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# Project Excalibur

This early artwork shows an Excalibur firing at three nearby targets. In most descriptions, each could fire at dozens of targets, which would be hundreds or thousands of kilometers away.

Project Excalibur was a Lawrence Livermore National Laboratory (LLNL) Cold War-era research program to develop an X-ray laser as a ballistic missile defense (BMD) for the United States.[1] The concept involved packing large numbers of expendable X-ray lasers around a nuclear device. When the device detonated, the X-rays released by the bomb would be focused by the lasers, each of which would be aimed at a target missile.[2] When detonated in space, the lack of atmosphere to block the X-rays allowed attacks on missiles thousands of kilometers away.

Excalibur appeared to offer an enormous leap forward in BMD performance. Previously, missile-based anti-ballistic missile (ABM) systems faced the problem that they attacked the warheads, not the ICBM missiles that launched them; a single ICBM could carry multiple warheads in a MIRV system, so if the attacker added a single new missile to their fleet, dozens of interceptors would have to be built to counter it. Excalibur would attack the missiles before the warheads separated, and a single Excalibur contained as many as 50 lasers and could potentially shoot down a corresponding number of missiles.[a] A single additional Excalibur would require dozens of ICBMs to counter it, dramatically reversing the cost-exchange ratio that had previously doomed ABM systems.

The basic concept behind Excalibur was conceived in the 1970s by George Chapline, Jr. and further developed by Peter L. Hagelstein, both part of Edward Teller's "O-Group" in LLNL. After a successful test in 1980, in 1981 Teller and Lowell Wood began talks with US President Ronald Reagan about the concept. These talks, combined with strong support from a like-minded group that met at the Heritage Foundation, were a major part of the series of events that ultimately led Reagan to announce the Strategic Defense Initiative (SDI) in 1983.[1] Further underground nuclear tests through the early 1980s suggested progress was being made, and this influenced the 1986 Reykjavík Summit, where Reagan refused to give up the possibility of proof-testing SDI technology with nuclear testing in space.[3]

Researchers at Livermore and Los Alamos began to raise concerns about the test results. Teller and Wood continued to state the program was proceeding well, even after a critical test in 1985 demonstrated it was not working as expected. This led to significant criticism within the US weapons laboratories. In 1987, the infighting became public, leading to an investigation on whether LLNL had misled the government about the Excalibur concept. In a 60 Minutes interview in 1988, Teller attempted to walk out rather than answer questions about the lab's treatment of a fellow worker who questioned the results.[4] Further tests revealed additional problems, and in 1988 the budget was cut dramatically. The project officially continued until 1992 when its last planned test, Greenwater, was cancelled.[5]

## History

### Conceptual development

The conceptual basis of short-wavelength lasers, using X-rays and gamma rays, are the same as their visible-light counterparts. There were discussions of such devices as early as 1960, the year the first ruby laser was demonstrated.[6]

The first announcement of a successful X-ray laser was made in 1972 by the University of Utah. Researchers spread thin layers of copper atoms on microscope slides and then heated them with pulses from a neodymium glass laser. This caused spots to appear on X-ray film in the direction of the layers and none in other directions. The announcement caused great excitement, but it was soon overshadowed by the fact that no other labs could reproduce the results, and the announcement was soon forgotten.[6] In 1974, the University of Paris-Sud announced lasing in an aluminum plasma created by a pulse of laser light, but, once again, the claimed results were regarded skeptically by other labs.[7]

DARPA had been funding low-level research into high-frequency lasers since the 1960s. By late 1976 they had all but given up on them. They commissioned a report by Physical Dynamics, which outlined possible uses of such a laser, including space-based weapons. None of these seemed promising, and DARPA dropped funding for X-ray laser research in favor of the more promising free electron laser.[8]

In June 1977, two well-known Soviet researchers, Igor Sobel'man, and Vladilen Letokhov, displayed a film exposed the output of plasmas of chlorine, calcium and titanium, similar to the Utah results. They were careful to point out that the results were very preliminary and that further study was required. Over the next few years, a small number of additional papers on the topic were presented. The most direct of these was Sobel'man's statements at a 1979 conference in Novosibirsk when he stated that he was observing lasing in a calcium plasma. As with earlier announcements, these results were met with skepticism.[8]

### First attempts at Livermore

George Chapline had been studying the X-ray laser concept through the 1970s. Chapline was a member of Teller's speculative-project "O-Group" and began to discuss the concept with fellow O-Group member Lowell Wood, Teller's protégé.[9] The two collaborated on a major review of the X-ray laser field in 1975. They suggested such a device would be a powerful tool in materials science, for making holograms of viruses where a conventional laser's longer wavelength did not provide the required optical resolution, and as a sort of flashbulb for taking images of the nuclear fusion process in their inertial confinement fusion devices. This review contained the calculations that demonstrated both the rapid reaction times needed in such a device and the extremely high energies required for pumping.[10]

I instantly put together the ideas I had gotten from Sobelman's talk with the results of the experiment, and in five minutes came up with the general idea of something that would most likely work to make an X-ray laser with a nuclear device.

George Chapline[10]

Chapline attended a meeting where Sobel'man's work on X-ray lasers was presented. He had learned of the unique underground nuclear tests made on behalf of the Defense Nuclear Agency (DNA), where the burst of X-rays produced by the nuclear reactions were allowed to travel down a long tunnel while the blast itself was cut off by large doors that slammed shut as the explosion approached. These tests were used to investigate the effects of X-rays from exoatmospheric nuclear explosions on reentry vehicles. He realized this was a perfect way to illuminate an X-ray laser.[10]

After a few weeks of work, he came up with a testable concept. At this time the DNA was making plans for another of its X-ray effects tests, and Chapline's device could easily be tested in the same "shot". The test shot, Diablo Hawk, was carried out on 13 September 1978 as part of the Operation Cresset series. However, the instrumentation on Chapline's device failed, and there was no way to know if the system had worked or not.[10]

It was at this time that Congress directed that $10 million be given to both LLNL and Los Alamos National Laboratory (LANL) for weapons tests on entirely new concepts. Chapline was given the go-ahead to plan for a new test that would be dedicated to the X-ray laser concept. In the DNA tests, the reentry vehicle had to be retrieved for study after the test, which demanded the complex system of protective doors and other techniques that made these tests very expensive. For the X-ray laser test, all of this could be ignored, as the laser was designed to be destroyed in the explosion. This allowed the laser to be placed at the top of the vertical access shaft, which greatly lowered the cost of the test from the typical$40 million needed in a DNA shot.[11] Given the schedule at the Nevada Test Site, their test would have to wait until 1980.[12]

### Dauphin success

George Chapline, Jr. (right) and George Maenchen (left) at the world's first X-ray laser prior to the Dauphin underground nuclear test.

Peter Hagelstein was in an undergraduate physics program at MIT in 1974 when he applied for a Hertz Foundation scholarship. Teller was on the Hertz board, and Hagelstein soon had an interview with Lowell Wood. Hagelstein won the scholarship, and Wood then went on to offer him a summer position at LLNL. He had never heard of the lab, and Wood explained they were working on lasers, fusion, and similar concepts. Hagelstein arrived in May 1975, but nearly left when he found the area to be "disgusting" and immediately surmised they were working on weapons research when he saw the barbed wire and armed guards. He stayed on only because he met interesting people.[13]

Hagelstein was given the task of simulating the X-ray laser process on LLNL's supercomputers. His program, known as XRASER for "X-Ray laser", eventually grew to about 40,000 lines of code.[14] He received his Masters in 1976 and took a full-time job at the lab intending to lead the development of a working laser. The idea was to use the lab's powerful fusion lasers as an energy source, as Hagelstein and Wood had suggested in their review paper. Hagelstein used XRASER to simulate about 45 such concepts before he found one that appeared to work.[10] These used the lasers to heat metal foils and give off X-rays, but by the late 1970s, none of these experiments had been successful.[14]

After the Diablo Hawk failure, Hagelstein reviewed Chapline's idea and came up with a new concept that should be much more efficient. Chapline had used a lightweight material, a fiber taken from a local weed, but Hagelstein suggested using a metal rod instead. Although initially skeptical, Wood came to support the idea and successfully argued that both concepts be tested in Chapline's shot.[10] The critical test was carried out on 14 November 1980 as Dauphin, part of Operation Guardian. Both lasers worked, but Hagelstein's design was much more powerful.[10] The lab soon decided to move forward with Hagelstein's version, forming the "R Program", led by another O-Group member, Tom Weaver.[15]

### Renewed interest

The Novette laser provided the energy needed for Hagelstein's first successful X-ray laser.

Hagelstein published his PhD thesis in January 1981 on the "Physics of short-wavelength-laser design".[16] In contrast to Chapline and Wood's earlier work which focused on civilian applications, the thesis' introduction mentions several potential uses, even weapons taken from science fiction works.[17]

Hagelstein soon returned to the civilian side of the X-ray laser development, initially developing a concept in which the lab's fusion lasers would produce a plasma whose photons would pump another material. This was initially based on fluorine gas confined inside a chromium foil film. This proved to be too difficult to manufacture, so a system more like the earlier Soviet concepts was developed. The laser would deposit enough energy in a selenium wire to cause 24 of the electrons to be ionized, leaving behind 10 electrons that would be pumped by collisions with the free electrons in the plasma.[10]

After several attempts using the Novette laser as an energy source, on 13 July 1984 the system worked for the first time. The team calculated that the system produced laser amplification of about 700, which they considered to be strong evidence of lasing. Dennis Matthews presented the success at the October 1984 American Physical Society Plasma Physics Meeting in Boston, where Szymon Suckewer of Princeton University presented their evidence of lasing in carbon using a much smaller laser and confined the plasma using magnets.[10]

### Teller in Washington, AvWeek "leaks"

The success of the Dauphin test presented a potential new solution to the BMD problem. The X-ray laser offered the possibility that many laser beams could be generated from a single nuclear weapon in orbit, meaning a single weapon would destroy many ICBMs. This would so blunt the attack that any US response would be overwhelming in comparison. Even if the Soviets launched a full-scale attack, it would limit US casualties to 30 million.[18] In February 1981, Teller and Wood traveled to Washington to present the technology to the policy makers and request greater financial support to pursue the development.[19]

This presented a problem. As fellow LLNL physicist Hugh DeWitt put it, "It has long been known that Teller and Wood are extreme technological optimists and super salesman for hypothetical new weapons systems",[20] or as Robert Park puts it, "Anyone who knows Teller's record recognizes that he is invariably optimistic about even the most improbable technological schemes."[21] Although this salesmanship had little effect in US military circles, it proved to be a continual annoyance in Congress, having a negative effect on the lab's credibility when these concepts failed to pan out. To avoid this, Roy Woodruff, the associate director of the weapons section, went with them to ensure that the two did not oversell the concept. In meetings with various congressional groups, Teller and Wood explained the technology but refused to give dates on when it might be available.[22]

Only days later, the 23 February 1981 edition of Aviation Week and Space Technology carried an article on the ongoing work.[23] It described the Dauphin shot in some detail, going on to mention the earlier 1978 test, but incorrectly ascribing that to a Krypton fluoride laser (KrF).[b] It went on to describe the battle-station concept in which a single bomb would be surrounded by laser rods that could shoot down as many as 50 missiles, and stated that "X-Ray lasers based on the successful Dauphin test are so small that a single payload bay on the Space Shuttle could carry to orbit a number sufficient to stop a Soviet nuclear weapons attack."[22] This was the first in a series of such articles in this and other sources based on a "steady leak of top secret information."[25]

### High Frontier

Karl Bendetsen chaired the efforts that would eventually present the basis for SDI to Reagan. Excalibur was one of three major concepts that were studied by the group.

By this time, LLNL was not the only group lobbying the government about space-based weapons. In 1979, Daniel O. Graham had been asked by Ronald Reagan to begin exploring the idea of missile defense, and in the years since had become a strong advocate of what was earlier known as Project BAMBI (Ballistic Missile Boost Intercept),[26] but now updated as "Smart Rocks". This required dozens of large satellites carrying many small, relatively simple missiles that would be launched at the ICBMs and track them like a conventional heat seeking missile.[27]

That same year, Malcolm Wallop and his aide Angelo Codevilla wrote an article on "Opportunities and Imperatives in Ballistic Missile Defense", which was to be published later that year in Strategic Review. They were later joined by Harrison Schmidt and Teller in forming what became known as the "laser lobby", advocating the building of laser-based BMD systems. Their concept, known simply as the Space Based Laser, used large chemical lasers placed in orbit.[28]

Graham was able to garner interest from other Republican supporters, and formed a group that would help advocate for his concept. The group was chaired by Karl Bendetsen and was provided space at the Heritage Foundation.[27] The group invited the laser lobby to join them to plan a strategy to introduce these concepts to the incoming president.[27]

At one of the Heritage meetings, Graham pointed out a serious problem for the Excalibur concept. He noted that if the Soviets launched a missile at the satellite, the US had only two choices – they could allow the missile to hit Excalibur and destroy it, or it could defend itself by shooting down the missile, which would also destroy Excalibur. In either case, a single missile would destroy the station, which invalidated the entire concept of the system in terms of having a single weapon that would destroy a large portion of the Soviet fleet.[29]

At the time, Teller was stumped. At the next meeting, he and Wood had an answer, apparently Teller's own concept; instead of being based on satellites, Excalibur would be placed in submarines and "pop-up" when the Soviets launched their missiles. This would also bypass another serious concern, that nuclear weapons in space were outlawed and it was unlikely the government or public would allow these.[29]

The group first met with the president on 8 January 1982. Planned to last 15 minutes, the meeting went on for an hour. Present were Teller, Bendetsen, William Wilson and Joseph Coors of the "Kitchen Cabinet". Graham and Wallop were not represented and the group apparently dismissed their concepts.[30] The same group met with the president another three times.[30][31]

Meanwhile, Teller continued to attack Graham's interceptor-based concept, as did other members of the group. There had been extensive studies on BAMBI in the 1960s and every few years since. These invariably reported the concept was simply too grandiose to work. Graham, seeing the others outmaneuver him after the first meetings, left the group and formed "High Frontier Inc.", publishing a glossy book on the topic in March 1982. Before publication, he had sent a copy to the US Air Force for comment. They responded with another report that stated the concept "had no technical merit and should be rejected."[32] In spite of this review, the High Frontier book was widely distributed and quickly found followers. This led to a curious situation in early 1982, later known as the "laser wars", with the House supporting Teller and the Senate supporting Wallop's group.[30]

### Further tests, instrumentation issues

Only a few days after Reagan's speech, on 26 March 1983, the second test of Hagelstein's design was carried out as part of the Cabra shot in the Operation Phalanx test series. Instrumentation again proved to be a problem and no good results were obtained. The identical experiment was carried out on 16 December 1983 in the Romano shot of the following Operation Fusileer series. This test showed gain and lasing.[48]

On 22 December 1983, Teller wrote a letter on LLNL letterhead to Keyworth stating that the system had concluded its scientific phase and was now "entering engineering phase".[49] When Woodruff learned of the letter he stormed into Teller's office and demanded that he send a retraction. Teller refused, so Woodruff wrote his own, only to be ordered not to send it by Roger Batzel, the lab's director.[50] Batzel rebuffed Woodruff's complaints, stating that Teller was meeting the President as a private citizen, not on behalf of Livermore.[51]

Shortly after, LLNL scientist George Maenchen circulated a memo noting that the instrument used to measure the laser output was subject to interactions with the explosion. The system worked by measuring the brightness of a series of beryllium reflectors when they were illuminated by the lasers. Maenchen noted that the reflectors themselves could give off their own signals when heated by the bomb, and unless they were separately calibrated, there was no way to know if the signal was from the laser or the bomb.[38] This calibration had not been carried out, rendering the results of all of these tests effectively useless.[52][53]

By this time, Los Alamos had been developing nuclear anti-missile weapons of its own, updated versions of the 1960s Casaba/Howitzer concepts. Given the constant stream of news about Excalibur, they added a laser to one of their own underground tests, shot Correo, also part of the Fusileer series. The 2 August 1984 test used different methods to measure the laser output, and these suggested that little or no lasing was taking place. George Miller received a "caustic" letter from Paul Robinson of Los Alamos, which stated they "doubted the existence of the X-ray laser had been demonstrated and that Livermore managers were losing their credibility because of their failure to stand up to Teller and Wood."[54]

### Concerned Scientists present concerns

The Union of Concerned Scientists presented a criticism of Excalibur in 1984 as part of a major report on the entire SDI concept. They noted that a key problem for all of the directed energy weapons was that they only worked in space, as the atmosphere quickly disperses the beams. This meant that the systems had to intercept the missiles when they were above the majority of the atmosphere. Additionally, all of the systems relied on using infrared tracking of the missiles, as radar tracking could be easily rendered unreliable using a wide variety of countermeasures. Thus, the interception had to take place in the period where the missile motor was still firing. This left only a brief period in which the directed energy weapons could be used.[55]

The report claimed that this could be countered by simply increasing the thrust of the missile. Existing missiles fired for about three to four minutes,[56] with at least half of that taking place outside the atmosphere.[c] They showed that it was possible to reduce this to about a minute, timing things so the motor was burning out just as the missile was reaching the upper atmosphere. If the warheads were quickly separated at that point, the defense would have to shoot at the individual warheads, thus facing the same poor cost-exchange ratios that had made the earlier ABM systems effectively useless. Once the rocket had stopped firing, tracking would be far more difficult.[55]

One of the key claims for the Excalibur concept was that a small number of weapons would be enough to counter a large Soviet fleet, whereas the other space-based systems would require huge fleets of satellites. The report singled out Excalibur as particularly vulnerable to the problem of quick-firing missiles because the only way to address this would be to build many more weapons so more would be available in the remaining short time window. At that point, it no longer had any advantage over the other systems, while still having all of the technical risks. The report concluded that the X-ray laser would "offer no prospect of being a useful component" of a BMD system.[55]

### Excalibur+ and Super-Excalibur

Faced with the twin problems of the original experiments apparently failing, and the publication of a report showing that it could be easily defeated even if it worked, Teller and Wood responded by announcing the Excalibur Plus concept, which would be 1,000 times more powerful than the original Excalibur. Soon after, they added Super-Excalibur, which was another thousand times more powerful than Excalibur Plus, making it a trillion times as bright as the bomb itself.[38][58][d]

Super-Excalibur would be so powerful that it would even be able to burn through the atmosphere, thereby countering the concerns about fast-firing missiles. The extra power also meant it could be divided up into more beams, making a single weapon able to be directed into as many as a hundred thousand beams. Instead of dozens of Excalibur weapons in pop-up launchers, Teller suggested that a single weapon in geostationary orbit "the size of an executive desk which applied this technology could potentially shoot down the entire Soviet land-based missile force if it were to be launched into the module's field of view".[38][59][e]

At this point, no detailed theoretical work on the concepts had been carried out, let alone any practical tests. In spite of this, Teller once again used LLNL letterhead to write to several politicians telling them of the great advance. This time Teller copied Batzel, but not Woodruff. Once again Woodruff asked to send a counterpoint letter, only to have Batzel refuse to let him send it.[38]

### Cottage test

Super-Excalibur was tested on the 23 March 1985 Cottage shot of Operation Grenadier, exactly two years after Reagan's speech. Once again the test appeared to be successful, and unnamed researchers at the lab were reported to have stated that the brightness of the beam had been increased six orders of magnitude (i.e. between 1 and 10 million times), a huge advance that would pave the way for a weapon.[61][62]

Teller immediately wrote another letter touting the success of the concept. This time he wrote to Paul Nitze, the head negotiator of START, and Robert McFarlane, head of the US National Security Council. Nitze was about to begin negotiations on the START arms limitations talks. Teller stated that Super-Excalibur would be so powerful that the US should not seriously negotiate on any sort of even footing and that the talks should be delayed because they included limits or outright bans on underground testing that would make further work on Super-Excalibur almost impossible.[49]

Commenting on the results, Wood set an optimistic tone, stating that "Where we stand between inception and production I can't tell you... [but] I am much more optimistic now about the utility of x-ray lasers in strategic defense that when we started." In contrast, George H. Miller, LLNL's new deputy associate director, set a much more cautious tone, stating that while the lasing action had been demonstrated, "what we have not proven is whether you can make a militarily useful x-ray laser. It's a research program where a lot of the physics and engineering issues are still be examined..."[62]

Several months later, physicists at Los Alamos reviewed the Cottage results and noted the same problem that Maenchen had pointed out earlier. They added such calibration to a test they were already carrying out and found that the results were indeed as bad as Maenchen has suggested. The targets contained oxygen that glowed when heated and produced spurious results.[38] On top of this, Livermore scientists studying the results noted that the explosion created sound waves in the rod before the lasing was complete, ruining the focus of the laser. A new lasing medium would be required.[62]

Livermore ordered an independent review of the program by Joseph Nilsen, who delivered a report on 27 June 1985 that agreed the system was not working.[53] Given the gravity of the situation, a further review by the JASONs was carried out on 26 and 27 September and came to the same conclusion. It now appeared that there was no conclusive evidence that any lasing had been seen in any of the tests, and if it had, it was simply not powerful enough.[53]

In July, Miller went to Washington to brief the SDI Office (SDIO) on their progress. While the instrumentation concerns had been publicly reported on multiple occasions by this point, he failed to mention these issues. Several sources noted this, and one stated they "were furious because Miller used the old view graphs on the experiment, which did not take into account the new disturbing findings."[53]

### Brilliant Pebbles begins

Brilliant Pebbles replaced Excalibur as LLNL's contribution to the SDI efforts. It became the centerpiece of post-SDI programs, until the majority of the original SDI concepts were canceled in 1993.

With Excalibur effectively dead, in 1987 Teller and Wood began pitching Wood's new concept, Brilliant Pebbles.[h] They first presented this to Abrahamsom in October and followed up with a March 1988 meeting with Reagan and his aides. The new concept used a fleet of about 100,000 small independent rockets that would weight about 5 pounds (2.3 kg) each and would destroy the missiles or warheads by colliding with them, no explosive required. Because they were independent, attacking them would require an equally huge number of interceptors. Better yet, the entire system could be developed in a few years and would cost $10 billion for a complete fleet.[92] Brilliant Pebbles was essentially an updated version of the Project BAMBI concepts Graham had been suggesting in 1981. At that time, Teller had continually derided the idea as "outlandish" and used his influence to ensure the concept did not receive serious attention. Ignoring his previous concerns with the concept, Teller went on to promote Brilliant Pebbles using arguments he had previously dismissed when raised about Excalibur; among them, he now stressed that the system did not place or explode nuclear weapons in space. When critics pointed out that the idea fell prey to the issues raised by the Union of Concerned Scientists, Teller simply ignored them.[92] In spite of all of these red flag issues, and the decades-long string of Air Force and DARPA reports suggesting the concept just would not work, Reagan once again enthusiastically embraced their latest concept. By 1989 the weight of each pebble had grown to 100 pounds (45 kg) and the cost of a small fleet of 4,600 of them had ballooned to$55 billion. It remained the centerpiece of the US BMD efforts into 1991 when the numbers were further cut to somewhere between 750 and 1,000. President Clinton indirectly canceled the project on 13 May 1993 when the SDI office was reorganized as the Ballistic Missile Defense Organization (BMDO)[92] and focused their efforts on theater ballistic missiles.[93]

### Teller, SDI and Reykjavík

Throughout SDI's history, journalist William Broad of the New York Times was highly critical of the program and Teller's role within it. His works have generally ascribed the entire basis for SDI to Teller's overselling of the Excalibur concept, convincing Reagan that a credible defensive system was only a few years away. He has repeatedly made the claim that "Over the protests of colleagues, Teller misled the highest officials of the United States Government into the deadly folly known as Star Wars".[94]

In particular, Broad points to the meeting between Teller and Reagan in September 1982 as the key moment in SDI's creation. Years later, Broad described the meeting this way: "For half an hour, Teller deployed x-ray lasers all over the Oval Office, reducing hundreds of incoming Soviet missiles to radioactive chaff, while Reagan, gazing up ecstatically, saw a crystal shield, covering the Last Hope of Man."[34]

This basic telling of the story is recounted in other contemporary sources; in their biography, Edward Teller: Giant of The Golden Age of Physics, Blumberg and Panos essentially make the same claim,[95] as does Robert Park in his Voodoo Science.[21]

Others give less credence to Teller's persuasive capabilities; Ray Pollock, who was present at the meeting, described in a 1986 letter that "I sat in on the mid-September 1982 meeting Teller had in the Oval Office... Teller got a warm reception but that is all. I had the feeling he confused the president."[96] In particular, he notes Teller's opening comment about "Third generation, third generation!"[i] as being a point of confusion. Keyworth was later quoted as calling the meeting "a disaster."[98] Others report that Teller's bypassing of official channels to arrange the meeting angered Caspar Weinberger and other members of the Department of Defense.[99]

Others debate Excalibur's role in SDI from the start. Park suggests that Reagan's "kitchen cabinet" was pushing for some sort of action on BMD even before this period.[21] Charles Townes suggested that the key impetus to move forward was not Teller, but a presentation by the Joint Chiefs of Staff made only a few weeks before his speech that suggested shifting some development funding to defensive systems. Reagan mentioned this during the speech introducing SDI. Nigel Hey points to Robert McFarlane and the United States National Security Council as a whole.[96] In a 1999 interview with Hey, Teller himself would suggest that he had little to do with the president's decision to announce SDI. He also did not want to talk about the X-ray laser and claimed that he did not even recognize the name "Excalibur".[94]

There is considerable debate on whether or not Excalibur had a direct effect on the failure of the Reykjavík Summit. During the October 1986 meeting, Reagan and Mikhail Gorbachev initially considered the issue of the destabilizing effect of intermediate-range missiles in Europe. As both proposed various ideas to eliminate them, they quickly began to ratchet up the numbers and types of weapons being considered. Gorbachev started with his acceptance of Reagan's 1981 "double zero option" for intermediate-range missiles but then countered with an additional offer to eliminate 50% of all nuclear-armed missiles. Reagan then countered with an offer to eliminate all such missiles within ten years, as long as the US was free to deploy defensive systems after that period. At that point, Gorbachev offered to eliminate all nuclear weapons of any sort within that same time period.[100]

It was at this point that SDI came into the negotiations. Gorbachev would only consider such a move if the US limited their SDI efforts to the laboratory for ten years. Excalibur, which Teller's letter of only a few days earlier once again claimed was ready to enter engineering,[25] would need to be tested in space before that point.[91] Reagan refused to back down on this issue, as did Gorbachev. Reagan attempted one last time to break the logjam, asking if he would really "turn down a historic opportunity because of a single word" ("laboratory"). Gorbachev stated it was a matter of principle; if the US continued real-world testing while the Soviets agreed to dismantle their weapons, he would return to Moscow to be considered a fool.[101]

## Physics

### Lasers

A ruby laser is a very simple device, consisting of the ruby (right), flash tube (left-center) and casing (top). An X-ray laser is similar in concept, with the ruby replaced by one or more metal rods, and the flash tube by a nuclear bomb.

Lasers rely on two physical phenomena to work, stimulated emission and population inversion.[102][103]

An atom is made of a nucleus and a number of electrons orbiting in shells around it. The latter particles can be found in many discrete energy states, defined by quantum mechanics. The energy levels depend on the structure of the nucleus, so they vary from element to element. Electrons can gain or lose energy by absorbing or emitting a photon with the same energy as the difference between two allowable energy states. This is why different elements have unique spectrums and gives rise to the science of spectroscopy.[104]

Electrons will naturally release photons if there is an unoccupied lower energy state. An isolated atom would normally be found in the ground state, with all of its electrons in their lowest possible state. But due to the surrounding environment adding energy, the electrons will be found in a range of energies at any given instant. Electrons that are not in the lowest possible energy state are known as "excited", as are the atoms that contain them.[104]

Stimulated emission occurs when an excited electron can drop by the same amount of energy as a passing photon. This causes a second photon to be emitted, closely matching the original's energy, momentum, and phase. Now there are two photons, doubling the chance that they will cause the same reaction in other atoms. As long as there is a large population of atoms with electrons in the matching energy state, the result is a chain reaction that releases a burst of single-frequency, highly collimated light.[102]

The process of gaining and losing energy is normally random, so under typical conditions, a large group of atoms is unlikely to be in a suitable state for this reaction. Lasers depend on some sort of setup that results in many electrons being in the desired states, a condition known as a population inversion. An easy to understand example is the ruby laser, where there is a metastable state where electrons will remain for a slightly longer period if they are first excited to even higher energy. This is accomplished through optical pumping, using the white light of a flash lamp to increase the electron energy to a blue-green or ultraviolet frequency. The electrons then rapidly lose energy until they reach the metastable energy level in the deep red. This results in a brief period where a large number of electrons lie at this medium energy level, resulting in a population inversion. At that point any one of the atoms can emit a photon at that energy, starting the chain reaction.[105][103]

### X-ray lasers

An X-ray laser works in the same general fashion as a ruby laser, but at much higher energy levels. The main problem in producing such a device is that the probability of any given transition between energy states depends on the cube of the energy. Comparing a ruby laser that operates at 694.3 nm to a hypothetical soft X-ray laser that might operate at 1 nm, this means the X-ray transition is 6943, or a little over 334 million times less likely. To provide the same total output energy, one needs a similar increase in input energy.[106]

Another problem is that the excited states are extremely short-lived: for a 1 nm transition, the electron will remain in the state for about 10-14 seconds. Without a metastable state to extend this time, this means there is only this fleeting time, much less than a shake, to carry out the reaction.[106] A suitable substance with a metastable state in the X-ray region is unknown in the open literature.[j]

Instead, X-ray lasers rely on the speed of various reactions to create the population inversion. When heated beyond a certain energy level, electrons dissociate from their atoms entirely, producing a gas of nuclei and electrons known as a plasma. Plasma is a gas, and its energy causes it to adiabatically expand according to the ideal gas law. As it does, its temperature drops, eventually reaching a point where the electrons can reconnect to nuclei. The cooling process causes the bulk of the plasma to reach this temperature at roughly the same time. Once reconnected to nuclei, the electrons lose energy through the normal process of releasing photons. Although rapid, this release process is slower than the reconnection process. This results in a brief period where there are a large number of atoms with the electrons in the high-energy just-reconnected state, causing a population inversion.[109]

To produce the required conditions, a huge amount of energy needs to be delivered extremely rapidly. It has been demonstrated that something on the order of 1 watt per atom is needed to provide the energy required to produce an X-ray laser.[109] Delivering so much energy to the lasing medium invariably means it will be vaporized, but the entire reaction occurs so rapidly this is not necessarily a problem. It does imply that such systems will be inherently one-shot devices.[109]

Finally, another complication is that there is no effective mirror for X-ray frequency light. In a common laser, the lasing medium is normally placed between two partial mirrors that reflect some of the output back into the media. This greatly increases the number of photons in the media and increases the chance that any given atom will be stimulated. More importantly, as the mirrors reflect only those photons traveling in a particular direction, and the stimulated photons will be released in the same direction, this causes the output to be highly focused.[109]

Lacking either of these effects, the X-ray laser has to rely entirely on stimulation as the photons travel through the media only once. To increase the odds that any given photon causes stimulation, and to focus the output, X-ray lasers are designed to be very long and skinny. In this arrangement, most of the photons being released naturally through conventional emissions in random directions will simply exit the media. Only those photons that happen to be released traveling down the long axis of the media have a reasonable chance of stimulating another release.[109] A suitable lasing medium would have an aspect ratio on the order of 10,000.[110]

### Excalibur

Although most details of the Excalibur concept remain classified, articles in Nature and Reviews of Modern Physics, along with those in optics-related journals, contain broad outlines of the underlying concepts and outline possible ways to build an Excalibur system.[111][74]

The basic concept would require one or more lasing rods arranged into a module along with a tracking camera. These would be arranged on a framework surrounding the nuclear weapon in the center. Nature's description shows multiple lasing rods embedded in a plastic matrix forming a cylinder around the bomb and tracking device, meaning that each device would be able to attack a single target. The accompanying text, however, describes it as having several aimable modules, perhaps four.[112] Most other descriptions show multiple modules arranged around the bomb that can be separately aimed, which more closely follows the suggestions of there being several dozen such lasers per device.[113]

In order to damage the airframe of an ICBM, it is estimated that about 3 kJ/cm² would need to hit it. The laser is essentially a focusing device, taking the radiation falling along the length of the rod and turning some small amount of that into a beam traveling out the end. One can consider the effect as increasing the brightness of the X-rays falling on the target compared to the X-rays released by the bomb itself. The enhancement of the brightness compared to the unfocused output from the bomb is ${\displaystyle \eta /d\theta }$, where ${\displaystyle \eta }$ is the efficiency of conversion from bomb X-rays to laser X-rays, and ${\displaystyle d\theta }$ is the dispersion angle.[114]

If a typical ICBM is 1 metre (3 ft 3 in) in diameter, at a distance of 1,000 kilometers (620 mi) represents a solid angle of 10−12 steradian (sr). Estimates of the dispersion angles from the Excalibur lasers were from 10−12 to 10−9. Estimates of ${\displaystyle \eta }$ vary from about 10−5 to 10−2; that is, they have laser gain less than one. In the worst-case scenario, with the widest dispersion angle and the lowest enhancement, the pump weapon would have to be approximately 1 MT for a single laser to deposit enough energy on the booster to be sure to destroy it at that range. Using best-case scenarios for both values, about 10 kT are required.[114]

The exact material of the lasing medium has not been specified. The only direct statement from one of the researchers was by Chapline, who described the medium on the original Diablo Hawk test being "an organic pith material" from a weed growing on a vacant lot in Walnut Creek, a town a short distance away from Livermore.[10] Various sources describe the later tests using metals; selenium,[115] zinc[112] and aluminum have been mentioned specifically.[25]

## BMD

### Missile-based systems

The US Army ran an ongoing BMD program dating from the 1940s. This was initially concerned with shooting down V-2-like targets, but an early study on the topic by Bell Labs suggested their short flight times would make it difficult to arrange an interception. The same report noted that the longer flight times of long-range missiles made this task simpler, in spite of various technical difficulties due to higher speeds and altitudes.[116]

This led to a series of systems that started with Nike Zeus, then Nike-X, Sentinel and finally the Safeguard Program. These systems used short and medium-range missiles equipped with nuclear warheads to attack incoming enemy ICBM warheads. The constantly changing concepts reflect their creation during a period of rapid changes in the opposing force as the Soviet ICBM fleet was expanded. The interceptor missiles had a limited range, less than 500 miles (800 km),[k] so interceptor bases had to be spread across the United States. Since the Soviet warheads could be aimed at any target, adding a single ICBM, which were becoming increasingly inexpensive in the 1960s, would (theoretically) require another interceptor at every base to counter it.[118]

This led to the concept of the cost-exchange ratio, the amount of money one had to spend on additional defenses to counter a dollar of new offensive capability.[118] Early estimates were around 20, meaning every dollar the Soviets spent on new ICBMs would require the US to spend \$20 to counter it. This implied the Soviets could afford to overwhelm the US's ability to build more interceptors.[118] With MIRV, the cost-exchange ratio was so one-sided that there was no effective defense that could not be overwhelmed for little cost, as pointed out in a famous 1968 article by Bethe and Garwin.[40] This is precisely what the US did when the Soviets installed their A-35 anti-ballistic missile system around Moscow; by adding MIRV to the Minuteman missile fleet, they could overwhelm the A-35 without adding a single new missile.[119]

### X-ray based attacks

Studies of high altitude nuclear explosions such as this Kingfish shot of Operation Fishbowl inspired the concept of X-rays attacks.

During high-altitude tests in the late 1950s and early 1960s, it was noticed that the burst of X-rays from a nuclear explosion were free to travel long distances, unlike low altitude bursts where the air interacted with the X-rays within a few tens of meters. This led to new and unexpected effects. It also led to the possibility of designing a bomb specifically to increase the X-ray release, which could be made so powerful that the rapid deposit of energy on a metal surface would cause it to explosively vaporize. At ranges on the order of 10 miles (16 km), this would have enough energy to destroy a warhead.[40]

The use of X-ray based attacks in earlier generation BMD systems had led to work to counter these attacks. In the US, these were carried out by placing a warhead (or parts of it) in a cavern connected by a long tunnel to a second cavern where an active warhead was placed. Before firing, the entire site was pumped into a vacuum. When the active warhead fired, the X-rays traveled down the tunnel to hit the target warhead. To protect the target from the blast itself, huge metal doors slammed shut in the tunnel in the short time between the X-rays arriving and the blast wave behind it. Such tests had been carried out continuously since the 1970s.[121][122]

### Boost-phase attacks

A potential solution to the problem of MIRV is to attack the ICBMs during the boost phase before the warheads have separated. This destroys all of the warheads with a single attack, rendering MIRV superfluous. Additionally, attacking during this phase allows the interceptors to track their targets using the large heat signature of the booster motor. These can be seen at distances on the order thousands of miles, with the proviso that they would be below the horizon for a ground-based sensor and thus require sensors being located in orbit.[123]

DARPA had considered this concept starting in the late 1950s, and by the early 1960s had settled on the Ballistic Missile Boost Intercept concept, Project BAMBI. BAMBI used small heat-seeking missiles launched from orbiting platforms to attack Soviet ICBMs as they launched. In order to keep enough BAMBI interceptors within the range of the Soviet missiles while the interceptor's launch platforms continued to move in orbit, an enormous number of platforms and missiles would be required.[123]

The basic concept continued to be studied through the 1960s and 1970s. A serious problem was that the interceptor missiles had to be very fast to reach the ICBM before its motor stopped firing, which required a larger motor on the interceptor, meaning higher weight to launch into orbit. As the difficulties of this problem became clear, the concept evolved into the "ascent phase" attack, which used more sensitive seekers that allowed the attack to continue after the ICBM's motor had stopped firing and the warhead bus was still ascending. In all of these studies, the system would require an enormous amount of weight to be lifted into orbit, typically hundreds of millions of pounds, well beyond any reasonable projections of US capability. The US Air Force repeatedly studied these various plans and rejected them all as essentially impossible.[32]

### Excalibur's promise and development issues

The bright spikes extending below the initial fireball of one of 1952's Operation Tumbler–Snapper test shots are known as the "rope trick effect" caused by the flash of X-rays released by the explosion heating the steel guy-wires white hot. Excalibur intended to focus these X-rays to allow attacks over long distances.

The Excalibur concept appeared to represent an enormous leap in BMD capability. By focusing the output of a nuclear explosion into X-rays, essentially a smaller version of the W71, the range and effective power of BMD was greatly enhanced. A single Excalibur could attack multiple targets across hundreds or even thousands of kilometers. Because the system was both small and relatively lightweight, the Space Shuttle could carry multiple Excalibur's into orbit in a single sortie.[22] Super Excalibur, a later design, would theoretically be able to shoot down the entire Soviet missile fleet singlehandedly.[38]

When first proposed, the plan was to place enough Excaliburs in orbit so that at least one would be over the Soviet Union at all times. But it was soon noted that this allowed the Excalibur platforms to be directly attacked; in this situation, the Excalibur would either have to allow itself to absorb the attack or sacrifice itself to shoot down the attacker. In either case, the Excalibur platform would likely be destroyed, allowing a subsequent and larger attack to occur unhindered.[29]

This led Teller to suggest a "pop-up" mode where an Excalibur would be placed on SLBM platforms on submarines patrolling off the Soviet coastline.[29] When a launch was detected, the missiles would be launched upward and then fire as they left the atmosphere. This plan also suffered from several problems. Most notable was the issue of timing; the Soviet missiles would be firing for only a few minutes, during which time the US had to detect the launch, order a counter-launch, and then wait for the missiles to climb to altitude.[124][125]

For practical reasons, submarines could only salvo their missiles over a period of minutes, which meant each one could only launch perhaps one or two Excaliburs before Soviet missiles were already on their way. Additionally, the launch would reveal the location of the submarine, leaving it a "sitting duck". These issues led the Office of Technology Assessment to conclude that "the practicality of a global scheme involving pop-up X-ray lasers of this type is doubtful."[126]

Another challenge was geometric in nature. For missiles launched close to the submarines, the laser would be shining through only the uppermost atmosphere. For ICBMs launched from Kazakhstan, some 3,000 kilometers (1,900 mi) from the Arctic Ocean, the curvature of the Earth meant that an Excalibur's laser beam would have a long path-length through the atmosphere. To obtain a shorter atmospheric path-length, Excalibur would have to climb much higher, during which time the target missile would be able to release its warheads.[127]

There was the possibility that a powerful enough laser could reach further into the atmosphere, perhaps as deep as 30 kilometers (19 mi) altitude if it was bright enough.[128] In this case, there would be so many X-ray photons that all of the air between the battle station and the target missiles would be completely ionized and there would still be enough X-rays left over to destroy the missile. This process, known as "bleaching", would require an extremely bright laser, more than 10 billion times brighter than the original Excalibur system.[129]

Finally, another problem was aiming the lasing rods before firing. For maximum performance, the laser rods needed to be long and skinny, but this would make them less robust mechanically. Moving them to point at their targets would cause them to bend, and some time would be required to allow this deformation to disappear. Complicating the issue was that the rods needed to be as skinny as possible to focus the output, a concept known as geometric broadening, but doing so caused the diffraction limit to decrease, offsetting this improvement.[127] Whether it was possible to meet the performance requirements within these competing limitations was never demonstrated.[38]

### Countermeasures

Excalibur worked during the boost phase and aimed at the booster itself. This meant that the hardening techniques developed for warheads were not applicable. While many of the other SDI weapons had simple countermeasures based on the weapon's required dwell time, like spinning the booster and polishing it mirror-bright, Excalibur's zero dwell time rendered these ineffective. Thus the primary way to defeat an Excalibur weapon is to use the atmosphere to block the progress of the beams. This can be accomplished using a missile that burns out while still in the atmosphere, thereby denying Excalibur the tracking system information needed for targeting.[55]

The Soviets conceived of a wide array of responses during the SDI era.[130] In 1997 Russia deployed the Topol-M ICBM which utilized a higher-thrust engine burn following take-off, and flew a relatively flat ballistic trajectory, both characteristics intended to complicate space-based sensor acquisition and interception.[131] The Topol fires its engine for only 150 seconds, about half the time of the SS-18, and has no bus, the warhead is released seconds after the engine stops. This makes it far more difficult to attack.[132]

In 1976, the organization now known as NPO Energia began development of two space-based platforms not unlike the SDI concepts; Skif was armed with a CO2 laser while Kaskad used missiles. These were abandoned, but with the announcement of SDI they were repurposed as anti-satellite weapons, with Skif being used against low-orbit objects and Kaskad against higher altitude and geostationary targets.[133]

Some of these systems were tested in 1987 on the Polyus spacecraft. What was mounted on this spacecraft remains unclear, but either a prototype Skif-DF or a mockup was part of the system. According to interviews conducted years later, mounting the Skif laser on the Polyus was more for propaganda purposes than as an effective defense technology, as the phrase "space based laser" carried political capital.[134] One of the claims is that Polyus would be the basis for the deployment of nuclear "mines" that might be fired from outside the range of the SDI components and reach the United States within six minutes.[134]

## Notes

1. ^ The later Super Excalibur concept theoretically supported thousands of lasers.
2. ^ Visible-spectrum gas lasers that were optically pumped by nuclear weapons had been developed and tested, and it is likely the Aviation Week article is confusing these earlier tests with the 1978 X-ray test.[24]
3. ^ A Department of Defense backgrounder report has a diagram showing such an MX-like missile firing for 180 seconds.[57]
4. ^ There is significant confusion in various sources about whether Excalibur+ and Super Excalibur refer to a single design, or two. Coffey and Stevens are examples of these different views.[38][58]
5. ^ Stevens' overview of known parameters calls this claim into question; he calculates that the effective range of the weapon would be on the order of 3,000 kilometers (1,900 mi), while working backward, Wood and Teller's own statements put the upper limit around 10,000 kilometers (6,200 mi). Neither is nearly enough to make it effective when fired from stationary orbit at ~36,000 kilometers (22,000 mi).[60]
6. ^ One SDIO official noted that Teller's claims of Soviet research were "5 percent information and 95 percent conjecture."[63]
7. ^ This basic line of reasoning, "but the Soviets are doing it", had been used repeatedly over the previous decades. It was used, sometimes based on fake stories leaked the press, to support the development of nuclear powered aircraft,[67] flying saucer aircraft[68] and was a major reason for the strong support of earlier ABM systems like Nike-X.[69] Ironically, in this case, it was the Aviation Week article in 1981 that prompted Soviet X-ray laser developments, which demonstrated only 20 kJ of output.[70]
8. ^ Or as congressman Charles Bennett insisted, "loose marbles",[73] a euphemism for insanity.
9. ^ "Third generation weapon" was a term Teller used to describe nuclear weapons that focused their output at particular targets, as opposed to traditional designs where the energy was released in all directions. The term was not widely used by others in the field, although it appears in later works.[97]
10. ^ Although chlorine atoms are reported to have such a state,[107] a dedicated X-ray laser using this technique does not appear in the literature. While atoms with such a state are unknown, metastable inner-shell molecular state molecules often have energy levels in the X-ray region and have been used for high-energy X-ray sources.[108]
11. ^ The Spartan, the longest-range US ABM, had a maximum range of about 450 miles (720 km).[117]

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• Goodchild, Peter (2004). Edward Teller, the Real Dr. Strangelove. Harvard University Press. ISBN 978-0-674-01669-9.
• Guyford, Steve (September 1985). Ballistic Missile Defense Technologies. Office of Technology Assessment.
• Hey, Nigel (2006). The Star Wars Enigma: Behind the Scenes of the Cold War Race for Missile Defense. Potomac Books. ISBN 978-1-57488-981-9.
• Hafemeister, David (2016). Nuclear Proliferation and Terrorism in the Post-9/11 World. Springer. Bibcode:2016nptp.book.....H. ISBN 978-3-319-25367-1.
• Hecht, Jeff (1984). Beam Weapons: The Next Arms Race. New York: Plenum Press. ISBN 978-0-306-41546-3.
• Hecht, Jeff (May 2008). "The History of the X-ray Laser". Optics & Photonics News. 19 (5). ISSN 1047-6938.
• Heppenheimer, Thomas (7 August 1989). "New Director Shifts Balance of Power at Livermore Lab". The Scientist. OCLC 18316428.
• Herken, Gregg (October 1987). "The earthly origins of Star Wars". Bulletin of the Atomic Scientists. OCLC 220821655.
• Jayne, Edward Randolph (1969). The ABM debate: strategic defense and national security (PDF) (Technical report). Massachusetts Institute of Technology. OCLC 19300718. Retrieved 13 December 2014.
• Kent, Glenn (2008). Thinking About America's Defense (PDF). RAND. ISBN 978-0-8330-4452-5.
• Lakoff, Sanford; York, Herbert (1989). A Shield in Space? Technology, Politics, and the Strategic Defense Initiative. University of California Press. ISBN 978-0-520-06650-2.
• Matlock, Jack (2004). Reagan and Gorbachev: How the Cold War Ended. Random House. ISBN 978-1-58836-425-8.
• Mohr, Charles (22 March 1984). "Study Assails Idea of Missile Defense". The New York Times.
• Park, Robert (2002). Voodoo Science: The Road from Foolishness to Fraud. Oxford University Press. ISBN 978-0-19-860443-3.
• Prelas, Mark (2015). Nuclear-Pumped Lasers. Springer. ISBN 978-3-319-19845-3.
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• Ritson, David (August 1987). "The Nuclear-Pumped X-ray Laser; A weapon for the twenty-first century". Nature. 328 (6130): 487–490. Bibcode:1987Natur.328..487R. doi:10.1038/328487a0.
• Scheer, Robert (12 November 1985). "Scientists Dispute Test of X-Ray Laser Weapon". Los Angeles Times.
• Schwartz, Stephen (2011). Atomic Audit: The Costs and Consequences of U.S. Nuclear Weapons Since 1940. Brookings Institution Press. ISBN 978-0-8157-2294-6.
• Smith, R. Jeffery (8 November 1985). "Experts Cast Doubts on X-ray Laser". Science. 230 (4726): 646–648. Bibcode:1985Sci...230..646S. doi:10.1126/science.230.4726.646. PMID 17797283.
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text:
mathlibdocumentation

data.fun_like.embedding

Typeclass for a type F with an injective map to A ↪ B#

This typeclass is primarily for use by embeddings such as rel_embedding.

Basic usage of embedding_like#

A typical type of embedding should be declared as:

structure my_embedding (A B : Type*) [my_class A] [my_class B] :=
(to_fun : A → B)
(injective' : function.injective to_fun)
(map_op' : ∀ {x y : A}, to_fun (my_class.op x y) = my_class.op (to_fun x) (to_fun y))

namespace my_embedding

variables (A B : Type*) [my_class A] [my_class B]

-- This instance is optional if you follow the "Embedding class" design below:
instance : embedding_like (my_embedding A B) A B :=
{ coe := my_embedding.to_fun,
coe_injective' := λ f g h, by cases f; cases g; congr',
injective' := my_embedding.injective' }

/-- Helper instance for when there's too many metavariables to directly
apply fun_like.to_coe_fn. -/
instance : has_coe_to_fun (my_embedding A B) (λ _, A → B) := ⟨my_embedding.to_fun⟩

@[simp] lemma to_fun_eq_coe {f : my_embedding A B} : f.to_fun = (f : A → B) := rfl

@[ext] theorem ext {f g : my_embedding A B} (h : ∀ x, f x = g x) : f = g := fun_like.ext f g h

/-- Copy of a my_embedding with a new to_fun equal to the old one. Useful to fix definitional
equalities. -/
protected def copy (f : my_embedding A B) (f' : A → B) (h : f' = ⇑f) : my_embedding A B :=
{ to_fun := f',
injective' := h.symm ▸ f.injective',
map_op' := h.symm ▸ f.map_op' }

end my_embedding

This file will then provide a has_coe_to_fun instance and various extensionality and simp lemmas.

Embedding classes extending embedding_like#

The embedding_like design provides further benefits if you put in a bit more work. The first step is to extend embedding_like to create a class of those types satisfying the axioms of your new type of morphisms. Continuing the example above:

/-- my_embedding_class F A B states that F is a type of my_class.op-preserving embeddings.
You should extend this class when you extend my_embedding. -/
class my_embedding_class (F : Type*) (A B : out_param $Type*) [my_class A] [my_class B] extends embedding_like F A B := (map_op : ∀ (f : F) (x y : A), f (my_class.op x y) = my_class.op (f x) (f y)) @[simp] lemma map_op {F A B : Type*} [my_class A] [my_class B] [my_embedding_class F A B] (f : F) (x y : A) : f (my_class.op x y) = my_class.op (f x) (f y) := my_embedding_class.map_op -- You can replace my_embedding.embedding_like with the below instance: instance : my_embedding_class (my_embedding A B) A B := { coe := my_embedding.to_fun, coe_injective' := λ f g h, by cases f; cases g; congr', injective' := my_embedding.injective', map_op := my_embedding.map_op' } -- [Insert has_coe_to_fun, to_fun_eq_coe, ext and copy here]  The second step is to add instances of your new my_embedding_class for all types extending my_embedding. Typically, you can just declare a new class analogous to my_embedding_class: structure cooler_embedding (A B : Type*) [cool_class A] [cool_class B] extends my_embedding A B := (map_cool' : to_fun cool_class.cool = cool_class.cool) class cooler_embedding_class (F : Type*) (A B : out_param$ Type*) [cool_class A] [cool_class B]
extends my_embedding_class F A B :=
(map_cool : ∀ (f : F), f cool_class.cool = cool_class.cool)

@[simp] lemma map_cool {F A B : Type*} [cool_class A] [cool_class B] [cooler_embedding_class F A B]
(f : F) : f cool_class.cool = cool_class.cool :=
my_embedding_class.map_op

-- You can also replace my_embedding.embedding_like with the below instance:
instance : cool_embedding_class (cool_embedding A B) A B :=
{ coe := cool_embedding.to_fun,
coe_injective' := λ f g h, by cases f; cases g; congr',
injective' := my_embedding.injective',
map_op := cool_embedding.map_op',
map_cool := cool_embedding.map_cool' }

-- [Insert has_coe_to_fun, to_fun_eq_coe, ext and copy here]

Then any declaration taking a specific type of morphisms as parameter can instead take the class you just defined:

-- Compare with: lemma do_something (f : my_embedding A B) : sorry := sorry
lemma do_something {F : Type*} [my_embedding_class F A B] (f : F) : sorry := sorry

This means anything set up for my_embeddings will automatically work for cool_embedding_classes, and defining cool_embedding_class only takes a constant amount of effort, instead of linearly increasing the work per my_embedding-related declaration.

@[class]
structure embedding_like (F : Sort u_1) (α : out_param (Sort u_2)) (β : out_param (Sort u_3)) :
Sort (max 1 (imax u_1 u_2 u_3))
• to_fun_like : α (λ (_x : α), β)
• injective' : ∀ (f : F),

The class embedding_like F α β expresses that terms of type F have an injective coercion to injective functions α ↪ β.

Instances of this typeclass
Instances of other typeclasses for embedding_like
• embedding_like.has_sizeof_inst
@[protected]
theorem embedding_like.injective {F : Sort u_1} {α : Sort u_2} {β : Sort u_3} [i : β] (f : F) :
@[simp]
theorem embedding_like.apply_eq_iff_eq {F : Sort u_1} {α : Sort u_2} {β : Sort u_3} [i : β] (f : F) {x y : α} :
f x = f y x = y
@[simp]
theorem embedding_like.comp_injective {α : Sort u_2} {β : Sort u_3} {γ : Sort u_4} {F : Sort u_1} [ γ] (f : α → β) (e : F) :

format: plain


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# Wannier functions on a ring

Let's say I have a single particle hamiltonian in a periodic potential, for example a 1D lattice such that: $$H = -\frac{\partial_x^2}{2m} + V(x)$$ with $V(x+a) = V(x)$ where $a$ is the lattice spacing between the atoms or sites. It is known by Bloch's theorem that a solution to such a system will have the form $$\psi_{k}(x)=e^{ikx}u_k(x)$$ where $u_k(x+a)=u_k(x)$.

My questions is about the boundary conditions. If we take $$\psi(x+Na) = \psi(x)$$ we get, if $N$ is large enough, a lot of different values for $k$ in the first Brillouin zone: $$k=\frac{2\pi n}{N} \text{ with }-\frac{\pi}{a}<k<\frac{\pi}{a},$$ so we get a band of possible states.

In this case we can define Wannier functions which using Fourier over the wave-functions: $$\phi(x-R) = \sum_k e^{-ik R} \psi_k(x)$$ where the summation is over all the $k$'s in the first Brillouin zone.

But if I take the B.C $$\psi(x+a) = \psi(x)$$ I get a single value for the momentum in each Brillouin zone $$k = 0, \pm 2\pi , \pm 4\pi,...$$ Is it still possible to define a Wannier function for such a state? I mean what will the Fourier be like if we have a single possible value of $k$??

• The relation $\psi(x + a) = \psi(a)$ is more than a boundary condition. A boundary condition tells you the values of the wave function at the edges of the system: for $N$ atoms at $x = a$, $2a$, $3a$, etc., the boundary consists of just two points, $x = 0$ and $x = Na$. But $\psi(x + a) = \psi(a)$ specifies the values of the wavefunction at more than just those two points: it tells you its value at $N$ points! The solutions you get are a subset of the $-\pi/a < k < \pi/a$ band of states which also satisfies the extra constraints. – Ted Pudlik Apr 30 '14 at 17:47

From the wikipedia article linked in your question, we see that Wannier functions can also be defined as $$\underbrace{\phi_{\textbf{R}}(\textbf{x})}_{\textbf{R} \text{ Bravais lattice vector}} \equiv \frac{1}{\sqrt{N}} \underbrace{\sum_{\textbf{k}}}_{\substack{\text{summed over first}\\\text{Brillouin zone}}} e^{-i\textbf{k}\cdot\textbf{R}} \psi_{\textbf{k}}(\textbf{x})$$ which is the same as "your" $\phi$ because thanks to the periodicity properties of Block wave functions it is easily seen that $$\underbrace{\phi_{\textbf{R}}(\textbf{x})}_{\text{"mine" }\phi} = \underbrace{\phi(\textbf{x}-\textbf{R})}_{\text{"your" }\phi}$$ Once we note this, it is easier to realize that going from $\psi_\textbf{k}$ to $\phi_\textbf{R}$ it's just a way to change the basis set of functions used to describe the system: instead of characterizing the states with the crystal momentum, we characterize them throught the Bravais lattice vectors. We can do this because the number of crystal momentum vectors $\textbf{k}$ it's the same as the number of Bravais lattice points $\textbf{R}$ (which is $N$).
Back to your question: I don't think it makes much sense to define a "Wannier function" of a particular $\psi_\textbf{k}$ because Wannier functions are defined as a "redistribution" of all Bloch wave functions, as you can also see rewriting them in functional form as $$\underbrace{\phi}_{\text{"your" }\phi} = \frac{1}{\sqrt{N}}\sum_\textbf{k} \psi_\textbf{k}$$ But even if you could do such thing, hence for some reason allowing your system to have a single crystal momentum vector $\textbf{k}$, you would have from the definition that $$\phi_\textbf{R}(\textbf{x}) = e^{-i\textbf{k}\cdot\textbf{R}} \psi_\textbf{k}(\textbf{x}), \qquad \text{in "my" notation}$$ $$\phi(\textbf{x}) = \psi_\textbf{k}(\textbf{x}), \qquad \text{in "your" notation}$$ which as you can probably see is really not that useful.


text:
# Challenge

This is one of the most beloved number sequence puzzles! No prerequisites! This should come after some easier number sequence puzzles, so we don't have to introduce the idea of a number sequence before introducing the puzzle.


text:
# Is there a number "a" such that the equation below exists? If so what is the value of "a" and its limit.

## ${\lim}_{x \to - 2} \frac{3 {x}^{2} + a x + a + 3}{{x}^{2} + x - 2}$

Oct 2, 2016

$a = 15$

${\lim}_{x \to - 2} \frac{3 {x}^{2} + 15 x + 18}{{x}^{2} + x - 2} = - 1$

#### Explanation:

${x}^{2} + x - 2 = \left(x + 2\right) \left(x - 1\right)$

So the denominator contains exactly one factor $\left(x + 2\right)$

So in order that $\frac{3 {x}^{2} + a x + a + 3}{{x}^{2} + x - 2}$ has a limit as $x \to - 2$, the only requirement is that:

$3 {x}^{2} + a x + \left(a + 3\right) \text{ }$ is divisible by $\left(x + 2\right)$

Let $f \left(x\right) = 3 {x}^{2} + a x + \left(a + 3\right)$

This is divisible by $\left(x + 2\right)$ if and only if $f \left(- 2\right) = 0$

Substituting $x = - 2$ we have:

$f \left(- 2\right) = 3 {\left(\textcolor{b l u e}{- 2}\right)}^{2} + a \left(\textcolor{b l u e}{- 2}\right) + a + 3$

$\textcolor{w h i t e}{f \left(- 2\right)} = 12 - 2 a + a + 3$

$\textcolor{w h i t e}{f \left(- 2\right)} = 15 - a$

So we require $a = 15$

With this value of $a$:

$f \left(x\right) = 3 {x}^{2} + 15 x + 18 = 3 \left({x}^{2} + 5 x + 6\right) = 3 \left(x + 2\right) \left(x + 3\right)$

$\frac{3 {x}^{2} + 15 x + 18}{{x}^{2} + x - 2} = \frac{3 \left(\textcolor{red}{\cancel{\textcolor{b l a c k}{x + 2}}}\right) \left(x + 3\right)}{\left(\textcolor{red}{\cancel{\textcolor{b l a c k}{x + 2}}}\right) \left(x - 1\right)} = \frac{3 \left(x + 3\right)}{x - 1}$

So:

${\lim}_{x \to - 2} \frac{3 {x}^{2} + 15 x + 18}{{x}^{2} + x - 2} = {\lim}_{x \to - 2} \frac{3 \left(x + 3\right)}{x - 1} = \frac{3 \left(\textcolor{b l u e}{- 2} + 3\right)}{\textcolor{b l u e}{- 2} - 1} = \frac{3}{- 3} = - 1$


text:
# Genus $0$ Modular curves of prime power level with a point defined over number fields other than $\Q$

Associated to an open subgroup $G$ of $\GL_2(\Zhat)$ satisfying conditions $-I \in G$ and $\det(G) \subsetneq (\Zhat)^{\times}$ there is a modular curve $X_G$ which is a smooth compact curve defined over an extension of $\Q.$ In this article, we give a complete list of all such prime power level genus $0$ modular curves with a point.

Number Theory

0
04 Aug 2022

# Streaming Tensor Train Approximation

4 Aug 2022

STTA accesses $\mathcal T$ exclusively via two-sided random sketches of the original data, making it streamable and easy to implement in parallel -- unlike existing deterministic and randomized tensor train approximations.

Numerical Analysis Numerical Analysis

0
04 Aug 2022

# Block Discrete Empirical Interpolation Methods

We present two block variants of the discrete empirical interpolation method (DEIM); as a particular application, we will consider a CUR factorization.

Numerical Analysis Numerical Analysis

0
03 Aug 2022

# An adaptive consensus based method for multi-objective optimization with uniform Pareto front approximation

2 Aug 2022

In this work we are interested in stochastic particle methods for multi-objective optimization.

Optimization and Control 35Q70, 35Q84, 35Q93, 90C29, 90C56

0
02 Aug 2022

# Metric Dimension of a Diagonal Family of Generalized Hamming Graphs

2 Aug 2022

Classical Hamming graphs are Cartesian products of complete graphs, and two vertices are adjacent if they differ in exactly one coordinate.

Combinatorics 05C69 (Primary) 05C12, 05B30, 05C15 (Secondary)

0
02 Aug 2022

# Implicit bulk-surface filtering method for node-based shape optimization and comparison of explicit and implicit filtering techniques

1 Aug 2022

This work studies shape filtering techniques, namely the convolution-based (explicit) and the PDE-based (implicit), and introduces an implicit bulk-surface filtering method to control the boundary smoothness and preserve the internal mesh quality simultaneously in the course of bulk (solid) shape optimization.

Numerical Analysis Numerical Analysis Optimization and Control

709
01 Aug 2022

# Data-driven solutions of ill-posed inverse problems arising from doping reconstruction in semiconductors

1 Aug 2022

We model a general class of such photovoltaic technologies by ill-posed global and local inverse problems based on a drift-diffusion system which describes charge transport in a self-consistent electrical field.

Numerical Analysis Numerical Analysis 68T07, 65N21, 35Q81

0
01 Aug 2022

# Confidence regions for the location of peaks of a smooth random field

30 Jul 2022

Local maxima of random processes are useful for finding important regions and are routinely used, for summarising features of interest (e. g. in neuroimaging).

Statistics Theory Statistics Theory

1
30 Jul 2022

# Rational Noncrossing Coxeter-Catalan Combinatorics

We solve two open problems in Coxeter-Catalan combinatorics.

Combinatorics Representation Theory Primary: 05A15. Secondary: 05E05, 05E10, 20C08

0
30 Jul 2022

# On the Maximum Gonality of a Curve over a Finite Field

28 Jul 2022

In general, the gonality of a curve of genus $g \ge 2$ is at most $2g - 2$.

Algebraic Geometry Number Theory

0
28 Jul 2022


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# Do microwaves break hydrogen bonds?

We're told that the photon energy of microwave frequency radiation ($$\sim 10^{-5}~\text{eV}$$) is not high enough to break hydrogen bonds. But if that's true, how does dielectric heating of water work? Liquid water is a network of polar molecules held together by H bonds so that they CAN'T rotate, in concert with the microwave beam or anything else ....

Seems like this is a problem of the two ways of looking at radiation -- classical wave vs photon stream -- being incompatible.

• Molecules of liquid water are certainly not held in place, unable to move. Otherwise it wouldn't be a liquid! – Al Nejati Nov 12 '19 at 2:55
• But people talk about "the well-known network of liquid water". – user247059 Nov 12 '19 at 2:59
• Specifically, Marcus (1995): "the well known three-dimensional network of liquid water". in Water binding by organic molecules. Cell Biochemistry and Function 13: 157-163. – user247059 Nov 12 '19 at 3:03
• Google doesn’t find that phrase. Reference please? – Bob Jacobsen Nov 12 '19 at 3:03
• Marcus (1995) Water binding by organic molecules. Cell Biochemistry and Function 13: 157-163 – user247059 Nov 12 '19 at 3:06

In a solid or liquid we have collective vibrations of the whole system. It can be useful to think of these as quasiparticles called phonons, that is when we add vibrational energy to the system as a whole it generates a phonon, or conversely a phonon can decay and emit energy.

Black body radiation is (mostly) the emission of photons from the collective vibrations i.e. the decay of phonons to emit photons, and dielectric heating is the reverse process i.e. the absorption of photons to create phonons. This is what happens in your microwave oven. The heating is due to the excitation of the collective vibrations, not to the interaction of photons with hydrogen bonds. The quanta of these collective excitations (i.e. their phonons) are generally very small so they can absorb photons of even very low energies.

In real materials the collective vibrational modes are anharmonic oscillators so they all interact with each other and the vibrational energy is distributed between them in accordance with the Boltzmann distribution. That means the vibration energy from the absorbed microwave photons is quickly equilibrated with higher energy modes such as the vibrational excitations of hydrogen bonds, and it can break those bonds. This means energy can be absorbed in small units from microwave photons and still build up sufficiently to break the much higher energy hydrogen bonds.

• John Rennie says: "This means energy can be absorbed in small units from microwave photons and still build up sufficiently to break the much higher energy hydrogen bonds." So does that mean it's wrong to say (as the cellphone industry does) that because single microwave photons do not have enough energy to break covalent bonds, microwaves cannot break molecules like DNA? – user247059 Nov 13 '19 at 1:53
• @user247059 in the example you discuss the microwave photons heat the water and the thermal energy in the water can break the hydrogen bonds. However even if the water is heated to 100°C there is still not enough thermal energy to break the covalent bonds in DNA. In any case you would cease to care after the microwaves had heated you by more than a few degrees :-) – John Rennie Nov 13 '19 at 5:28
• Sorry, this doesn't make sense to me. The generally accepted model for how microwaves heat water is that they make the water molecules (dipoles) rotate, and this rotation causes friction, which manifests as heat. My original question was how this can be correct, when the presence of H-bonds between water molecules would prevent the water molecules from rotating. You now appear to be saying that the heat PRODUCED BY the rotation of water molecules causes breakage of the H-bonds that prevent the rotation of water molecules. As I said, this does not seem to me to make sense. – user247059 Nov 13 '19 at 8:04
• The generally accepted model for how microwaves heat water is that they make the water molecules (dipoles) rotate - that is incorrect. – John Rennie Nov 13 '19 at 8:49
• Well that's what it says in Chapter 2 of Kappe CO Stadler A and Dallinger D (2012 ) Microwaves in Organic and Medicinal Chemistry, Second Edition Wiley-VCH Verlag GmbH & Co. KGaA. What do you think is the generally accepted model of how microwaves heat water? – user247059 Nov 13 '19 at 9:08

I would think that microwaves are only sufficient to factor into the rotational modes of the water molecule and that they couldn't "break" per se, the hydrogen bond. Even if they did, I think the Van Der Walls forces would make a quick repair!

You don’t need an individual quantum of energy to break any waves; you just need the energy to be absorbed.

It’s like pushing someone on a swing (with rigid arms on the swing). You don’t need to be strong enough to push them so they make a complete loop after a single push: it would be enough to keep pushing at the right time, and if there is sufficiently low friction in the system you will eventually push them “over the top”.

The vibration / rotation modes of water molecules are coupled - excite one, and over time some of the energy transfers to other modes. So the entire system absorbs energy, until bonds start to break. In fact, even without adding energy bonds are breaking (and forming) all the time - again, because the energy of an individual water molecule will change as they “jiggle”.

We can say something about the average energy of a molecule - but usually not about the energy of the individual molecule.

Does that make sense?

• Floris -- I presume you mean you don't need an individual quantum of energy to break a bond (not a wave). But that actually is the exact reason why people say low intensity microwaves (as in cell phone emissions) are not harmful. They say microwave photons don't have enough energy to ionize molecules and non-ionizing is non-harmful. The accepted model is that it's all down to single-photon interactions with matter. – user247059 Nov 12 '19 at 3:27
• @user247059 In microwave ovens energy is given to the water in food and raises its temperature. vibrational and rotational levels. It is only when the temperature reaches the boiling point that the liquid phase is broken and H2O becomes steam – anna v Nov 12 '19 at 5:48
• @user247059 “harm” can happen in two ways. Heating is a real effect, and sufficient heating can denature proteins. This is the reason that an MRI scanner (with 10’s of kW of RF transmit power) computes the SAR (specific absorption ratio) to make sure the patient doesn’t overheat (body temperature DOES go up... just not enough to harm you. Limits are set at 2 W/kg (“normal mode”) but can be higher for specific applications and with sufficient care and supervision). Ionization usually requires higher energy quanta (although a flame is ionized just because of the energy of endothermic reaction) – Floris Nov 12 '19 at 15:39
• Note also that “breaking a hydrogen bond” is not at all the same thing as “ionizing”. The hydrogen bond is between the hydrogen atom of one molecule and the oxygen (electrons) of the other molecule; when the bond is broken the hydrogen “returns to” its parent molecule, usually. And anyway, water molecules are always ionizing (OH-, H3O+) and recombining; their ionization is not what is damaging. It’s complex molecules that change their configuration / create new reaction products, that are “damaged” by radiation. – Floris Nov 12 '19 at 15:41


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# Exhausting the entropy of a hash function

In the case of password storage, consider the following:

I have an idea that one can exhaust the entropy of input to the MD5 function by using a 128 bit random value as the password (indeed, any hash function, using the output length as input). Is this a correct assumption, or is the entropy exhausted at 123.4 bit, this being the best attack to date? Or does this only apply to hash functions that for every value in the interval $[0, 2^{L}]$ provide another unique value in the same interval?

I hope you understand what I'm trying to ask here - I see that I have a hard time explaining it clearly. What I want to do with this idea is argue that in the case of MD5 stored passwords, there is no reason to use passwords with a higher entropy than the hash itself.

-
MD5 restricted to 128-bit inputs is likely not injective (and thus also not surjective), independently of any attacks on the hash function. A hash function models a random function, not a random permutation. But you are right, there is no point in having passwords with higher entropy than the hash output size. –  Paŭlo Ebermann Nov 30 '12 at 8:34

In the case of hashing passwords, entropy is a measure of what the password could have been. A password with "$n$ bits of entropy" is a password such that breaking it by dictionary attack (trying potential passwords until the right one is found) has average cost $2^{n-1}$.
It is useless to have a password entropy much beyond the output length of the employed hash function, because if you hash to $k$ bits, then trying random passwords will succeed with probability $2^{-k}$, hence average cost $2^{k}$. Thus, no need to go beyond $k+1$ bits or entropy for the password.


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# What does mechanism design constitute? (Game Theory)

I've been reading about mechanism design and the field seems a little abstract, in that I cannot ascertain whether a certain approach towards solving games falls under mechanism design.

Let me clarify with the help of a specific example of an evolutionary game.

I have a population within which agents are either mutants or incumbents. The parameters to be decided are:

1. the fraction of the mutants ($\epsilon$)
2. the fitness $F$ of the agents

Now, if I came up with algorithms to define/calculate the above in such a way that the population ends up being evolutionarily stable, would my solution be called a mechanism?

Thanks


text:
1. Other
2. Other
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# Question: im confused about the wording but for part a my...

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I'm confused about the wording, but for part a my domain is [0,10] for part b my domain is [0, 10) then I got stuck on parts c-e


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Question:GoPro Hero HD first version- Making full spectrum( Removing IR filter)

suman asked on January 16, 2017 09:43

Greetings wonderful people,

I have first version of gopro Hero 1 HD ( without screen version). I want to take out the IR filter from the camera to make it work in full spectrum. I disassembled the gopro (see picture below) and found that the lens is fixed. The only option is cutting the lens using dremel tool. But i am confused on where to cut?

Can anyone having previous experience help me on how to do so. The purpose of my work is to monitor crop health by attaching the modified gopro in the DJI Phantom 3 Professional.

I look forward to your suggestions.

Thanks a lot

Suman Ghimire Researcher SIIT, Thammasat University Thailand

Via @wildintellect on the plots-infrared discussion list:

Replace the lens, there are several aftermarket GoPro lens made for IR. http://stuntcams.com/shop/gopro-hd-infrared-replacement-ir-lens-170-degrees-p-425.html Last time I researched this (Hero 2) the IR filter was not a removable component it was a coating.

Hope that helps!

Dear Warren,

This is the first version of GoPro Hero HD without screen not Hero 2 (although it looks quite similar). Yeah the lens is fixed (no any way to open the lens) . In order, to open it the only option i have is to cut the lens with a dremel. But i am confused on which part to cut in the lens . I have attached the picture of the lens so you can see how it looks like and could potentially give me an idea.


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Have you looked at this paper? https://arxiv.org/pdf/1804.02918&ved=2ahUKEwj46ZyDrajjAhXkIzQIHdt5AqYQFjACegQIAhAB&usg=AOvVaw3OoDyPWKVPW7fDitFpn_cH&cshid=1562693079932

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What sort of algorithm would effectively notice this? What you are describing, is an online algorithm that uses a small set of "past" measurements to make an inference, in this case, about the slope of the signal. Since you are keeping track of the current beginning and end of the rolling buffer, it is essentially like having access to a linear buffer of N ...

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To be honest, I don't think CNNs, RNNs and LSTM are useful for this kind of problem – a bandpass filter followed by a threshold would be. Now, that would have three parameters: Lower cutoff frequency Upper cutoff frequency threshold value and what is usually called "Machine Learning" is nothing but finding local minima over some (loss) function with real ...

1

The term LTI system is a bit broad, so perhaps restricting ourselves to single input-single output systems makes sense. Let's just look at $s$ (Laplace) for now. The $Z$ transform follows in a straightforward way. Also, if the system is not stable, a practical inverse filter will not recover the input. Convolution commutes so following a stable filter with ...

2

You have to be clear what you mean by "invertible". Commonly, you want the inverse system to be causal and stable, and that puts certain restrictions on the original system. In the case of systems with rational transfer functions, you just have to look at the zeros of the transfer function, because they become the poles of the inverse system. If all zeros ...

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In general LTI System is invertible if it has neither zeros nor poles in the Fourier Domain (Its spectrum). The way to prove it is to calculate the Fourier Transform of its Impulse Response. The intuition is simple, if it has no zeros in the frequency domain one could calculate its inverse (Element wise inverse) in the frequency domain. Few remarks for the ...

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