(\lceil\be\rceil-\be)}\\
& =F(\,\cdot\,,t)+\int_0^t\!\!\int_0^\tau\pa_\tau^{\lceil\be\rceil+1}v(\,\cdot\,,\tau;r)\,\rd r\rd\tau-\f{\I_3}{\Ga(\be-\lfloor\be\rfloor)\Ga(\lceil\be\rceil-\be)}=F(\,\cdot\,,t).
\end{align*}
Therefore, it is verified that the function $u$ defined by \eqref{eq-Duhamel} indeed satisfies \eqref{eq-Duhamel-u}, and the proof of Lemma \ref{lem-Duhamel} is completed.
\end{proof}

\begin{proof}[Proof of Lemma $\ref{lem-FP-V}$]
If $\Om$ is a bounded domain, the results follow immediately by the same argument as that in Sakamoto and Yamamoto \cite{SY11}. Henceforth we only deal with the unbounded case of $\Om=\BR^d$.

Recalling the definition of $S(\bm\xi)$ in Lemma \ref{lem-FP-V}, formally we have $\cF(\cL V(\,\cdot\,,t))(\bm\xi)=S(\bm\xi)\wt V(\bm\xi,t)$. Then taking Fourier transform in \eqref{eq-IBVP-V} with respect to the spatial variables yields a fractional ordinary differential equation with a parameter $\bm\xi$:
\[
\begin{cases}
(\pa_t^\al+S(\bm\xi))\wt V(\bm\xi,t)=0, & t>0,\\
\begin{cases}
\wt V(\bm\xi,0)=\wt v_0(\bm\xi) & \mbox{if }0<\al\le1,\\
\wt V(\bm\xi,0)=0,\ \pa_t\wt V(\bm\xi,0)=\wt v_1(\bm\xi) & \mbox{if }1<\al\le2.
\end{cases}
\end{cases}
\]
The solution to the above equation turns out to be
\begin{equation}\label{eq-sol-ODE}
\wt V(\bm\xi,t)=\begin{cases}
\wt v_{\lceil\al\rceil-1}(\bm\xi)t^{\lceil\al\rceil-1}E_{\al,\lceil\al\rceil}(-S(\bm\xi)t^\al), & 0<\al<2,\\
\wt v_1(\bm\xi)S(\bm\xi)^{-1/2}\sin(S(\bm\xi)^{1/2}t), & \al=2,
\end{cases}
\end{equation}
where $S(\bm\xi)\ge0$ for $\al=2$ because we assumed $\bm b=\bm0$ and $c\ge0$ in this case.

For any fixed $t\ge0$, our aim is to verify the boundedness of $\|V(\,\cdot\,,t)\|_{H^p(\BR^d)}$. In the case of $0<\al<2$, we have to estimate $|E_{\al,\lceil\al\rceil}(-S(\bm\xi)t^\al)|$. Denoting by $\ka>0$ the smallest eigenvalue of the strict positive definite matrix $\bm A$, we see
\[
\mathrm{Re}\,S(\bm\xi)=\bm A\bm\xi\cdot\bm\xi+c\ge\f\ka2|\bm\xi|^2\mbox{ for }|\bm\xi|\gg1,\quad|\mathrm{Im}\,S(\bm\xi)|=|\bm b\cdot\bm\xi|\le|\bm b||\bm\xi|.
\]
Hence, there exists a constant $R=R(\al)>0$ such that $\f{\pi\al}2<|\mathrm{arg}(-S(\bm\xi)t^\al)|\le\pi$ for $|\bm\xi|\ge R$. Then we can employ \eqref{eq-est-ML} to estimate
\[
|E_{\al,\lceil\al\rceil}(-S(\bm\xi)t^\al)|\le\f C{1+|S(\bm\xi)|t^\al}\le\f C{1+|\bm\xi|^2t^\al},\quad|\bm\xi|\ge R.
\]
For $|\bm\xi|<R$, it is readily seen that $|E_{\al,\lceil\al\rceil}(-S(\bm\xi)t^\al)|$ is uniformly bounded.

For $0<\al\le1$, we divide $\BR^d=B_R\cup(\BR^d\setminus B_R)$ and use \eqref{eq-sol-ODE} to estimate
\begin{align*}
\|V(\,\cdot\,,t)\|_{H^p(\BR^d)}^2 & =\left(\int_{B_R}+\int_{\BR^d\setminus B_R}\right)(1+|\bm\xi|^2)^p|\wt v_0(\bm\xi)|^2|E_{\al,1}(-S(\bm\xi)t^\al)|^2\,\rd\bm\xi\\
& \le C\int_{B_R}(1+|\bm\xi|^2)^p|\wt v_0(\bm\xi)|^2\,\rd\bm\xi+\int_{\BR^d\setminus B_R}(1+|\bm\xi|^2)^p|\wt v_0(\bm\xi)|^2\left(\f C{1+|\bm\xi|^2t^\al}\right)^2\,\rd\bm\xi\\
& \le C\int_{\BR^d}(1+|\bm\xi|^2)^p|\wt v_0(\bm\xi)|^2\,\rd\bm\xi=C\|v_0\|_{H^p(\BR^d)}^2.
\end{align*}
For $1<\al<2$, we utilize the same argument as above and the uniform boundedness of $\f{\ze^{1/\al}}{1+\ze}$ for $\ze\ge0$ to deduce
\begin{align*}
\|V(\,\cdot\,,t)\|_{H^p(\BR^d)}^2 & =t^2\left(\int_{B_R}+\int_{\BR^d\setminus B_R}\right)(1+|\bm\xi|^2)^p|\wt v_1(\bm\xi)|^2|E_{\al,2}(-S(\bm\xi)t^\al)|^2\,\rd\bm\xi\\
& \le C\,t^2\int_{B_R}(1+|\bm\xi|^2)^{2/\al}(1+|\bm\xi|^2)^{p-2/\al}|\wt v_1(\bm\xi)|^2\,\rd\bm\xi\\
& \quad\,+\int_{\BR^d\setminus B_R}(1+|\bm\xi|^2)^{p-2/\al}|\wt v_1(\bm\xi)|^2\left(\f{C((1+|\bm\xi|^2)t^\al)^{1/\al}}{1+|\bm\xi|^2t^\al}\right)^2\,\rd\bm\xi\\
& \le C(t^2+1)\int_{\BR^d}(1+|\bm\xi|^2)^{p-2/\al}|\wt v_1(\bm\xi)|^2\,\rd\bm\xi=C(t^2+1)\|v_1\|_{H^{p-2/\al}(\BR^d)}^2.
\end{align*}

In the case of $\al=2$, thanks to the assumption $\bm b=\bm0$ and $c\ge0$, we have $S(\bm\xi)\ge\ka|\bm\xi|^2$. Then we estimate \eqref{eq-sol-ODE} as
\begin{align*}
\|V(\,\cdot\,,t)\|_{H^p(\BR^d)}^2 & =\left(\int_{B_1}+\int_{\BR^d\setminus B_1}\right)\f{(1+|\bm\xi|^2)^p|\wt v_1(\bm\xi)||\sin(S(\bm\xi)^{1/2}t)|}{S(\bm\xi)}\,\rd\bm\xi\\
& \le\int_{B_1}(1+|\bm\xi|^2)^{p-1}|\wt v_1(\bm\xi)|^2(1+|\bm\xi|^2)\f{S(\bm\xi)t^2}{S(\bm\xi)}\,\rd\bm\xi+\int_{\BR^d\setminus B_1}(1+|\bm\xi|^2)^{p-1}|\wt v_1(\bm\xi)|^2\f{1+|\bm\xi|^2}{\ka|\bm\xi|^2}\,\rd\bm\xi\\
& \le C\,t^2\int_{B_1}(1+|\bm\xi|^2)^{p-1}|\wt v_1(\bm\xi)|^2\,\rd\bm\xi+C\int_{\BR^d\setminus B_1}(1+|\bm\xi|^2)^{p-1}|\wt v_1(\bm\xi)|^2\,\rd\bm\xi\\
& \le C(t^2+1)\|v_1\|_{H^{p-1}(\BR^d)}^2.
\end{align*}
The proof of Lemma \ref{lem-FP-V} is completed.
\end{proof}

\section{Proofs of the Main Results}\label{sec:4}

\begin{proof}[Proof of Theorem $\ref{thm-local}$]
Let $u_1,u_2$ be the solutions to \eqref{eq-IBVP-u} with orbits $\bm\ga_1,\bm\ga_2\in\cU_1$, respectively. Setting $w:=u_1-u_2$, it is easy to observe that $w$ satisfies the following initial(-boundary) value problem
\begin{equation}\label{eq-IVP-w}
\begin{cases}
(\pa_t^\al+\cL)w=G & \mbox{in }\Om\times(0,T),\\
\begin{cases}
w=0 & \mbox{if }0<\al\le1,\\
w=\pa_t w=0 & \mbox{if }1<\al\le2
\end{cases} & \mbox{in }\Om\times\{0\},\\
w=0\quad\mbox{if }\Om\mbox{ is bounded} & \mbox{on }\pa\Om\times(0,T),
\end{cases}
\end{equation}
where $G(\bm x,t):=g(\bm x-\bm\ga_1(t))-g(\bm x-\bm\ga_2(t))$. According to the mean value theorem, there exists a smooth function $\bm\eta:\Om\times(0,T)\longrightarrow\BR^d$ such that
\[
G(\bm x,t)=\nb g(\bm\eta(\bm x,t))\cdot\bm\rho(t)=\sum_{k=1}^d G_k(\bm x,t)\rho_k(t),
\]
where $\bm\eta(\bm x,t)$ is a point lying on the segment between $\bm x-\bm\ga_1(t)$ and $\bm x-\bm\ga_2(t)$, and
\[
\bm\rho:=\bm\ga_2-\bm\ga_1=(\rho_1,\ldots,\rho_d)^\T,\quad G_k(\bm x,t):=\pa_k g(\bm\eta(\bm x,t))\ (k=1,\ldots,d).
\]
Substituting the observation points $\bm x=\bm x^j$ ($j=1,\ldots,d$) into the governing equation of \eqref{eq-IVP-w}, we obtain
\begin{equation}\label{eq-linear1}
\sum_{k=1}^d G_k(\bm x^j,t)\rho_k(t)=\pa_t^\al w(\bm x^j,t)+\cL w(\bm x^j,t),\quad j=1,\ldots,d.
\end{equation}

In order to give a representation of $\cL w(\bm x^j,t)$, we take advantage of Lemma \ref{lem-Duhamel} to write $\cL w$ as
\begin{equation}\label{eq-rep-w}
\cL w(\,\cdot\,,t)=\int_0^t D_t^{\lceil\al\rceil-\al}\cL v(\,\cdot\,,t;s)\,\rd s,\quad0<t\le T,
\end{equation}
where $v$ satisfies the following homogeneous initial(-boundary) value problem with a parameter $s\in(0,T)$:
\[
\begin{cases}
(\pa_t^\al+\cL)v=0 & \mbox{in }\Om\times(s,T),\\
\begin{cases}
v=G(\,\cdot\,,s) & \mbox{if }0<\al\le1,\\
v=0,\ \pa_t v=G(\,\cdot\,,s) & \mbox{if }1<\al\le2
\end{cases} & \mbox{in }\Om\times\{s\},\\
v=0\quad\mbox{if }\Om\mbox{ is bounded} & \mbox{on }\pa\Om\times(s,T).
\end{cases}
\]
In the case of a bounded domain $\Om$, it follows from \eqref{eq-sol-IBVP-V} that
\[
\cL v(\,\cdot\,,t;s)=\sum_{n=1}^\infty\la_n(G(\,\cdot\,,s),\vp_n)(t-s)^{\lceil\al\rceil-1}E_{\al,\lceil\al\rceil}(-\la_n(t-s)^\al)\vp_n.
\]
Using Lemma \ref{lem-RL-ML}, we substitute the above equality into \eqref{eq-rep-w} with $\bm x=\bm x^j$ to represent
\begin{align}
\cL w(\bm x^j,t) & =\int_0^t\sum_{n=1}^\infty\la_n(G(\,\cdot\,,s),\vp_n)(t-s)^{\al-1}E_{\al,\al}(-\la_n(t-s)^\al)\vp_n(\bm x^j)\,\rd s\nonumber\\
& =\int_0^t\sum_{k=1}^d q_{jk}(t,s)\rho_k(s)\,\rd s,\quad j=1,\ldots,d,\label{eq-rep-Aw}
\end{align}
where
\begin{equation}\label{eq-IBVP-q}
q_{jk}(t,s):=(t-s)^{\al-1}\sum_{n=1}^\infty\la_n(G_k(\,\cdot\,,s),\vp_n)\vp_n(\bm x^j)E_{\al,\al}(-\la_n(t-s)^\al).
\end{equation}
In the case of $\Om=\BR^d$, we turn to the Fourier transform to see $\cF(\cL v(\,\cdot\,,t;s))=S\,\wt v(\,\cdot\,,t;s)$, where we recall $S(\bm\xi)=\bm A\bm\xi\cdot\bm\xi+\ri\,\bm b\cdot\bm\xi+c$. Then it follows from \eqref{eq-sol-IVP-V} that
\[
\cF(\cL v(\,\cdot\,,t;s))(\bm\xi)=\begin{cases}
S(\bm\xi)\wt G(\bm\xi,s)(t-s)^{\lceil\al\rceil-1}E_{\al,\lceil\al\rceil}(-S(\bm\xi)t^\al), & 0<\al<2,\\
S(\bm\xi)^{1/2}\wt G(\bm\xi,s)\sin(S(\bm\xi)^{1/2}(t-s)), & \al=2.
\end{cases}
\]
Taking the inverse Fourier transform in the above equality and applying Lemma \ref{lem-RL-ML} to \eqref{eq-rep-w} again, we obtain
\begin{align*}
\cL w(\,\cdot\,,t) & =\cF^{-1}\left(\int_0^t D_t^{\lceil\al\rceil-\al}\cF(\cL v(\,\cdot\,,t;s))\,\rd s\right)\\
& =\left\{\!\begin{alignedat}{2}
& \cF^{-1}\left(\int_0^t S(\bm\xi)\wt G(\bm\xi,s)(t-s)^{\al-1}E_{\al,\al}(-S(\bm\xi)(t-s)^\al)\,\rd s\right), & \quad & 0<\al<2,\\
& \cF^{-1}\left(\int_0^t S(\bm\xi)^{1/2}\wt G(\bm\xi,s)\sin(S(\bm\xi)^{1/2}(t-s))\,\rd s\right), & \quad & \al=2.
\end{alignedat}\right.
\end{align*}
By $\wt G(\bm\xi,s)=\sum_{k=1}^d\wt G_k(\bm\xi,s)\rho_k(s)$ and taking $\bm x=\bm x^j$ ($j=1,\ldots,d$), again we arrive at the expression \eqref{eq-rep-Aw}, where $q_{jk}(t,s)$ is defined by
\begin{equation}\label{eq-IVP-q}
q_{jk}(t,s):=\begin{cases}
(t-s)^{\al-1}\cF^{-1}(S(\bm\xi)\wt G_k(\bm\xi,s)E_{\al,\al}(-S(\bm\xi)(t-s)^\al))(\bm x^j), & 0<\al<2,\\
\cF^{-1}(S(\bm\xi)^{1/2}\wt G_k(\bm\xi,s)\sin(S(\bm\xi)^{1/2}(t-s)))(\bm x^j), & \al=2.
\end{cases}
\end{equation}

Since the expression \eqref{eq-rep-Aw} is valid for both bounded and unbounded $\Om$, we plug \eqref{eq-rep-Aw} into \eqref{eq-linear1} and rewrite it in form of a linear system as
\begin{equation}\label{eq-linear2}
\bm P(t)\bm\rho(t)=\pa_t^\al\bm h(t)+\int_0^t\bm Q(t,s)\bm\rho(s)\,\rd s,
\end{equation}
where $\bm h(t):=(w(\bm x^1,t),\ldots,w(\bm x^d,t))^\T$ and
\[
\bm P(t):=(G_k(\bm x^j,t))_{1\le j,k\le d},\quad\bm Q(t,s):=(q_{jk}(t,s))_{1\le j,k\le d}
\]
are $d\times d$ matrices. Recalling the admissible set $\cU_1$ for $\bm\ga_1$ and $\bm\ga_2$, we see that $\bm\eta(\bm x^j,t)\in B_\ve(\bm x^j)$ for all $t\in[0,T]$. Therefore, by $G_k(\bm x,t)=\pa_k g(\bm\eta(\bm x,t))$, the key assumption \eqref{eq-assume-obs} indicates that the matrix
\[
\bm P(t)=\begin{pmatrix}
\nb g(\eta(\bm x^1,t)) & \nb g(\eta(\bm x^2,t)) & \cdots & \nb g(\eta(\bm x^d,t))
\end{pmatrix}^\T
\]
is invertible for all $t\in[0,T]$. In other words, there exists a constant $C>0$ such that
\begin{equation}\label{eq-inv-P}
|\bm P(t)^{-1}|\le C,\quad\forall\,t\in[0,T].
\end{equation}

As for the matrix $\bm Q(t,s)$, it suffices {to estimate $|q_{jk}(t,s)|$ appearing} in \eqref{eq-IBVP-q} and \eqref{eq-IVP-q} separately. In the case of \eqref{eq-IBVP-q}, the uniform boundedness of $E_{\al,\al}(-\ze)$ for $\ze\ge0$ yields
\begin{align*}
|q_{jk}(t,s)| & \le C(t-s)^{\al-1}\left|\sum_{n=1}^\infty\la_n(G_k(\,\cdot\,,s),\vp_n)\vp_n(\bm x^j)\right|\\
& =C(t-s)^{\al-1}|\cL G_k(\bm x^j,s)|=C(t-s)^{\al-1}|\cL\pa_k g(\bm\eta(\bm x^j,t))|\\
& \le C(t-s)^{\al-1}\|\cL\pa_k(g\circ\bm\eta)(\,\cdot\,,s)\|_{C(\ov\Om)}.
\end{align*}
Since $g$ is a given smooth function and $\bm\eta$ is also smooth and depends only on the admissible set $\cU_1$, it turns out that $\|\cL\pa_k(g\circ\bm\eta)\|_{C(\ov\Om\times[0,T])}$ are uniformly bounded for $k=1,\ldots,d$, implying
\begin{equation}\label{eq-est-q}
|q_{jk}(t,s)|\le C(t-s)^{\al-1},\quad1\le j,k\le d.
\end{equation}
For \eqref{eq-IVP-q}, we deal with the cases of $0<\al<2$ and $\al=2$ separately. For $0<\al<2$, the similar argument to that in the proof of Lemma \ref{lem-FP-V} guarantees a constant $R=R(\al)>0$ such that $\f{\pi\al}2<|\mathrm{arg}(-S(\bm\xi)t^\al)|\le\pi$ for $|\bm\xi|\ge R$. Then we employ \eqref{eq-est-ML} to estimate
\begin{align*}
|E_{\al,\al}(-S(\bm\xi)(t-s)^\al)| & \le\left\{\!\begin{alignedat}{2}
& C, & \quad & |\bm\xi|\le R,\\
& \f C{1+|S(\bm\xi)|(t-s)^\al}, & \quad & |\bm\xi|\ge R
\end{alignedat}\right.\\
& \le C,\quad\bm\xi\in\BR^d,\ 0\le s<t\le T.
\end{align*}
On the other hand, it is readily seen that $|S(\bm\xi)|\le|\bm A\bm\xi\cdot\bm\xi|+|\bm b\cdot\bm\xi|+|c|\le C(1+|\bm\xi|^2)$. Thus, based on the definition of the inverse Fourier transform, we can estimate
\begin{align}
|q_{jk}(t,s)| & =(t-s)^{\al-1}\left|\cF^{-1}\left(S(\bm\xi)\wt G_k(\bm\xi,s)E_{\al,\al}(-S(\bm\xi)(t-s)^\al)\right)(\bm x^j)\right|\nonumber\\
& \le\f{(t-s)^{\al-1}}{(2\pi)^{d/2}}\int_{\BR^d}|S(\bm\xi)||\wt G_k(\bm\xi,s)||E_{\al,\al}(-S(\bm\xi)(t-s)^\al)||\e^{\ri\,\bm\xi\cdot\bm x^j}|\,\rd\bm\xi\nonumber\\
& \le C(t-s)^{\al-1}\int_{\BR^d}\left((1+|\bm\xi|^2)^{(d+5)/4}|\wt G_k(\bm\xi,s)|\right)(1+|\bm\xi|^2)^{-(d+1)/4}\,\rd\bm\xi\nonumber\\
& \le C(t-s)^{\al-1}\left(\int_{\BR^d}(1+|\bm\xi|^2)^{(d+5)/2}|\wt G_k(\bm\xi,s)|^2\right)^{1/2}\left(\int_{\BR^d}\f{\rd\bm\xi}{(1+|\bm\xi|^2)^{(d+1)/2}}\right)^{1/2}\label{eq-ineq-CS}\\
& \le C\|G_k(\,\cdot\,,s)\|_{H^{(d+5)/2}(\BR^d)}(t-s)^{\al-1},\nonumber
\end{align}
where we used the Cauchy-Schwarz inequality in \eqref{eq-ineq-CS}. Since $g,\bm\eta$ are smooth and $g$ is compactly supported, we see that $G_k=\pa_k(g\circ\bm\eta)$ is also smooth and compactly supported, indicating the uniform boundedness of $\|G_k(\,\cdot\,,s)\|_{H^{(d+5)/2}(\BR^d)}$ for $0<s<T$ and $k=1,\ldots,d$. Therefore, again we arrive at \eqref{eq-est-q} in the case $\Om=\BR^d$ with $0<\al<2$. Finally, for $\al=2$ we estimate in the same manner as
\begin{align*}
|q_{jk}(t,s)| & \le\f1{(2\pi)^{d/2}}\int_{\BR^d}S(\bm\xi)^{1/2}|\wt G_k(\bm\xi,s)||\sin(S(\bm\xi)^{1/2}(t-s))||\e^{\ri\,\bm\xi\cdot\bm x^j}|\,\rd\bm\xi\\
& \le C(t-s)\int_{\BR^d}S(\bm\xi)|\wt G_k(\bm\xi,s)|\,\rd\bm\xi\le C(t-s)\int_{\BR^d}(1+|\bm\xi|^2)|\wt G_k(\bm\xi,s)|\,\rd\bm\xi\\
& \le C(t-s)\left(\int_{\BR^d}(1+|\bm\xi|^2)^{(d+5)/2}|\wt G_k(\bm\xi,s)|^2\,\rd\bm\xi\right)^{1/2}\\
& \le C(t-s)\|G_k(\,\cdot\,,s)\|_{H^{(d+5)/2}(\BR^d)}\le C(t-s),\quad1\le j,k\le d,
\end{align*}
which is consistent with \eqref{eq-est-q}. Consequently, it reveals that the upper bound \eqref{eq-est-q} holds for both cases of domains and remains valid for any $0<\al\le2$. This together with \eqref{eq-norm} implies the estimate
\[
|\bm Q(t,s)|\le C\|\bm Q(t,s)\|_\F=C\left(\sum_{j,k=1}^d|q_{jk}(t,s)|\right)^{1/2}\le C(t-s)^{\al-1}.
\]
The combination of \eqref{eq-linear2}, \eqref{eq-inv-P} and the above estimate yields
\[
|\bm\rho(t)|\le|\bm P(t)^{-1}|\left(|\pa_t^\al\bm h(t)|+\int_0^t|\bm Q(t,s)||\bm\rho(s)|\,\rd s\right)\le C\left(|\pa_t^\al\bm h(t)|+\int_0^t(t-s)^{\al-1}|\bm\rho(s)|\,\rd s\right).
\]
Eventually, we employ {Gr\"onwall's} inequality with a weakly singular kernel (see Henry \cite[Lemma 7.1.1]{H81}) to conclude
\begin{align*}
|\bm\rho(t)| & \le C\left(|\pa_t^\al\bm h(t)|+C\int_0^t\left.\f\rd{\rd\ze}E_{\al,1}(\ze^\al)\right|_{\ze=C(t-s)}|\pa_s^\al\bm h(s)|\,\rd s\right)\\
& \le C\|\pa_t^\al\bm h\|_{C[0,T]}\left(1+\int_0^t s^{\al-1}E_{\al,\al}(C s^\al)\,\rd s\right)\le C\|\pa_t^\al\bm h\|_{C[0,T]}
\end{align*}
for $0<t\le T$, and hence
\begin{align*}
\|\bm\ga_1-\bm\ga_2\|_{C[0,T]} & =\|\bm\rho\|_{C[0,T]}\le C\|\pa_t^\al\bm h\|_{C[0,T]}\le C\left(\sum_{j=1}^d\|\pa_t^\al w(\bm x^j,\,\cdot\,)\|_{C[0,T]}^2\right)^{1/2}\\
& \le C\sum_{j=1}^d\|\pa_t^\al(u_1-u_2)(\bm x^j,\,\cdot\,)\|_{C[0,T]}.
\end{align*}
This completes the proof of Theorem \ref{thm-local}.
\end{proof}

\begin{remark} {In three dimensions, if the source moves on the plane $\{x_3=C\}$ where $C\in \BR$ is known, then two observation points are sufficient to imply the stability. Analogously,
if $d'\ (1\le d'<d, d>1)$ components of the orbit function are known, then the number of observation points can be reduced to $d-d'$.}
\end{remark}

\begin{proof}[Proof of Corollary $\ref{cor-global}$]
Fix the set $X$ of observation points and the constant $\ve$ in the assumption of Corollary \ref{cor-global}. For any $\bm\ga\in\cU_0$, by $\|\bm\ga'\|_{C[0,T]}\le K$ we know that $\bm\ga(t)\in\ov{B_\ve}$ for $t\le\ve/K$. Then we define
\[
T_\ell=\begin{cases}
\ve\ell/K, & \ell=0,1,\ldots,\lceil K T/\ve\rceil-1,\\
T, & \ell=\lceil K T/\ve\rceil
\end{cases}
\]
and consider the intervals $[T_{\ell-1},T_\ell]$ ($\ell=1,\ldots,\lceil K T/\ve\rceil$) successively.

We adopt an inductive argument and start from $\ell=1$. On $[T_0,T_1]=[0,T_1]$, the above observation implies $\|\bm\ga_i\|_{C[0,T_1]}\le\ve$ ($i=1,2$). Taking $\bm y=\bm0$ in \eqref{eq-assume-obs'}, we see that there exist $d$ observation points $\{\bm x^j(\bm0)\}_{j=1}^d\subset X$ satisfying \eqref{eq-assume-obs}. {Observing} that all assumptions in Theorem \ref{thm-local} are fulfilled, we utilize the uniqueness result in Theorem \ref{thm-local} to conclude that {the relation} $u_1(\bm x^j(\bm0),\,\cdot\,)=u_2(\bm x^j(\bm0),\,\cdot\,)$ ($j=1,\ldots,d$) on $[0,T_1]$ implies $\bm\ga_1=\bm\ga_2$ on $[0,T_1]$.

For general $\ell=2,\ldots,\lceil K T/\ve\rceil$, we make the induction hypothesis that the relation $u_1(\bm x^j,\,\cdot\,)=u_2(\bm x^j,\,\cdot\,)$ ($j=1,\ldots,N$) on $[0,T_{\ell-1}]$ implies $\bm\ga_1=\bm\ga_2$ on $[0,T_{\ell-1}]$. By the well-posedness of the forward problem, we have $u_1=u_2$ in $\Om\times[0,T_{\ell-1}]$. Introducing
\[
w_\ell(\bm x,t):=(u_1-u_2)(\bm x,t+T_{\ell+1}),
\]
we immediately see that $w_\ell$ satisfies the equation
\[
(\pa_t^\al+\cL)w_\ell=g(\bm x-\bm\ga_1(t+T_{\ell-1}))-g(\bm x-\bm\ga_2(t+T_{\ell-1}))
\]
in $(\bm\ga_1(T_{\ell-1})+\Om)\times(0,T-T_{\ell-1})$ with the homogeneous initial(-boundary) condition. Repeating the same argument as that in the proofs of Theorem \ref{thm-local} and the case of $\ell=1$, we can take $\bm y=\bm y_\ell:=\bm\ga_1(T_{\ell-1})$ in \eqref{eq-assume-obs'}, so that again we can find $\{\bm x^j(\bm y_\ell)\}_{j=1}^d\subset X$ such that \eqref{eq-assume-obs} is fulfilled with $\bm x^j$ replaced by $\bm x^j(\bm y_\ell)-\bm y_\ell$ ($j=1,\ldots,d$). Since all assumptions in Theorem \ref{thm-local} are satisfied in $(\bm\ga_1(T_{\ell-1})+\Om)\times(0,T_\ell-T_{\ell-1})$, we conclude that $w_\ell(\bm x^j(\bm y_\ell),\,\cdot\,)=0$ ($j=1,\ldots,d$) on $[0,T_\ell-T_{\ell-1}]$ implies $\bm\ga_1-\bm\ga_2=\bm0$ on $[T_{\ell-1},T_\ell]$ or equivalently,
\begin{equation}\label{eq-induction}
u_1(\bm x^j(\bm y_\ell),\,\cdot\,)=u_2(\bm x^j(\bm y_\ell),\,\cdot\,)\ (j=1,\ldots,d)\mbox{ on }[T_{\ell-1},T_\ell]\mbox{ implies }\bm\ga_1=\bm\ga_2\mbox{ on }[T_{\ell-1},T_\ell].
\end{equation}
By the {inductive argument,  for any } $\ell=1,\ldots,\lceil K T/\ve\rceil$ there exists a set of $d$ observation points $\{\bm x^j(\bm y_\ell)\}_{\ell=1}^d\subset X$ ($\bm y_\ell=\bm\ga_1(T_{\ell-1})$) such that \eqref{eq-induction} holds. Consequently, the proof is completed by collecting the uniqueness on all intervals $[T_{\ell-1},T_\ell]$.
\end{proof}

\section*{Acknowledgement}
This work is partly supported by the A3 Foresight Program ``Modeling and Computation of Applied Inverse Problems'', Japan Society for the Promotion of Science (JSPS) and National Natural Science Foundation of China (NSFC).
G. Hu is supported by the NSFC grant (No. 11671028) and NSAF grant (No. U1930402).
Y. Liu and M. Yamamoto are supported by JSPS KAKENHI Grant Number JP15H05740.
M. Yamamoto is partly supported by NSFC (Nos. 11771270, 91730303) and RUDN University Program 5-100.

redpajama_set_name: RedPajamaArXiv


The Lord s blessing so rested upon the ministry of his servants among a society which they had found in a distracted and dwindling condition, that, at the end of the year, they had the gratification to report not only the return of many of the wanderers but an accession of more than a hundred members. The congregations, too, had become immense. The people of the towns, and the marine population of the ships in the Hamoaze, came in crowds to hear the word of God.


text:
Q: Empty sys/class/leds On my 8.1 oreo android tablet, I have a front RGB led. I'm trying to change the color of it by using ADB commands, however I can't find the led in the system. I searched for everywhere (under sys/class, sys/devices, virtual, platform, software directories but nothing)
There is a suggested method described below as all leds info are stored under '/sys/class/leds/' directory, but this folder is totally empty in my case.
Control Android LED from shell
I contacted the manufacturer, but they refused to answer as don't want to disclose internal system structures to anybody.
I'm asking for your experience. Is there any command I can use to find and control it with ADB?

A: Handling LEDs via /sys/class/leds folder is a part of Linux sysfs. Most likely the phone vendor (you didn't provide this info, but it matters in this case) didn't register phone's LEDs with sysfs, see LED registration API.
I have tested on my Xiaomi Redmi Note 8 (Android 10, MIUI 12.0, Linux kernel 4.14) and it does have /sys/class/leds, I even was able to turn on/off flashlight there. But - it needed root access which I got after jailbreaking my phone, and it means that you can't deal with LEDs in stock firmware.
You probably can get that LEDs sysfs functionality on your phone installing a custom Android firmware such as LineageOS, but it's not guaranteed, and with that firmware you may get other problems related to its stability and functionality on your phone.
So you can hardly rely on this solution in general case, may be in some cases only, and probably with rooting the phone.

redpajama_set_name: RedPajamaStackExchange


I'm not at all surprised to see Marina and the Diamonds at the top. It was on pretty much constantly whilst I was emptying my family home as it was (mostly) uplifting, and had a decent tempo to keep me going.
It's interesting to see Stephen Fry in the Top Artists, and the only reason for that was because his auto-biography, which I listened to as an audio book, was broken into tiny chunks, I'm guessing that there were 131 tracks to it.
It's quite amusing to me to find the Seekers in the Top Albums. This was an album I bought whilst emptying the house as I found a lot of Seekers records amongst Mum's record collection and was curious.
Metric (just), Vampire Weekend, Ra Ra Riot and Muse are still in the top artist lists.


Every day somewhere on the metro there is a problem.
You can't arrange to go anywhere because you don't know if you are going to get there. It's about time you started getting rebates no matter what the problem is.
Bring back the buses and do away with the Metro because it's as much use as a chocolate poker.


text:
multiPlan arhitekti
The Ring Stadium / OFIS arhitekti + multiPlan arhitekti
OFIS Architects, multiPlan arhitekti
Maribor, Slovenia
OFIS arhitekti & Multiplan arhitekti
Competition Year
Concrete, Steel, Glass
Football Stadium wich includes: Covered tribunes, public programs such as Gyms, Fitness club with swimming pool, shops, restaurants
Text description provided by the architects. The project is a result of the winning competition back in 1998. The plot that was used as a multi-functional sport field is located in the centre of the city. In the sixties a small tribune was build along one site of the field that was covered with huge concrete arched roof.
The brief was to convert the field into a football stadium and extend the existing building with covered tribunes (12.500 spectators, VIP and press facilities) and additional public programme such as 4 big gymnasiums, fitness-club with swimming pools, shops and restaurants.
The project proposed a ring of tribunes weaving above enclosed base with public programs.
The lowest and the highest point of the tribunes are defined by the quality of the view of the spectator. In the corners of the field, where the views are restricted, the entrances to the tribunes are displaced. The ring is pulled down here to the level of the entry plateau. Then it rises gently and it reaches the highest points in the middle of the field. There the maximum number of the seats is provided, offering the best views to the field. The corridor of the ring has double skin and is providing rooms for VIP, Press and refreshments.
In the base sport halls and shops are displaced. The roof of the base forms an entry plateau to the tribunes and also offers views to the smaller sport field at the side of the stadium.
OFIS Architects
Glass Steel Concrete
Projects Built Projects Selected Projects Sports Architecture Stadiums Sports ArchitectureMariborSlovenia
Cite: "The Ring Stadium / OFIS arhitekti + multiPlan arhitekti" 14 Jun 2008. ArchDaily. Accessed . <https://www.archdaily.com/2391/the-ring-stadium-ofis-arhitekti-multiplan-arhitekti/> ISSN 0719-8884


A BIG thank you to all our lovely Raffle Donors (singbentleyheath.org.uk/raffle) who gave us gifts and vouchers for prizes. Our raffle team raised over £1,000 at Spring Sing! 2019 which will be shared with Parkinson's UK and Admiral Nurses.
We have some prizes unclaimed at the end of the night so please do check all your raffle tickets before you throw them away – if you have tickets WHITE 10, WHITE 101, WHITE 140, PEACH 235 or PINK 225 then please contact the choir through the website or social media as soon as possible.
We have already started our planning for Spring Sing! 2020 and we would appreciate any feedback – good or bad – from audience and choir members. Please take a moment to send us your thoughts through our website (singbentleyheath.org.uk/feedback) or a section rep.