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Discovery of nuclear fission
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1938 achievement in physics
Nuclear fission was discovered in December 1938 by chemists Otto Hahn and Fritz Strassmann and physicists Lise Meitner and Otto Robert Frisch. Fission is a nuclear reaction or radioactive decay process in which the nucleus of an atom splits into two or more smaller, lighter nuclei and often other particles. The fission process often produces gamma rays and releases a very large amount of energy, even by the energetic standards of radioactive decay. Scientists already knew about alpha decay and beta decay, but fission assumed great importance because the discovery that a nuclear chain reaction was possible led to the development of nuclear power and nuclear weapons. Hahn was awarded the 1944 Nobel Prize in Chemistry for the discovery of nuclear fission.
Hahn and Strassmann at the Kaiser Wilhelm Institute for Chemistry in Berlin bombarded uranium with slow neutrons and discovered that barium had been produced. Hahn suggested a bursting of the nucleus, but he was unsure of what the physical basis for the results were. They reported their findings by mail to Meitner in Sweden, who a few months earlier had fled Nazi Germany. Meitner and her nephew Frisch theorised, and then proved, that the uranium nucleus had been split and published their findings in "Nature". Meitner calculated that the energy released by each disintegration was approximately 200 megaelectronvolts, and Frisch observed this. By analogy with the division of biological cells, he named the process "fission".
The discovery came after forty years of investigation into the nature and properties of radioactivity and radioactive substances. The discovery of the neutron by James Chadwick in 1932 created a new means of nuclear transmutation. Enrico Fermi and his colleagues in Rome studied the results of bombarding uranium with neutrons, and Fermi concluded that his experiments had created new elements with 93 and 94 protons, which his group dubbed "ausenium" and "hesperium". Fermi won the 1938 Nobel Prize in Physics for his "demonstrations of the existence of new radioactive elements produced by neutron irradiation, and for his related discovery of nuclear reactions brought about by slow neutrons". However, not everyone was convinced by Fermi's analysis of his results. Ida Noddack suggested that instead of creating a new, heavier element 93, it was conceivable that the nucleus had broken up into large fragments, and Aristid von Grosse suggested that what Fermi's group had found was an isotope of protactinium.
This spurred Hahn and Meitner, the discoverers of the most stable isotope of protactinium, to conduct a four-year-long investigation into the process with their colleague Strassmann. After much hard work and many discoveries, they determined that what they were observing was fission, and that the new elements that Fermi had found were fission products. Their work overturned long-held beliefs in physics and paved the way for the discovery of the real elements 93 (neptunium) and 94 (plutonium), for the discovery of fission in other elements, and for the determination of the role of the uranium-235 isotope in that of uranium. Niels Bohr and John Wheeler reworked the liquid drop model to explain the mechanism of fission.
Background.
Radioactivity.
In the last years of the 19th century, scientists frequently experimented with the cathode-ray tube, which by then had become a standard piece of laboratory equipment. A common practice was to aim the cathode rays at various substances and to see what happened. Wilhelm Röntgen had a screen coated with barium platinocyanide that would fluoresce when exposed to cathode rays. On 8 November 1895, he noticed that even though his cathode-ray tube was not pointed at his screen, which was covered in black cardboard, the screen still fluoresced. He soon became convinced that he had discovered a new type of rays, which are today called X-rays. The following year Henri Becquerel was experimenting with fluorescent uranium salts, and wondered if they too might produce X-rays. On 1 March 1896 he discovered that they did indeed produce rays, but of a different kind, and even when the uranium salt was kept in a dark drawer, it still made an intense image on an X-ray plate, indicating that the rays came from within, and did not require an external energy source.
Unlike Röntgen's discovery, which was the object of widespread curiosity from scientists and lay people alike for the ability of X-rays to make visible the bones within the human body, Becquerel's discovery made little impact at the time, and Becquerel himself soon moved on to other research. Marie Curie tested samples of as many elements and minerals as she could find for signs of Becquerel rays, and in April 1898 also found them in thorium. She gave the phenomenon the name "radioactivity". Along with Pierre Curie and Gustave Bémont, she began investigating pitchblende, a uranium-bearing ore, which was found to be more radioactive than the uranium it contained. This indicated the existence of additional radioactive elements. One was chemically akin to bismuth, but strongly radioactive, and in July 1898 they published a paper in which they concluded that it was a new element, which they named "polonium". The other was chemically like barium, and in a December 1898 paper they announced the discovery of a second hitherto unknown element, which they called "radium". Convincing the scientific community was another matter. Separating radium from the barium in the ore proved very difficult. It took three years for them to produce a tenth of a gram of radium chloride, and they never did manage to isolate polonium.
In 1898, Ernest Rutherford noted that thorium gave off a radioactive gas. In examining the radiation, he classified Becquerel radiation into two types, which he called α (alpha) and β (beta) radiation. Subsequently, Paul Villard discovered a third type of Becquerel radiation which, following Rutherford's scheme, were called "gamma rays", and Curie noted that radium also produced a radioactive gas. Identifying the gas chemically proved frustrating; Rutherford and Frederick Soddy found it to be inert, much like argon. It later came to be known as radon. Rutherford identified beta rays as cathode rays (electrons), and hypothesised—and in 1909 with Thomas Royds proved—that alpha particles were helium nuclei. Observing the radioactive disintegration of elements, Rutherford and Soddy classified the radioactive products according to their characteristic rates of decay, introducing the concept of a half-life. In 1903, Soddy and Margaret Todd applied the term "isotope" to atoms that were chemically and spectroscopically identical but had different radioactive half-lives. Rutherford proposed a model of the atom in which a very small, dense and positively charged nucleus of protons was surrounded by orbiting, negatively charged electrons (the Rutherford model). Niels Bohr improved upon this in 1913 by reconciling it with the quantum behaviour of electrons (the Bohr model).
Protactinium.
Soddy and Kasimir Fajans independently observed in 1913 that alpha decay caused atoms to shift down two places in the periodic table, while the loss of two beta particles restored it to its original position. In the resulting reorganisation of the periodic table, radium was placed in group II, actinium in group III, thorium in group IV and uranium in group VI. This left a gap between thorium and uranium. Soddy predicted that this unknown element, which he referred to (after Dmitri Mendeleev) as "ekatantalium", would be an alpha emitter with chemical properties similar to tantalium (now known as tantalum). It was not long before Fajans and Oswald Helmuth Göhring discovered it as a decay product of a beta-emitting product of thorium. Based on the radioactive displacement law of Fajans and Soddy, this was an isotope of the missing element, which they named "brevium" after its short half-life. However, it was a beta emitter, and therefore could not be the mother isotope of actinium. This had to be another isotope.
Two scientists at the Kaiser Wilhelm Institute (KWI) in Berlin-Dahlem took up the challenge of finding the missing isotope. Otto Hahn had graduated from the University of Marburg as an organic chemist, but had been a post-doctoral researcher at University College London under Sir William Ramsay, and under Rutherford at McGill University, where he had studied radioactive isotopes. In 1906, he returned to Germany, where he became an assistant to Emil Fischer at the University of Berlin. At McGill he had become accustomed to working closely with a physicist, so he teamed up with Lise Meitner, who had received her doctorate from the University of Vienna in 1906, and had then moved to Berlin to study physics under Max Planck at the Friedrich-Wilhelms-Universität. Meitner found Hahn, who was her own age, less intimidating than older, more distinguished colleagues. Hahn and Meitner moved to the recently established Kaiser Wilhelm Institute for Chemistry in 1913, and by 1920 had become the heads of their own laboratories there, with their own students, research programs and equipment. The new laboratories offered new opportunities, as the old ones had become too contaminated with radioactive substances to investigate feebly radioactive substances. They developed a new technique for separating the tantalum group from pitchblende, which they hoped would speed the isolation of the new isotope.
The work was interrupted by the outbreak of the First World War in 1914. Hahn was called up into the German Army, and Meitner became a volunteer radiographer in Austrian Army hospitals. She returned to the Kaiser Wilhelm Institute in October 1916. Hahn joined the new gas command unit at Imperial Headquarters in Berlin in December 1916 after travelling between the western and eastern fronts, Berlin and Leverkusen between the summer of 1914 and late 1916.
Most of the students, laboratory assistants and technicians had been called up, so Hahn, who was stationed in Berlin between January and September 1917, and Meitner had to do everything themselves. By December 1917 she was able to isolate the substance, and after further work were able to prove that it was indeed the missing isotope. Meitner submitted her and Hahn's findings for publication in March 1918 to the scientific paper "Physikalischen Zeitschrift" under the title .
Although Fajans and Göhring had been the first to discover the element, custom required that an element was represented by its longest-lived and most abundant isotope, and brevium did not seem appropriate. Fajans agreed to Meitner and Hahn naming the element protactinium, and assigning it the chemical symbol Pa. In June 1918, Soddy and John Cranston announced that they had extracted a sample of the isotope, but unlike Hahn and Meitner were unable to describe its characteristics. They acknowledged Hahn's and Meitner's priority, and agreed to the name. The connection to uranium remained a mystery, as neither of the known isotopes of uranium decayed into protactinium. It remained unsolved until uranium-235 was discovered in 1929.
For their discovery Hahn and Meitner were repeatedly nominated for the Nobel Prize in Chemistry in the 1920s by several scientists, among them Max Planck, Heinrich Goldschmidt, and Fajans himself. In 1949, the International Union of Pure and Applied Chemistry (IUPAC) named the new element definitively protactinium, and confirmed Hahn and Meitner as discoverers.
Transmutation.
Patrick Blackett was able to accomplish nuclear transmutation of nitrogen into oxygen in 1925, using alpha particles directed at nitrogen. In modern notation for the atomic nuclei, the reaction was:
147N + 42He → 178O + p
This was the first observation of a nuclear reaction, that is, a reaction in which particles from one decay are used to transform another atomic nucleus. A fully artificial nuclear reaction and nuclear transmutation was achieved in April 1932 by Ernest Walton and John Cockcroft, who used artificially accelerated protons against lithium, to break this nucleus into two alpha particles. The feat was popularly known as "splitting the atom", but was not nuclear fission; as it was not the result of initiating an internal radioactive decay process.
Just a few weeks before Cockcroft and Walton's feat, another scientist at the Cavendish Laboratory, James Chadwick, discovered the neutron, using an ingenious device made with sealing wax, through the reaction of beryllium with alpha particles:
94Be + 42He → 126C + n
Irène Curie and Frédéric Joliot irradiated aluminium foil with alpha particles and found that this results in a short-lived radioactive isotope of phosphorus with a half-life of around three minutes:
2713Al + 42He → 3015P + n
which then decays to a stable isotope of silicon
3015P → 3014Si + e+
They noted that radioactivity continued after the neutron emissions ceased. Not only had they discovered a new form of radioactive decay in the form of positron emission, they had transmuted an element into a hitherto unknown radioactive isotope of another, thereby inducing radioactivity where there had been none before. Radiochemistry was now no longer confined to certain heavy elements, but extended to the entire periodic table.
Chadwick noted that being electrically neutral, neutrons would be able to penetrate the nucleus more easily than protons or alpha particles. Enrico Fermi and his colleagues in Rome—Edoardo Amaldi, Oscar D'Agostino, Franco Rasetti and Emilio Segrè—picked up on this idea. Rasetti visited Meitner's laboratory in 1931, and again in 1932 after Chadwick's discovery of the neutron. Meitner showed him how to prepare a polonium-beryllium neutron source. On returning to Rome, Rasetti built Geiger counters and a cloud chamber modelled after Meitner's. Fermi initially intended to use polonium as a source of alpha particles, as Chadwick and Curie had done. Radon was a stronger source of alpha particles than polonium, but it also emitted beta and gamma rays, which played havoc with the detection equipment in the laboratory. But Rasetti went on his Easter vacation without preparing the polonium-beryllium source, and Fermi realised that since he was interested in the products of the reaction, he could irradiate his sample in one laboratory and test it in another down the hall. The neutron source was easy to prepare by mixing with powdered beryllium in a sealed capsule. Moreover, radon was easily obtained; Giulio Cesare Trabacchi had more than a gram of radium and was happy to supply Fermi with radon. With a half-life of only 3.82 days it would only go to waste otherwise, and the radium continually produced more.
Working in assembly-line fashion, they started by irradiating water, and then progressed up the periodic table through lithium, beryllium, boron and carbon, without inducing any radioactivity. When they got to aluminium and then fluorine, they had their first successes. Induced radioactivity was ultimately found through the neutron bombardment of 22 different elements. Meitner was one of the select group of physicists to whom Fermi mailed advance copies of his papers, and she was able to report that she had verified his findings with respect to aluminium, silicon, phosphorus, copper and zinc. When a new copy of "La Ricerca Scientifica" arrived at the Niels Bohr's Institute for Theoretical Physics at the University of Copenhagen, her nephew, Otto Frisch, as the only physicist there who could read Italian, found himself in demand from colleagues wanting a translation. The Rome group had no samples of the rare earth metals, but at Bohr's institute George de Hevesy had a complete set of their oxides that had been given to him by Auergesellschaft, so de Hevesy and Hilde Levi carried out the process with them.
When the Rome group reached uranium, they had a problem: the radioactivity of natural uranium was almost as great as that of their neutron source. What they observed was a complex mixture of half-lives. Following the displacement law, they checked for the presence of lead, bismuth, radium, actinium, thorium and protactinium (skipping the elements whose chemical properties were unknown), and (correctly) found no indication of any of them. Fermi noted three types of reactions were caused by neutron irradiation: emission of an alpha particle (n, α); proton emission (n, p); and gamma emission (n, γ). Invariably, the new isotopes decayed by beta emission, which caused elements to move up the periodic table.
Based on the periodic table of the time, Fermi believed that element 93 was ekarhenium—the element below rhenium—with characteristics similar to manganese and rhenium. Such an element was found, and Fermi tentatively concluded that his experiments had created new elements with 93 and 94 protons, which he dubbed "ausenium" and "hesperium". The results were published in "Nature" in June 1934. However, in this paper Fermi cautioned that "a careful search for such heavy particles has not yet been carried out, as they require for their observation that the active product should be in the form of a very thin layer. It seems therefore at present premature to form any definite hypothesis on the chain of disintegrations involved." In retrospect, what they had detected was indeed an unknown rhenium-like element, technetium, which lies between manganese and rhenium on the periodic table.
Leo Szilard and Thomas A. Chalmers reported that neutrons generated by gamma rays acting on beryllium were captured by iodine, a reaction that Fermi had also noted. When Meitner repeated their experiment, she found that neutrons from the gamma-beryllium sources were captured by heavy elements like iodine, silver and gold, but not by lighter ones like sodium, aluminium and silicon. She concluded that slow neutrons were more likely to be captured than fast ones, a finding she reported to "Naturwissenschaften" in October 1934. Everyone had been thinking that energetic neutrons were required, as was the case with alpha particles and protons, but that was required to overcome the Coulomb barrier; the neutrally charged neutrons were more likely to be captured by the nucleus if they spent more time in its vicinity. A few days later, Fermi considered a curiosity that his group had noted: uranium seemed to react differently in different parts of the laboratory; neutron irradiation conducted on a wooden table induced more radioactivity than on a marble table in the same room. Fermi thought about this and tried placing a piece of paraffin wax between the neutron source and the uranium. This resulted in a dramatic increase in activity. He reasoned that the neutrons had been slowed by collisions with hydrogen atoms in the paraffin and wood. The departure of D'Agostino meant that the Rome group no longer had a chemist, and the subsequent loss of Rasetti and Segrè reduced the group to just Fermi and Amaldi, who abandoned the research into transmutation to concentrate on exploring the physics of slow neutrons.
The current model of the nucleus in 1934 was the liquid drop model first proposed by George Gamow in 1930. His simple and elegant model was refined and developed by Carl Friedrich von Weizsäcker and, after the discovery of the neutron, by Werner Heisenberg in 1935 and Niels Bohr in 1936, it agreed closely with observations. In the model, the nucleons were held together in the smallest possible volume (a sphere) by the strong nuclear force, which was capable of overcoming the longer ranged Coulomb electrical repulsion between the protons. The model remained in use for certain applications into the 21st century, when it attracted the attention of mathematicians interested in its properties, but in its 1934 form it confirmed what physicists thought they already knew: that nuclei were static, and that the odds of a collision chipping off more than an alpha particle were practically zero.
Discovery.
Objections.
Fermi won the 1938 Nobel Prize in Physics for his "demonstrations of the existence of new radioactive elements produced by neutron irradiation, and for his related discovery of nuclear reactions brought about by slow neutrons". However, not everyone was convinced by Fermi's analysis of his results. Ida Noddack suggested in September 1934 that instead of creating a new, heavier element 93, that: <templatestyles src="Template:Blockquote/styles.css" />One could assume equally well that when neutrons are used to produce nuclear disintegrations, some distinctly new nuclear reactions take place which have not been observed previously with proton or alpha-particle bombardment of atomic nuclei. In the past one has found that transmutations of nuclei only take place with the emission of electrons, protons, or helium nuclei, so that the heavy elements change their mass only a small amount to produce near neighbouring elements. When heavy nuclei are bombarded by neutrons, it is conceivable that the nucleus breaks up into several large fragments, which would of course be isotopes of known elements but would not be neighbours of the irradiated element.
Noddack's article was read by Fermi's team in Rome, Curie and Joliot in Paris, and Meitner and Hahn in Berlin. However, the quoted objection comes some distance down, and is but one of several gaps she noted in Fermi's claim. Bohr's liquid drop model had not yet been formulated, so there was no theoretical way to calculate whether it was physically possible for the uranium atoms to break into large pieces. Noddack and her husband, Walter Noddack, were renowned chemists who had been nominated for the Nobel Prize in Chemistry for the discovery of rhenium, although at the time they were also embroiled in a controversy over the discovery of element 43, which they called "masurium". The discovery of technetium by Emilio Segrè and Carlo Perrier put an end to their claim, but did not occur until 1937. It is unlikely that Meitner or Curie had any prejudice against Noddack because of her sex, but Meitner was not afraid to tell Hahn "Hähnchen, von Physik verstehst Du Nichts" ("Hahn dear, of physics you understand nothing"). The same attitude carried over to Noddack, who did not propose an alternative nuclear model, nor conduct experiments to support her claim. Although Noddack was a renowned analytical chemist, she lacked the background in physics to appreciate the enormity of what she was proposing.
Noddack was not the only critic of Fermi's claim. Aristid von Grosse suggested that what Fermi had found was an isotope of protactinium. Meitner was eager to investigate Fermi's results, but she recognised that a highly skilled chemist was required, and she wanted the best one she knew: Hahn, although they had not collaborated for many years. Initially, Hahn was not interested, but von Grosse's mention of protactinium changed his mind. "The only question", Hahn later wrote, "seemed to be whether Fermi had found isotopes of transuranian elements, or isotopes of the next-lower element, protactinium. At that time Lise Meitner and I decided to repeat Fermi's experiments in order to find out whether the 13-minute isotope was a protactinium isotope or not. It was a logical decision, having been the discoverers of protactinium."
Hahn and Meitner were joined by Fritz Strassmann. Strassmann had received his doctorate in analytical chemistry from the Technical University of Hannover in 1929, and had come to the Kaiser Wilhelm Institute for Chemistry to study under Hahn, believing that this would improve his employment prospects. He enjoyed the work and the people so much that he stayed on after his stipend expired in 1932. After the Nazi Party came to power in Germany in 1933, he declined a lucrative offer of employment because it required political training and Nazi Party membership, and he resigned from the Society of German Chemists when it became part of the Nazi German Labour Front. As a result, he could neither work in the chemical industry nor receive his habilitation, which was required to become an independent researcher in Germany. Meitner persuaded Hahn to hire Strassmann using money from the director's special circumstances fund. In 1935, Strassmann became an assistant on half pay. Soon he would be credited as a collaborator on the papers they produced.
The 1933 Law for the Restoration of the Professional Civil Service removed Jewish people from the civil service, which included academia. Meitner never tried to conceal her Jewish descent, but initially was exempt from its impact on multiple grounds: she had been employed before 1914, had served in the military during the World War, was an Austrian rather than a German citizen, and the Kaiser Wilhelm Institute was a government-industry partnership. However, she was dismissed from her adjunct professorship at the University of Berlin on the grounds that her World War I service was not at the front, and she had not completed her habilitation until 1922. Carl Bosch, the director of IG Farben, a major sponsor of the Kaiser Wilhelm Institute for Chemistry, assured Meitner that her position there was safe, and she agreed to stay. Meitner, Hahn and Strassmann drew closer together personally as their anti-Nazi politics increasingly alienated them from the rest of the organisation, but it gave them more time for research, as administration was devolved to Hahn's and Meitner's assistants.
Research.
The Berlin group started by irradiating uranium salt with neutrons from a radon-beryllium source similar to the one that Fermi had used. They dissolved it and added potassium perrhenate, platinum chloride and sodium hydroxide. What remained was then acidified with hydrogen sulphide, resulting in platinum sulphide and rhenium sulphide precipitation. Fermi had noted four radioactive isotopes with the longest-lived having 13- and 90-minute half-lives, and these were detected in the precipitate. The Berlin group then tested for protactinium by adding protactinium-234 to the solution. When this was precipitated, it was found to be separated from the 13- and 90-minute half-life isotopes, demonstrating that von Grosse was incorrect, and they were not isotopes of protactinium. Moreover, the chemical reactions involved ruled out all elements from mercury and above on the periodic table. They were able to precipitate the 90-minute activity with osmium sulphide and the 13-minute one with rhenium sulphide, which ruled out them being isotopes of the same element. All this provided strong evidence that they were indeed transuranium elements, with chemical properties similar to osmium and rhenium.
Fermi had also reported that fast and slow neutrons had produced different activities. This indicated that more than one reaction was taking place. When the Berlin group could not replicate the Rome group's findings, they commenced their own research into the effects of fast and slow neutrons. To minimise radioactive contamination if there were an accident, different phases were carried out in different rooms, all in Meitner's section on the ground floor of the Kaiser Wilhelm Institute. Neutron irradiation was carried out in one laboratory, chemical separation in another, and measurements were conducted in a third. The equipment they used was simple and mostly hand made.
By March 1936, they had identified ten different half-lives, with varying degrees of certainty. To account for them, Meitner had to hypothesise a new (n, 2n) class of reaction and the alpha decay of uranium, neither of which had ever been reported before, and for which physical evidence was lacking. So while Hahn and Strassmann refined their chemical procedures, Meitner devised new experiments to shine more light on the reaction processes. In May 1937, they issued parallel reports, one in "Zeitschrift für Physik" with Meitner as the principal author, and one in "Chemische Berichte" with Hahn as the principal author. Hahn concluded his by stating emphatically: "Vor allem steht ihre chemische Verschiedenheit von allen bisher bekannten Elementen außerhalb jeder Diskussion" ("Above all, their chemical distinction from all previously known elements needs no further discussion.")
Meitner was increasingly uncertain. They had now constructed three (n, γ) reactions:
Meitner was certain that these had to be (n, γ) reactions, as slow neutrons lacked the energy to chip off protons or alpha particles. She considered the possibility that the reactions were from different isotopes of uranium; three were known: uranium-238, uranium-235 and uranium-234. However, when she calculated the neutron cross section it was too large to be anything other than the most abundant isotope, uranium-238. She concluded that it must be a case of nuclear isomerism, which had been discovered in protactinium by Hahn in 1922. Nuclear isomerism had been given a physical explanation by von Weizsäcker, who had been Meitner's assistant in 1936, but had since taken a position at the Kaiser Wilhelm Institute for Physics. Different nuclear isomers of protactinium had different half-lives, and this could be the case for uranium too, but if so it was somehow being inherited by the daughter and granddaughter products, which seemed to be stretching the argument to breaking point. Then there was the third reaction, an (n, γ) one, which occurred only with slow neutrons. Meitner therefore ended her report on a very different note to Hahn, reporting that: "The process must be neutron capture by uranium-238, which leads to three isomeric nuclei of uranium-239. This result is very difficult to reconcile with current concepts of the nucleus."
After this, the Berlin group moved on to working with thorium, as Strassmann put it, "to recover from, the horror of the work with uranium". However, thorium was not easier to work with than uranium. For a start, it had a decay product, radiothorium (22890Th) that overwhelmed weaker neutron-induced activity. But Hahn and Meitner had a sample from which they had regularly removed its mother isotope, mesothorium (22888Ra), over a period of several years, allowing the radiothorium to decay away. Even then, it was still more difficult to work with because its induced decay products from neutron irradiation were isotopes of the same elements produced by thorium's own radioactive decay. What they found was three different decay series, all alpha emitters—a form of decay not found in any other heavy element, and for which Meitner once again had to postulate multiple isomers. They did find an interesting result: these (n, α) decay series occurred simultaneously when the energy of the incident neutrons was less than 2.5 MeV; when they had more, an (n, γ) reaction that formed 23390Th was favoured.
In Paris, Irene Curie and Pavel Savitch had also set out to replicate Fermi's findings. In collaboration with Hans von Halban and Peter Preiswerk, they irradiated thorium and produced the isotope with a 22-minute half-life that Fermi had noted. In all, Curie's group detected eight different half-lives in their irradiated thorium. Curie and Savitch detected a radioactive substance with a 3.5-hour half-life. The Paris group proposed that it might be an isotope of thorium. Meitner asked Strassmann, who was now doing most of the chemistry work, to check. He detected no sign of thorium. Meitner wrote to Curie with their results, and suggested a quiet retraction. Nonetheless, Curie persisted. They investigated the chemistry, and found that the 3.5-hour activity was coming from something that seemed to be chemically similar to lanthanum (which in fact it was), which they attempted unsuccessfully to isolate with a fractional crystallization process. (It is possible that their precipitate was contaminated with yttrium, which is chemically similar.) By using Geiger counters and skipping the chemical precipitation, Curie and Savitch detected the 3.5-hour half-life in irradiated uranium.
With the "Anschluss", Germany's unification with Austria on 12 March 1938, Meitner lost her Austrian citizenship. James Franck offered to sponsor her immigration to the United States, and Bohr offered a temporary place at his institute, but when she went to the Danish embassy for a visa, she was told that Denmark no longer recognised her Austrian passport as valid. On 13 July 1938, Meitner departed for the Netherlands with Dutch physicist Dirk Coster. Before she left, Otto Hahn gave her a diamond ring he had inherited from his mother to sell if necessary. She reached safety, but with only her summer clothes. Meitner later said that she left Germany forever with 10 marks in her purse. With the help of Coster and Adriaan Fokker, she flew to Copenhagen, where she was greeted by Frisch, and stayed with Niels and Margrethe Bohr at their holiday house in Tisvilde. On 1 August she took the train to Stockholm, where she was met by Eva von Bahr.
Interpretation.
The Paris group published their results in September 1938. Hahn dismissed the isotope with the 3.5-hour half-life as contamination, but after looking at the details of the Paris group's experiments and the decay curves, Strassmann was worried. He decided to repeat the experiment, using his more efficient method of separating radium. This time, they found what they thought was radium, which Hahn suggested resulted from two alpha decays:
23892U + n → α + 23590Th → α + 23588Ra
Meitner found this very hard to believe.
In November, Hahn travelled to Copenhagen, where he met with Bohr and Meitner. They told him that they were very unhappy about the proposed radium isomers. On Meitner's instructions, Hahn and Strassmann began to redo the experiments, even as Fermi was collecting his Nobel Prize in Stockholm. Assisted by Clara Lieber and Irmgard Bohne, Hahn and Strassmann isolated the three radium isotopes (verified by their half-lives) and used fractional crystallisation to separate them from the barium carrier by adding barium bromide crystals in four steps. Since radium precipitates preferentially in a solution of barium bromide, at each step the fraction drawn off would contain less radium than the one before. However, they found no difference between each of the fractions. In case their process was faulty in some way, they verified it with known isotopes of radium; the process was fine. Hahn and Strassmann found a fourth radium isotope. Their half-lives were formulated as such by Hahn and Strassmann:
formula_0
formula_1
formula_2
formula_3
On 19 December, Hahn wrote to Meitner, informing her that the radium isotopes behaved chemically like barium. Anxious to finish up before the Christmas break, Hahn and Strassmann submitted their findings to "Naturwissenschaften" on 22 December without waiting for Meitner to reply. Hahn understood that a "burst" of the atomic nuclei had occurred, but he was unsure about that interpretation. Hahn concluded the article in "Naturwissenschaften" with: "As chemists... we should substitute the symbols Ba, La, Ce for Ra, Ac, Th. As 'nuclear chemists' fairly close to physics we cannot yet bring ourselves to take this step which contradicts all previous experience in physics."
Frisch normally celebrated Christmas with Meitner in Berlin, but in 1938 she accepted an invitation from Eva von Bahr to spend it with her family at Kungälv, and Meitner asked Frisch to join her there. Meitner received the letter from Hahn describing his chemical proof that some of the product of the bombardment of uranium with neutrons was barium. Barium had an atomic mass 40% less than uranium, and no previously known methods of radioactive decay could account for such a large difference in the mass of the nucleus.
Nonetheless, she had immediately written back to Hahn to say: "At the moment the assumption of such a thoroughgoing breakup seems very difficult to me, but in nuclear physics we have experienced so many surprises, that one cannot unconditionally say: 'It is impossible.'" Meitner felt that Hahn was too careful a chemist to make an elementary blunder, but found the results difficult to explain. All the nuclear reactions that had been documented involved chipping protons or alpha particles from the nucleus. Breaking it up seemed far more difficult. However the liquid drop model that Gamow had postulated suggested the possibility that an atomic nucleus could become elongated and overcome the surface tension that held it together.
According to Frisch:<templatestyles src="Template:Blockquote/styles.css" />At that point we both sat down on a tree trunk (all that discussion had taken place while we walked through the wood in the snow, I with my skis on, Lise Meitner making good her claim that she could walk just as fast without), and started to calculate on scraps of paper. The charge of a uranium nucleus, we found, was indeed large enough to overcome the effect of the surface tension almost completely; so the uranium nucleus might indeed resemble a very wobbly unstable drop, ready to divide itself at the slightest provocation, such as the impact of a single neutron.But there was another problem. After separation, the two drops would be driven apart by their mutual electric repulsion and would acquire high speed and hence a very large energy, about 200 MeV in all; where could that energy come from? Fortunately Lise Meitner remembered the empirical formula for computing the masses of nuclei and worked out that the two nuclei formed by the division of a uranium nucleus together would be lighter than the original uranium nucleus by about one-fifth the mass of a proton. Now whenever mass disappears energy is created, according to Einstein's formula formula_4, and one-fifth of a proton mass was just equivalent to 200 MeV. So here was the source for that energy; it all fitted!
Meitner and Frisch had correctly interpreted Hahn's results to mean that the nucleus of uranium had split roughly in half. The first two reactions that the Berlin group had observed were light elements created by the breakup of uranium nuclei; the third, the 23-minute one, was a decay into the real element 93. On returning to Copenhagen, Frisch informed Bohr, who slapped his forehead and exclaimed "What idiots we have been!" Bohr promised not to say anything until they had a paper ready for publication. To speed the process, they decided to submit a one-page note to "Nature". At this point, the only evidence that they had was the barium. Logically, if barium was formed, the other element must be krypton, although Hahn mistakenly believed that the atomic masses had to add up to 239 rather than the atomic numbers adding up to 92, and thought it was masurium (technetium), and so did not check for it:
23592U + n → 56Ba + 36Kr + some n
Over a series of long-distance phone calls, Meitner and Frisch came up with a simple experiment to bolster their claim: to measure the recoil of the fission fragments, using a Geiger counter with the threshold set above that of the alpha particles. Frisch conducted the experiment on 13 January 1939, and found the pulses caused by the reaction just as they had predicted. He decided he needed a name for the newly discovered nuclear process. He spoke to William A. Arnold, an American biologist working with de Hevesy and asked him what biologists called the process by which living cells divided into two cells. Arnold told him that biologists called it fission. Frisch then applied that name to the nuclear process in his paper. Frisch mailed both the jointly-authored note on fission and his paper on the recoil experiment to "Nature" on 16 January 1939; the former appeared in print on 11 February and the latter on 18 February. In their second publication on nuclear fission in February 1939, Hahn and Strassmann used the term "Uranspaltung" (uranium fission) for the first time, and predicted the existence and liberation of additional neutrons during the fission process, opening up the possibility of a nuclear chain reaction.
In an 8 March 1959 interview, Meitner said: "It [the discovery of nuclear fission] was achieved with an unusually good chemistry by Hahn and Strassmann, with a fantastically good chemistry that nobody else could do at that time. Later, the Americans learned it. But at that time Hahn and Strassmann were really the only ones who could do it at all because they were such good chemists. They really demonstrated a physical process with chemistry, so to speak."
Reception.
Bohr brings the news to the United States.
Before departing for the United States on 7 January 1939 with his son Erik to attend the Fifth Washington Conference on Theoretical Physics, Bohr promised Frisch that he would not mention fission until the papers appeared in print, but during the Atlantic crossing on the , Bohr discussed the mechanism of fission with Leon Rosenfeld, and failed to inform him that the information was confidential. On arrival in New York City on 16 January, they were met by Fermi and his wife Laura Capon, and by John Wheeler, who had been a fellow at Bohr's institute in 1934–1935. As it happened, there was a meeting of Princeton University's Physics Journal Club that evening, and when Wheeler asked Rosenfeld if he had any news to report, Rosenfeld told them. An embarrassed Bohr fired off a note to "Nature" defending Meitner and Frisch's claim to the priority of the discovery. Hahn was annoyed that while Bohr mentioned his and Strassmann's work in the note, he cited only Meitner and Frisch.
News spread quickly of the new discovery, which was correctly seen as an entirely novel physical effect with great scientific—and potentially practical—possibilities. Isidor Isaac Rabi and Willis Lamb, two Columbia University physicists working at Princeton, heard the news and carried it back to Columbia. Rabi said he told Fermi; Fermi gave credit to Lamb. For Fermi, the news came as a profound embarrassment, as the transuranic elements that he had partly been awarded the Nobel Prize for discovering had not been transuranic elements at all, but fission products. He added a footnote to this effect to his Nobel Prize acceptance speech. Bohr soon thereafter went from Princeton to Columbia to see Fermi. Not finding Fermi in his office, Bohr went down to the cyclotron area and found Herbert L. Anderson. Bohr grabbed him by the shoulder and said: "Young man, let me explain to you about something new and exciting in physics."
Further research.
It was clear to many scientists at Columbia that they should try to detect the energy released in the nuclear fission of uranium from neutron bombardment. On 25 January 1939, a Columbia University group conducted the first nuclear fission experiment in the United States, which was done in the basement of Pupin Hall. The experiment involved placing uranium oxide inside of an ionization chamber and irradiating it with neutrons, and measuring the energy thus released. The next day, the Fifth Washington Conference on Theoretical Physics began in Washington, D.C., under the joint auspices of The George Washington University and the Carnegie Institution of Washington. From there, the news on nuclear fission spread even further, which fostered many more experimental demonstrations.
Bohr and Wheeler overhauled the liquid drop model to explain the mechanism of nuclear fission, with conspicuous success. Their paper appeared in "Physical Review" on 1 September 1939, the day Germany invaded Poland, starting World War II in Europe. As the experimental physicists studied fission, they uncovered more puzzling results. George Placzek asked Bohr why uranium fissioned with both very fast and very slow neutrons. Walking to a meeting with Wheeler, Bohr had an insight that the fission at low energies was due to the uranium-235 isotope, while at high energies it was mainly due to the far more abundant uranium-238 isotope. This was based on Meitner's 1937 measurements of the neutron capture cross-sections. This would be experimentally verified in February 1940, after Alfred Nier was able to produce sufficient pure uranium-235 for John R. Dunning, Aristid von Grosse and Eugene T. Booth to test.
Other scientists resumed the search for the elusive element 93, which seemed to be straightforward, as they now knew it resulted from the 23-minute half-life. At the Radiation Laboratory in Berkeley, California, Emilio Segrè and Edwin McMillan used the cyclotron to create the isotope. They then detected a beta activity with a 2-day half-life, but it had rare-earth element chemical characteristics, and element 93 was supposed to have chemistry akin to rhenium. It was therefore overlooked as just another fission product. Another year passed before McMillan and Philip Abelson determined that the 2-day half-life element was that of the elusive element 93, which they named "neptunium". They paved the way for the discovery by Glenn Seaborg, Emilio Segrè and Joseph W. Kennedy of element 94, which they named "plutonium" in 1941.
Another avenue of research, spearheaded by Meitner, was to determine if other elements could fission after being irradiated with neutrons. It was soon determined that thorium and protactinium could. Measurements were also made of the amount of energy released. Hans von Halban, Frédéric Joliot-Curie and Lew Kowarski demonstrated that uranium bombarded by neutrons emitted more neutrons than it absorbed, suggesting the possibility of a nuclear chain reaction. Fermi and Anderson did so too a few weeks later. It was apparent to many scientists that, in theory at least, an extremely powerful energy source could be created, although most still considered an atomic bomb an impossibility.
Nobel Prize.
Both Hahn and Meitner had been nominated for the chemistry and the physics Nobel Prizes many times even before the discovery of nuclear fission for their work on radioactive isotopes and protactinium. Several more nominations followed for the discovery of fission between 1940 and 1943. Nobel Prize nominations were vetted by committees of five, one for each award. Although both Hahn and Meitner received nominations for physics, radioactivity and radioactive elements had traditionally been seen as the domain of chemistry, and so the Nobel Committee for Chemistry evaluated the nominations in 1944.
The committee received reports from Theodor Svedberg in 1941 and Arne Westgren in 1942. These chemists were impressed by Hahn's work, but felt that the experimental work of Meitner and Frisch was not extraordinary. They did not understand why the physics community regarded their work as seminal. As for Strassmann, although his name was on the papers, there was a long-standing policy of conferring awards on the most senior scientist in a collaboration. In 1944 the Nobel Committee for Chemistry voted to recommend that Hahn alone be given the Nobel Prize in Chemistry for 1944. However, Germans had been forbidden to accept Nobel Prizes after the Nobel Peace Prize had been awarded to Carl von Ossietzky in 1936. The committee's recommendation was rejected by the Royal Swedish Academy of Sciences, which decided to defer the award for one year.
The war was over when the academy reconsidered the award in September 1945. The Nobel Committee for Chemistry had now become more cautious, as it was apparent that much research had been undertaken by the Manhattan Project in the United States in secret, and it suggested deferring the 1944 Nobel Prize in Chemistry for another year. The academy was swayed by Göran Liljestrand, who argued that it was important for the academy to assert its independence from the Allies of World War II, and award the Nobel Prize in Chemistry to a German, as it had done after World War I when it had awarded it to Fritz Haber. Hahn therefore became the sole recipient of the 1944 Nobel Prize in Chemistry "for his discovery of the fission of heavy nuclei".
Meitner wrote in a letter to her friend Birgit Broomé-Aminoff on 20 November 1945:<templatestyles src="Template:Blockquote/styles.css" />Surely Hahn fully deserved the Nobel Prize in chemistry. There is really no doubt about it. But I believe that Otto Robert Frisch and I contributed something not insignificant to the clarification of the process of uranium fission – how it originates and that it produces so much energy, and that was something very remote from Hahn. For this reason I found it a bit unjust that in the newspapers I was called a "Mitarbeiterin" [subordinate] of Hahn's in the same sense that Strassmann was.
In 1946, the Nobel Committee for Physics considered nominations for Meitner and Frisch received from Max von Laue, Niels Bohr, Oskar Klein, Egil Hylleraas and James Franck. Reports were written for the committee by Erik Hulthén, who held the chair of experimental physics at Stockholm University, in 1945 and 1946. Hulthén argued that theoretical physics should be considered award-worthy only if it inspired great experiments. The role of Meitner and Frisch in being the first to understand and explain fission was not understood. There may also have been personal factors: the chairman of the committee, Manne Siegbahn, disliked Meitner, and had a professional rivalry with Klein. Meitner and Frisch would continue to be nominated regularly for many years, but would never be awarded a Nobel Prize.
In history and memory.
At the end of the war in Europe, Hahn was taken into custody and incarcerated at Farm Hall with nine other senior scientists, all of whom except Max von Laue had been involved with the German nuclear weapons program, and all except Hahn and Paul Harteck were physicists. It was here that they heard the news of the atomic bombings of Hiroshima and Nagasaki. Unwilling to accept that they were years behind the Americans, and unaware that their conversations were being recorded, many of them said in conversations, that they had never wanted their nuclear weapons program to succeed in the first place. Hahn did not believe them. Hahn was still there when his Nobel Prize was announced in November 1945. The Farm Hall scientists would spend the rest of their lives attempting to rehabilitate the image of German science that had been tarnished by the Nazi period. Inconvenient details like the thousands of female slave labourers from Sachsenhausen concentration camp who mined uranium ore for their experiments were swept under the rug.
For Hahn, this necessarily involved asserting his claim of the discovery of fission for himself, for chemistry, and for Germany. He used his Nobel Prize acceptance speech to assert this narrative, so he mentioned both Meitner's and Straßmann's involvements in his Nobel lecture. Hahn's message resonated strongly in Germany, where he was revered as the proverbial good German, a decent man who had been a staunch opponent of the Nazi regime, but had remained in Germany where he had pursued pure science. As president of the Max Planck Society from 1946 to 1960, he projected an image of German science as undiminished in brilliance and untainted by Nazism to an audience that wanted to believe it. After the Second World War, Hahn came out strongly against the use of nuclear energy for military purposes. He saw the application of his scientific discoveries to such ends as a misuse, or even a crime. Lawrence Badash wrote: "His wartime recognition of the perversion of science for the construction of weapons and his postwar activity in planning the direction of his country's scientific endeavours now inclined him increasingly toward being a spokesman for social responsibility."
In contrast, in the immediate aftermath of the war Meitner and Frisch were hailed as the discoverers of fission in English-speaking countries. Japan was seen as a puppet state of Germany and the destruction of Hiroshima and Nagasaki as poetic justice for the persecution of the Jewish people. In January 1946, Meitner toured the United States, where she gave lectures and received honorary degrees. She attended a cocktail party for Lieutenant General Leslie Groves, the director of the Manhattan Project (who gave her sole credit for the discovery of fission in his 1962 memoirs), and was named Woman of the Year by the Women's National Press Club. At the reception for this award, she sat next to the President of the United States, Harry S. Truman. But Meitner did not enjoy public speaking, especially in English, nor did she relish the role of a celebrity, and she declined the offer of a visiting professorship at Wellesley College. Hahn nominated Meitner and Frisch for the Nobel Prize in Physics in 1948. He and Meitner remained close friends after the war.
In 1966, the United States Atomic Energy Commission jointly awarded the Enrico Fermi Award to Hahn, Strassmann and Meitner for their discovery of fission. The ceremony was held in the Hofburg palace in Vienna. It was the first time that the Enrico Fermi Prize had been awarded to non-Americans, and the first time it was presented to a woman. Meitner's diploma bore the words: "For pioneering research in the naturally occurring radioactivities and extensive experimental studies leading to the discovery of fission". Hahn's diploma was slightly different: "For pioneering research in the naturally occurring radioactivities and extensive experimental studies culminating in the discovery of fission." Hahn and Strassmann were present, but Meitner was too ill to attend, so Frisch accepted the award on her behalf.
During combined celebrations in Germany of the 100th birthdays of Einstein, Hahn, Meitner and von Laue in 1978, Hahn's narrative of the discovery of fission began to crumble. Hahn and Meitner had died in 1968, but Strassmann was still alive, and he asserted the importance of his analytical chemistry and Meitner's physics in the discovery, and their role as more than just assistants. A detailed biography of Strassmann appeared in 1981, a year after his death, and a prize-winning one of Meitner for young adults in 1986. Scientists questioned the focus on chemistry, historians challenged the accepted narrative of the Nazi period, and feminists saw Meitner as yet another example of the Matilda effect, where a woman had been airbrushed from the pages of history. By 1990, Meitner had been restored to the narrative, although her role remained contested, particularly in Germany.
Weizsäcker, a colleague of Hahn and Meitner during their time in Berlin, and a fellow inmate with Hahn in Farm Hall, strongly supported Hahn's role in the discovery of nuclear fission. He told an audience that had gathered for the ceremonial inclusion of a bust of Meitner in the "Ehrensaal" (Hall of Fame) at the "Deutsches Museum" in Munich on 4 July 1991 that neither Meitner nor physics had contributed to the discovery of fission, which, he declared, was "a discovery of Hahn's and not of Lise Meitner's."
Notes.
<templatestyles src="Reflist/styles.css" />
References.
<templatestyles src="Refbegin/styles.css" />
Further reading.
<templatestyles src="Refbegin/styles.css" />
|
[
{
"math_id": 0,
"text": "\\mathrm{Ra \\ I?}^{}_{} \\ \\xrightarrow[\\mathrm{<} \\ \\text {1 min.}]{\\text{ }\\mathrm{\\beta}} \\mathrm {\\ Ac \\ I \\ } \\xrightarrow[\\mathrm{<} \\ \\text{30 min.}]{\\text{ }\\mathrm{\\beta}} \\mathrm {\\ Th?} "
},
{
"math_id": 1,
"text": "\\mathrm{Ra \\ II}^{}_{} \\ \\xrightarrow[\\text {14}\\ \\mathrm{\\pm} \\ {2 min.}]{\\text{ }\\mathrm{\\beta}} \\mathrm {\\ Ac \\ II \\ } \\xrightarrow[\\mathrm{\\sim} \\ \\text{2,5 h.}]{\\text{ }\\mathrm{\\beta}} \\mathrm {\\ Th?} "
},
{
"math_id": 2,
"text": "\\mathrm{Ra \\ III}^{}_{} \\ \\xrightarrow[\\text {86}\\ \\mathrm{\\pm} \\ {6 min.}]{\\text{ }\\mathrm{\\beta}} \\mathrm {\\ Ac \\ III \\ } \\xrightarrow[\\mathrm{\\sim}\\ \\text{couple of days?}]{\\text{ }\\mathrm{\\beta}} \\mathrm {\\ Th?} "
},
{
"math_id": 3,
"text": "\\mathrm{Ra \\ IV}^{}_{} \\ \\xrightarrow[\\text {250-300 hrs.}]{\\text{ }\\mathrm{\\beta}} \\mathrm {\\ Ac \\ IV \\ } \\xrightarrow[\\text{40 hrs.}]{\\text{ }\\mathrm{\\beta}} \\mathrm {\\ Th?} "
},
{
"math_id": 4,
"text": "E=m\\,c^2"
}
] |
https://en.wikipedia.org/wiki?curid=64011351
|
64013419
|
Octanol-water partition coefficient
|
Measure of lipophilicity and hydrophilicity
The n"-octanol-water partition coefficient, "K"ow is a partition coefficient for the two-phase system consisting of "n"-octanol and water. "K"ow is also frequently referred to by the symbol P, especially in the English literature. It is also called n"-octanol-water partition ratio.
"K"ow serves as a measure of the relationship between lipophilicity (fat solubility) and hydrophilicity (water solubility) of a substance. The value is greater than one if a substance is more soluble in fat-like solvents such as n-octanol, and less than one if it is more soluble in water.
If a substance is present as several chemical species in the octanol-water system due to association or dissociation, each species is assigned its own "K"ow value. A related value, D, does not distinguish between different species, only indicating the concentration ratio of the substance between the two phases.
History.
In 1899, Charles Ernest Overton and Hans Horst Meyer independently proposed that the tadpole toxicity of non-ionizable organic compounds depends on their ability to partition into lipophilic compartments of cells. They further proposed the use of the partition coefficient in an olive oil/water mixture as an estimate of this lipophilic associated toxicity. Corwin Hansch later proposed the use of n-octanol as an inexpensive synthetic alcohol that could be obtained in a pure form as an alternative to olive oil.
Applications.
"K"ow values are used, among others, to assess the environmental fate of persistent organic pollutants. Chemicals with high partition coefficients, for example, tend to accumulate in the fatty tissue of organisms (bioaccumulation). Under the Stockholm Convention, chemicals with a log "K"ow greater than 5 are considered to bioaccumulate.
Furthermore, the parameter plays an important role in drug research (Rule of Five) and toxicology. Ernst Overton and Hans Meyer discovered as early as 1900 that the efficacy of an anaesthetic increased with increasing "K"ow value (the so-called Meyer-Overton rule).
"K"ow values also provide a good estimate of how a substance is distributed within a cell between the lipophilic biomembranes and the aqueous cytosol.
Estimation.
Since it is not possible to measure "K"ow for all substances, various models have been developed to allow for their prediction, e.g. Quantitative structure–activity relationships (QSAR) or linear free energy relationships (LFER) such as the Hammett equation.
A variant of the UNIFAC system can also be used to estimate octanol-water partition coefficients.
The "K"ow or P-value always only refers to a single species or substance:
formula_0
with:
*formula_1 concentration of species "i" of a substance in the octanol-rich phase
*formula_2 concentration of species "i" of a substance in the water-rich phase
If different species occur in the octanol-water system by dissociation or association, several P-values and one D-value exist for the system. If, on the other hand, the substance is only present in a single species, the P and D values are identical.
P is usually expressed as a common logarithm, i.e. Log P (also Log Pow or, less frequently, Log pOW):
formula_3 Log P is positive for lipophilic and negative for hydrophilic substances or species.
The P-value only correctly refers to the concentration ratio of a single substance distributed between the octanol and water phases. In the case of a substance that occurs as multiple species, it can therefore be calculated by summing the concentrations of all "n" species in the octanol phase and the concentrations of all "n" species in the aqueous phase:
formula_4
with:
* formula_5 concentration of the substance in the octanol-rich phase
* formula_6 concentration of the substance in the water-rich phase
D values are also usually given in the form of the common logarithm as Log D:
formula_7
Like Log P, Log D is positive for lipophilic and negative for hydrophilic substances. While P values are largely independent of the pH value of the aqueous phase due to their restriction to only one species, D values are often strongly dependent on the pH value of the aqueous phase.
Example values.
Values for log "K"ow typically range between -3 (very hydrophilic) and +10 (extremely lipophilic/hydrophobic).
The values listed here are sorted by the partition coefficient. Acetamide is hydrophilic, and 2,2′,4,4′,5-Pentachlorobiphenyl is lipophilic.
References.
<templatestyles src="Reflist/styles.css" />
Further reading.
<templatestyles src="Refbegin/styles.css" />
|
[
{
"math_id": 0,
"text": "K_\\mathrm{ow} = P = \\frac{c_o^{S_i}}{c_w^{S_i}}"
},
{
"math_id": 1,
"text": "c_o^{S_i}"
},
{
"math_id": 2,
"text": "c_w^{S_i}"
},
{
"math_id": 3,
"text": "\\log{P} = \\log \\frac{c_o^{S_i}}{c_w^{S_i}} = \\log c_o^{S_i} - \\log c_w^{S_i}"
},
{
"math_id": 4,
"text": "D = \\frac{c_o}{c_w} = \\frac{c_o^{S_1} + c_o^{S_2} + \\dots + c_o^{S_n}}{c_w^{S_1} + c_w^{S_2} + \\dots + c_w^{S_n}}"
},
{
"math_id": 5,
"text": "c_o"
},
{
"math_id": 6,
"text": "c_w"
},
{
"math_id": 7,
"text": "\\log{D} = \\log \\frac{c_o}{c_w} = \\log c_o - \\log c_w"
}
] |
https://en.wikipedia.org/wiki?curid=64013419
|
64014128
|
Lattice disjoint
|
In mathematics, specifically in order theory and functional analysis, two elements "x" and "y" of a vector lattice "X" are lattice disjoint or simply disjoint if formula_0, in which case we write formula_1, where the absolute value of "x" is defined to be formula_2.
We say that two sets "A" and "B" are lattice disjoint or disjoint if "a" and "b" are disjoint for all "a" in "A" and all "b" in "B", in which case we write formula_3.
If "A" is the singleton set formula_4 then we will write formula_5 in place of formula_6.
For any set "A", we define the disjoint complement to be the set formula_7.
Characterizations.
Two elements "x" and "y" are disjoint if and only if formula_8.
If "x" and "y" are disjoint then formula_9 and formula_10, where for any element "z", formula_11 and formula_12.
Properties.
Disjoint complements are always bands, but the converse is not true in general.
If "A" is a subset of "X" such that formula_13 exists, and if "B" is a subset lattice in "X" that is disjoint from "A", then "B" is a lattice disjoint from formula_14.
Representation as a disjoint sum of positive elements.
For any "x" in "X", let formula_15 and formula_16, where note that both of these elements are formula_17 and formula_18 with formula_19.
Then formula_20 and formula_21 are disjoint, and formula_18 is the unique representation of "x" as the difference of disjoint elements that are formula_17.
For all "x" and "y" in "X", formula_22 and formula_23.
If "y ≥ 0" and "x" ≤ "y" then "x"+ ≤ "y".
Moreover, formula_24 if and only if formula_25 and formula_26.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\inf \\left\\{ |x|, |y| \\right\\} = 0"
},
{
"math_id": 1,
"text": "x \\perp y"
},
{
"math_id": 2,
"text": "|x| := \\sup \\left\\{ x, - x \\right\\}"
},
{
"math_id": 3,
"text": "A \\perp B"
},
{
"math_id": 4,
"text": "\\{ a \\}"
},
{
"math_id": 5,
"text": "a \\perp B"
},
{
"math_id": 6,
"text": "\\{ a \\} \\perp B"
},
{
"math_id": 7,
"text": "A^{\\perp} := \\left\\{ x \\in X : x \\perp A \\right\\}"
},
{
"math_id": 8,
"text": "\\sup\\{ | x |, | y | \\} = | x | + | y |"
},
{
"math_id": 9,
"text": "| x + y | = | x | + | y |"
},
{
"math_id": 10,
"text": "\\left(x + y \\right)^{+} = x^{+} + y^{+}"
},
{
"math_id": 11,
"text": "z^{+} := \\sup \\left\\{ z, 0 \\right\\}"
},
{
"math_id": 12,
"text": "z^{-} := \\sup \\left\\{ -z, 0 \\right\\}"
},
{
"math_id": 13,
"text": "x = \\sup A"
},
{
"math_id": 14,
"text": "\\{ x \\}"
},
{
"math_id": 15,
"text": "x^{+} := \\sup \\left\\{ x, 0 \\right\\}"
},
{
"math_id": 16,
"text": "x^{-} := \\sup \\left\\{ -x, 0 \\right\\}"
},
{
"math_id": 17,
"text": "\\geq 0"
},
{
"math_id": 18,
"text": "x = x^{+} - x^{-}"
},
{
"math_id": 19,
"text": "| x | = x^{+} + x^{-}"
},
{
"math_id": 20,
"text": "x^{+}"
},
{
"math_id": 21,
"text": "x^{-}"
},
{
"math_id": 22,
"text": "\\left| x^{+} - y^{+} \\right| \\leq | x - y |"
},
{
"math_id": 23,
"text": "x + y = \\sup\\{ x, y \\} + \\inf\\{ x, y \\}"
},
{
"math_id": 24,
"text": "x \\leq y"
},
{
"math_id": 25,
"text": "x^{+} \\leq y^{+}"
},
{
"math_id": 26,
"text": "x^{-} \\leq x^{-1}"
}
] |
https://en.wikipedia.org/wiki?curid=64014128
|
64024339
|
Addressed fiber Bragg structure
|
Optical frequency response of which includes two narrowband components
An addressed fiber Bragg structure (AFBS) is a fiber Bragg grating, the optical frequency response of which includes two narrowband components with the frequency spacing between them (which is the address frequency of the AFBS) being in the radio frequency (RF) range. The frequency spacing (the address frequency) is unique for every AFBS in the interrogation circuit and does not change when the AFBS is subjected to strain or temperature variation. An addressed fiber Bragg structure can perform triple function in fiber-optic sensor systems: a sensor, a shaper of double-frequency probing radiation, and a multiplexor. The key feature of AFBS is that it enables the definition of its central wavelength without scanning its spectral response, as opposed to conventional fiber Bragg gratings (FBG), which are probed using optoelectronic interrogators. An interrogation circuit of AFBS is significantly simplified in comparison with conventional interrogators and consists of a broadband optical source (such as a superluminescent diode), an optical filter with a predefined linear inclined frequency response, and a photodetector. The AFBS interrogation principle intrinsically allows to include several AFBSs with the same central wavelength and different address frequencies into a single measurement system.
History.
The concept of addressed fiber Bragg structures was introduced in 2018 by Airat Sakhabutdinov and developed in collaboration with his scientific adviser, Oleg Morozov. The idea emerged from the earlier works of Morozov and his colleagues, where the double-frequency optical radiation from an electro-optic modulator was used for the definition of the FBG central wavelength based on the amplitude and phase analysis of the beating signal at the frequency equal to the spacing between the two components of the probing radiation. This eliminates the need for scanning the FBG spectral response while providing high accuracy of measurements and reducing the system cost. AFBS has been developed as a further step towards simplification of FBG interrogation systems by transferring the shaping of double-frequency probing radiation from the source modulator to the sensor itself.
Types of AFBS.
Thus far, two types of AFBS with different mechanisms of forming double-frequency radiation have been presented: 2π-FBG and 2λ-FBG.
2π-FBG.
A 2π-FBG is an FBG with two discrete phase π-shifts. It comprises three sequential uniform FBGs with gaps equal to one grating period between them (see Fig. 1). In the system, several 2π-FBGs must be connected in parallel so that the photodetector receives the light propagated through the structures.
2λ-FBG.
A 2λ-FBG consists of two identical ultra-narrow FBGs, the central wavelengths of which are separated by an address frequency. Several 2λ-FBGs in the system can be connected in series, so that the photodetector receives the light reflected from the structures.
Interrogation principle.
Fig. 2 presents the block diagram of the interrogation system for two AFBSs (2π-FBG-type) with different address frequencies Ω1 and Ω2. A broadband light source 1 generates continuous light radiation (diagram a), which corresponds to the measurement bandwidth. The light is transmitted through the fiber-optic coupler 9, then enters the two AFBSs 2.1 and 2.2. Both AFBSs transmit two-frequency radiations that are summed into a combined radiation (diagram b) using another coupler 10. At the output of the coupler, a four-frequency radiation (diagram c) is formed, which is sent through a fiber-optic splitter 6. The splitter divides the optical signal into two channels – the measuring channel and the reference channel. In the measuring channel, an optical filter 3 with a pre-defined linear inclined frequency response is installed, which modifies the amplitudes of the four-frequency radiation into the asymmetrical radiation (diagram d). After that, the signal is sent to the photodetector 4 and is received by the measuring analog-to-digital converter (ADC) 5. The signal from the ADC is used to define the measurement information from the AFBS. In the reference channel, the signal (diagram e) is sent to the reference photodetector 7 for the optical power output control, and then it is received by the reference ADC 8. Thus, the normalization of output signal intensity is achieved, and all subsequent calculations are performed using the relations of the intensities in the measuring and reference channels.
Assume that the response from each spectral components of AFBSs is represented by a single harmonic, then the total optical response from the two AFBSs can be expressed as:
formula_0
where "Ai", "Bi" are the amplitudes of the frequency components of the "i"-th AFBS; ω"i", is the frequency of the left spectral components of the "i"-th AFBS; Ω"i" is the address frequency of the "i"-th AFBS.
The luminous power received by the photodetector can be described by the following expression:
formula_1
By narrowband filtering of the signal "P"("t") at the address frequencies, a system of equations can be obtained, using which the central frequencies of the AFBSs can be defined:
formula_2
where "Dj" is the amplitude of the signal at the address frequencies Ω"j", and the exponential multipliers describe the bandpass filters at the address frequencies.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "F(t)=\\left[{\\sum_{i=1}^NA_i\\sin(\\omega_it)+B_i\\sin((\\omega_i+\\Omega_i)t)}\\right]^2,"
},
{
"math_id": 1,
"text": "P(t)=\\sum_{i=1}^N\\sum_{k=1}^N\\left({A_iA_k\\cos((\\omega_i-\\omega_k)t)+A_iB_k\\cos((\\omega_i-\\omega_k-\\Omega_k)t)+ \\atop B_iA_k\\cos((\\omega_i-\\omega_k+\\Omega_i)t)+B_iB_k\\cos((\\omega_i-\\omega_k+\\Omega_i-\\Omega_k)t)}\\right)."
},
{
"math_id": 2,
"text": "\\sum_{i=1}^N\\sum_{k=1}^N\\left({A_iA_k\\exp\\left ( -\\frac{(\\Omega_j-|\\omega_i-\\omega_k|)^2}{2\\sigma^2} \\right )+A_iB_k\\exp\\left ( -\\frac{(\\Omega_j-|\\omega_i-\\omega_k-\\Omega_k|)^2}{2\\sigma^2} \\right )+ \\atop B_iA_k\\exp\\left ( -\\frac{(\\Omega_j-|\\omega_i-\\omega_k+\\Omega_i|)^2}{2\\sigma^2} \\right )+B_iB_k\\exp\\left ( -\\frac{(\\Omega_j-|\\omega_i-\\omega_k+\\Omega_i-\\Omega_k|)^2}{2\\sigma^2} \\right )}\\right)=D_j, j=\\overline{1,N},"
}
] |
https://en.wikipedia.org/wiki?curid=64024339
|
640249
|
Saddle point
|
Critical point on a surface graph which is not a local extremum
In mathematics, a saddle point or minimax point is a point on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a critical point), but which is not a local extremum of the function. An example of a saddle point is when there is a critical point with a relative minimum along one axial direction (between peaks) and at a relative maximum along the crossing axis. However, a saddle point need not be in this form. For example, the function formula_0 has a critical point at formula_1 that is a saddle point since it is neither a relative maximum nor relative minimum, but it does not have a relative maximum or relative minimum in the formula_2-direction.
The name derives from the fact that the prototypical example in two dimensions is a surface that "curves up" in one direction, and "curves down" in a different direction, resembling a riding saddle. In terms of contour lines, a saddle point in two dimensions gives rise to a contour map with a pair of lines intersecting at the point. Such intersections are rare in actual ordnance survey maps, as the height of the saddle point is unlikely to coincide with the integer multiples used in such maps. Instead, the saddle point appears as a blank space in the middle of four sets of contour lines that approach and veer away from it. For a basic saddle point, these sets occur in pairs, with an opposing high pair and an opposing low pair positioned in orthogonal directions. The critical contour lines generally do not have to intersect orthogonally.
Mathematical discussion.
A simple criterion for checking if a given stationary point of a real-valued function "F"("x","y") of two real variables is a saddle point is to compute the function's Hessian matrix at that point: if the Hessian is indefinite, then that point is a saddle point. For example, the Hessian matrix of the function formula_3 at the stationary point formula_4 is the matrix
formula_5
which is indefinite. Therefore, this point is a saddle point. This criterion gives only a sufficient condition. For example, the point formula_6 is a saddle point for the function formula_7 but the Hessian matrix of this function at the origin is the null matrix, which is not indefinite.
In the most general terms, a saddle point for a smooth function (whose graph is a curve, surface or hypersurface) is a stationary point such that the curve/surface/etc. in the neighborhood of that point is not entirely on any side of the tangent space at that point.
In a domain of one dimension, a saddle point is a point which is both a stationary point and a point of inflection. Since it is a point of inflection, it is not a local extremum.
Saddle surface.
A saddle surface is a smooth surface containing one or more saddle points.
Classical examples of two-dimensional saddle surfaces in the Euclidean space are second order surfaces, the hyperbolic paraboloid formula_3 (which is often referred to as ""the" saddle surface" or "the standard saddle surface") and the hyperboloid of one sheet. The Pringles potato chip or crisp is an everyday example of a hyperbolic paraboloid shape.
Saddle surfaces have negative Gaussian curvature which distinguish them from convex/elliptical surfaces which have positive Gaussian curvature. A classical third-order saddle surface is the monkey saddle.
Examples.
In a two-player zero sum game defined on a continuous space, the equilibrium point is a saddle point.
For a second-order linear autonomous system, a critical point is a saddle point if the characteristic equation has one positive and one negative real eigenvalue.
In optimization subject to equality constraints, the first-order conditions describe a saddle point of the Lagrangian.
Other uses.
In dynamical systems, if the dynamic is given by a differentiable map "f" then a point is hyperbolic if and only if the differential of "ƒ" "n" (where "n" is the period of the point) has no eigenvalue on the (complex) unit circle when computed at the point. Then
a "saddle point" is a hyperbolic periodic point whose stable and unstable manifolds have a dimension that is not zero.
A saddle point of a matrix is an element which is both the largest element in its column and the smallest element in its row.
References.
Citations.
<templatestyles src="Reflist/styles.css" />
Sources.
<templatestyles src="Refbegin/styles.css" />
|
[
{
"math_id": 0,
"text": "f(x,y) = x^2 + y^3"
},
{
"math_id": 1,
"text": "(0, 0)"
},
{
"math_id": 2,
"text": "y"
},
{
"math_id": 3,
"text": "z=x^2-y^2"
},
{
"math_id": 4,
"text": "(x, y, z)=(0, 0, 0)"
},
{
"math_id": 5,
"text": "\\begin{bmatrix}\n2 & 0\\\\\n0 & -2 \\\\\n\\end{bmatrix}\n"
},
{
"math_id": 6,
"text": "(0, 0, 0)"
},
{
"math_id": 7,
"text": "z=x^4-y^4,"
}
] |
https://en.wikipedia.org/wiki?curid=640249
|
64035593
|
Marsaglia's theorem
|
Describes flaws with the pseudorandom numbers from a linear congruential generator
In computational number theory, Marsaglia's theorem connects modular arithmetic and analytic geometry to describe the flaws with the pseudorandom numbers resulting from a linear congruential generator. As a direct consequence, it is now widely considered that linear congruential generators are weak for the purpose of generating random numbers. Particularly, it is inadvisable to use them for simulations with the Monte Carlo method or in cryptographic settings, such as issuing a public key certificate, unless specific numerical requirements are satisfied. Poorly chosen values for the modulus and multiplier in a Lehmer random number generator will lead to a short period for the sequence of random numbers. Marsaglia's result may be further extended to a mixed linear congruential generator.
For example, with RANDU, we have formula_0, and in three dimensions, it shows that all the points fall into at most formula_1 planes. The actual RANDU algorithm, which uses formula_2, is much worse. All the points in fact fall into 15 planes.
Main statement.
Consider a Lehmer random number generator with
formula_3
for any modulus formula_4 and multiplier formula_5 where each formula_6, and define a sequence
formula_7
Define the points
formula_8
on a unit formula_9-cube formed from successive terms of the sequence of formula_10. With such a multiplicative number generator, all formula_9-tuples of resulting random numbers lie in at most formula_11 hyperplanes. Additionally, for a choice of constants formula_12 which satisfy the congruence
formula_13
there are at most formula_14 parallel hyperplanes which contain all formula_9-tuples produced by the generator. Proofs for these claims may be found in Marsaglia's original paper.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "m = 2^{32}"
},
{
"math_id": 1,
"text": "floor((2^{31}\\times 3!)^{1/3}) = 2344"
},
{
"math_id": 2,
"text": "k = 65539"
},
{
"math_id": 3,
"text": "r_{i+1} \\equiv kr_i\\mod m "
},
{
"math_id": 4,
"text": "m"
},
{
"math_id": 5,
"text": " k"
},
{
"math_id": 6,
"text": "0< r_i < m "
},
{
"math_id": 7,
"text": "u_1 = \\frac{r_1} m, u_2 = \\frac{r_2} m, u_3 = \\frac{r_3} m , \\ldots "
},
{
"math_id": 8,
"text": "\\pi_1 = (u_1, \\ldots, u_n), \\pi_2 = (u_2, \\ldots, u_{n+1}), \\pi_3 = (u_3, \\ldots, u_{n+2}), \\ldots"
},
{
"math_id": 9,
"text": "n"
},
{
"math_id": 10,
"text": "u"
},
{
"math_id": 11,
"text": "(n!m)^{1/n}"
},
{
"math_id": 12,
"text": "c_1,c_2, \\ldots, c_n"
},
{
"math_id": 13,
"text": "c_1 + c_2k+c_3k^2 + \\cdots + c_n k^{n-1} \\equiv 0 \\mod m,"
},
{
"math_id": 14,
"text": "|c_1| + |c_2| + \\cdots + |c_n| "
}
] |
https://en.wikipedia.org/wiki?curid=64035593
|
64038584
|
Saturated family
|
Concept in functional analysis
In mathematics, specifically in functional analysis, a family formula_0 of subsets a topological vector space (TVS) formula_1 is said to be saturated if formula_0 contains a non-empty subset of formula_1 and if for every formula_2 the following conditions all hold:
Definitions.
If formula_7 is any collection of subsets of formula_1 then the smallest saturated family containing formula_7 is called the saturated hull of formula_8
The family formula_0 is said to cover formula_1 if the union formula_9 is equal to formula_1;
it is total if the linear span of this set is a dense subset of formula_10
Examples.
The intersection of an arbitrary family of saturated families is a saturated family.
Since the power set of formula_1 is saturated, any given non-empty family formula_0 of subsets of formula_1 containing at least one non-empty set, the saturated hull of formula_0 is well-defined.
Note that a saturated family of subsets of formula_1 that covers formula_1 is a bornology on formula_10
The set of all bounded subsets of a topological vector space is a saturated family.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\mathcal{G}"
},
{
"math_id": 1,
"text": "X"
},
{
"math_id": 2,
"text": "G \\in \\mathcal{G},"
},
{
"math_id": 3,
"text": "G"
},
{
"math_id": 4,
"text": "a,"
},
{
"math_id": 5,
"text": "aG"
},
{
"math_id": 6,
"text": "\\mathcal{G}."
},
{
"math_id": 7,
"text": "\\mathcal{S}"
},
{
"math_id": 8,
"text": "\\mathcal{S}."
},
{
"math_id": 9,
"text": "\\bigcup_{G \\in \\mathcal{G}} G"
},
{
"math_id": 10,
"text": "X."
}
] |
https://en.wikipedia.org/wiki?curid=64038584
|
64039706
|
Comparison of Gaussian process software
|
Comparison of statistical analysis software that allows doing inference with Gaussian processes
This is a comparison of statistical analysis software that allows doing inference with Gaussian processes often using approximations.
This article is written from the point of view of Bayesian statistics, which may use a terminology different from the one commonly used in kriging. The next section should clarify the mathematical/computational meaning of the information provided in the table independently of contextual terminology.
Description of columns.
This section details the meaning of the columns in the table below.
Solvers.
These columns are about the algorithms used to solve the linear system defined by the prior covariance matrix, i.e., the matrix built by evaluating the kernel.
Input.
These columns are about the points on which the Gaussian process is evaluated, i.e. formula_0 if the process is formula_1.
Output.
These columns are about the values yielded by the process, and how they are connected to the data used in the fit.
Hyperparameters.
These columns are about finding values of variables which enter somehow in the definition of the specific problem but that can not be inferred by the Gaussian process fit, for example parameters in the formula of the kernel.
If both the "Prior" and "Posterior" cells contain "Manually", the software provides an interface for computing the marginal likelihood and its gradient w.r.t. hyperparameters, which can be feed into an optimization/sampling algorithm, e.g., gradient descent or Markov chain Monte Carlo.
Linear transformations.
These columns are about the possibility of fitting datapoints simultaneously to a process and to linear transformations of it.
Notes.
<templatestyles src="Reflist/styles.css" />
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "x"
},
{
"math_id": 1,
"text": "f(x)"
},
{
"math_id": 2,
"text": "\\mathbb R^n \\to \\mathbb R^m"
}
] |
https://en.wikipedia.org/wiki?curid=64039706
|
64041384
|
Europium(II) fluoride
|
<templatestyles src="Chembox/styles.css"/>
Chemical compound
Europium(II) fluoride is an inorganic compound with a chemical formula EuF2. It was first synthesized in 1937.
Production.
Europium(II) fluoride can be produced by reducing europium(III) fluoride with metallic europium or hydrogen gas.
formula_0
formula_1
Properties.
Europium(II) fluoride is a bright yellowish solid with a fluorite structure.
EuF2 can be used to dope a trivalent rare-earth fluoride, such as LaF3, to create a vacancy-filled structure with increased conductivity over a pure crystal. Such a crystal can be used as a fluoride-specific semipermeable membrane in a fluoride selective electrode to detect trace quantities of fluoride.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\mathrm{2 \\ EuF_3 + Eu \\longrightarrow 3 \\ EuF_2}"
},
{
"math_id": 1,
"text": "\\mathrm{2 \\ EuF_3 + H_2 \\longrightarrow 2 \\ EuF_2 + 2 \\ HF}"
}
] |
https://en.wikipedia.org/wiki?curid=64041384
|
640422
|
Approximation error
|
Mathematical concept
The approximation error in a data value is the discrepancy between an exact value and some approximation to it. This error can be expressed as an absolute error (the numerical amount of the discrepancy) or as a relative error (the absolute error divided by the data value).
An approximation error can occur for a variety of reasons, among them a computing machine precision or measurement error (e.g. the length of a piece of paper is 4.53 cm but the ruler only allows you to estimate it to the nearest 0.1 cm, so you measure it as 4.5 cm).
In the mathematical field of numerical analysis, the numerical stability of an algorithm indicates the extent to which errors in the input of the algorithm will lead to large errors of the output; numerically stable algorithms do not yield a significant error in output when the input is malformed and vice versa.
Formal definition.
Given some value "v", we say that "v"approx approximates "v" with absolute error "ε">0 if
formula_0
where the vertical bars denote the absolute value.
We say that "v"approx approximates "v" with relative error "η">0 ifformula_1.If "v" ≠ 0, then
formula_2.
The percent error (an expression of the relative error) is
formula_3
An error bound is an upper limit on the relative or absolute size of an approximation error.
Examples.
As an example, if the exact value is 50 and the approximation is 49.9, then the absolute error is 0.1 and the relative error is 0.1/50 = 0.002 = 0.2%. As a practical example, when measuring a 6 mL beaker, the value read was 5 mL. The correct reading being 6 mL, this means the percent error in that particular situation is, rounded, 16.7%.
The relative error is often used to compare approximations of numbers of widely differing size; for example, approximating the number 1,000 with an absolute error of 3 is, in most applications, much worse than approximating the number 1,000,000 with an absolute error of 3; in the first case the relative error is 0.003 while in the second it is only 0.000003.
There are two features of relative error that should be kept in mind. First, relative error is undefined when the true value is zero as it appears in the denominator (see below). Second, relative error only makes sense when measured on a ratio scale, (i.e. a scale which has a true meaningful zero), otherwise it is sensitive to the measurement units. For example, when an absolute error in a temperature measurement given in Celsius scale is 1 °C, and the true value is 2 °C, the relative error is 0.5. But if the exact same approximation is made with the Kelvin scale, a 1 K absolute error with the same true value of 275.15 K = 2 °C gives a relative error of 3.63×10-3.
Comparison.
Statements about "relative errors" are sensitive to addition of constants, but not to multiplication by constants. For "absolute errors", the opposite is true: are sensitive to multiplication by constants, but not to addition of constants.34
Polynomial-time approximation of real numbers.
We say that a real value "v" is polynomially computable with absolute error from an input if, for every rational number "ε">0, it is possible to compute a rational number "v"approx that approximates "v" with absolute error "ε", in time polynomial in the size of the input and the encoding size of "ε" (which is O(log(1/"ε")). Analogously, "v" is polynomially computable with relative error if, for every rational number "η">0, it is possible to compute a rational number "v"approx that approximates "v" with relative error "η", in time polynomial in the size of the input and the encoding size of "η".
If "v" is polynomially computable with relative error (by some algorithm called REL), then it is also polynomially computable with absolute error. "Proof". Let "ε">0 be the desired absolute error. First, use REL with relative error "η="1/2; find a rational number "r"1 such that |"v"-"r"1| ≤ |"v"|/2, and hence "|v|" ≤ 2 |"r"1|. If "r"1=0, then "v"=0 and we are done. Since REL is polynomial, the encoding length of "r"1 is polynomial in the input. Now, run REL again with relative error "η=ε/"(2 "|r"1|). This yields a rational number "r"2 that satisfies |"v"-"r"2| ≤ "ε|v"| / (2"r"1) ≤ "ε", so it has absolute error "ε" as wished.34
The reverse implication is usually not true. But, if we assume that some positive lower bound on |v| can be computed in polynomial time, e.g. |"v"| > "b" > 0, and "v" is polynomially computable with absolute error (by some algorithm called ABS), then it is also polynomially computable with relative error, since we can simply call ABS with absolute error "ε = η b."
An algorithm that, for every rational number "η">0, computes a rational number "v"approx that approximates "v" with relative error "η", in time polynomial in the size of the input and 1/"η" (rather than log(1/"η")), is called an FPTAS.
Instruments.
In most indicating instruments, the accuracy is guaranteed to a certain percentage of full-scale reading. The limits of these deviations from the specified values are known as limiting errors or guarantee errors.
Generalizations.
The definitions can be extended to the case when formula_4 and formula_5 are "n"-dimensional vectors, by replacing the absolute value with an "n"-norm.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "|v-v_\\text{approx}| \\leq \\varepsilon"
},
{
"math_id": 1,
"text": "|v-v_\\text{approx}| \\leq \\eta\\cdot v"
},
{
"math_id": 2,
"text": " \\eta = \\frac{\\epsilon}{|v|}\n = \\left| \\frac{v-v_\\text{approx}}{v} \\right|\n = \\left| 1 - \\frac{v_\\text{approx}}{v} \\right|\n"
},
{
"math_id": 3,
"text": "\\delta = 100\\%\\times\\eta = 100\\%\\times\\left| \\frac{v-v_\\text{approx}}{v} \\right|."
},
{
"math_id": 4,
"text": "v"
},
{
"math_id": 5,
"text": "v_{\\text{approx}}"
}
] |
https://en.wikipedia.org/wiki?curid=640422
|
64042964
|
Redundancy principle (biology)
|
Principle in biology
The redundancy principle in biology expresses the need of many copies of the same entity (cells, molecules, ions) to fulfill a biological function. Examples are numerous: disproportionate numbers of spermatozoa during fertilization compared to one egg, large number of neurotransmitters released during neuronal communication compared to the number of receptors, large numbers of released calcium ions during transient in cells, and many more in molecular and cellular transduction or gene activation and cell signaling. This redundancy is particularly relevant when the sites of activation are physically separated from the initial position of the molecular messengers. The redundancy is often generated for the purpose of resolving the time constraint of fast-activating pathways. It can be expressed in terms of the theory of extreme statistics to determine its laws and quantify how the shortest paths are selected. The main goal is to estimate these large numbers from physical principles and mathematical derivations.
When a large distance separates the source and the target (a small activation site), the redundancy principle explains that this geometrical gap can be compensated by large number. Had nature used less copies than normal, activation would have taken a much longer time, as finding a small target by chance is a rare event and falls into narrow escape problems.
Molecular rate.
The time for the fastest particles to reach a target in the context of redundancy depends on the numbers and the local geometry of the target. In most of the time, it is the rate of activation. This rate should be used instead of the classical Smoluchowski's rate describing the mean arrival time, but not the fastest. The statistics of the minimal time to activation set kinetic laws in biology, which can be quite different from the ones associated to average times.
Physical models.
Stochastic process.
The motion of a particle located at position formula_0 can be described by the Smoluchowski's limit of the Langevin equation:
formula_1
where formula_2 is the diffusion coefficient of the particle, formula_3 is the friction coefficient per unit of mass, formula_4 the force per unit of mass, and formula_5 is a Brownian motion. This model is classically used in molecular dynamics simulations.
Jump processes.
formula_6, which is for example a model of telomere length dynamics. Here formula_7 , with formula_8.
Directed motion process.
formula_9 where formula_10 is a unit vector chosen from a uniform distribution. Upon hitting an obstacle at a boundary point formula_11, the velocity changes to formula_12 where formula_13 is chosen on the unit sphere in the supporting half space at formula_14 from a uniform distribution, independently of formula_10. This rectilinear with constant velocity is a simplified model of spermatozoon motion in a bounded domain formula_15. Other models can be diffusion on graph, active graph motion.
Mathematical formulation: Computing the rate of arrival time for the fastest.
The mathematical analysis of large numbers of molecules, which are obviously redundant in the traditional activation theory, is used to compute the in vivo time scale of stochastic chemical reactions. The computation relies on asymptotics or probabilistic approaches to estimate the mean time of the fastest to reach a small target in various geometries.
With N non-interacting i.i.d. Brownian trajectories (ions) in a bounded domain Ω that bind at a site, the shortest arrival time is by definition
formula_16 where formula_17 are the independent arrival times of the N ions in the medium. The survival distribution of arrival time of the fastest formula_18 is expressed in terms of a single particle, formula_19. Here formula_20 is the survival probability of a single particle prior to binding at the target.This probability is computed from the solution of the diffusion equation in a domain formula_21:
formula_22
formula_23
where the boundary formula_24 contains NR binding sites formula_25 (formula_26). The single particle survival probability is
formula_27 so that formula_28where
formula_29and formula_30.
The probability density function (pdf) of the arrival time is
formula_31 which gives the MFPT
formula_32 The probability formula_33 can be computed using short-time asymptotics of the diffusion equation as shown in the next sections.
Explicit computation in dimension 1.
The short-time asymptotic of the diffusion equation is based on the ray method approximation. For an semi-interval formula_34, the survival pdf is solution of
formula_35
that isformula_36
The survival probability with D=1 is formula_37. To compute the MFPT, we expand the complementary error function
formula_38 which givesformula_39,
leading (the main contribution of the integral is near 0) to formula_40
This result is reminiscent of using the Gumbel's law. Similarly, escape from the interval [0,a] is computed from the infinite sum
formula_41.The conditional survival probability is approximated by
formula_42, where the maximum occurs at formula_43 min[y,a-y] for 0<y (the shortest ray from y to the boundary). All other integrals can be computed explicitly, leading to
formula_44
Arrival times of the fastest in higher dimensions.
The arrival times of the fastest among many Brownian motions are expressed in terms of the shortest distance from the source S to the absorbing window A, measured by the distance formula_45where d is the associated Euclidean distance. Interestingly, trajectories followed by the fastest are as close as possible from the optimal trajectories. In technical language, the associated trajectories of the fastest among N, concentrate near the optimal trajectory (shortest path) when the number N of particles increases. For a diffusion coefficient D and a window of size a, the expected first arrival times of N identically independent distributed Brownian particles initially positioned at the source S are expressed in the following asymptotic formulas :
formula_46
formula_47
formula_48
These formulas show that the expected arrival time of the fastest particle is in dimension 1 and 2, O(1/\log(N)). They should be used instead of the classical forward rate in models of activation in biochemical reactions. The method to derive formulas is based on short-time asymptotic and the Green's function representation of the Helmholtz equation. Note that other distributions could lead to other decays with respect N.
Optimal Paths.
Minimizing The optimal path in large N.
The optimal paths for the fastest can be found using the Wencell-Freidlin functional in the Large-deviation theory. These paths correspond to the short-time asymptotics of the diffusion equation from a source to a target. In general, the exact solution is hard to find, especially for a space containing various distribution of obstacles.
The Wiener integral representation of the pdf for a pure Brownian motion is obtained for a zero drift and diffusion tensor formula_49 constant, so that it is given by the probability of a sampled path until it exits at the small window formula_50at the random time T
formula_51
formula_52
where
formula_53 in the product and T is the exit time in the narrow absorbing window formula_54 Finally,
formula_55
where formula_56 is the ensemble of shortest paths selected among n Brownian trajectories, starting at point y and exiting between time t and t+dt from the domain formula_21. The probabilityformula_57 is used to show that the empirical stochastic trajectories of formula_58 concentrate near the shortest paths starting from y and ending at the small absorbing window formula_59, under the condition that formula_60. The paths of formula_56 can be approximated using discrete broken lines among a finite number of points and we denote the associated ensemble by formula_61. Bayes' rule leads toformula_62 where formula_63 is the probability that a path of formula_61 exits in m-discrete time steps. A path made of broken lines (random walk with a time stepformula_64) can be expressed using Wiener path-integral. The probability of a Brownian path x(s) can be expressed in the limit of a path-integral with the functional:
formula_65
The Survival probability conditioned on starting at y is given by the Wiener representation:
formula_66
where formula_67 is the limit Wiener measure: the exterior integral is taken over all end points x and the path integral is over all paths starting from x(0). When we consider n-independent paths formula_68 (made of points with a time step formula_64 that exit in m-steps, the probability of such an event is
formula_69formula_70.Indeed, when there are n paths of m steps, and the fastest one escapes in m-steps, they should all exit in m steps. Using the limit of path integral, we get heuristically the representation
formula_71
formula_72
where the integral is taken over all paths starting at y(0) and exiting at time formula_73. This formula suggests that when n is large, only the paths that minimize the integrant will contribute. For large n, this formula suggests that paths that will contribute the most are the ones that will minimize the exponent, which allows selecting the paths for which the energy functional is minimal, that is
formula_74
where the integration is taken over the ensemble of regular pathsformula_75 inside formula_21 starting at y and exiting in formula_59, defined as
formula_76
This formal argument shows that the random paths associated to the fastest exit time are concentrated near the shortest paths. Indeed, the Euler-Lagrange equations for the extremal problem are the classical geodesics between y and a point in the narrow window formula_59.
Fastest escape from a cusp in two dimensions.
The formula for the fastest escape can generalize to the case where the absorbing window is located in funnel cusp and the initial particles are distributed outside the cusp. The cusp has a size formula_77 in the opening and a curvature R. The diffusion coefficient is D. The shortest arrival time, valid for large n is given by formula_78 Hereformula_79and c is a constant that depends on the diameter of the domain. The time taken by the first arrivers is proportional to the reciprocal of the size of the narrow target formula_77 . This formula is derived for fixed geometry and large n and not in the opposite limit of large n and small epsilon.
Concluding remarks.
How nature sets the disproportionate numbers of particles remain unclear, but can be found using the theory of diffusion. One example is the number of neurotransmitters around 2000 to 3000 released during synaptic transmission, that are set to compensate the low copy number of receptors, so the probability of activation is restored to one.
In natural processes these large numbers should not be considered wasteful, but are necessary for generating the fastest possible response and make possible rare events that otherwise would never happen. This property is universal, ranging from the molecular scale to the population level.
Nature's strategy for optimizing the response time is not necessarily defined by the physics of the motion of an individual particle, but rather by the extreme statistics, that select the shortest paths. In addition, the search for a small activation site selects the particle to arrive first: although these trajectories are rare, they are the ones that set the time scale. We may need to reconsider our estimation toward numbers when punctioning nature in agreement with the redundant principle that quantifies the request to achieve the biological function.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "X_t"
},
{
"math_id": 1,
"text": "dX_t=\\sqrt{2D} \\, dB_t+\\frac{1}{\\gamma}F(x)dt,"
},
{
"math_id": 2,
"text": "D"
},
{
"math_id": 3,
"text": "\\gamma"
},
{
"math_id": 4,
"text": "F(x)"
},
{
"math_id": 5,
"text": "B_t"
},
{
"math_id": 6,
"text": "\\begin{align}\nx_{n+1}=\n\\begin{cases} x_n-a, & \\text{with probability } l(x_n) \\\\ x_n+b, & \n\\text{ with probability } r(x_n) \n\\end{cases}\n\\end{align}"
},
{
"math_id": 7,
"text": "r(x)=\\frac{1}{1+\\beta x},"
},
{
"math_id": 8,
"text": "r(x)+l(x)=1"
},
{
"math_id": 9,
"text": "\\dot{X}=v_0 \\bf u,"
},
{
"math_id": 10,
"text": " \\bf u"
},
{
"math_id": 11,
"text": "X_0 \\in \\partial \\Omega"
},
{
"math_id": 12,
"text": "\\dot{X}=v_0 \\bf v,"
},
{
"math_id": 13,
"text": "\\bf v"
},
{
"math_id": 14,
"text": "X_0"
},
{
"math_id": 15,
"text": " \\Omega"
},
{
"math_id": 16,
"text": "\n\\tau^{1}=\\min (t_1,\\ldots,t_N),\n"
},
{
"math_id": 17,
"text": "t_i"
},
{
"math_id": 18,
"text": "Pr(\\tau^{1}>t)"
},
{
"math_id": 19,
"text": "\nPr(\\tau^{1}>t)=Pr^N(t_1>t)\n"
},
{
"math_id": 20,
"text": "Pr\\{t_{1}>t \\}"
},
{
"math_id": 21,
"text": "\\Omega"
},
{
"math_id": 22,
"text": "\n\\frac{\\partial p(x,t)}{\\partial t} =D \\Delta p(x,t) \\hbox { for } x \\in \\Omega, t>0\n"
},
{
"math_id": 23,
"text": "\n\\begin{align} \np(x,0)=&p_0(x) \\hbox{ for } x \\in \\Omega \\\\\n\\frac{\\partial p}{\\partial n}(x,t) &=0 \\hbox{ for } x \\in \\partial \\Omega_r\\\\\np(x,t)&=0 \\hbox{ for } x \\in \\partial \\Omega_a,\n\\end{align}\n"
},
{
"math_id": 24,
"text": "\\partial \\Omega"
},
{
"math_id": 25,
"text": "\\partial \\Omega_i\\subset\\partial \\Omega"
},
{
"math_id": 26,
"text": "\\partial \\Omega_a=\\bigcup\\limits_{i=1}^{N_R}\\partial\\Omega_i,\\ \\partial\\Omega_r=\\partial\\Omega-\\partial\\Omega_a"
},
{
"math_id": 27,
"text": "\n\\Pr\\{t_{1}>t \\} =\\int\\limits_{\\Omega} p(x,t)dx,\n"
},
{
"math_id": 28,
"text": "\n\\Pr\\{\\tau^{1}=t \\} = \\frac{d}{dt}\\Pr\\{\\tau^{1}<t \\}=N(\\Pr\\{t_{1}>t \\})^{N-1}\\Pr\\limits\\{t_{1}=t\n\\},\n"
},
{
"math_id": 29,
"text": "\n\\Pr\\{t_{1}=t \\}= {\\oint_{\\partial \\Omega_a}} \\frac{\\partial p(x,t)}{\\partial n}\\, dS_{x}\n"
},
{
"math_id": 30,
"text": "\\Pr\\{t_{1}=t \\}= N_R {\\oint_{\\partial \\Omega_1}} \\frac{\\partial p(x,t)}{\\partial n}\\,dS_{x} "
},
{
"math_id": 31,
"text": "\n\\Pr\\{\\tau^{1}=t \\} =N N_R \\left[\\int\\limits_{\\Omega} p(x,t)dx \\right]^{N-1}\\oint\\limits_{\\partial \\Omega_1} \\frac{\\partial p(x,t)}{\\partial n} dS_{x},\n"
},
{
"math_id": 32,
"text": "\n\\bar{\\tau}^{1}=\\int\\limits\\limits_0 ^{\\infty}\\Pr\\{\\tau^{1}>t\\} dt = \\int\\limits_0 ^{\\infty} \\left[ \\Pr\\{t_{1}>t\\} \\right]^N dt.\n"
},
{
"math_id": 33,
"text": "\\Pr\\{t_{1}>t \\}"
},
{
"math_id": 34,
"text": "[0,\\infty["
},
{
"math_id": 35,
"text": "\\begin{align}\n\\frac{\\partial (x,t)}{\\partial t}& =D \\frac{\\partial^2 p(x,t)}{\\partial x^2}\n\\quad\\mbox{ for } x>0,\\ t>0 \\\\\np(x,0)&=\\delta(x-a)\\quad\\mbox{ for }\\ x>0,\\quad p(0,t)=0\\quad\\mbox{ for } t>0,\n\\end{align}"
},
{
"math_id": 36,
"text": "p(x,t) =\\frac{1}{\\sqrt{4D \\pi t}}\\left[\\exp\\left\\{ - \\frac{(x-a)^2}{4Dt}\\right\\}- \\exp\\left\\{ - \\frac{(x+a)^2}{4Dt}\\right\\}\\right]. "
},
{
"math_id": 37,
"text": "\\Pr\\{t_{1}>t \\}=\\int\\limits\\limits_{0}^{\\infty} p(x,t)\\,dx=1-\\frac{2}{\\sqrt{\\pi}} \\int\\limits\\limits_{a/\\sqrt{4t}}^{\\infty}e^{-u^2}\\,du "
},
{
"math_id": 38,
"text": "\\frac{2}{\\sqrt{\\pi}} \\int\\limits\\limits_{x}^{\\infty}e^{-u^2}\\,du =\\frac{e^{-x^2}}{x\\sqrt{\\pi}}\\left(1-\\frac{1}{2x^2}+O(x^{-4})\\right)\\quad\\mbox{for}\\ x\\gg1, "
},
{
"math_id": 39,
"text": "\\bar{\\tau}^{1}=\\int\\limits\\limits_0 ^{\\infty} \\left[ \\Pr\\{t_{1}>t\\} \\right]^N dt \\approx \\int\\limits\\limits_0 ^{\\infty} \\exp\\left\\{ N\\ln\\left(1-\\frac{e^{-(a/\\sqrt{4t})^2}}{(a/\\sqrt{4t})\\sqrt{\\pi}}\\right)\\right\\}\\, dt \\approx \\frac{a^2}{4}\\int\\limits\\limits_0^{\\infty} \\exp \\left\\{ -N\\frac{\\sqrt{u}e^{-\\frac{1}{u}}}{\\sqrt{\\pi}} \\right\\}du "
},
{
"math_id": 40,
"text": "\\bar{\\tau}^{1} \\approx \\frac{a^2}{4D\\ln \\frac{N}{\\sqrt{\\pi}}}\\quad\\mbox{for}\\ N\\gg1. "
},
{
"math_id": 41,
"text": "p(x,t\\,|\\,y) =\\frac{1}{\\sqrt{ 4 D \\pi t}}\\sum\\limits_{n=-\\infty}^{\\infty} \\left[\\exp \\left\\{ -\\frac{(x-y+2na)^2}{4t} \\right\\} -\\exp \\left\\{ -\\frac{(x+y+2na)^2}{4t} \\right\\} \\right] "
},
{
"math_id": 42,
"text": "\\Pr\\{t_{1}>t\\,|\\,y \\}=\\int\\limits\\limits_{0}^{a} p(x,t\\,|\\,y)\\,dx ds\\sim1-\\max\\frac{2\\sqrt{t}}{\\sqrt{\\pi}}\\left[\\frac{e^{-y^2/4t}}{y},\\frac{e^{-(a-y)^2/4t}}{a-y}\\right] \\quad\\mbox{as}\\ t\\to0\n "
},
{
"math_id": 43,
"text": "\\delta= "
},
{
"math_id": 44,
"text": "\\bar{\\tau}^{1}= \\int\\limits\\limits_0 ^{\\infty} \\left[ \\Pr\\{t_{1}>t\\} \\right]^N dt \\approx \n\\int\\limits\\limits_0 ^{\\infty} \\exp\\left\\{ N\\ln\\left(1-\\frac{8\\sqrt{t}}{\\delta\\sqrt{\\pi}}\ne^{-\\delta^2/16t}\\right) \\right\\}dt \\approx \\frac{\\delta^2}{16D\\ln\\frac{2N}{\\sqrt{\\pi}}}\\quad\\mbox{for}\\ N\\gg1. "
},
{
"math_id": 45,
"text": "\\delta_{min}=d(S,A),"
},
{
"math_id": 46,
"text": "\n\\bar\\tau^{d1} \\approx \\frac{\\delta^2_{min}}{4D\\ln\\left(\\frac{N}{\\sqrt{\\pi}}\\right)}, \\hbox{in dim 1, valid for} N \\gg1\n,\n"
},
{
"math_id": 47,
"text": "\n \\bar \\tau^{d2} \\approx \\frac{\\delta^2_{min}}{ 4 D \\log \\left(\\frac{\\pi\n\\sqrt{2}N}{8\\log\\left(\\frac{1}{a}\\right)}\\right)}, \\hbox{ in dim 2 for } \\frac{N}{\\log (\\frac{1}{\\epsilon})}\\gg1, \n"
},
{
"math_id": 48,
"text": "\n\\bar\\tau^{d3} \\approx \\frac{\\delta^2_{min}}{4D{\\log\\left(\nN\\frac{4a^2}{\\pi^{1/2}\\delta^2_{min}}\\right)}}, \\hbox{ in dim } 3, \\hbox{ for } \\frac{Na^2}{\\delta^2_{min}}\\gg1.\n"
},
{
"math_id": 49,
"text": "\\sigma=D"
},
{
"math_id": 50,
"text": "\\partial\\Omega_a"
},
{
"math_id": 51,
"text": "Pr\\{ x_N(t_{1,M})\\in\\Omega,{x}_N(t_{2,M})\\in\\Omega,\\dots, x_M(t)=x, t\\leq T\\leq t+\\Delta t |x(0)=y\\}"
},
{
"math_id": 52,
"text": "=[\\int\\limits_{\\Omega} \\cdots \\int\\limits\\limits_{\\Omega}\\prod_{j=1}^{M} \\frac{d{y}_j}{\\sqrt{(2\\pi \\Delta t)^n\\det {\\sigma}(x)(t_{j-1,M}))}}\n\\exp \\{ -\\frac{1}{2\\Delta t} \\left[{y}_j-x(t_{j-1,N})- {a}({x}(t_{j-1,N}))\\Delta t \\right]^T{\\sigma}^{-1}(x(t_{j-1,N}))\\left[{y}_j-x(t_{j-1,N})-{a}(x(t_{j-1,N}))\\Delta t \\right]\\}\n \n"
},
{
"math_id": 53,
"text": "\\Delta t=t/M, t_{j,N}=j\\Delta t,\\ x(t_{0,N})=y \\hbox{ and } {y}_j=x(t_{j,N})"
},
{
"math_id": 54,
"text": "\\partial\\Omega_a."
},
{
"math_id": 55,
"text": "\\langle\\tau^{(n)}\\rangle=\\int\\limits\\limits_0 ^{\\infty}\\exp \\left\\{ n \\log \\int\\limits_{\\Omega} p(x,t|y)\\,dx\\right\\} dt =\\int_0 ^{\\infty} \\tau_{\\sigma} Pr\\{ \\hbox{ Path }\\sigma \\in S_n (y), \\tau_{\\sigma}=t \\} dt,"
},
{
"math_id": 56,
"text": "S_n(y)"
},
{
"math_id": 57,
"text": "Pr\\{ \\hbox{ Path }\\sigma \\in S_n \\}"
},
{
"math_id": 58,
"text": "S_n"
},
{
"math_id": 59,
"text": "\\partial \\Omega_a"
},
{
"math_id": 60,
"text": "\\epsilon=\\frac{|\\partial \\Omega_a|}{|\\partial\\Omega|} \\ll 1"
},
{
"math_id": 61,
"text": "\\tilde S_n(y)"
},
{
"math_id": 62,
"text": "Pr\\{ \\hbox{ Path }\\sigma \\in \\tilde S_n(y)| t<\\tau_{\\sigma}<t+dt \\}=\\sum_{m=0}^{\\infty}\n\nPr\\{ \\hbox{ Path }\\sigma \\in \\tilde S_n(y)|m, t<\\tau_{\\sigma}<t+dt \\}Pr\\{ m \\mbox{ steps}\\}"
},
{
"math_id": 63,
"text": "Pr\\{ m \\mbox{ steps}\\}=Pr\\{ \\mbox{the paths of }\\tilde S_n(y) \\mbox{exit in m steps} \\}"
},
{
"math_id": 64,
"text": "\\Delta t"
},
{
"math_id": 65,
"text": "Pr\\{ x(s)| s\\in[0,t] \\} \\approx \\exp \\left(-\\int_{0}^t |\\dot x|^2ds \\right)."
},
{
"math_id": 66,
"text": "S(t|x_0)= \\int_{x\\in \\Omega} dx \\int_{x(0)}^{x(t)=x} {\\mathcal D} (x)\\exp \\left(-\\int_{0}^t |\\dot x|^2ds \\right),"
},
{
"math_id": 67,
"text": "{\\mathcal D} (x)"
},
{
"math_id": 68,
"text": "(\\sigma_1,..\\sigma_n)"
},
{
"math_id": 69,
"text": " Pr \\{ \\sigma_1,..\\sigma_n \\in S_n(y)|m, \\tau_{\\sigma}=m \\Delta t \\}= \n \n \\left(\\int\\limits_{y_0=y} \\cdots \\int\\limits_{{y}_j \\in\\Omega}\n \\int\\limits_{{y}_n\\in \\partial \\Omega_a} \\frac{1}{(4D\\Delta t)^{dm/2}}\\prod_{j=1}^{m} \\exp \n\\Bigg \\{ -\\frac{1}{4D\\Delta t} \\left[|{y}_j-{y}_{j-1})|^2 \\right] \\}\n\\right)^n"
},
{
"math_id": 70,
"text": "\\approx \\left(\\frac{1}{(4D\\Delta t)^{dm/2}}\\right)^n\\int_{x} {\\mathcal D}\n(x)\\exp \\Bigg \\{-n\\int\\limits_0^{m \\Delta t} \\dot{x}^2ds \\Bigg \\} "
},
{
"math_id": 71,
"text": "Pr \\{ \\hbox{ Path }\\sigma \\in \\tilde S_n(y)|m, \\tau_{\\sigma}=m \\Delta t \\}= \\left(\\int\\limits_{{y}_0=y} \\cdots \\int\\limits_{{y}_j \\in\\Omega}\\int\\limits_{{y}_n\\in\\partial\\Omega_a} \\frac{1}{(4D\\Delta t)^{dm/2}}\\prod_{j=1}^{m} \\exp ( -\\frac{1}{4D\\Delta t} \\left[|{y}_j-{y}_{j-1})|^2 \\right])\\right)^n \n"
},
{
"math_id": 72,
"text": "\\approx \\int_{ x \\in \\Omega } dx \\int_{x(0)=y}^{x(t)=x} {D} (x)\\exp (-n \\int\\limits_0^{m \\Delta t} \\dot{x}^2ds ) ,"
},
{
"math_id": 73,
"text": "m\\Delta t"
},
{
"math_id": 74,
"text": "E=\\min_{X\\in \\mathcal P_t}\\int\\limits_0^T \\dot{x}^2ds,"
},
{
"math_id": 75,
"text": "\\mathcal P_t"
},
{
"math_id": 76,
"text": "\\mathcal P_T=\\{ P(0)=y, P(T)\\in \\partial \\Omega_a \\hbox{ and } P(s) \\in \\Omega \\hbox{ and } 0\\leq s\\leq T\\}."
},
{
"math_id": 77,
"text": "\\epsilon"
},
{
"math_id": 78,
"text": "\\tau^{(n)} \\approx \\frac{\\pi^2 R^3}{4\\epsilon D (\\frac{1-\\cos(c\\sqrt{\\tilde \\epsilon})}{\\tilde\\epsilon })^2 \\log(\\frac{2n}{\\sqrt{\\pi}})}. "
},
{
"math_id": 79,
"text": "\\tilde \\epsilon=\\frac{\\epsilon}{R}"
}
] |
https://en.wikipedia.org/wiki?curid=64042964
|
64045357
|
Positive linear operator
|
Concept in functional analysis
In mathematics, more specifically in functional analysis, a positive linear operator from an preordered vector space formula_0 into a preordered vector space formula_1 is a linear operator formula_2 on formula_3 into formula_4 such that for all positive elements formula_5 of formula_6 that is formula_7 it holds that formula_8
In other words, a positive linear operator maps the positive cone of the domain into the positive cone of the codomain.
Every positive linear functional is a type of positive linear operator.
The significance of positive linear operators lies in results such as Riesz–Markov–Kakutani representation theorem.
Definition.
A linear function formula_2 on a preordered vector space is called positive if it satisfies either of the following equivalent conditions:
The set of all positive linear forms on a vector space with positive cone formula_12 called the dual cone and denoted by formula_13 is a cone equal to the polar of formula_14
The preorder induced by the dual cone on the space of linear functionals on formula_3 is called the <templatestyles src="Template:Visible anchor/styles.css" />dual preorder.
The order dual of an ordered vector space formula_3 is the set, denoted by formula_15 defined by formula_16
Canonical ordering.
Let formula_0 and formula_1 be preordered vector spaces and let formula_17 be the space of all linear maps from formula_3 into formula_18
The set formula_19 of all positive linear operators in formula_17 is a cone in formula_17 that defines a preorder on formula_17.
If formula_20 is a vector subspace of formula_17 and if formula_21 is a proper cone then this proper cone defines a <templatestyles src="Template:Visible anchor/styles.css" />canonical partial order on formula_20 making formula_20 into a partially ordered vector space.
If formula_0 and formula_1 are ordered topological vector spaces and if formula_22 is a family of bounded subsets of formula_3 whose union covers formula_3 then the positive cone formula_23 in formula_24, which is the space of all continuous linear maps from formula_3 into formula_25 is closed in formula_24 when formula_24 is endowed with the formula_22-topology.
For formula_23 to be a proper cone in formula_24 it is sufficient that the positive cone of formula_3 be total in formula_3 (that is, the span of the positive cone of formula_3 be dense in formula_3).
If formula_4 is a locally convex space of dimension greater than 0 then this condition is also necessary.
Thus, if the positive cone of formula_3 is total in formula_3 and if formula_4 is a locally convex space, then the canonical ordering of formula_24 defined by formula_23 is a regular order.
Properties.
Proposition: Suppose that formula_3 and formula_4 are ordered locally convex topological vector spaces with formula_3 being a Mackey space on which every positive linear functional is continuous. If the positive cone of formula_4 is a weakly normal cone in formula_4 then every positive linear operator from formula_3 into formula_4 is continuous.
Proposition: Suppose formula_3 is a barreled ordered topological vector space (TVS) with positive cone formula_26 that satisfies formula_27 and formula_4 is a semi-reflexive ordered TVS with a positive cone formula_28 that is a normal cone. Give formula_24 its canonical order and let formula_29 be a subset of formula_24 that is directed upward and either majorized (that is, bounded above by some element of formula_24) or simply bounded. Then formula_30 exists and the section filter formula_31 converges to formula_32 uniformly on every precompact subset of formula_33
References.
<templatestyles src="Reflist/styles.css" />
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "(X, \\leq)"
},
{
"math_id": 1,
"text": "(Y, \\leq)"
},
{
"math_id": 2,
"text": "f"
},
{
"math_id": 3,
"text": "X"
},
{
"math_id": 4,
"text": "Y"
},
{
"math_id": 5,
"text": "x"
},
{
"math_id": 6,
"text": "X,"
},
{
"math_id": 7,
"text": "x \\geq 0,"
},
{
"math_id": 8,
"text": "f(x) \\geq 0."
},
{
"math_id": 9,
"text": "x \\geq 0"
},
{
"math_id": 10,
"text": "x \\leq y"
},
{
"math_id": 11,
"text": "f(x) \\leq f(y)."
},
{
"math_id": 12,
"text": "C,"
},
{
"math_id": 13,
"text": "C^*,"
},
{
"math_id": 14,
"text": "-C."
},
{
"math_id": 15,
"text": "X^+,"
},
{
"math_id": 16,
"text": "X^+ := C^* - C^*."
},
{
"math_id": 17,
"text": "\\mathcal{L}(X; Y)"
},
{
"math_id": 18,
"text": "Y."
},
{
"math_id": 19,
"text": "H"
},
{
"math_id": 20,
"text": "M"
},
{
"math_id": 21,
"text": "H \\cap M"
},
{
"math_id": 22,
"text": "\\mathcal{G}"
},
{
"math_id": 23,
"text": "\\mathcal{H}"
},
{
"math_id": 24,
"text": "L(X; Y)"
},
{
"math_id": 25,
"text": "Y,"
},
{
"math_id": 26,
"text": "C"
},
{
"math_id": 27,
"text": "X = C - C"
},
{
"math_id": 28,
"text": "D"
},
{
"math_id": 29,
"text": "\\mathcal{U}"
},
{
"math_id": 30,
"text": "u = \\sup \\mathcal{U}"
},
{
"math_id": 31,
"text": "\\mathcal{F}(\\mathcal{U})"
},
{
"math_id": 32,
"text": "u"
},
{
"math_id": 33,
"text": "X."
}
] |
https://en.wikipedia.org/wiki?curid=64045357
|
64053581
|
Virasoro group
|
In abstract algebra, the Virasoro group or Bott–Virasoro group (often denoted by "Vir") is an infinite-dimensional Lie group defined as the universal central extension of the group of diffeomorphisms of the circle. The corresponding Lie algebra is the Virasoro algebra, which has a key role in conformal field theory (CFT) and string theory.
The group is named after Miguel Ángel Virasoro and Raoul Bott.
Background.
An orientation-preserving diffeomorphism of the circle formula_0, whose points are labelled by a real coordinate formula_1 subject to the identification formula_2, is a smooth map formula_3 such that formula_4 and formula_5. The set of all such maps spans a group, with multiplication given by the composition of diffeomorphisms. This group is the universal cover of the group of orientation-preserving diffeomorphisms of the circle, denoted as formula_6.
Definition.
The Virasoro group is the universal central extension of formula_6. The extension is defined by a specific two-cocycle, which is a real-valued function formula_7 of pairs of diffeomorphisms. Specifically, the extension is defined by the Bott–Thurston cocycle:
formula_8
In these terms, the Virasoro group is the set formula_9 of all pairs formula_10, where formula_11 is a diffeomorphism and formula_12 is a real number, endowed with the binary operation
formula_13
This operation is an associative group operation. This extension is the only central extension of the universal cover of the group of circle diffeomorphisms, up to trivial extensions. The Virasoro group can also be defined without the use explicit coordinates or an explicit choice of cocycle to represent the central extension, via a description the universal cover of the group.
Virasoro algebra.
The Lie algebra of the Virasoro group is the Virasoro algebra. As a vector space, the Lie algebra of the Virasoro group consists of pairs formula_14, where formula_15 is a vector field on the circle and formula_12 is a real number as before. The vector field, in particular, can be seen as an infinitesimal diffeomorphism formula_16. The Lie bracket of pairs formula_14 then follows from the multiplication defined above, and can be shown to satisfy
formula_17
where the bracket of vector fields on the right-hand side is the usual one: formula_18. Upon defining the complex generators
formula_19
the Lie bracket takes the standard textbook form of the Virasoro algebra:
formula_20
The generator formula_21 commutes with the whole algebra. Since its presence is due to a central extension, it is subject to a superselection rule which guarantees that, in any physical system having Virasoro symmetry, the operator representing formula_21 is a multiple of the identity. The coefficient in front of the identity is then known as a central charge.
Properties.
Since each diffeomorphism formula_11 must be specified by infinitely many parameters (for instance the Fourier modes of the periodic function formula_22), the Virasoro group is infinite-dimensional.
Coadjoint representation.
The Lie bracket of the Virasoro algebra can be viewed as a differential of the adjoint representation of the Virasoro group. Its dual, the coadjoint representation of the Virasoro group, provides the transformation law of a CFT stress tensor under conformal transformations. From this perspective, the Schwarzian derivative in this transformation law emerges as a consequence of the Bott–Thurston cocycle; in fact, the Schwarzian is the so-called Souriau cocycle (referring to Jean-Marie Souriau) associated with the Bott–Thurston cocycle.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "S^1"
},
{
"math_id": 1,
"text": "x"
},
{
"math_id": 2,
"text": "x\\sim x+2\\pi"
},
{
"math_id": 3,
"text": "f:\\mathbb{R}\\to\\mathbb{R}:x\\mapsto f(x)"
},
{
"math_id": 4,
"text": "f(x+2\\pi)=f(x)+2\\pi"
},
{
"math_id": 5,
"text": "f'(x)>0"
},
{
"math_id": 6,
"text": "\\widetilde{\\text{Diff}}{}^+(S^1)"
},
{
"math_id": 7,
"text": "\\mathsf{C}(f,g)"
},
{
"math_id": 8,
"text": "\n\\mathsf{C}(f,g)\n\\equiv\n-\\frac{1}{48\\pi}\\int_0^{2\\pi}\n\\log\\big[f'\\big(g(x)\\big)\\big]\n\\frac{g''(x)\\,\\text{d}x}{g'(x)}.\n"
},
{
"math_id": 9,
"text": "\\widetilde{\\text{Diff}}{}^+(S^1)\\times\\mathbb{R}"
},
{
"math_id": 10,
"text": "(f,\\alpha)"
},
{
"math_id": 11,
"text": "f"
},
{
"math_id": 12,
"text": "\\alpha"
},
{
"math_id": 13,
"text": "\n(f,\\alpha)\\cdot(g,\\beta)\n=\n\\big(f\\circ g,\\alpha+\\beta+\\mathsf{C}(f,g)\\big).\n"
},
{
"math_id": 14,
"text": "(\\xi,\\alpha)"
},
{
"math_id": 15,
"text": "\\xi=\\xi(x)\\partial_x"
},
{
"math_id": 16,
"text": "f(x)=x+\\epsilon\\xi(x)"
},
{
"math_id": 17,
"text": "\n\\big[(\\xi,\\alpha),(\\zeta,\\beta)\\big]\n=\n\\bigg([\\xi,\\zeta],-\\frac{1}{24\\pi}\\int_0^{2\\pi}\\text{d}x\\,\\xi(x)\\zeta'''(x)\\bigg)\n"
},
{
"math_id": 18,
"text": "[\\xi,\\zeta]=(\\xi(x)\\zeta'(x)-\\zeta(x)\\xi'(x))\\partial_x"
},
{
"math_id": 19,
"text": "\nL_m\\equiv\\Big(-ie^{imx}\\partial_x,-\\frac{i}{24}\\delta_{m,0}\\Big),\n\\qquad\nZ\\equiv (0,-i),\n"
},
{
"math_id": 20,
"text": "\n[L_m,L_n]\n=\n(m-n)L_{m+n}+\\frac{Z}{12}m(m^2-1)\\delta_{m+n}.\n"
},
{
"math_id": 21,
"text": "Z"
},
{
"math_id": 22,
"text": "f(x)-x"
}
] |
https://en.wikipedia.org/wiki?curid=64053581
|
64057898
|
2 Kings 14
|
2 Kings, chapter 14
2 Kings 14 is the fourteenth chapter of the second part of the Books of Kings in the Hebrew Bible or the Second Book of Kings in the Old Testament of the Christian Bible. The book is a compilation of various annals recording the acts of the kings of Israel and Judah by a Deuteronomic compiler in the seventh century BCE, with a supplement added in the sixth century BCE. This chapter records the events during the reigns of Amaziah the son of Joash, king of Judah, as well as of Joash, and his son, Jeroboam (II) in the kingdom of Israel. The narrative is a part of a major section – covering the period of Jehu's dynasty.
Text.
This chapter was originally written in the Hebrew language. It is divided into 29 verses.
Textual witnesses.
Some early manuscripts containing the text of this chapter in Hebrew are of the Masoretic Text tradition, which includes the Codex Cairensis (895), Aleppo Codex (10th century), and Codex Leningradensis (1008).
There is also a translation into Koine Greek known as the Septuagint, made in the last few centuries BCE. Extant ancient manuscripts of the Septuagint version include Codex Vaticanus (B; formula_0B; 4th century) and Codex Alexandrinus (A; formula_0A; 5th century).
Analysis.
This chapter as a whole (as many other parts of 1–2 Kings) functions as a ‘parable and allegory’, and in particular includes a ‘proverb’ given by Jehoash king of Israel to Amaziah king of Judah (). Some examples of the parabolic or allegoric style are provided in form of the ‘history repeating itself’. During the time of Rehoboam the son of Solomon, after the division of the kingdom of Israel (), Shisak the king of Egypt plundered the temple in Jerusalem () and this event has a similar pattern in this chapter when Jehoash the king of Israel plundered the temple and broke down a large portion of the walls of the city of Jerusalem (). Another parallel commences at the end of the chapter when another Jeroboam started to reign in Israel and the subsequent chapters reveal a ‘providential chronological and historical symmetry’ with the first Jeroboam. While Jeroboam I initiated the separation of the united kingdom to form the northern kingdom of Israel, Jeroboam II started the countdown to the end of this northern kingdom. There is an indication that the kingdoms were reunited briefly under Jehu's dynasty is supported by some details in Jeroboam II's reign: the Israel king extended the borders of his kingdom from Hamath in the north to the Sea of the Arabah in the south, far into the territory of the kingdom of Judah (), which ‘echoes the ideal boundaries of the original united kingdom’ (). can also be translated as “he recovered Damascus and Hamath to Judah in Israel” as if Jeroboam II recovered the territory of Judah back to “the kingdom of Israel”, forming a (semi)united kingdom.
Amaziah, king of Judah (14:1–22).
The historical records of Amaziah the king of Judah might be taken exclusively from the Judean annals. He took revenge for his father's murder (verse 5, cf. ; verses 6–7 are an educated scribe's addition according to , cf. also Ezekiel 18) only to fall victim to murder himself (verses 19–20). Amaziah also defeated the Edomites in the Arabah ("Valley of Salt", verse 7, cf. , also verse 22), highlighting a struggle between Edom and Judah at the time (cf. ; ). However, the most detail is about the war with Israel which Amaziah initiated but ultimately lost (verses 8–14). Amaziah outlived Joash by at least fifteen years, but his violent death in the reign of Jeroboam II, the son of Joash, (verses 15–16) probably still related back to the events of his defeat. Amaziah's successor, Azariah (later, Uzziah), was chosen by 'the people of Judah' (verse 21), probably meaning 'the people of the land', who had an 'increasingly influential role in Judean politics' since the end of Athaliah's reign. Azariah () managed to consolidate his father's conquest of Edom by claiming the port of Elath for Judah (cf. ).
"In the second year of Joash son of Jehoahaz, king of Israel, Amaziah the son of Joash, king of Judah, became king"
"He was twenty and five years old when he began to reign, and reigned twenty and nine years in Jerusalem. And his mother's name was Jehoaddan of Jerusalem."
War between Israel and Judah.
Historical records show that Adad-nirari III of Assyria claims a successful westward campaign in 806 BCE, defeating, among others, 'Omri-Land' (the name Assyrian uses for Israel) and also Edom ("ANET" 281-2). This might encourage Amaziah to wage wars against Edom and Israel. He was successful to defeat Edom, but he miscalculated the strength of Israel. Joash the king of Israel, had warned Amaziah, using a parable: “A thistle in Lebanon sent a message to a cedar in Lebanon, 'Give your daughter to my son in marriage.' Then a wild beast in Lebanon came along and trampled the thistle underfoot. You have indeed defeated Edom and now you are arrogant. Glory in your victory, but stay at home! Why ask for trouble and cause your own downfall and that of Judah also?" However, Amaziah insisted on the war. Joash's army defeated Amaziah's at Beth-shemesh, on the borders of Dan and Philistia, then plundered Judah's palace and the temple, also broke down 200 meters of the 'particularly sensitive northern wall of Jerusalem', leaving the city defenseless.
"And they brought him on horses: and he was buried at Jerusalem with his fathers in the city of David."
Jeroboam (II), king of Israel (14:23–29).
Jeroboam's reign outshines that of Joash, his father, as the northern kingdom enjoys a glorious period, when Aram-Damascus was ensnared between Israel and Assyria (cf. verse 28), that apparently allowed Jeroboam to control the territories northwards to Hamath on the Orontes, and also to the east and south as far as the Dead Sea (verse 25). This implies a hegemony over Judah, or at least the Jordan valley and the regions east of the Jordan, Gilead and Gad. The Book of Amos provides highlights to Israel's momentary political success: 'they were proud of the land they gained (), the higher classes at least enjoyed the incoming wealth (), the people believed they were God's favorites ()', although Amos prophesied that this period of happiness would be short. The prophet Jonah ben Amittai was active in Israel at the time and had forecast Jeroboam's successes, so this can be seen as God's will, like other previous political events, 'according to the word of the LORD, which he spoke by the hand of...' (cf. the 'underlying rule'in and the examples in ; ; ; ). It is thought that God saw how Israel had suffered so much in the past, so God took pity, as connected to . The mention of Jonah supports the historical basis of his claim in the Book of Jonah that God's mercy extended to peoples beyond Israel, including Assyria.
"In the fifteenth year of Amaziah the son of Joash king of Judah Jeroboam the son of Joash king of Israel began to reign in Samaria, and reigned forty and one years."
Archeology.
The excavation at Tell al-Rimah yields a stele of Adad-nirari III which mentioned "Jehoash the Samarian" and contains the first cuneiform mention of Samaria by that name. The inscriptions of this "Tell al-Rimah Stele" may provide evidence of the existence of King Jehoash (=Joash) of Israel, attest to the weakening of Syrian kingdom (cf. ), and show the vassal status of the northern kingdom of Israel to the Assyrians.
A postulated image of Joash is reconstructed from plaster remains recovered at Kuntillet Ajrud. The ruins were from a temple built by the northern Israel kingdom when Jehoash of Israel gained control over the kingdom of Judah during the reign of Amaziah of Judah.
Notes.
<templatestyles src="Reflist/styles.css" />
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": " \\mathfrak{G}"
}
] |
https://en.wikipedia.org/wiki?curid=64057898
|
64057968
|
2 Chronicles 24
|
Second Book of Chronicles, chapter 24
2 Chronicles 24 is the twenty-fourth chapter of the Second Book of Chronicles the Old Testament in the Christian Bible or of the second part of the Books of Chronicles in the Hebrew Bible. The book is compiled from older sources by an unknown person or group, designated by modern scholars as "the Chronicler", and had the final shape established in late fifth or fourth century BCE. This chapter belongs to the section focusing on the kingdom of Judah until its destruction by the Babylonians under Nebuchadnezzar and the beginning of restoration under Cyrus the Great of Persia (2 Chronicles 10 to 36). The focus of this chapter is the reign of Joash, king of Judah.
Text.
This chapter was originally written in the Hebrew language and is divided into 27 verses.
Textual witnesses.
Some early manuscripts containing the text of this chapter in Hebrew are of the Masoretic Text tradition, which includes the Aleppo Codex (10th century), and Codex Leningradensis (1008).
There is also a translation into Koine Greek known as the Septuagint, made in the last few centuries BCE. Extant ancient manuscripts of the Septuagint version include Codex Vaticanus (B; formula_0B; 4th century), and Codex Alexandrinus (A; formula_0A; 5th century).
Joash repairs the Temple (24:1–16).
The Chronicles divide the reign of Joash into two periods: before and after the death of Jehoiada (verse 2: 'all the days of the priest Jehoiada'; cf. 2 Kings 12:2 : 'all his days, because the priest Jehoiada instructed him'). During his good period, Joash displayed strong leadership in rebuilding the neglected Temple in Jerusalem. This efforts occurred as long as Jehoiada is alive, the only priest recorded to live longer than Aaron (verses 15–16; cf. Numbers 33:39) and to be buried 'among the kings', a clear expression of Jehoiada's status as a "regal priest".
"Joash was seven years old when he began to reign, and he reigned forty years in Jerusalem. His mother's name also was Zibiah of Beersheba."
The wickedness of Joash (24:17–22).
The Chronicles use the phrases 'abandoned the house of the LORD', 'sacred poles', and 'idols'. to describe Joah's wickedness, followed by the important theological statement in the books: 'the Lord gives sinners the opportunity to return to his way by sending prophets to them' (verse 19), punctuated by the word of Zechariah, the son of Jehoiada, 'because you have forsaken the LORD, he has also forsaken you' (verse 20). Joash reacted shockingly by ordering Zechariah to be stoned to death in the forecourt of the temple, showing no gratitude to Jehoiada. Zechariah's dying words resembles the lines of Exodus 5:21.
Death of Joash (24:23–27).
This section parallels to 2 Kings 12:17–18 but with more emphasize to theological aspect: the Arameans were greatly outnumbered by the Judeans (who abandoned God), yet they prevailed over Judah, which is in contrast to the theme of a small Judean force defeating powerful armies with the help of God in the past. Joash was buried in the city of David (on account of his earlier good behavior), but not amongst the kings (because of his sins; verses 25–26).
"Now concerning his sons, and the greatness of the burdens laid upon him, and the repairing of the house of God, behold, they are written in the story of the book of the kings. And Amaziah his son reigned in his stead."
Notes.
<templatestyles src="Reflist/styles.css" />
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": " \\mathfrak{G}"
}
] |
https://en.wikipedia.org/wiki?curid=64057968
|
64057980
|
2 Chronicles 25
|
Second Book of Chronicles, chapter 25
2 Chronicles 25 is the twenty-fifth chapter of the Second Book of Chronicles the Old Testament in the Christian Bible or of the second part of the Books of Chronicles in the Hebrew Bible. The book is compiled from older sources by an unknown person or group, designated by modern scholars as "the Chronicler", and had the final shape established in late fifth or fourth century BCE. This chapter belongs to the section focusing on the kingdom of Judah until its destruction by the Babylonians under Nebuchadnezzar and the beginning of restoration under Cyrus the Great of Persia (2 Chronicles 10 to 36). The focus of this chapter is the reign of Amaziah, king of Judah.
Text.
This chapter was originally written in the Hebrew language and is divided into 28 verses.
Textual witnesses.
Some early manuscripts containing the text of this chapter in Hebrew are of the Masoretic Text tradition, which includes the Aleppo Codex (10th century), and Codex Leningradensis (1008).
There is also a translation into Koine Greek known as the Septuagint, made in the last few centuries BCE. Extant ancient manuscripts of the Septuagint version include Codex Vaticanus (B; formula_0B; 4th century), and Codex Alexandrinus (A; formula_0A; 5th century).
Amaziah, king of Judah (25:1–16).
Verses 1–4 and verse 11 in this section parallel to 2 Kings 14, along by two parts unique to the Chronicler: verses 5–10 and verses 12–16, both involving a prophetic figure. Amaziah's reign could be divided into a period of obedience to YHWH and success (verses 1–13), then a period of idolatry and defeat (verses 14–28).
"Amaziah was twenty-five years old when he began to reign, and he reigned twenty-nine years in Jerusalem. His mother's name was Jehoaddan of Jerusalem."
Jehoash of Israel defeats Amaziah (25:17–28).
This section records the consequences of Amaziah worshipping Edomite deities (verses 15, 20: 'it was God's doing'; cf. 2 Chronicles 10:15; 22:7) in form of his defeat to Jehoash of the northern kingdom.
"And they brought him upon horses, and buried him with his fathers in the city of Judah."
Notes.
<templatestyles src="Reflist/styles.css" />
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": " \\mathfrak{G}"
}
] |
https://en.wikipedia.org/wiki?curid=64057980
|
640580
|
Romberg's method
|
Numerical integration method
In numerical analysis, Romberg's method is used to estimate the definite integral formula_0 by applying Richardson extrapolation repeatedly on the trapezium rule or the rectangle rule (midpoint rule). The estimates generate a triangular array. Romberg's method is a Newton–Cotes formula – it evaluates the integrand at equally spaced points.
The integrand must have continuous derivatives, though fairly good results
may be obtained if only a few derivatives exist.
If it is possible to evaluate the integrand at unequally spaced points, then other methods such as Gaussian quadrature and Clenshaw–Curtis quadrature are generally more accurate.
The method is named after Werner Romberg, who published the method in 1955.
Method.
Using formula_1, the method can be inductively defined by
formula_2
where formula_3 and formula_4.
In big O notation, the error for "R"("n", "m") is: formula_5
The zeroeth extrapolation, "R"("n", 0), is equivalent to the trapezoidal rule with 2"n" + 1 points; the first extrapolation, "R"("n", 1), is equivalent to Simpson's rule with 2"n" + 1 points. The second extrapolation, "R"("n", 2), is equivalent to Boole's rule with 2"n" + 1 points. The further extrapolations differ from Newton-Cotes formulas. In particular further Romberg extrapolations expand on Boole's rule in very slight ways, modifying weights into ratios similar as in Boole's rule. In contrast, further Newton-Cotes methods produce increasingly differing weights, eventually leading to large positive and negative weights. This is indicative of how large degree interpolating polynomial Newton-Cotes methods fail to converge for many integrals, while Romberg integration is more stable.
By labelling our formula_6 approximations as formula_7 instead of formula_8, we can perform Richardson extrapolation with the error formula defined below:
formula_9
Once we have obtained our formula_10 approximations formula_11, we can label them as formula_12.
When function evaluations are expensive, it may be preferable to replace the polynomial interpolation of Richardson with the rational interpolation proposed by .
A geometric example.
To estimate the area under a curve the trapezoid rule is applied first to one-piece, then two, then four, and so on.
After trapezoid rule estimates are obtained, Richardson extrapolation is applied.
<templatestyles src="Template:Table alignment/tables.css" />
Example.
As an example, the Gaussian function is integrated from 0 to 1, i.e. the error function erf(1) ≈ 0.842700792949715. The triangular array is calculated row by row and calculation is terminated if the two last entries in the last row differ less than 10−8.
0.77174333
0.82526296 0.84310283
0.83836778 0.84273605 0.84271160
0.84161922 0.84270304 0.84270083 0.84270066
0.84243051 0.84270093 0.84270079 0.84270079 0.84270079
The result in the lower right corner of the triangular array is accurate to the digits shown.
It is remarkable that this result is derived from the less accurate approximations
obtained by the trapezium rule in the first column of the triangular array.
Implementation.
Here is an example of a computer implementation of the Romberg method (in the C programming language):
void print_row(size_t i, double *R) {
printf("R[%2zu] = ", i);
for (size_t j = 0; j <= i; ++j) {
printf("%f ", R[j]);
printf("\n");
INPUT:
(*f) : pointer to the function to be integrated
a : lower limit
b : upper limit
max_steps: maximum steps of the procedure
acc : desired accuracy
OUTPUT:
Rp[max_steps-1]: approximate value of the integral of the function f for x in [a,b] with accuracy 'acc' and steps 'max_steps'.
double romberg(double (*f)(double), double a, double b, size_t max_steps, double acc)
double R1[max_steps], R2[max_steps]; // buffers
double *Rp = &R1[0], *Rc = &R2[0]; // Rp is previous row, Rc is current row
double h = b-a; //step size
Rp[0] = (f(a) + f(b))*h*0.5; // first trapezoidal step
print_row(0, Rp);
for (size_t i = 1; i < max_steps; ++i) {
h /= 2.;
double c = 0;
size_t ep = 1 « (i-1); //2^(n-1)
for (size_t j = 1; j <= ep; ++j) {
c += f(a + (2*j-1) * h);
Rc[0] = h*c + .5*Rp[0]; // R(i,0)
for (size_t j = 1; j <= i; ++j) {
double n_k = pow(4, j);
Rc[j] = (n_k*Rc[j-1] - Rp[j-1]) / (n_k-1); // compute R(i,j)
// Print ith row of R, R[i,i] is the best estimate so far
print_row(i, Rc);
if (i > 1 && fabs(Rp[i-1]-Rc[i]) < acc) {
return Rc[i];
// swap Rn and Rc as we only need the last row
double *rt = Rp;
Rp = Rc;
Rc = rt;
return Rp[max_steps-1]; // return our best guess
Here is an implementation of the Romberg method (in the Python programming language):
def print_row(i, R):
"""Prints a row of the Romberg table."""
print(f"R[{i:2d}] = ", end="")
for j in range(i + 1):
print(f"{R[j]:f} ", end="")
print()
def romberg(f, a, b, max_steps, acc):
Calculates the integral of a function using Romberg integration.
Args:
f: The function to integrate.
a: Lower limit of integration.
b: Upper limit of integration.
max_steps: Maximum number of steps.
acc: Desired accuracy.
Returns:
The approximate value of the integral.
R1, R2 = [0] * max_steps, [0] * max_steps # Buffers for storing rows
Rp, Rc = R1, R2 # Pointers to previous and current rows
h = b - a # Step size
Rp[0] = 0.5 * h * (f(a) + f(b)) # First trapezoidal step
print_row(0, Rp)
for i in range(1, max_steps):
h /= 2.
c = 0
ep = 1 « (i - 1) # 2^(i-1)
for j in range(1, ep + 1):
c += f(a + (2 * j - 1) * h)
Rc[0] = h * c + 0.5 * Rp[0] # R(i,0)
for j in range(1, i + 1):
n_k = 4**j
Rc[j] = (n_k * Rc[j - 1] - Rp[j - 1]) / (n_k - 1) # Compute R(i,j)
# Print ith row of R, R[i,i] is the best estimate so far
print_row(i, Rc)
if i > 1 and abs(Rp[i - 1] - Rc[i]) < acc:
return Rc[i]
# Swap Rn and Rc for next iteration
Rp, Rc = Rc, Rp
return Rp[max_steps - 1] # Return our best guess
References.
Citations.
<templatestyles src="Reflist/styles.css" />
Bibliography.
<templatestyles src="Refbegin/styles.css" />
|
[
{
"math_id": 0,
"text": " \\int_a^b f(x) \\, dx "
},
{
"math_id": 1,
"text": "h_n = \\frac{(b-a)}{2^n}"
},
{
"math_id": 2,
"text": "\\begin{align}\nR(0,0) &= h_1 (f(a) + f(b)) \\\\\nR(n,0) &= \\tfrac{1}{2} R(n-1,0) + h_n \\sum_{k=1}^{2^{n-1}} f(a + (2k-1)h_n) \\\\\nR(n,m) &= R(n,m-1) + \\tfrac{1}{4^m-1} (R(n,m-1) - R(n-1,m-1)) \\\\\n&= \\frac{1}{4^m-1} ( 4^m R(n,m-1) - R(n-1, m-1))\n\\end{align}"
},
{
"math_id": 3,
"text": " n \\ge m "
},
{
"math_id": 4,
"text": " m \\ge 1 \\, "
},
{
"math_id": 5,
"text": " O\\left(h_n^{2m+2}\\right)."
},
{
"math_id": 6,
"text": "O(h^2)"
},
{
"math_id": 7,
"text": "A_0\\big(\\frac{h}{2^n}\\big)"
},
{
"math_id": 8,
"text": "R(n,0)"
},
{
"math_id": 9,
"text": " \\int_a^b f(x) \\, dx = A_0\\bigg(\\frac{h}{2^n}\\bigg)+a_0\\bigg(\\frac{h}{2^n}\\bigg)^{2} + a_1\\bigg(\\frac{h}{2^n}\\bigg)^{4} + a_2\\bigg(\\frac{h}{2^n}\\bigg)^{6} + \\cdots "
},
{
"math_id": 10,
"text": "O(h^{2(m+1)})"
},
{
"math_id": 11,
"text": "A_m\\big(\\frac{h}{2^n}\\big)"
},
{
"math_id": 12,
"text": "R(n,m)"
},
{
"math_id": 13,
"text": "2^n+1"
}
] |
https://en.wikipedia.org/wiki?curid=640580
|
6406095
|
K-medoids
|
Clustering algorithm minimizing the sum of distances to k representatives
The k-medoids problem is a clustering problem similar to k-means. The name was coined by Leonard Kaufman and Peter J. Rousseeuw with their PAM (Partitioning Around Medoids) algorithm. Both the k-means and k-medoids algorithms are partitional (breaking the dataset up into groups) and attempt to minimize the distance between points labeled to be in a cluster and a point designated as the center of that cluster. In contrast to the k-means algorithm, k-medoids chooses actual data points as centers (medoids or exemplars), and thereby allows for greater interpretability of the cluster centers than in k-means, where the center of a cluster is not necessarily one of the input data points (it is the average between the points in the cluster). Furthermore, k-medoids can be used with arbitrary dissimilarity measures, whereas k-means generally requires Euclidean distance for efficient solutions. Because k-medoids minimizes a sum of pairwise dissimilarities instead of a sum of squared Euclidean distances, it is more robust to noise and outliers than k-means.
k-medoids is a classical partitioning technique of clustering that splits the data set of n objects into k clusters, where the number k of clusters assumed known "a priori" (which implies that the programmer must specify k before the execution of a k-medoids algorithm). The "goodness" of the given value of k can be assessed with methods such as the silhouette method.
The medoid of a cluster is defined as the object in the cluster whose sum (and, equivalently, the average) of dissimilarities to all the objects in the cluster is minimal, that is, it is a most centrally located point in the cluster.
Algorithms.
In general, the k-medoids problem is NP-hard to solve exactly. As such, many heuristic solutions exist.
Partitioning Around Medoids (PAM).
PAM uses a greedy search which may not find the optimum solution, but it is faster than exhaustive search. It works as follows:
The runtime complexity of the original PAM algorithm per iteration of (3) is formula_2, by only computing the change in cost. A naive implementation recomputing the entire cost function every time will be in formula_3. This runtime can be further reduced to formula_4, by splitting the cost change into three parts such that computations can be shared or avoided (FastPAM). The runtime can further be reduced by eagerly performing swaps (FasterPAM), at which point a random initialization becomes a viable alternative to BUILD.
Alternating Optimization.
Algorithms other than PAM have also been suggested in the literature, including the following Voronoi iteration method known as the "Alternating" heuristic in literature, as it alternates between two optimization steps:
"k"-means-style Voronoi iteration tends to produce worse results, and exhibit "erratic behavior". Because it does not allow re-assigning points to other clusters while updating means it only explores a smaller search space. It can be shown that even in simple cases this heuristic finds inferior solutions the swap based methods can solve.
Hierarchical Clustering.
Multiple variants of hierarchical clustering with a "medoid linkage" have been proposed. The Minimum Sum linkage criterion directly uses the objective of medoids, but the Minimum Sum Increase linkage was shown to produce better results (similar to how Ward linkage uses the increase in squared error). Earlier approaches simply used the distance of the cluster medoids of the previous medoids as linkage measure, but which tends to result in worse solutions, as the distance of two medoids does not ensure there exists a good medoid for the combination. These approaches have a run time complexity of formula_5, and when the dendrogram is cut at a particular number of clusters "k", the results will typically be worse than the results found by PAM. Hence these methods are primarily of interest when a hierarchical tree structure is desired.
Other Algorithms.
Other approximate algorithms such as CLARA and CLARANS trade quality for runtime. CLARA applies PAM on multiple subsamples, keeping the best result. CLARANS works on the entire data set, but only explores a subset of the possible swaps of medoids and non-medoids using sampling. BanditPAM uses the concept of multi-armed bandits to choose candidate swaps instead of uniform sampling as in CLARANS.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "m_{\\text{best}}"
},
{
"math_id": 1,
"text": "o_{\\text{best}}"
},
{
"math_id": 2,
"text": "O(k (n-k)^2)"
},
{
"math_id": 3,
"text": "O(n^2k^2)"
},
{
"math_id": 4,
"text": "O(n^2)"
},
{
"math_id": 5,
"text": "O(n^3)"
}
] |
https://en.wikipedia.org/wiki?curid=6406095
|
64064108
|
ITF-6
|
ITF-6 is the implementation of an Interleaved 2 of 5 (ITF) barcode to encode a addon to ITF-14 and ITF-16 barcodes. Originally was developed as a part of JIS specification for Physical Distribution Center. Instead of ITF-14, it wasn’t standardized by ISO Committee but it is widely used to encode additional data to Global Trade Item Number such as items quantity or container weight.
History.
In 1983, the Logistics Symbol Committee proposed the Interleaved 2 of 5 barcode as a method to improve the JAN code. In 1985, a logistics symbol JIS drafting committee was set up at the Distribution System Development Center, and the final examination was started toward JIS. Then in 1987 it was standardized as JIS-X-0502, a standard physical distribution barcode symbol ITF-14/16/6.
The ITF barcode has an add-on version for displaying the weight, etc., and it is possible to encode a 5-digits numerical value and 6-th check character as ITF-6 after ITF-14 or ITF-16(obsolete in 2010).
Currently ITF-6 isn’t standardized by ISO Committee and it is used only as a part of JIS standards. However, it is widely used by manufacturers to encode additional data and it is supported by wide range of barcode scanners
Uses.
Despite the fact that ITF-6 barcode isn’t included into ISO standards, it is widely used as add-on to encode items quantity in package or item weight. At this time, it is used only with ITF-14 (Global Trade Item Number), but up to 2010 it was used with standardized only in Japan ITF-16 (Extended Symbology for Physical Distribution).
From the left, ITF-6 contains 5 significant digits and the last one is control digit, which is calculated same way as UPC checksums. If a decimal point is required, the decimal point is between the 3rd and 4th digits:
<br>NNNNN(C/D) - without decimal point;
<br>NNN.NN(C/D) - with decimal point.
ITF-6 is supported by various barcode generating software and barcode scanners.
Checksum.
Checksum is calculated as other UPC checksums:
<br>formula_0
Example for the first 5 digits 12345:
<br>10 - ((3*1 + 2 + 3*3 + 4 + 3*5) mod 10) = 7. Check digit is 7.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "x_6 = 10 - ((3x_1 + x_2 + 3x_3 + x_4 + 3x_5)\\pmod{10})"
}
] |
https://en.wikipedia.org/wiki?curid=64064108
|
64067273
|
Unified strength theory
|
The unified strength theory (UST). proposed by Yu Mao-Hong is a series of yield criteria (see yield surface) and failure criteria (see Material failure theory). It is a generalized classical strength theory which can be used to describe the yielding or failure of material begins when the combination of principal stresses reaches a critical value.
Mathematical Formulation.
Mathematically, the formulation of UST is expressed in principal stress state as
formula_0(1a)<br>
formula_1(1b)<br>
where formula_2 are three principal stresses, formula_3is the uniaxial tensile strength and formula_4 is tension-compression strength ratio (formula_5).
The unified yield criterion (UYC) is the simplification of UST when formula_6, i.e.
formula_7(2a)
formula_8(2b)
Limit surfaces of Unified Strength Theory.
The limit surfaces of the unified strength theory in principal stress space are usually a semi-infinite dodecahedron cone with unequal sides. The shape and size of the limiting dodecahedron cone depends on the parameter b and formula_4. The limit surfaces of UST and UYC are shown as follows.
Derivation of Unified Strength Theory.
Due to the relation (formula_9), the principal stress state (formula_2) may be converted to the twin-shear stress state (formula_10) or (formula_11). Twin-shear element models proposed by Mao-Hong Yu are used for representing the twin-shear stress state. Considering all the stress components of the twin-shear models and their different effects yields the unified strength theory as
formula_12(3a)<br>
formula_13(3b)<br>
The relations among the stresses components and principal stresses read
formula_14, formula_15(4a)<br>
formula_16, formula_17(4b)<br>
formula_18, formula_19(4c)<br>
The formula_20 and "C" should be obtained by uniaxial failure state<br>
formula_21(5a)<br>
formula_22(5b)<br>
By substituting Eqs.(4a), (4b) and (5a) into the Eq.(3a), and substituting Eqs.(4a), (4c) and (5b) into Eq.(3b), the formula_20 and "C" are introduced as<br>
formula_23,
formula_24(6)<br>
History of Unified Strength Theory.
The development of the unified strength theory can be divided into three stages as follows.<br>
1. Twin-shear yield criterion (UST with formula_25 and formula_26)<br>
formula_27(7a)<br>
formula_28(7b)<br>
2. Twin-shear strength theory (UST with formula_26).<br>
formula_29(8a)<br>
formula_30(8b)<br>
3. Unified strength theory.<br>
Applications of the Unified Strength theory.
Unified strength theory has been used in Generalized Plasticity, Structural Plasticity, Computational Plasticity and many other fields
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "F = {\\sigma _1} - \\frac{\\alpha }{{1 + b}}(b{\\sigma _2} + {\\sigma _3}) = {\\sigma _t},\\, {\\text{ when }}{\\sigma _2} \\leqslant \\frac{{{\\sigma _1} + \\alpha {\\sigma _3}}}{{1 + \\alpha }}"
},
{
"math_id": 1,
"text": "F' = \\frac{1}{{1 + b}}({\\sigma _1} + b{\\sigma _3}) - \\alpha {\\sigma _3} = {\\sigma _t},\\, {\\text{ when }} {\\sigma _2} \\geqslant \\frac{{{\\sigma _1} + \\alpha {\\sigma _3}}}{{1 + \\alpha }}"
},
{
"math_id": 2,
"text": "{\\sigma _1},{\\sigma _2},{\\sigma _3}"
},
{
"math_id": 3,
"text": "{\\sigma _{t}}"
},
{
"math_id": 4,
"text": "\\alpha"
},
{
"math_id": 5,
"text": "\\alpha = {\\sigma _t}/{\\sigma _c}"
},
{
"math_id": 6,
"text": "\\alpha = 1"
},
{
"math_id": 7,
"text": "f = {\\sigma _1} - \\frac{1}{{1 + b}}(b{\\sigma _2} + {\\sigma _3}) = {\\sigma _s},{\\text{ when }} {\\sigma _2} \\leqslant \\frac{1}{2}({\\sigma _1} + {\\sigma _3})"
},
{
"math_id": 8,
"text": "f' = \\frac{1}{{1 + b}}({\\sigma _1} + b{\\sigma _2}) - {\\sigma _3} = {\\sigma _s},{\\text{ when }} {\\sigma _2} \\geqslant \\frac{1}{2}({\\sigma _1} + {\\sigma _3})"
},
{
"math_id": 9,
"text": "{\\tau _{13}} = {\\tau _{12}} + {\\tau _{23}}"
},
{
"math_id": 10,
"text": "{\\tau _{13}},{\\tau _{12}};{\\sigma _{13}},{\\sigma _{12}}"
},
{
"math_id": 11,
"text": "{\\tau _{13}},{\\tau _{23}};{\\sigma _{13}},{\\sigma _{23}}"
},
{
"math_id": 12,
"text": "F = {\\tau _{13}} + b{\\tau _{12}} + \\beta ({\\sigma _{13}} + b{\\sigma _{12}}) = C, {\\text{ when }} {\\tau _{12}} + \\beta {\\sigma _{12}} \\geqslant {\\tau _{23}} + \\beta {\\sigma _{23}}"
},
{
"math_id": 13,
"text": "F' = {\\tau _{13}} + b{\\tau _{23}} + \\beta ({\\sigma _{13}} + b{\\sigma _{23}}) = C, {\\text{ when }}{\\tau _{12}} + \\beta {\\sigma _{12}} \\leqslant {\\tau _{23}} + \\beta {\\sigma _{23}}"
},
{
"math_id": 14,
"text": "{\\tau _{13}} = \\frac{1}{2}\\left( {{\\sigma _1} - {\\sigma _3}} \\right)"
},
{
"math_id": 15,
"text": "{\\sigma _{13}} = \\frac{1}{2}\\left( {{\\sigma _1} + {\\sigma _3}} \\right)"
},
{
"math_id": 16,
"text": "{\\tau _{12}} = \\frac{1}{2}\\left( {{\\sigma _1} - {\\sigma _2}} \\right)"
},
{
"math_id": 17,
"text": "{\\sigma _{12}} = \\frac{1}{2}\\left( {{\\sigma _1} + {\\sigma _2}} \\right)"
},
{
"math_id": 18,
"text": "{\\tau _{23}} = \\frac{1}{2}\\left( {{\\sigma _2} - {\\sigma _3}} \\right)"
},
{
"math_id": 19,
"text": "{\\sigma _{23}} = \\frac{1}{2}\\left( {{\\sigma _2} + {\\sigma _3}} \\right)"
},
{
"math_id": 20,
"text": "\\beta "
},
{
"math_id": 21,
"text": "{\\sigma _1} = {\\sigma _t},{\\sigma _2} = {\\sigma _3} = 0"
},
{
"math_id": 22,
"text": "{\\sigma _1} = {\\sigma _2} = 0,{\\sigma _3} = - {\\sigma _{\\text{c}}}"
},
{
"math_id": 23,
"text": "\\beta = \\frac{{{\\sigma _{\\text{c}}} - {\\sigma _{\\text{t}}}}}{{{\\sigma _{\\text{c}}} + {\\sigma _{\\text{t}}}}} = \\frac{{1 - \\alpha }}{{1 + \\alpha }}"
},
{
"math_id": 24,
"text": "C = \\frac{{1 + b{\\sigma _{\\text{c}}}{\\sigma _{\\text{t}}}}}{{{\\sigma _{\\text{c}}} + {\\sigma _{\\text{t}}}}} = \\frac{{1 + b}}{{1 + \\alpha }}{\\sigma _t}"
},
{
"math_id": 25,
"text": "\\alpha = 1 "
},
{
"math_id": 26,
"text": " b = 1"
},
{
"math_id": 27,
"text": "f = {\\sigma _1} - \\frac{1}{2}({\\sigma _2} + {\\sigma _3}) = {\\sigma _t},{\\text{ when }} {\\sigma _2} \\leqslant \\frac{{{\\sigma _1} + {\\sigma _3}}}{2}"
},
{
"math_id": 28,
"text": "f = \\frac{1}{2}({\\sigma _1} + {\\sigma _2}) - {\\sigma _3} = {\\sigma _t},{\\text{ when }} {\\sigma _2} \\geqslant \\frac{{{\\sigma _1} + {\\sigma _3}}}{2}"
},
{
"math_id": 29,
"text": "F = {\\sigma _1} - \\frac{\\alpha }{2}({\\sigma _2} + {\\sigma _3}) = {\\sigma _t}, {\\text{ when }}{\\sigma _2} \\leqslant \\frac{{{\\sigma _1} + \\alpha {\\sigma _3}}}{{1 + \\alpha }}"
},
{
"math_id": 30,
"text": "F = \\frac{1}{2}({\\sigma _1} + {\\sigma _2}){\\text{ - }}\\alpha {\\sigma _3} = {\\sigma _t},{\\text{ when }} {\\sigma _2} \\geqslant \\frac{{{\\sigma _1} + \\alpha {\\sigma _3}}}{{1 + \\alpha }}"
}
] |
https://en.wikipedia.org/wiki?curid=64067273
|
640697
|
Reactive oxygen species
|
Highly reactive molecules formed from diatomic oxygen (O₂)
In chemistry and biology, reactive oxygen species (ROS) are highly reactive chemicals formed from diatomic oxygen (), water, and hydrogen peroxide. Some prominent ROS are hydroperoxide (O2H), superoxide (O2-), hydroxyl radical (OH.), and singlet oxygen. ROS are pervasive because they are readily produced from O2, which is abundant. ROS are important in many ways, both beneficial and otherwise. ROS function as signals, that turn on and off biological functions. They are intermediates in the redox behavior of O2, which is central to fuel cells. ROS are central to the photodegradation of organic pollutants in the atmosphere. Most often however, ROS are discussed in a biological context, ranging from their effects on aging and their role in causing dangerous genetic mutations.
Inventory of ROS.
ROS are not uniformly defined. All sources include superoxide, singlet oxygen, and hydroxyl radical. Hydrogen peroxide is not nearly as reactive as these species, but is readily activated and is thus included. Peroxynitrite and nitric oxide are reactive oxygen-containing species as well.
In its fleeting existence, the hydroxyl radical reacts rapidly irreversibly with all organic compounds.
Competing with its formation, superoxide is destroyed by the action of superoxide dismutases, enzymes that catalyze its disproportionation:
Biological function.
In a biological context, ROS are byproducts of the normal metabolism of oxygen. ROS have roles in cell signaling and homeostasis. ROS are intrinsic to cellular functioning, and are present at low and stationary levels in normal cells. In plants, ROS are involved in metabolic processes related to photoprotection and tolerance to various types of stress. However, ROS can cause irreversible damage to DNA as they oxidize and modify some cellular components and prevent them from performing their original functions. This suggests that ROS has a dual role; whether they will act as harmful, protective or signaling factors depends on the balance between ROS production and disposal at the right time and place. In other words, oxygen toxicity can arise both from uncontrolled production and from the inefficient elimination of ROS by the antioxidant system. ROS were also demonstrated to modify the visual appearance of fish. This potentially affects their behavior and ecology, such as their temperature control, their visual communication, their reproduction and survival.
During times of environmental stress (e.g., UV or heat exposure), ROS levels can increase dramatically. This may result in significant damage to cell structures. Cumulatively, this is known as oxidative stress. The production of ROS is strongly influenced by stress factor responses in plants, these factors that increase ROS production include drought, salinity, chilling, defense of pathogens, nutrient deficiency, metal toxicity and UV-B radiation. ROS are also generated by exogenous sources such as ionizing radiation generating irreversible effects in the development of tissues in both animals and plants.
Sources of ROS production.
Endogenous sources.
ROS are produced during the processes of respiration and photosynthesis in organelles such as mitochondria, peroxisomes and chloroplasts. During the respiration process the mitochondria convert energy for the cell into a usable form, adenosine triphosphate (ATP). The process of ATP production in the mitochondria, called oxidative phosphorylation, involves the transport of protons (hydrogen ions) across the inner mitochondrial membrane by means of the electron transport chain. In the electron transport chain, electrons are passed through a series of proteins via oxidation-reduction reactions, with each acceptor protein along the chain having a greater reduction potential than the previous. The last destination for an electron along this chain is an oxygen molecule. In normal conditions, the oxygen is reduced to produce water; however, in about 0.1–2% of electrons passing through the chain (this number derives from studies in isolated mitochondria, though the exact rate in live organisms is yet to be fully agreed upon), oxygen is instead prematurely and incompletely reduced to give the superoxide radical (•O2−), most well documented for Complex I and Complex III.
Another source of ROS production in animal cells is the electron transfer reactions catalyzed by the mitochondrial P450 systems in steroidogenic tissues.
These P450 systems are dependent on the transfer of electrons from NADPH to P450. During this process, some electrons "leak" and react with O2 producing superoxide. To cope with this natural source of ROS, the steroidogenic tissues, ovary and testis, have a large concentration of antioxidants such as vitamin C (ascorbate) and β-carotene and anti-oxidant enzymes.
If too much damage is present in mitochondria, a cell undergoes apoptosis or programmed cell death.
In addition, ROS are produced in immune cell signaling via the NOX pathway. Phagocytic cells such as neutrophils, eosinophils, and mononuclear phagocytes produce ROS when stimulated.
In chloroplasts, the carboxylation and oxygenation reactions catalyzed by rubisco ensure that the functioning of the electron transport chain (ETC) occurs in an environment rich in O2. The leakage of electrons in the ETC will inevitably produce ROS within the chloroplasts.
ETC in photosystem I (PSI) was once believed to be the only source of ROS in chloroplasts. The flow of electrons from the excited reaction centers is directed to the NADP and these are reduced to NADPH, and then they enter the Calvin cycle and reduce the final electron acceptor, CO2. In cases where there is an ETC overload, part of the electron flow is diverted from ferredoxin to O2, forming the superoxide free radical (by the Mehler reaction). In addition, electron leakage to O2 can also occur from the 2Fe-2S and 4Fe-4S clusters in the PSI ETC. However, PSII also provides electron leakage locations (QA, QB) for O2-producing O2-. Superoxide (O2-) is generated from PSII, instead of PSI; QB is shown as the location for the generation of O2•-.
Exogenous sources.
The formation of ROS can be stimulated by a variety of agents such as pollutants, heavy metals, tobacco, smoke, drugs, xenobiotics, microplastics, or radiation. In plants, in addition to the action of dry abiotic factors, high temperature, interaction with other living beings can influence the production of ROS.
Ionizing radiation can generate damaging intermediates through the interaction with water, a process termed radiolysis. Since water comprises 55–60% of the human body, the probability of radiolysis is quite high under the presence of ionizing radiation. In the process, water loses an electron and becomes highly reactive. Then through a three-step chain reaction, water is sequentially converted to hydroxyl radical (•OH), hydrogen peroxide (H2O2), superoxide radical (•O2−), and ultimately oxygen (O2).
The hydroxyl radical is extremely reactive and immediately removes electrons from any molecule in its path, turning that molecule into a free radical and thus propagating a chain reaction. However, hydrogen peroxide is actually more damaging to DNA than the hydroxyl radical, since the lower reactivity of hydrogen peroxide provides enough time for the molecule to travel into the nucleus of the cell, subsequently reacting with macromolecules such as DNA.
In plants, the production of ROS occurs during events of abiotic stress that lead to a reduction or interruption of metabolic activity. For example, the increase in temperature, drought are factors that limit the availability of CO2 due to stomatal closure, increasing the production of ROS, such as O2·- and 1O2 in chloroplasts. The production of 1O2 in chloroplasts can cause reprogramming of the expression of nucleus genes leading to chlorosis and programmed cell death.
In cases of biotic stress, the generation of ROS occurs quickly and weakly initially and then becomes more solid and lasting. The first phase of ROS accumulation is associated with plant infection and is probably independent of the synthesis of new ROS-generating enzymes. However, the second phase of ROS accumulation is associated only with infection by non-virulent pathogens and is an induced response dependent on increased mRNA transcription encoding enzymes.
Antioxidant enzymes.
Superoxide dismutase.
Superoxide dismutases (SOD) are a class of enzymes that catalyzes the dismutation of superoxide into oxygen and hydrogen peroxide. As such, they are an important antioxidant defense in nearly all cells exposed to oxygen. In mammals and most chordates, three forms of superoxide dismutase are present. SOD1 is located primarily in the cytoplasm, SOD2 in the mitochondria and SOD3 is extracellular. The first is a dimer (consists of two units), while the others are tetramers (four subunits). SOD1 and SOD3 contain copper and zinc ions, while SOD2 has a manganese ion in its reactive centre. The genes are located on chromosomes 21, 6, and 4, respectively (21q22.1, 6q25.3 and 4p15.3-p15.1).
The SOD-catalysed dismutation of superoxide may be written with the following half-reactions:
formula_0
where M = Cu ("n" = 1); Mn ("n" = 2); Fe ("n" = 2); Ni ("n" = 2). In this reaction the oxidation state of the metal cation oscillates between n and "n" + 1.
Catalase, which is concentrated in peroxisomes located next to mitochondria, reacts with the hydrogen peroxide to catalyze the formation of water and oxygen. Glutathione peroxidase reduces hydrogen peroxide by transferring the energy of the reactive peroxides to a sulfur-containing tripeptide called glutathione. The sulfur contained in these enzymes acts as the reactive center, carrying reactive electrons from the peroxide to the glutathione. Peroxiredoxins also degrade , within the mitochondria, cytosol, and nucleus.
formula_1
Damaging effects.
Effects of ROS on cell metabolism are well documented in a variety of species. These include not only roles in apoptosis (programmed cell death) but also positive effects such as the induction of host defence genes and mobilization of ion transporters. This implicates them in control of cellular function. In particular, platelets involved in wound repair and blood homeostasis release ROS to recruit additional platelets to sites of injury. These also provide a link to the adaptive immune system via the recruitment of leukocytes.
Reactive oxygen species are implicated in cellular activity to a variety of inflammatory responses including cardiovascular disease. They may also be involved in hearing impairment via cochlear damage induced by elevated sound levels, in ototoxicity of drugs such as cisplatin, and in congenital deafness in both animals and humans. ROS are also implicated in mediation of apoptosis or programmed cell death and ischaemic injury. Specific examples include stroke and heart attack.
In general, the harmful effects of reactive oxygen species on the cell are the damage of DNA or RNA, oxidation of polyunsaturated fatty acids in lipids (lipid peroxidation), oxidation of amino acids in proteins, and oxidative deactivation of specific enzymes by oxidation co-factors.
Pathogen response.
When a plant recognizes an attacking pathogen, one of the first induced reactions is to rapidly produce superoxide (O2-) or hydrogen peroxide (H2O2) to strengthen the cell wall. This prevents the spread of the pathogen to other parts of the plant, essentially forming a net around the pathogen to restrict movement and reproduction.
In the mammalian host, ROS is induced as an antimicrobial defense. To highlight the importance of this defense, individuals with chronic granulomatous disease who have deficiencies in generating ROS, are highly susceptible to infection by a broad range of microbes including "Salmonella enterica", "Staphylococcus aureus", "Serratia marcescens", and "Aspergillus" spp.
Studies on the homeostasis of the "Drosophila melanogaster"’s intestines have shown the production of ROS as a key component of the immune response in the gut of the fly. ROS acts both as a bactericide, damaging the bacterial DNA, RNA and proteins, as well as a signalling molecule that induces repair mechanisms of the epithelium. The uracil released by microorganism triggers the production and activity of DUOX, the ROS-producing enzyme in the intestine. DUOX activity is induced according to the level of uracil in the gut; under basal conditions, it is down-regulated by the protein kinase MkP3. The tight regulation of DUOX avoids excessive production of ROS and facilitates differentiation between benign and damage-inducing microorganisms in the gut.
The manner in which ROS defends the host from invading microbe is not fully understood. One of the more likely modes of defense is damage to microbial DNA. Studies using "Salmonella" demonstrated that DNA repair mechanisms were required to resist killing by ROS. A role for ROS in antiviral defense mechanisms has been demonstrated via Rig-like helicase-1 and mitochondrial antiviral signaling protein. Increased levels of ROS potentiate signaling through this mitochondria-associated antiviral receptor to activate interferon regulatory factor (IRF)-3, IRF-7, and nuclear factor kappa B (NF-κB), resulting in an antiviral state. Respiratory epithelial cells induce mitochondrial ROS in response to influenza infection. This induction of ROS led to the induction of type III interferon and the induction of an antiviral state, limiting viral replication. In host defense against mycobacteria, ROS play a role, although direct killing is likely not the key mechanism; rather, ROS likely affect ROS-dependent signalling controls, such as cytokine production, autophagy, and granuloma formation.
Reactive oxygen species are also implicated in activation, anergy and apoptosis of T cells.
Oxidative damage.
In aerobic organisms the energy needed to fuel biological functions is produced in the mitochondria via the electron transport chain. Reactive oxygen species (ROS) with the potential to cause cellular damage are produced along with the release of energy. ROS can damage lipids, DNA, RNA, and proteins, which, in theory, contributes to the physiology of aging.
ROS are produced as a normal product of cellular metabolism. In particular, one major contributor to oxidative damage is hydrogen peroxide (H2O2), which is converted from superoxide that leaks from the mitochondria. Catalase and superoxide dismutase ameliorate the damaging effects of hydrogen peroxide and superoxide, respectively, by converting these compounds into oxygen and hydrogen peroxide (which is later converted to water), resulting in the production of benign molecules. However, this conversion is not 100% efficient, and residual peroxides persist in the cell. While ROS are produced as a product of normal cellular functioning, excessive amounts can cause deleterious effects.
Impairment of cognitive function.
Memory capabilities decline with age, evident in human degenerative diseases such as Alzheimer's disease, which is accompanied by an accumulation of oxidative damage. Current studies demonstrate that the accumulation of ROS can decrease an organism's fitness because oxidative damage is a contributor to senescence. In particular, the accumulation of oxidative damage may lead to cognitive dysfunction, as demonstrated in a study in which old rats were given mitochondrial metabolites and then given cognitive tests. Results showed that the rats performed better after receiving the metabolites, suggesting that the metabolites reduced oxidative damage and improved mitochondrial function. Accumulating oxidative damage can then affect the efficiency of mitochondria and further increase the rate of ROS production. The accumulation of oxidative damage and its implications for aging depends on the particular tissue type where the damage is occurring. Additional experimental results suggest that oxidative damage is responsible for age-related decline in brain functioning. Older gerbils were found to have higher levels of oxidized protein in comparison to younger gerbils. Treatment of old and young mice with a spin trapping compound caused a decrease in the level of oxidized proteins in older gerbils but did not have an effect on younger gerbils. In addition, older gerbils performed cognitive tasks better during treatment but ceased functional capacity when treatment was discontinued, causing oxidized protein levels to increase. This led researchers to conclude that oxidation of cellular proteins is potentially important for brain function.
Cause of aging.
According to the free radical theory of aging, oxidative damage initiated by reactive oxygen species is a major contributor to the functional decline that is characteristic of aging. While studies in invertebrate models indicate that animals genetically engineered to lack specific antioxidant enzymes (such as SOD), in general, show a shortened lifespan (as one would expect from the theory), the converse manipulation, increasing the levels of antioxidant enzymes, has yielded inconsistent effects on lifespan (though some studies in "Drosophila" do show that lifespan can be increased by the overexpression of MnSOD or glutathione biosynthesizing enzymes). Also contrary to this theory, deletion of mitochondrial SOD2 can extend lifespan in "Caenorhabditis elegans".
In mice, the story is somewhat similar. Deleting antioxidant enzymes, in general, yields shorter lifespan, although overexpression studies have not (with some exceptions) consistently extended lifespan. Study of a rat model of premature aging found increased oxidative stress, reduced antioxidant enzyme activity and substantially greater DNA damage in the brain neocortex and hippocampus of the prematurely aged rats than in normally aging control rats. The DNA damage 8-OHdG is a product of ROS interaction with DNA. Numerous studies have shown that 8-OHdG increases with age (see DNA damage theory of aging).
Cancer.
ROS are constantly generated and eliminated in the biological system and are required to drive regulatory pathways. Under normal physiological conditions, cells control ROS levels by balancing the generation of ROS with their elimination by scavenging systems. But under oxidative stress conditions, excessive ROS can damage cellular proteins, lipids and DNA, leading to fatal lesions in the cell that contribute to carcinogenesis.
Cancer cells exhibit greater ROS stress than normal cells do, partly due to oncogenic stimulation, increased metabolic activity and mitochondrial malfunction. ROS is a double-edged sword. On one hand, at low levels, ROS facilitates cancer cell survival since cell-cycle progression driven by growth factors and receptor tyrosine kinases (RTK) require ROS for activation and chronic inflammation, a major mediator of cancer, is regulated by ROS. On the other hand, a high level of ROS can suppress tumor growth through the sustained activation of cell-cycle inhibitor and induction of cell death as well as senescence by damaging macromolecules. In fact, most of the chemotherapeutic and radiotherapeutic agents kill cancer cells by augmenting ROS stress. The ability of cancer cells to distinguish between ROS as a survival or apoptotic signal is controlled by the dosage, duration, type, and site of ROS production. Modest levels of ROS are required for cancer cells to survive, whereas excessive levels kill them.
Metabolic adaptation in tumours balances the cells' need for energy with equally important need for macromolecular building blocks and tighter control of redox balance. As a result, production of NADPH is greatly enhanced, which functions as a cofactor to provide reducing power in many enzymatic reactions for macromolecular biosynthesis and at the same time rescuing the cells from excessive ROS produced during rapid proliferation. Cells counterbalance the detrimental effects of ROS by producing antioxidant molecules, such as reduced glutathione (GSH) and thioredoxin (TRX), which rely on the reducing power of NADPH to maintain their activities.
Most risk factors associated with cancer interact with cells through the generation of ROS. ROS then activate various transcription factors such as nuclear factor kappa-light-chain-enhancer of activated B cells (NF-κB), activator protein-1 (AP-1), hypoxia-inducible factor-1α and signal transducer and activator of transcription 3 (STAT3), leading to expression of proteins that control inflammation; cellular transformation; tumor cell survival; tumor cell proliferation; and invasion, angiogenesis as well as metastasis. And ROS also control the expression of various tumor suppressor genes such as p53, retinoblastoma gene (Rb), and phosphatase and tensin homolog (PTEN).
Carcinogenesis.
ROS-related oxidation of DNA is one of the main causes of mutations, which can produce several types of DNA damage, including non-bulky (8-oxoguanine and formamidopyrimidine) and bulky (cyclopurine and etheno adducts) base modifications, abasic sites, non-conventional single-strand breaks, protein-DNA adducts, and intra/interstrand DNA crosslinks. It has been estimated that endogenous ROS produced via normal cell metabolism modify approximately 20,000 bases of DNA per day in a single cell. 8-oxoguanine is the most abundant among various oxidized nitrogeneous bases observed. During DNA replication, DNA polymerase mispairs 8-oxoguanine with adenine, leading to a G→T transversion mutation. The resulting genomic instability directly contributes to carcinogenesis. Cellular transformation leads to cancer and interaction of atypical PKC-ζ isoform with p47phox controls ROS production and transformation from apoptotic cancer stem cells through blebbishield emergency program.
Cell proliferation.
Uncontrolled proliferation is a hallmark of cancer cells. Both exogenous and endogenous ROS have been shown to enhance proliferation of cancer cells. The role of ROS in promoting tumor proliferation is further supported by the observation that agents with potential to inhibit ROS generation can also inhibit cancer cell proliferation. Although ROS can promote tumor cell proliferation, a great increase in ROS has been associated with reduced cancer cell proliferation by induction of G2/M cell cycle arrest; increased phosphorylation of ataxia telangiectasia mutated (ATM), checkpoint kinase 1 (Chk 1), Chk 2; and reduced cell division cycle 25 homolog c (CDC25).
Cell death.
A cancer cell can die in three ways: apoptosis, necrosis, and autophagy. Excessive ROS can induce apoptosis through both the extrinsic and intrinsic pathways. In the extrinsic pathway of apoptosis, ROS are generated by Fas ligand as an upstream event for Fas activation via phosphorylation, which is necessary for subsequent recruitment of Fas-associated protein with death domain and caspase 8 as well as apoptosis induction. In the intrinsic pathway, ROS function to facilitate cytochrome c release by activating pore-stabilizing proteins (Bcl-2 and Bcl-xL) as well as inhibiting pore-destabilizing proteins (Bcl-2-associated X protein, Bcl-2 homologous antagonist/killer). The intrinsic pathway is also known as the caspase cascade and is induced through mitochondrial damage which triggers the release of cytochrome c. DNA damage, oxidative stress, and loss of mitochondrial membrane potential lead to the release of the pro-apoptotic proteins mentioned above stimulating apoptosis. Mitochondrial damage is closely linked to apoptosis and since mitochondria are easily targeted there is potential for cancer therapy.
The cytotoxic nature of ROS is a driving force behind apoptosis, but in even higher amounts, ROS can result in both apoptosis and necrosis, a form of uncontrolled cell death, in cancer cells.
Numerous studies have shown the pathways and associations between ROS levels and apoptosis, but a newer line of study has connected ROS levels and autophagy. ROS can also induce cell death through autophagy, which is a self-catabolic process involving sequestration of cytoplasmic contents (exhausted or damaged organelles and protein aggregates) for degradation in lysosomes. Therefore, autophagy can also regulate the cell's health in times of oxidative stress. Autophagy can be induced by ROS levels through many pathways in the cell in an attempt to dispose of harmful organelles and prevent damage, such as carcinogens, without inducing apoptosis. Autophagic cell death can be prompted by the over expression of autophagy where the cell digests too much of itself in an attempt to minimize the damage and can no longer survive. When this type of cell death occurs, an increase or loss of control of autophagy regulating genes is commonly co-observed. Thus, once a more in-depth understanding of autophagic cell death is attained and its relation to ROS, this form of programmed cell death may serve as a future cancer therapy.
Autophagy and apoptosis are distinct mechanisms for cell death brought on by high levels of ROS. Aautophagy and apoptosis, however, rarely act through strictly independent pathways. There is a clear connection between ROS and autophagy and a correlation seen between excessive amounts of ROS leading to apoptosis. The depolarization of the mitochondrial membrane is also characteristic of the initiation of autophagy. When mitochondria are damaged and begin to release ROS, autophagy is initiated to dispose of the damaging organelle. If a drug targets mitochondria and creates ROS, autophagy may dispose of so many mitochondria and other damaged organelles that the cell is no longer viable. The extensive amount of ROS and mitochondrial damage may also signal for apoptosis. The balance of autophagy within the cell and the crosstalk between autophagy and apoptosis mediated by ROS is crucial for a cell's survival. This crosstalk and connection between autophagy and apoptosis could be a mechanism targeted by cancer therapies or used in combination therapies for highly resistant cancers.
Tumor cell invasion, angiogenesis and metastasis.
After growth factor stimulation of RTKs, ROS can trigger activation of signaling pathways involved in cell migration and invasion such as members of the mitogen activated protein kinase (MAPK) family – extracellular regulated kinase (ERK), c-jun NH-2 terminal kinase (JNK) and p38 MAPK. ROS can also promote migration by augmenting phosphorylation of the focal adhesion kinase (FAK) p130Cas and paxilin.
Both in vitro and in vivo, ROS have been shown to induce transcription factors and modulate signaling molecules involved in angiogenesis (MMP, VEGF) and metastasis (upregulation of AP-1, CXCR4, AKT and downregulation of PTEN).
Chronic inflammation and cancer.
Experimental and epidemiologic research over the past several years has indicated close associations among ROS, chronic inflammation, and cancer. ROS induces chronic inflammation by the induction of COX-2, inflammatory cytokines (TNFα, interleukin 1 (IL-1), IL-6), chemokines (IL-8, CXCR4) and pro-inflammatory transcription factors (NF-κB). These chemokines and chemokine receptors, in turn, promote invasion and metastasis of various tumor types.
Cancer therapy.
Both ROS-elevating and ROS-eliminating strategies have been developed with the former being predominantly used. Cancer cells with elevated ROS levels depend heavily on the antioxidant defense system. ROS-elevating drugs further increase cellular ROS stress level, either by direct ROS-generation (e.g. motexafin gadolinium, elesclomol) or by agents that abrogate the inherent antioxidant system such as SOD inhibitor (e.g. ATN-224, 2-methoxyestradiol) and GSH inhibitor (e.g. PEITC, buthionine sulfoximine (BSO)). The result is an overall increase in endogenous ROS, which when above a cellular tolerability threshold, may induce cell death. On the other hand, normal cells appear to have, under lower basal stress and reserve, a higher capacity to cope with additional ROS-generating insults than cancer cells do. Therefore, the elevation of ROS in all cells can be used to achieve the selective killing of cancer cells.
Radiotherapy also relies on ROS toxicity to eradicate tumor cells. Radiotherapy uses X-rays, γ-rays as well as heavy particle radiation such as protons and neutrons to induce ROS-mediated cell death and mitotic failure.
Due to the dual role of ROS, both prooxidant and antioxidant-based anticancer agents have been developed. However, modulation of ROS signaling alone seems not to be an ideal approach due to adaptation of cancer cells to ROS stress, redundant pathways for supporting cancer growth and toxicity from ROS-generating anticancer drugs. Combinations of ROS-generating drugs with pharmaceuticals that can break the redox adaptation could be a better strategy for enhancing cancer cell cytotoxicity.
James Watson and others have proposed that lack of intracellular ROS due to a lack of physical exercise may contribute to the malignant progression of cancer, because spikes of ROS are needed to correctly fold proteins in the endoplasmatic reticulum and low ROS levels may thus aspecifically hamper the formation of tumor suppressor proteins. Since physical exercise induces temporary spikes of ROS, this may explain why physical exercise is beneficial for cancer patient prognosis. Moreover, high inducers of ROS such as 2-deoxy-D-glucose and carbohydrate-based inducers of cellular stress induce cancer cell death more potently because they exploit the cancer cell's high avidity for sugars.
Positive role of ROS in memory.
ROS are critical in memory formation. ROS also have a central role in epigenetic DNA demethylation, which is relevant to learning and memory
In mammalian nuclear DNA, a methyl group can be added, by a DNA methyltransferase, to the 5th carbon of cytosine to form 5mC (see red methyl group added to form 5mC near the top of the first figure). The DNA methyltransferases most often form 5mC within the dinucleotide sequence "cytosine-phosphate-guanine" to form 5mCpG. This addition is a major type of epigenetic alteration and it can silence gene expression. Methylated cytosine can also be demethylated, an epigenetic alteration that can increase the expression of a gene. A major enzyme involved in demethylating 5mCpG is TET1. However, TET1 is only able to act on 5mCpG if an ROS has first acted on the guanine to form 8-hydroxy-2'-deoxyguanosine (8-OHdG), resulting in a 5mCp-8-OHdG dinucleotide . However, TET1 is only able to act on the 5mC part of the dinucleotide when the base excision repair enzyme OGG1 binds to the 8-OHdG lesion without immediate excision. Adherence of OGG1 to the 5mCp-8-OHdG site recruits TET1 and TET1 then oxidizes the 5mC adjacent to 8-OHdG, as shown in the first figure, initiating a demethylation pathway shown in the second figure.
The thousands of CpG sites being demethylated during memory formation depend on ROS in an initial step. The altered protein expression in neurons, controlled in part by ROS-dependent demethylation of CpG sites in gene promoters within neuron DNA, are central to memory formation.
References.
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Further reading.
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|
[
{
"math_id": 0,
"text": "\\begin{align}\n& \\ce{M}^{(n+1)+} + \\ce{O2- ->[SOD]}\\ \\ce{M}^{n+} + \\ce{O2} \\\\\n& \\ce{M}^{n+} + \\ce{O2- + 2H+ ->[][SOD]}\\ \\ce{M}^{(n+1)+} + \\ce{H2O2}\n\\end{align}"
},
{
"math_id": 1,
"text": "\\begin{align}\n& \\ce{2 H2O2 ->[\\text{catalase}] 2 H2O{} + O2} \\\\\n& \\ce{2GSH{} + H2O2 ->[][\\text{glutathione} \\atop \\text{peroxidase}] GS-SG{} + 2H2O} \n\\end{align}"
}
] |
https://en.wikipedia.org/wiki?curid=640697
|
640746
|
Secant method
|
Root-finding method
In numerical analysis, the secant method is a root-finding algorithm that uses a succession of roots of secant lines to better approximate a root of a function "f". The secant method can be thought of as a finite-difference approximation of Newton's method. However, the secant method predates Newton's method by over 3000 years.
The method.
For finding a zero of a function f, the secant method is defined by the recurrence relation.
formula_0
As can be seen from this formula, two initial values "x"0 and "x"1 are required. Ideally, they should be chosen close to the desired zero.
Derivation of the method.
Starting with initial values "x"0 and "x"1, we construct a line through the points ("x"0, "f"("x"0)) and ("x"1, "f"("x"1)), as shown in the picture above. In slope–intercept form, the equation of this line is
formula_1
The root of this linear function, that is the value of x such that "y"
0 is
formula_2
We then use this new value of x as "x"2 and repeat the process, using "x"1 and "x"2 instead of "x"0 and "x"1. We continue this process, solving for "x"3, "x"4, etc., until we reach a sufficiently high level of precision (a sufficiently small difference between "x""n" and "x""n"−1):
formula_3
Convergence.
The iterates formula_4 of the secant method converge to a root of formula_5 if the initial values formula_6 and formula_7 are sufficiently close to the root. The order of convergence is formula_8, where
formula_9
is the golden ratio. In particular, the convergence is super linear, but not quite quadratic.
This result only holds under some technical conditions, namely that formula_5 be twice continuously differentiable and the root in question be simple (i.e., with multiplicity 1).
If the initial values are not close enough to the root, then there is no guarantee that the secant method converges. There is no general definition of "close enough", but the criterion has to do with how "wiggly" the function is on the interval formula_10. For example, if formula_5 is differentiable on that interval and there is a point where formula_11 on the interval, then the algorithm may not converge.
Comparison with other root-finding methods.
The secant method does not require that the root remain bracketed, like the bisection method does, and hence it does not always converge. The false position method (or ) uses the same formula as the secant method. However, it does not apply the formula on formula_12 and formula_13, like the secant method, but on formula_12 and on the last iterate formula_14 such that formula_15 and formula_16 have a different sign. This means that the false position method always converges; however, only with a linear order of convergence. Bracketing with a super-linear order of convergence as the secant method can be attained with improvements to the false position method (see Regula falsi § Improvements in "regula falsi") such as the ITP method or Illinois method.
The recurrence formula of the secant method can be derived from the formula for Newton's method
formula_17
by using the finite-difference approximation, for a small formula_18:
formula_19
The secant method can be interpreted as a method in which the derivative is replaced by an approximation and is thus a quasi-Newton method.
If we compare Newton's method with the secant method, we see that Newton's method converges faster (order 2 against "φ" ≈ 1.6). However, Newton's method requires the evaluation of both formula_5 and its derivative formula_20 at every step, while the secant method only requires the evaluation of formula_5. Therefore, the secant method may occasionally be faster in practice. For instance, if we assume that evaluating formula_5 takes as much time as evaluating its derivative and we neglect all other costs, we can do two steps of the secant method (decreasing the logarithm of the error by a factor "φ"2 ≈ 2.6) for the same cost as one step of Newton's method (decreasing the logarithm of the error by a factor 2), so the secant method is faster. If, however, we consider parallel processing for the evaluation of the derivative, Newton's method proves its worth, being faster in time, though still spending more steps.
Generalization.
Broyden's method is a generalization of the secant method to more than one dimension.
The following graph shows the function "f" in red and the last secant line in bold blue. In the graph, the "x" intercept of the secant line seems to be a good approximation of the root of "f".
Computational example.
Below, the secant method is implemented in the Python programming language.
It is then applied to find a root of the function "f"("x")
"x"2 − 612 with initial points formula_21 and formula_22
def secant_method(f, x0, x1, iterations):
"""Return the root calculated using the secant method."""
for i in range(iterations):
x2 = x1 - f(x1) * (x1 - x0) / float(f(x1) - f(x0))
x0, x1 = x1, x2
# Apply a stopping criterion here (see below)
return x2
def f_example(x):
return x ** 2 - 612
root = secant_method(f_example, 10, 30, 5)
print(f"Root: {root}") # Root: 24.738633748750722
It is very important to have a good stopping criterion above, otherwise, due to limited numerical precision of floating point numbers, the algorithm can return inaccurate results if running for too many iterations. For example, the loop above can stop when one of these is reached first: abs(x0 - x1) < tol, or abs(x0/x1-1) < tol, or abs(f(x1)) < tol.
Notes.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\nx_n\n = x_{n-1} - f(x_{n-1}) \\frac{x_{n-1} - x_{n-2}}{f(x_{n-1}) - f(x_{n-2})}\n = \\frac{x_{n-2} f(x_{n-1}) - x_{n-1} f(x_{n-2})}{f(x_{n-1}) - f(x_{n-2})}.\n"
},
{
"math_id": 1,
"text": "y = \\frac{f(x_1) - f(x_0)}{x_1 - x_0}(x - x_1) + f(x_1)."
},
{
"math_id": 2,
"text": "x = x_1 - f(x_1) \\frac{x_1 - x_0}{f(x_1) - f(x_0)}."
},
{
"math_id": 3,
"text": "\n\\begin{align}\nx_2 & = x_1 - f(x_1) \\frac{x_1 - x_0}{f(x_1) - f(x_0)}, \\\\[6pt]\nx_3 & = x_2 - f(x_2) \\frac{x_2 - x_1}{f(x_2) - f(x_1)}, \\\\[6pt]\n& \\,\\,\\,\\vdots \\\\[6pt]\nx_n & = x_{n-1} - f(x_{n-1}) \\frac{x_{n-1} - x_{n-2}}{f(x_{n-1}) - f(x_{n-2})}.\n\\end{align}\n"
},
{
"math_id": 4,
"text": "x_n"
},
{
"math_id": 5,
"text": "f"
},
{
"math_id": 6,
"text": "x_0"
},
{
"math_id": 7,
"text": "x_1"
},
{
"math_id": 8,
"text": "\\varphi"
},
{
"math_id": 9,
"text": "\\varphi = \\frac{1+\\sqrt{5}}{2} \\approx 1.618"
},
{
"math_id": 10,
"text": "[x_0, x_1]"
},
{
"math_id": 11,
"text": "f' = 0"
},
{
"math_id": 12,
"text": "x_{n-1}"
},
{
"math_id": 13,
"text": "x_{n-2}"
},
{
"math_id": 14,
"text": "x_k"
},
{
"math_id": 15,
"text": "f(x_k)"
},
{
"math_id": 16,
"text": "f(x_{n-1})"
},
{
"math_id": 17,
"text": "x_n = x_{n-1} - \\frac{f(x_{n-1})}{f'(x_{n-1})}"
},
{
"math_id": 18,
"text": "\\epsilon"
},
{
"math_id": 19,
"text": "f'(x_{n-1}) \\approx \\frac{f(x_{n-1}) - f(x_{n-2})}{x_{n-1} - x_{n-2}} \\approx {\\frac {f(x_{n-1}+{\\frac {\\epsilon }{2}})-f(x_{n-1}-{\\frac {\\epsilon }{2}})}{\\epsilon }}"
},
{
"math_id": 20,
"text": "f'"
},
{
"math_id": 21,
"text": "x_0 = 10"
},
{
"math_id": 22,
"text": "x_1 = 30"
}
] |
https://en.wikipedia.org/wiki?curid=640746
|
640846
|
Coefficient of relationship
|
Mathematical guess about inbreeding
The coefficient of relationship is a measure of the degree of consanguinity (or biological relationship) between two individuals. The term coefficient of relationship was defined by Sewall Wright in 1922, and was derived from his definition of the coefficient of inbreeding of 1921. The measure is most commonly used in genetics and genealogy. A coefficient of inbreeding can be calculated for an individual, and is typically one-half the coefficient of relationship between the parents.
In general, the higher the level of inbreeding the closer the coefficient of relationship between the parents approaches a value of 1, expressed as a percentage, and approaches a value of 0 for individuals with arbitrarily remote common ancestors.
Coefficient of relationship.
The coefficient of relationship (formula_0) between two individuals B and C is obtained by a summation of coefficients calculated for every line by which they are connected to their common ancestors. Each such line connects the two individuals via a common ancestor, passing through no individual which is not a common ancestor more than once. A path coefficient between an ancestor A and an offspring O separated by formula_1 generations is given as:
formula_2
where formula_3 and formula_4 are the coefficients of inbreeding for A and O, respectively.
The coefficient of relationship formula_5is now obtained by summing over all path coefficients:
formula_6
By assuming that the pedigree can be traced back to a sufficiently remote population of perfectly random-bred stock ("f"A = 0 for all "A" in the sum) the definition of "r" may be simplified to
formula_7
where "p" enumerates all paths connecting B and C with unique common ancestors (i.e. all paths terminate at a common ancestor and may not pass through a common ancestor to a common ancestor's ancestor), and "L(p)" is the length of the path "p".
To give an (artificial) example:
Assuming that two individuals share the same 32 ancestors of "n" = 5 generations ago, but do not have any common ancestors at four or fewer generations ago, their coefficient of relationship would be
formula_8, which for n = 5, is, formula_9, equal to 0.03125 or approximately 3%.
Individuals for which the same situation applies for their 1024 ancestors of ten generations ago would have a coefficient of "r" = 2−10 = 0.1%.
If follows that the value of "r" can be given to an accuracy of a few percent if the family tree of both individuals is known for a depth of five generations, and to an accuracy of a tenth of a percent if the known depth is at least ten generations. The contribution to "r" from common ancestors of 20 generations ago (corresponding to roughly 500 years in human genealogy, or the contribution from common descent from a medieval population) falls below one part-per-million.
Human relationships.
The coefficient of relationship is sometimes used to express degrees of kinship in numeric terms in human genealogy.
In human relationships, the value of the coefficient of relationship is usually calculated based on the knowledge of a full family tree extending to a comparatively small number of generations, perhaps of the order of three or four. As explained above, the value for the coefficient of relationship so calculated is thus a lower bound, with an actual value that may be up to a few percent higher. The value is accurate to within 1% if the full family tree of both individuals is known to a depth of seven generations.
A first-degree relative (FDR) is a person's parent (father or mother), full sibling (brother or sister) or offspring. It constitutes a category of family members that largely overlaps with the term nuclear family, but without spouses. If the persons are related by blood, the first degree relatives share approximately 50% of their genes. First-degree relatives are a common measure used to diagnose risks for common diseases by analyzing family history.
A second-degree relative (SDR) is someone who shares 25% of a person's genes. It includes uncles, aunts, nephews, nieces, grandparents, grandchildren, half-siblings, and double cousins.
Third-degree relatives are a segment of the extended family and includes first cousins, great-grandparents and great-grandchildren. Third-degree relatives are generally defined by the expected amount of genetic overlap that exists between two people, with the third-degree relatives of an individual sharing approximately 12.5% of their genes. The category includes great-grandparents, great-grandchildren, grand-uncles, grand-aunts, first cousins, half-uncles, half-aunts, half-nieces and half-nephews.
Most incest laws concern the relationships where "r" = 25% or higher, although many ignore the rare case of double first cousins. Some jurisdictions also prohibit sexual relations or marriage between cousins of various degree, or individuals related only through adoption or affinity. Whether there is any likelihood of conception is generally considered irrelevant.
Kinship coefficient.
The kinship coefficient is a simple measure of relatedness, defined as the probability that a pair of randomly sampled homologous alleles are identical by descent. More simply, it is the probability that an allele selected randomly from an individual, i, and an allele selected at the same autosomal locus from another individual, j, are identical and from the same ancestor.
The coefficient of relatedness is equal to twice the kinship coefficient.
Calculation.
The kinship coefficient between two individuals, i and j, is represented as Φij. The kinship coefficient between a non-inbred individual and itself, Φii, is equal to 1/2. This is due to the fact that humans are diploid, meaning the only way for the randomly chosen alleles to be identical by descent is if the same allele is chosen twice (probability 1/2). Similarly, the relationship between a parent and a child is found by the chance that the randomly picked allele in the child is from the parent (probability 1/2) and the probability of the allele that is picked from the parent being the same one passed to the child (probability 1/2). Since these two events are independent of each other, they are multiplied Φij = 1/2 X 1/2 = 1/4.
See also.
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Notes.
<templatestyles src="Reflist/styles.css" />
References.
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|
[
{
"math_id": 0,
"text": " r "
},
{
"math_id": 1,
"text": " n "
},
{
"math_id": 2,
"text": "p_{AO} = 2^{-n} \\cdot \\sqrt{\\frac{1 + f_A}{1 + f_O}}"
},
{
"math_id": 3,
"text": " f_A "
},
{
"math_id": 4,
"text": " f_O "
},
{
"math_id": 5,
"text": " r_{BC} "
},
{
"math_id": 6,
"text": "r_{BC} = \\sum{p_{AB} \\cdot p_{AC}}"
},
{
"math_id": 7,
"text": "r_{BC} = \\sum_{p}{2^{-L(p)}}"
},
{
"math_id": 8,
"text": "r = 2^n \\cdot 2^{-2n} = 2^{-n}"
},
{
"math_id": 9,
"text": "2^{-5} = \\frac{1}{32}"
}
] |
https://en.wikipedia.org/wiki?curid=640846
|
640949
|
Continuously variable slope delta modulation
|
Continuously variable slope delta modulation (CVSD or CVSDM) is a voice coding method. It is a delta modulation with variable step size (i.e., special case of adaptive delta modulation), first proposed by Greefkes and Riemens in 1970.
CVSD encodes at 1 bit per sample, so that audio sampled at 16 kHz is encoded at 16 kbit/s.
The encoder maintains a reference sample and a step size. Each input sample is compared to the reference sample. If the input sample is larger, the encoder emits a "1" bit and adds the step size to the reference sample. If the input sample is smaller, the encoder emits a "0" bit and subtracts the step size from the reference sample. The encoder also keeps the previous "N" bits of output ("N" = 3 or "N" = 4 are very common) to determine adjustments to the step size; if the previous "N" bits are all 1s or 0s, the step size is increased. Otherwise, the step size is decreased (usually in an exponential manner, with formula_0 being in the range of 5 ms). The step size is adjusted for every input sample processed.
To allow for bit errors to fade out and to allow (re)synchronization to an ongoing bitstream, the output register (which keeps the reference sample) is normally realized as a leaky integrator with a time constant (formula_0) of about 1 ms.
The decoder reverses this process, starting with the reference sample, and adding or subtracting the step size according to the bit stream. The sequence of adjusted reference samples are the reconstructed waveform, and the step size is adjusted according to the same all-1s-or-0s logic as in the encoder.
Adaptation of step size allows one to avoid slope overload (step of quantization increases when the signal rapidly changes) and decreases granular noise when the signal is constant (decrease of step of quantisation).
CVSD is sometimes called a compromise between simplicity, low bitrate, and quality. Common bitrates are 9.6–128 kbit/s.
Like other delta-modulation techniques, the output of the decoder does not exactly match the original input to the encoder.
Applications.
12 kbit/s CVSD is used by Motorola's SECURENET line of digitally encrypted two-way radio products.
16 and 32 kbit/s CVSD were used by military TRI-TAC digital telephones (DNVT, DSVT) for use in deployed areas to provide voice recognition quality audio. 16 kbit/s rates were typically used by US Army forces to conserve bandwidth over tactical links. 32 kbit/s rates were typically used by US Air Force forces for improved voice quality.
64 kbit/s CVSD is one of the options to encode voice signals in telephony-related Bluetooth service profiles; e.g., between mobile phones and wireless headsets. The other options are PCM with logarithmic a-law or μ-law quantization, as well as mSBC codec featuring 16 kHz sample rate and best quality.
Numerous arcade games, such as "Sinistar" and "Smash TV", and pinball machines, such as "Gorgar" or "Space Shuttle", play pre-recorded speech through an HC-55516 CVSD decoder.
SBS application 24 kbit/s delta modulation.
Delta modulation was used by Satellite Business Systems or SBS for its voice ports to provide long-distance phone service to large domestic corporations with a significant inter-corporation communications need (such as IBM). This system was in service throughout the 1980s. The voice ports used "digitally implemented 24 kbit/s delta modulation" with voice activity compression (VAC) and echo suppressors to control the half second echo path through the satellite. Listening tests were conducted to verify that the "24 kbit/s Delta Modulator" achieved "full voice quality" with no discernible degradation as compared to a high quality phone line or the standard 64 kbit/s μ-law companded PCM. This provided an 8:3 improvement in satellite channel capacity. IBM developed the Satellite Communications Controller and the voice port functions.
The original proposal in 1974 used a state-of-the-art 24 kbit/s Delta Modulator with a single integrator and a Shindler compander modified for gain error recovery. This proved to have less than full phone line speech quality. In 1977, one engineer with two assistants in the IBM Research Triangle Park, NC laboratory was assigned to improve the quality.
The final implementation replaced the integrator with a predictor implemented with a two-pole complex-pair low-pass filter designed to approximate the long-term average speech spectrum. The theory was that ideally the integrator should be a predictor designed to match the signal spectrum. A nearly perfect Shindler compander replaced the modified version. It was found the modified compander resulted in a less than perfect step size at most signal levels and the fast gain error recovery increased the noise as determined by actual listening tests as compared to simple signal to noise measurements. The final compander achieved a very mild gain error recovery due to the natural truncation rounding error caused by 12-bit arithmetic.
The complete function of delta modulation, VAC, and echo control for 6 ports was implemented in a single digital integrated circuit chip with 12-bit arithmetic. A single DAC was shared by all 6 ports providing voltage compare functions for the modulators and feeding sample and hold circuits for the demodulator outputs. A single card held the chip, DAC, and all the analog circuits for the phone line interface including transformers.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\tau"
}
] |
https://en.wikipedia.org/wiki?curid=640949
|
64095544
|
Guided filter
|
Edge-preserving smoothing image filter
A guided filter is an edge-preserving smoothing image filter. As with a bilateral filter, it can filter out noise or texture while retaining sharp edges.
Comparison.
Compared to the bilateral filter, the guided image filter has two advantages: bilateral filters have high computational complexity, while the guided image filter uses simpler calculations with linear computational complexity. Bilateral filters sometimes include unwanted gradient reversal artifacts and cause image distortion. The guided image filter is based on linear combination, making the output image consistent with the gradient direction of the guidance image, preventing gradient reversal.
Definition.
One key assumption of the guided filter is that the relation between guidance formula_0 and the filtering output formula_1 is linear. Suppose that formula_1 is a linear transformation of formula_0 in a window formula_2 centered at the pixel formula_3.
In order to determine the linear coefficient formula_4, constraints from the filtering input formula_5 are required. The output formula_1 is modeled as the input formula_5 with unwanted components formula_6, such as noise/textures subtracted.
The basic model：
(1) formula_7
(2) formula_8
in which:
formula_9 is the formula_10 output pixel;
formula_11 is the formula_10 input pixel;
formula_12 is the formula_10 pixel of noise components;
formula_13 is the formula_10 guidance image pixel;
formula_4 are some linear coefficients assumed to be constant in formula_2.
The reason to use a linear combination is that the boundary of an object is related to its gradient. The local linear model ensures that formula_1 has an edge only if formula_0 has an edge, since formula_14.
Subtract (1) and (2) to get formula (3)；At the same time, define a cost function (4)：
(3) formula_15
(4) formula_16
in which
formula_17 is a regularization parameter penalizing large formula_18;
formula_19 is a window centered at the pixel formula_3.
And the cost function's solution is：
(5) formula_20
(6) formula_21
in which
formula_22 and formula_23 are the mean and variance of formula_0 in formula_19;
formula_24 is the number of pixels in formula_19;
formula_25 is the mean of formula_5 in formula_19.
After obtaining the linear coefficients formula_4, the filtering output formula_26 is provided by the following algorithm:
Algorithm.
By definition, the algorithm can be written as:
Algorithm 1. Guided Filter.
input： filtering input image formula_5 ，guidance image formula_0 ，window radius formula_27 ，regularization formula_17
output： filtering output formula_1
1.
formula_28 = formula_29
formula_30 = formula_31
formula_32 = formula_33
formula_34 = formula_35
2.
formula_36 = formula_37
formula_38 = formula_39
3.
formula_40 = formula_41
formula_42 = formula_43
4.
formula_44 = formula_45
formula_46 = formula_47
5.
formula_1 = formula_48
formula_49 is a mean filter with a wide variety of O(N) time methods.
Properties.
Edge-preserving filtering.
When the guidance image formula_0 is the same as the filtering input formula_5. The guided filter removes noise in the input image while preserving clear edges.
Specifically, a “flat patch” or a “high variance patch” can be specified by the parameter formula_17 of the guided filter. Patches with variance much lower than the parameter formula_17 will be smoothed, and those with variances much higher than formula_17 will be preserved. The role of the range variance formula_50 in the bilateral filter is similar to formula_17 in the guided filter. Both of them define the edge/high variance patches that should be kept and noise/flat patches that should be smoothed.”
Gradient-preserving filtering.
When using the bilateral filter to filter an image, artifacts may appear on the edges. This is because of the pixel value's abrupt change on the edge. These artifacts are inherent and hard to avoid, because edges appear in all kinds of pictures.
The guided filter performs better in avoiding gradient reversal. Moreover, in some cases, it can be ensured that gradient reversal does not occur.
Structure-transferring filtering.
Due to the local linear model of formula_51, it is possible to transfer the structure from the guidance formula_0 to the output formula_1. This property enables some special filtering-based applications, such as feathering, matting and dehazing.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "I"
},
{
"math_id": 1,
"text": "q"
},
{
"math_id": 2,
"text": "\\omega_k"
},
{
"math_id": 3,
"text": "k"
},
{
"math_id": 4,
"text": "(a_k, b_k)"
},
{
"math_id": 5,
"text": "p"
},
{
"math_id": 6,
"text": "n"
},
{
"math_id": 7,
"text": "q_i = a_k I_i + b_k, \\forall i \\in \\omega_k"
},
{
"math_id": 8,
"text": "q_{i} = p_{i} - n_{i}"
},
{
"math_id": 9,
"text": "q_{i}"
},
{
"math_id": 10,
"text": "i_{th}"
},
{
"math_id": 11,
"text": "p_i"
},
{
"math_id": 12,
"text": "n_{i}"
},
{
"math_id": 13,
"text": "I_i"
},
{
"math_id": 14,
"text": " \\nabla q = a \\nabla I"
},
{
"math_id": 15,
"text": "n_{i} = p_{i} - a_k I_{i} - b_k "
},
{
"math_id": 16,
"text": "E(a_{k},b_{k})=\\sum_{i{\\epsilon}{\\omega}_{k}}^{}((a_{k}I_{i} + b_{k} - p_{i})^{2} + {\\epsilon}a_{k}^{2})"
},
{
"math_id": 17,
"text": "\\epsilon"
},
{
"math_id": 18,
"text": "a_{k}"
},
{
"math_id": 19,
"text": "\\omega_{k}"
},
{
"math_id": 20,
"text": "a_{k} = \\frac{\\frac{1}{\\left|\\omega\\right|}\\sum_{i\\epsilon\\omega_{k}}I_{i}p_{i} - \\mu_{k}\\bar{p_{k}}}{\\sigma^{2}_{k}+\\epsilon}"
},
{
"math_id": 21,
"text": "b_{k} = \\bar{p_{k}} - a_{k}\\mu_{k}"
},
{
"math_id": 22,
"text": "\\mu_{k}"
},
{
"math_id": 23,
"text": "\\sigma^{2}_{k}"
},
{
"math_id": 24,
"text": "\\left|\\omega\\right|"
},
{
"math_id": 25,
"text": "\\bar{p}_{k} = \\frac{1}{\\left|\\omega\\right|}\\sum_{i\\epsilon\\omega_{k}}p_{i}"
},
{
"math_id": 26,
"text": "q_i"
},
{
"math_id": 27,
"text": "r"
},
{
"math_id": 28,
"text": "mean_{I}"
},
{
"math_id": 29,
"text": "f_{mean}(I)"
},
{
"math_id": 30,
"text": "mean_{p}"
},
{
"math_id": 31,
"text": "f_{mean}(p)"
},
{
"math_id": 32,
"text": "corr_{I}"
},
{
"math_id": 33,
"text": "f_{mean}(I.*I)"
},
{
"math_id": 34,
"text": "corr_{Ip}"
},
{
"math_id": 35,
"text": "f_{mean}(I.*p)"
},
{
"math_id": 36,
"text": "var_{I}"
},
{
"math_id": 37,
"text": "corr_{I} - mean_{I.} * mean_{I}"
},
{
"math_id": 38,
"text": "cov_{Ip}"
},
{
"math_id": 39,
"text": "corr_{Ip} - mean_{I.} * mean_{p}"
},
{
"math_id": 40,
"text": "a"
},
{
"math_id": 41,
"text": "cov_{Ip}./(var_{I} + \\epsilon)"
},
{
"math_id": 42,
"text": "b"
},
{
"math_id": 43,
"text": "mean_{p} - a. * mean_{I}"
},
{
"math_id": 44,
"text": "mean_{a}"
},
{
"math_id": 45,
"text": "f_{mean}(a)"
},
{
"math_id": 46,
"text": "mean_{b}"
},
{
"math_id": 47,
"text": "f_{mean}(b)"
},
{
"math_id": 48,
"text": "mean_{a.} * I + mean_{b}"
},
{
"math_id": 49,
"text": "f_{mean}"
},
{
"math_id": 50,
"text": "\\sigma_r^2"
},
{
"math_id": 51,
"text": "q = aI + b"
}
] |
https://en.wikipedia.org/wiki?curid=64095544
|
6409655
|
Maryam Mirzakhani
|
Iranian mathematician (1977–2017)
Maryam Mirzakhani (, ; 12 May 1977 – 14 July 2017) was an Iranian mathematician and a professor of mathematics at Stanford University. Her research topics included Teichmüller theory, hyperbolic geometry, ergodic theory, and symplectic geometry. On 13 August 2014, Mirzakhani was honored with the Fields Medal, the most prestigious award in mathematics, becoming the first woman to win the prize, as well as the first Iranian. The award committee cited her work in "the dynamics and geometry of Riemann surfaces and their moduli spaces".
Throughout her career, Maryam Mirzakhani achieved remarkable milestones that cemented her reputation as one of the most brilliant mathematicians of her time. After completing her PhD at Harvard University in 2004, she became a research fellow at the Clay Mathematics Institute and later joined Princeton University as a professor. In 2009, she moved to Stanford University, where she continued her pioneering research until her passing. Mirzakhani's work focused on the intricate and complex dynamics of geometric structures, with particular emphasis on moduli spaces and Riemann surfaces. Her innovative approaches and profound insights significantly advanced the field, earning her widespread acclaim and recognition, including the Fields Medal, the highest honor in mathematics.
Born and raised in Tehran, Mirzakhani's passion for mathematics began at a young age. She earned her undergraduate degree from Sharif University of Technology and went on to pursue her PhD at Harvard University under the mentorship of Fields Medalist Curtis T. McMullen. Her academic journey led her to positions at Princeton University and Stanford University, where she became a full professor in 2009. Despite her untimely death at the age of 40 due to breast cancer, her legacy endures through numerous accolades in her honor, including the Maryam Mirzakhani New Frontiers Prize and the 12 May Initiative, both dedicated to promoting women in mathematics.
Early life and education.
Mirzakhani was born on 12 May 1977 in Tehran, Iran. As a child, she attended Tehran Farzanegan School, part of the National Organization for Development of Exceptional Talents (NODET). In her junior and senior years of high school, she won the gold medal for mathematics in the Iranian National Olympiad, thus allowing her to bypass the national college entrance exam. In 1994, Mirzakhani became the first Iranian woman to win a gold medal at the International Mathematical Olympiad in Hong Kong, scoring 41 out of 42 points. The following year, in Toronto, she became the first Iranian to achieve the full score and to win two gold medals in the International Mathematical Olympiad. Later in her life, she collaborated with friend, colleague, and Olympiad silver medalist, Roya Beheshti Zavareh (), on their book "'Elementary Number Theory, Challenging Problems'," (in Persian) which was published in 1999. Mirzakhani and Zavareh together were the first women to compete in the Iranian National Mathematical Olympiad and won gold and silver medals in 1995, respectively.
On 17 March 1998, after attending a conference consisting of gifted individuals and former Olympiad competitors, Mirzakhani and Zavareh, along with other attendees, boarded a bus in Ahvaz en route to Tehran. The bus was involved in an accident wherein it fell off a cliff, killing seven of the passengers—all Sharif University students. This incident is widely considered a national tragedy in Iran. Mirzakhani and Zavareh were two of the few survivors.
In 1999, she obtained a Bachelor of Science in mathematics from the Sharif University of Technology. During her time there, she received recognition from the American Mathematical Society for her work in developing a simple proof of the theorem of Schur. She then went to the United States for graduate work, earning a PhD in 2004 from Harvard University, where she worked under the supervision of the Fields Medalist, Curtis T. McMullen. At Harvard, she is said to have been "distinguished by determination and relentless questioning". She used to take her class notes in her native language Persian.
Career.
Mirzakhani was a 2004 research fellow of the Clay Mathematics Institute and a professor at Princeton University. In 2009, she became a professor at Stanford University.
Research work.
Mirzakhani made several contributions to the theory of moduli spaces of Riemann surfaces. Mirzakhani's early work solved the problem of counting simple closed geodesics on hyperbolic Riemann surfaces by finding a relationship to volume calculations on moduli space. Geodesics are the natural generalization of the idea of a "straight line" to "curved spaces". Slightly more formally, a curve is a geodesic if no slight deformation can make it shorter. Closed geodesics are geodesics which are also closed curves—that is, they are curves that close up into loops. A closed geodesic is simple if it does not cross itself.
A previous result, known as the "prime number theorem for geodesics", established that the number of closed geodesics of length less than formula_0 grows exponentially with formula_0 – it is asymptotic to formula_1. However, the analogous counting problem for simple closed geodesics remained open, despite being "the key object to unlocking the structure and geometry of the whole surface," according to University of Chicago topologist Benson Farb. Mirzakhani's 2004 PhD thesis solved this problem, showing that the number of simple closed geodesics of length less than formula_0 is polynomial in formula_0. Explicitly, it is asymptotic to formula_2, where formula_3 is the genus (roughly, the number of "holes") and formula_4 is a constant depending on the hyperbolic structure. This result can be seen as a generalization of the theorem of the three geodesics for spherical surfaces.
Mirzakhani solved this counting problem by relating it to the problem of computing volumes in moduli space—a space whose points correspond to different complex structures on a surface genus formula_3. In her thesis, Mirzakhani found a volume formula for the moduli space of bordered Riemann surfaces of genus formula_3 with formula_5 geodesic boundary components. From this formula followed the counting for simple closed geodesics mentioned above, as well as a number of other results. This led her to obtain a new proof for the formula discovered by Edward Witten and Maxim Kontsevich on the intersection numbers of tautological classes on moduli space.
Her subsequent work focused on Teichmüller dynamics of moduli space. In particular, she was able to prove the long-standing conjecture that William Thurston's earthquake flow on Teichmüller space is ergodic. One can construct a simple earthquake map by cutting a surface along a finite number of disjoint simple closed geodesics, sliding the edges of each of these cut past each other by some amount, and closing the surface back up. One can imagine the surface being cut by strike-slip faults. An earthquake is a sort of limit of simple earthquakes, where one has an infinite number of geodesics, and instead of attaching a positive real number to each geodesic, one puts a measure on them.
In 2014, with Alex Eskin and with input from Amir Mohammadi, Mirzakhani proved that complex geodesics and their closures in moduli space are surprisingly regular, rather than irregular or fractal. The closures of complex geodesics are algebraic objects defined in terms of polynomials and therefore, they have certain rigidity properties, which is analogous to a celebrated result that Marina Ratner arrived at during the 1990s. The International Mathematical Union said in its press release that "It is astounding to find that the rigidity in homogeneous spaces has an echo in the inhomogeneous world of moduli space."
Awarding of Fields Medal.
Mirzakhani was awarded the Fields Medal in 2014 for "her outstanding contributions to the dynamics and geometry of Riemann surfaces and their moduli spaces". The award was made in Seoul at the International Congress of Mathematicians on 13 August. At the time of the award, Jordan Ellenberg explained her research to a popular audience:
<templatestyles src="Template:Blockquote/styles.css" />[Her] work expertly blends dynamics with geometry. Among other things, she studies billiards. But now, in a move very characteristic of modern mathematics, it gets kind of meta: She considers not just one billiard table, but the universe of all "possible" billiard tables. And the kind of dynamics she studies doesn't directly concern the motion of the billiards on the table, but instead a transformation of the billiard table itself, which is changing its shape in a rule-governed way; if you like, the table itself moves like a strange planet around the universe of all possible tables ... This isn't the kind of thing you do to win at pool, but it's the kind of thing you do to win a Fields Medal. And it's what you need to do in order to expose the dynamics at the heart of geometry; for there's no question that they're there.
In 2014, President Hassan Rouhani of Iran congratulated her for winning the award.
Personal life.
In 2008, Mirzakhani married Jan Vondrák, a Czech theoretical computer scientist and applied mathematician who currently is a professor at Stanford University. They had a daughter. Mirzakhani lived in Palo Alto, California.
Mirzakhani described herself as a "slow" mathematician, saying that "you have to spend some energy and effort to see the beauty of math." To solve problems, Mirzakhani would draw doodles on sheets of paper and write mathematical formulas around the drawings. Her daughter described her mother's work as "painting".
She declared:
<templatestyles src="Template:Blockquote/styles.css" />I don't have any particular recipe [for developing new proofs] ... It is like being lost in a jungle and trying to use all the knowledge that you can gather to come up with some new tricks, and with some luck, you might find a way out.
Death and legacy.
Mirzakhani was diagnosed with breast cancer in 2013. In 2016, the cancer spread to her bones and liver, and she died on 14 July 2017 at the age of 40 at Stanford Hospital in Stanford, California.
Iranian president Hassan Rouhani and other officials offered their condolences and praised Mirzakhani's scientific achievements. Rouhani said in his message that "the unprecedented brilliance of this creative scientist and modest human being, who made Iran's name resonate in the world's scientific forums, was a turning point in showing the great will of Iranian women and young people on the path towards reaching the peaks of glory and in various international arenas."
Upon her death, several Iranian newspapers, along with President Hassan Rouhani, broke taboo and published photographs of Mirzakhani with her hair uncovered. Although most newspapers used photographs with a dark background, digital manipulation, and even paintings to "hide" her hair, this gesture was widely noted in the western press and on social media. Mirzakhani's death has also renewed debates within Iran regarding matrilineal citizenship for children of mixed-nationality parentage; Fars News Agency reported that, subsequent to Mirzakhani's death, 60 Iranian MPs urged the speeding up of an amendment to a law that would allow children of Iranian mothers married to foreigners to be given Iranian nationality, in order to make it easier for Mirzakhani's daughter to visit Iran.
Numerous obituaries and tributes were published in the days following Mirzakhani's death. As a result of advocacy carried out by the Women's Committee within the Iranian Mathematical Society (), the International Council for Science agreed to declare Mirzakhani's birthday, 12 May, as International Women in Mathematics Day in respect of her memory.
Various establishments have also been named after Mirzakhani to honor her life and achievements. In 2017, Farzanegan High School – the high school Mirzakhani formerly attended – named their amphitheater and library after her. Additionally, Sharif University of Technology, the institute wherein Mirzakhani obtained her bachelor's, has since named their main library in the College of Mathematics after her. Further, the House of Mathematics in Isfahan, in collaboration with the mayor, named a conference hall in the city after her.
In 2014, students at the University of Oxford founded the Mirzakhani Society, a society for women and non-binary students studying mathematics at the University of Oxford. Mirzakhani met the society in September 2015, when she visited Oxford.
In 2022, following a £2.48m donation from XTX Markets, the University of Oxford launched the Maryam Mirzakhani Scholarships, which provide support for female mathematicians pursuing doctoral studies at the university.
In 2016, Mirzakhani was made a member of the National Academy of Sciences (of the United States), making her the first Iranian woman to be officially accepted as a member of the academy.
On 2 February 2018, Satellogic, a high-resolution Earth observation imaging and analytics company, launched a ÑuSat type micro-satellite named in honor of Mirzakhani.
On 4 November 2019, The Breakthrough Prize Foundation announced that the <templatestyles src="Template:Visible anchor/styles.css" />Maryam Mirzakhani New Frontiers Prize has been created to be awarded to outstanding women in the field of mathematics each year. The $50,000 award will be presented to early-career mathematicians who have completed their PhDs within the past two years.
In February 2020, on International Day of Women and Girls in STEM, Mirzakhani was honored by UN Women as one of seven female scientists dead or alive who have shaped the world.
In 2020, George Csicsery featured her in the documentary film "Secrets of the Surface: The Mathematical Vision of Maryam Mirzakhani".
The 12 May Initiative was created in Mirzakhani's honor to celebrate women in mathematics. The Initiative is coordinated by the European Women in Mathematics, Association for Women in Mathematics, African Women in Mathematics Association, Colectivo de Mujeres Matemáticas de Chile, and the Women's Committee of the Iranian Mathematical Society. In 2020, 152 events were held.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "L"
},
{
"math_id": 1,
"text": " e^L/L"
},
{
"math_id": 2,
"text": " cL^{6g-6}"
},
{
"math_id": 3,
"text": "g"
},
{
"math_id": 4,
"text": "c"
},
{
"math_id": 5,
"text": "n"
}
] |
https://en.wikipedia.org/wiki?curid=6409655
|
64099901
|
Continuous spontaneous localization model
|
Quantum mechanical theory of spontaneous collapse
The continuous spontaneous localization (CSL) model is a spontaneous collapse model in quantum mechanics, proposed in 1989 by Philip Pearle. and finalized in 1990 Gian Carlo Ghirardi, Philip Pearle and Alberto Rimini.
Introduction.
The most widely studied among the dynamical reduction (also known as collapse) models is the CSL model. Building on the Ghirardi-Rimini-Weber model, the CSL model describes the collapse of the wave function as occurring continuously in time, in contrast to the Ghirardi-Rimini-Weber model.
Some of the key features of the model are:
Dynamical equation.
The CSL dynamical equation for the wave function is stochastic and non-linear:formula_2Here formula_3 is the Hamiltonian describing the quantum mechanical dynamics, formula_4 is a reference mass taken equal to that of a nucleon, formula_5, and the noise field formula_6 has zero average and correlation equal toformula_7where formula_8 denotes the stochastic average over the noise. Finally, we writeformula_9where formula_10 is the mass density operator, which readsformula_11where formula_12 and formula_13 are, respectively, the second quantized creation and annihilation operators of a particle of type formula_14 with spin formula_15 at the point formula_16 of mass formula_17. The use of these operators satisfies the conservation of the symmetry properties of identical particles. Moreover, the mass proportionality implements automatically the amplification mechanism. The choice of the form of formula_10 ensures the collapse in the position basis.
The action of the CSL model is quantified by the values of the two phenomenological parameters formula_0 and formula_1. Originally, the Ghirardi-Rimini-Weber model proposed formula_18sformula_19 at formula_20m, while later Adler considered larger values: formula_21sformula_19 for formula_20m, and formula_22sformula_19 for formula_23m. Eventually, these values have to be bounded by experiments.
From the dynamics of the wave function one can obtain the corresponding master equation for the statistical operator formula_24:formula_25Once the master equation is represented in the position basis, it becomes clear that its direct action is to diagonalize the density matrix in position. For a single point-like particle of mass formula_26, it readsformula_27where the off-diagonal terms, which have formula_28, decay exponentially. Conversely, the diagonal terms, characterized by formula_29, are preserved. For a composite system, the single-particle collapse rate formula_0 should be replaced with that of the composite systemformula_30where formula_31 is the Fourier transform of the mass density of the system.
Experimental tests.
Contrary to most other proposed solutions of the measurement problem, collapse models are experimentally testable. Experiments testing the CSL model can be divided in two classes: interferometric and non-interferometric experiments, which respectively probe direct and indirect effects of the collapse mechanism.
Interferometric experiments.
Interferometric experiments can detect the direct action of the collapse, which is to localize the wavefunction in space. They include all experiments where a superposition is generated and, after some time, its interference pattern is probed. The action of CSL is a reduction of the interference contrast, which is quantified by the reduction of the off-diagonal terms of the statistical operatorformula_32where formula_33 denotes the statistical operator described by quantum mechanics, and we defineformula_34Experiments testing such a reduction of the interference contrast are carried out with cold-atoms, molecules and entangled diamonds.
Similarly, one can also quantify the minimum collapse strength to solve the measurement problem at the macroscopic level. Specifically, an estimate can be obtained by requiring that a superposition of a single-layered graphene disk of radius formula_35m collapses in less than formula_36s.
Non-interferometric experiments.
Non-interferometric experiments consist in CSL tests, which are not based on the preparation of a superposition. They exploit an indirect effect of the collapse, which consists in a Brownian motion induced by the interaction with the collapse noise. The effect of this noise amounts to an effective stochastic force acting on the system, and several experiments can be designed to quantify such a force. They include:
formula_39where formula_40 is the vacuum dielectric constant and formula_41 is the light speed. This prediction of CSL can be tested by analyzing the X-ray emission spectrum from a bulk Germanium test mass.
Dissipative and colored extensions.
The CSL model consistently describes the collapse mechanism as a dynamical process. It has, however, two weak points.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\lambda"
},
{
"math_id": 1,
"text": "r_C"
},
{
"math_id": 2,
"text": "\\operatorname{d}\\!|\\psi_t\\rangle=\\left[\t-\\frac i\\hbar \\hat H\\operatorname{d}\\! t+\\frac{\\sqrt{\\lambda}}{m_0}\\int \\operatorname{d}\\! {\\bf x}\\,\\hat N_t({\\bf x})\\operatorname{d}\\! W_t({\\bf x})\\right.\n\\left.-\\frac\\lambda{2m_0^2}\\int\\operatorname{d}\\!{\\bf x}\\int\\operatorname{d}\\!{\\bf y}\\,g({\\bf x}-{\\bf y})\\hat N_t({\\bf x})\\hat N_t({\\bf y})\\operatorname{d}\\! t\t\\right]|\\psi_t\\rangle."
},
{
"math_id": 3,
"text": "\\hat H"
},
{
"math_id": 4,
"text": "m_0"
},
{
"math_id": 5,
"text": "g({\\bf x}-{\\bf y})=e^{-{({\\bf x}-{\\bf y})^2}/{4r_C^2}}"
},
{
"math_id": 6,
"text": "w_t({\\bf x})=\\operatorname{d}\\! W_t({\\bf x})/\\operatorname{d}\\! t"
},
{
"math_id": 7,
"text": "\n\\mathbb E[w_t({\\bf x}) w_s({\\bf y})]=g({\\bf x}-{\\bf y})\\delta(t-s),\n"
},
{
"math_id": 8,
"text": "\\mathbb E [\\ \\cdot\\ ]"
},
{
"math_id": 9,
"text": "\\hat N_t({\\bf x})=\\hat M({\\bf x})-\\langle\\psi_t|\\hat M({\\bf x})|\\psi_t\\rangle,"
},
{
"math_id": 10,
"text": "\\hat M({\\bf x})"
},
{
"math_id": 11,
"text": "\n\\hat M({\\bf x})=\\sum_j m_j\\sum_s\\hat a^\\dagger_j({\\bf x},s)\\hat a_j({\\bf x},s),\n"
},
{
"math_id": 12,
"text": "\\hat a^\\dagger_j({\\bf y},s)"
},
{
"math_id": 13,
"text": "\\hat a_j({\\bf y},s)"
},
{
"math_id": 14,
"text": "j"
},
{
"math_id": 15,
"text": "s"
},
{
"math_id": 16,
"text": "{\\bf y}"
},
{
"math_id": 17,
"text": "m_j"
},
{
"math_id": 18,
"text": "\\lambda=10^{-17}\\,"
},
{
"math_id": 19,
"text": "^{-1}"
},
{
"math_id": 20,
"text": "r_C=10^{-7}\\,"
},
{
"math_id": 21,
"text": "\\lambda=10^{-8\\pm2}\\,"
},
{
"math_id": 22,
"text": "\\lambda=10^{-6\\pm2}\\,"
},
{
"math_id": 23,
"text": "r_C=10^{-6}\\,"
},
{
"math_id": 24,
"text": "\\hat \\rho_t"
},
{
"math_id": 25,
"text": "\n\\frac{\\operatorname{d}\\! \\hat\\rho_t}{\\operatorname{d}\\! t}\n=-\\frac{i}{\\hbar}\\left[{\\hat H},{\\hat \\rho_t}\\right]\n-\\frac{\\lambda}{2m_0^2}\\int\\operatorname{d}\\!{\\bf x}\\int\\operatorname{d}\\!{\\bf y}\\,g({\\bf x}-{\\bf y})\n\\left[{\\hat M({\\bf x})},\\left[{{\\hat M({\\bf y})},{\\hat \\rho_t}}\\right]\\right].\n"
},
{
"math_id": 26,
"text": "m"
},
{
"math_id": 27,
"text": "\n\\frac{\\partial \\langle{{\\bf x}|\\hat \\rho_t|{\\bf y}}\\rangle}{\\partial t}=-\\frac{i}{\\hbar}\\langle{{\\bf x}|\\left[{\\hat H},{\\hat \\rho_t}\\right]|{\\bf y}}\\rangle-\\lambda\\frac{m^2}{m_0^2}\\left(1-e^{-\\tfrac{({\\bf x}-{\\bf y})^2}{4r_C^2}}\\right)\\langle{{\\bf x}|\\hat \\rho_t|{\\bf y}}\\rangle,\n"
},
{
"math_id": 28,
"text": "{\\bf x}\\neq{\\bf y}"
},
{
"math_id": 29,
"text": "{\\bf x}={\\bf y}"
},
{
"math_id": 30,
"text": "\n\\lambda\\frac{m^2}{m_0^2}\\to\\lambda\\frac{r_C^3}{\\pi^{3/2}m_0^2}\\int\\operatorname{d}\\!{\\bf k}|\\tilde\\mu({\\bf k})|^2e^{-k^2r_C^2},\n"
},
{
"math_id": 31,
"text": "\\tilde \\mu(k)"
},
{
"math_id": 32,
"text": "\\rho(x,x',t) = \\frac{1}{2\\pi\\hbar} \\int_{-\\infty}^{+ \\infty} \\operatorname{d}\\! k \\int_{-\\infty}^{+ \\infty} \\operatorname{d}\\! w\\, e^{-ikw/\\hbar} F_{{ CSL}}(k, x-x',t) \\rho^{{ QM}}(x+w, x'+w, t),"
},
{
"math_id": 33,
"text": "\\rho^{{QM}}"
},
{
"math_id": 34,
"text": "F_{{ CSL}}(k,q,t)= \\exp\\bigg[-\\lambda \\frac{m^2}{m_0^2} t \\left(1-\\frac{1}{t}\\int_0^t \\operatorname{d}\\!\\tau \\,e^{-{(q-\\frac{k\\tau}{m})^2}/{4r_C^2}} \\right) \\bigg]."
},
{
"math_id": 35,
"text": "\\simeq 10^{-5}"
},
{
"math_id": 36,
"text": "\\simeq 10^{-2}"
},
{
"math_id": 37,
"text": "\\omega"
},
{
"math_id": 38,
"text": "Q"
},
{
"math_id": 39,
"text": "\n\\frac{\\operatorname{d}\\! \\Gamma(\\omega)}{\\operatorname{d}\\! \\omega}=\\frac{\\hbar Q^2\\lambda}{2\\pi^2\\epsilon_0c^3m_0^2r_C^2\\omega},\n"
},
{
"math_id": 40,
"text": "\\epsilon_0"
},
{
"math_id": 41,
"text": "c"
},
{
"math_id": 42,
"text": "E"
},
{
"math_id": 43,
"text": "\nE(t)=E(0)+\\frac{3m\\lambda\\hbar^{2}}{4m_{0}^{2}r_C^{2}}t,\n"
},
{
"math_id": 44,
"text": "E(0)"
},
{
"math_id": 45,
"text": "\\simeq 10^{-14}"
},
{
"math_id": 46,
"text": "\\lambda=10^{-16}"
},
{
"math_id": 47,
"text": "r_C=10^{-7}"
},
{
"math_id": 48,
"text": "\n\\langle{\\hat x^2}\\rangle_t=\\langle{\\hat x^2}\\rangle_t^{ (QM)}+\\frac{\\hbar^2\\eta t^3}{3 m^2},\n"
},
{
"math_id": 49,
"text": "\\langle{\\hat x^2}\\rangle_t^{ (QM)}"
},
{
"math_id": 50,
"text": "\\eta"
},
{
"math_id": 51,
"text": "\n\\eta=\\frac{\\lambda r_C^3}{2\\pi^{3/2}m_0^2}\\int\\operatorname{d}\\!{\\bf k}\\,e^{-{\\bf k}^2r_C^2}k_x^2|\\tilde \\mu({\\bf k})|^2,\n"
},
{
"math_id": 52,
"text": "x"
},
{
"math_id": 53,
"text": "\\tilde \\mu({\\bf k})"
},
{
"math_id": 54,
"text": "\\mu({\\bf r})"
},
{
"math_id": 55,
"text": "\\gamma"
},
{
"math_id": 56,
"text": "T"
},
{
"math_id": 57,
"text": "\\omega_0"
},
{
"math_id": 58,
"text": "\n\\langle{\\hat x^2}\\rangle_{ eq}=\\frac{k_B T}{m\\omega_0^2}+\\frac{ \\hbar^2 \\eta}{2m^2 \\omega_0^2 \\gamma },\n"
},
{
"math_id": 59,
"text": "\nk_B\n"
},
{
"math_id": 60,
"text": "T_{ CSL}"
},
{
"math_id": 61,
"text": "\nE(t)=e^{-\\beta t}(E(0)-E_{ as})+E_{ as},\n"
},
{
"math_id": 62,
"text": "\nE_{ as}=\\tfrac32 k_B T_{ CSL}"
},
{
"math_id": 63,
"text": "\\beta=4 \\chi \\lambda /(1+\\chi)^5"
},
{
"math_id": 64,
"text": "\\chi=\\hbar^2/(8 m_0 k_B T_{ CSL}r_C^2)"
},
{
"math_id": 65,
"text": "T_{ CSL}=1"
},
{
"math_id": 66,
"text": "T_{CSL}"
},
{
"math_id": 67,
"text": "w_t({\\bf x})"
},
{
"math_id": 68,
"text": "f(t)"
},
{
"math_id": 69,
"text": "F_{{ CSL}}(k,q,t)"
},
{
"math_id": 70,
"text": "\nF_{{cCSL}}(k,q,t)=F_{{ CSL}}(k,q,t) \\exp\\left[\t\\frac{\\lambda \\bar\\tau}{2}\\left(\te^{-(q-k t/m)^2/4r_C^2}-e^{-q^2/4r_C^2}\t\\right)\t\\right],\n"
},
{
"math_id": 71,
"text": " \\bar\\tau=\\int_0^t\\operatorname{d}\\! s\\,f(s)"
},
{
"math_id": 72,
"text": "f(t)=\\tfrac12\\Omega_{ C}e^{-\\Omega_{ C}|t|}"
},
{
"math_id": 73,
"text": "\\Omega_{C}"
},
{
"math_id": 74,
"text": "\\Omega_{ C}"
},
{
"math_id": 75,
"text": "\\Omega_{ C}=10^{12}\\,"
}
] |
https://en.wikipedia.org/wiki?curid=64099901
|
6410389
|
Logarithmic convolution
|
In mathematics, the scale convolution of two functions formula_0 and formula_1, also known as their logarithmic convolution is defined as the function
formula_2
when this quantity exists.
Results.
The logarithmic convolution can be related to the ordinary convolution by changing the variable from formula_3 to formula_4:
formula_5
Define formula_6 and formula_7 and let formula_4, then
formula_8
"This article incorporates material from logarithmic convolution on PlanetMath, which is licensed under the ."
|
[
{
"math_id": 0,
"text": "s(t)"
},
{
"math_id": 1,
"text": "r(t)"
},
{
"math_id": 2,
"text": " s *_l r(t) = r *_l s(t) = \\int_0^\\infty s\\left(\\frac{t}{a}\\right)r(a) \\, \\frac{da}{a}"
},
{
"math_id": 3,
"text": "t"
},
{
"math_id": 4,
"text": "v = \\log t"
},
{
"math_id": 5,
"text": "\\begin{align}\n s *_l r(t) & = \\int_0^\\infty s \\left(\\frac{t}{a}\\right)r(a) \\, \\frac{da}{a} \\\\\n& =\n\\int_{-\\infty}^\\infty s\\left(\\frac{t}{e^u}\\right) r(e^u) \\, du \\\\\n& = \\int_{-\\infty}^\\infty s \\left(e^{\\log t - u}\\right)r(e^u) \\, du.\n\\end{align}"
},
{
"math_id": 6,
"text": "f(v) = s(e^v)"
},
{
"math_id": 7,
"text": "g(v) = r(e^v)"
},
{
"math_id": 8,
"text": " s *_l r(v) = f * g(v) = g * f(v) = r *_l s(v). "
}
] |
https://en.wikipedia.org/wiki?curid=6410389
|
6410624
|
Geostrophic current
|
Oceanic flow in which the pressure gradient force is balanced by the Coriolis effect
A geostrophic current is an oceanic current in which the pressure gradient force is balanced by the Coriolis effect. The direction of geostrophic flow is parallel to the isobars, with the high pressure to the right of the flow in the Northern Hemisphere, and the high pressure to the left in the Southern Hemisphere. This concept is familiar from weather maps, whose isobars show the direction of geostrophic winds. Geostrophic flow may be either barotropic or baroclinic. A geostrophic current may also be thought of as a rotating shallow water wave with a frequency of zero.
The principle of "geostrophy" or "geostrophic balance" is useful to oceanographers because it allows them to infer ocean currents from measurements of the sea surface height (by combined satellite altimetry and gravimetry) or from vertical profiles of seawater density taken by ships or autonomous buoys. The major currents of the world's oceans, such as the Gulf Stream, the Kuroshio Current, the Agulhas Current, and the Antarctic Circumpolar Current, are all approximately in geostrophic balance and are examples of geostrophic currents.
Simple explanation.
Sea water naturally tends to move from a region of high pressure (or high sea level) to a region of low pressure (or low sea level). The force pushing the water towards the low pressure region is called the pressure gradient force. In a geostrophic flow, instead of water moving from a region of high pressure (or high sea level) to a region of low pressure (or low sea level), it moves along the lines of equal pressure (isobars). This occurs because the Earth is rotating. The rotation of the earth results in a "force" being felt by the water moving from the high to the low, known as Coriolis force. The Coriolis force acts at right angles to the flow, and when it balances the pressure gradient force, the resulting flow is known as geostrophic.
As stated above, the direction of flow is with the high pressure to the right of the flow in the Northern Hemisphere, and the high pressure to the left in the Southern Hemisphere. The direction of the flow depends on the hemisphere, because the direction of the Coriolis force is opposite in the different hemispheres.
Formulation.
The geostrophic equations are a simplified form of the Navier–Stokes equations in a rotating reference frame. In particular, it is assumed that there is no acceleration (steady-state), that there is no viscosity, and that the pressure is hydrostatic. The resulting balance is (Gill, 1982):
formula_0
formula_1
where formula_2 is the Coriolis parameter, formula_3 is the density, formula_4 is the pressure and formula_5 are the velocities in the formula_6-directions respectively.
One special property of the geostrophic equations, is that they satisfy the steady-state version of the continuity equation. That is:
formula_7
Rotating waves of zero frequency.
The equations governing a linear, rotating shallow water wave are:
formula_8
formula_9
The assumption of steady-state made above (no acceleration) is:
formula_10
Alternatively, we can assume a wave-like, periodic, dependence in time:
formula_11
In this case, if we set formula_12, we have reverted to the geostrophic equations above. Thus a geostrophic current
can be thought of as a rotating shallow water wave with a frequency of zero.
|
[
{
"math_id": 0,
"text": " fv = \\frac{1}{\\rho} \\frac{\\partial p}{\\partial x}"
},
{
"math_id": 1,
"text": " fu = -\\frac{1}{\\rho} \\frac{\\partial p}{\\partial y}"
},
{
"math_id": 2,
"text": "f"
},
{
"math_id": 3,
"text": "\\rho"
},
{
"math_id": 4,
"text": "p"
},
{
"math_id": 5,
"text": "u,v"
},
{
"math_id": 6,
"text": "x,y"
},
{
"math_id": 7,
"text": " \\frac{\\partial u}{\\partial x} + \\frac{\\partial v}{\\partial y} = 0 "
},
{
"math_id": 8,
"text": " \\frac{\\partial u}{\\partial t} - fv = -\\frac{1}{\\rho} \\frac{\\partial p}{\\partial x}"
},
{
"math_id": 9,
"text": " \\frac{\\partial v}{\\partial t} + fu = -\\frac{1}{\\rho} \\frac{\\partial p}{\\partial y}"
},
{
"math_id": 10,
"text": " \\frac{\\partial u}{\\partial t} = \\frac{\\partial v}{\\partial t} =0 "
},
{
"math_id": 11,
"text": " u \\propto v \\propto e^{i \\omega t} "
},
{
"math_id": 12,
"text": " \\omega = 0 "
}
] |
https://en.wikipedia.org/wiki?curid=6410624
|
64107061
|
Conway circle theorem
|
Geometrical construction based on extending the sides of a triangle
In plane geometry, the Conway circle theorem states that when the sides meeting at each vertex of a triangle are extended by the length of the opposite side, the six endpoints of the three resulting line segments lie on a circle whose centre is the incentre of the triangle. The circle on which these six points lie is called the Conway circle of the triangle. The theorem and circle are named after mathematician John Horton Conway.
Proof.
Let "I" be the center of the incircle of triangle "ABC", "r" its radius and "Fa", "Fb" and "Fc" the three points where the incircle touches the triangle sides "a", "b" and "c". Since the (extended) triangle sides are tangents of the incircle it follows that "IFa", "IFb" and "IFc" are perpendicular to "a", "b" and "c". Furthermore the following equalities for line segments hold. |AFc|=|AFb|, |BFc|=|BFa|, |CFa|=|CFb|. With that the six triangles "IFcPa", "IFcQb", "IFaPb", "IFaQc", "IFbQa" and "IFbPc" all have a side of length |"AFc"|+|"BFc"|+|"CFa"| and a side of length "r" with a right angle between them. This means that due SAS congruence theorem for triangles all six triangles are congruent, which yields |"IPa"|=|"IQa"|=|"IPb"|=|"IQb"|=|"IPc"|=|"IQc"|. So the six points "Pa", "Qa", "Pb", "Qb", "Pc" and "Qc" have all the same distance from the triangle incenter "I", that is they lie on a common circle with center "I".
Additional properties.
The radius of the Conway circle is
formula_0
where formula_1 and formula_2 are the inradius and semiperimeter of the triangle.
Generalisation.
Conway's circle is a special case of a more general circle for a triangle that can be obtained as follows: Given any △ABC with an arbitrary point P on line AB. Construct BQ = BP, CR = CQ, AS = AR, BT = BS, CU = CT. Then AU = AP, and PQRSTU is cyclic.
If you you place P on the extended triangle side AB such that BP=b and BP being completely outside the triangle then the above constructions yield Conway's circle theorem.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\sqrt{r^2 + s^2}=\\sqrt{\\frac{a^2b+ab^2+b^2c+bc^2+a^2c+ac^2+abc}{a+b+c}}"
},
{
"math_id": 1,
"text": "r"
},
{
"math_id": 2,
"text": "s"
}
] |
https://en.wikipedia.org/wiki?curid=64107061
|
641073
|
Vertical bar
|
Typographic symbol ()
The vertical bar, |, is a glyph with various uses in mathematics, computing, and typography. It has many names, often related to particular meanings: Sheffer stroke (in logic), pipe, bar, or (literally, the word "or"), vbar, and others.
Usage.
Mathematics.
The vertical bar is used as a mathematical symbol in numerous ways.
If used as a pair of brackets, it suggests the notion of the word "size". These are:
Likewise, the vertical bar is also used singly in many different ways:
The double vertical bar, formula_21, is also employed in mathematics.
In LaTeX mathematical mode, the ASCII vertical bar produces a vertical line, and codice_0 creates a double vertical line (codice_1 is set as formula_32). This has different spacing from codice_2 and codice_3, which are relational operators: codice_4 is set as formula_33. See below about LaTeX in text mode.
Chemistry.
In chemistry, the vertical line is used in cell notation of electrochemical cells.
Example,
Zn | Zn2+ || Cu2+ | Cu
Single vertical lines show components of the cell which do not mix, usually being in different phases. The double vertical line ( || ) is used to represent salt bridge; which is used to allow free moving ions to move.
Physics.
The vertical bar is used in bra–ket notation in quantum physics. Examples:
Computing.
Pipe.
A pipe is an inter-process communication mechanism originating in Unix, which directs the output (standard out and, optionally, standard error) of one process to the input (standard in) of another. In this way, a series of commands can be "piped" together, giving users the ability to quickly perform complex multi-stage processing from the command line or as part of a Unix shell script ("bash file"). In most Unix shells (command interpreters), this is represented by the vertical bar character. For example:
codice_5
where the output from the grep process (all lines containing 'blair') is piped to the more process (which allows a command line user to read through results one page at a time).
The same "pipe" feature is also found in later versions of DOS and Microsoft Windows.
This usage has led to the character itself being called "pipe".
Disjunction.
In many programming languages, the vertical bar is used to designate the logic operation "or", either bitwise "or" or logical "or".
Specifically, in C and other languages following C syntax conventions, such as C++, Perl, Java and C#, codice_6 denotes a bitwise "or"; whereas a double vertical bar codice_7 denotes a (short-circuited) logical "or". Since the character was originally not available in all code pages and keyboard layouts, ANSI C can transcribe it in form of the trigraph codice_8, which, outside string literals, is equivalent to the codice_9 character.
In regular expression syntax, the vertical bar again indicates logical "or" (alternation). For example: the Unix command codice_10 matches lines containing 'fu' or 'bar'.
Concatenation.
The double vertical bar operator "||" denotes string concatenation in PL/I, standard ANSI SQL, and theoretical computer science (particularly cryptography).
Delimiter.
Although not as common as commas or tabs, the vertical bar can be used as a delimiter in a flat file. Examples of a pipe-delimited standard data format are LEDES 1998B and HL7. It is frequently used because vertical bars are typically uncommon in the data itself.
Similarly, the vertical bar may see use as a delimiter for regular expression operations (e.g. in sed). This is useful when the regular expression contains instances of the more common forward slash (codice_11) delimiter; using a vertical bar eliminates the need to escape all instances of the forward slash. However, this makes the bar unusable as the regular expression "alternative" operator.
Backus–Naur form.
In Backus–Naur form, an expression consists of sequences of symbols and/or sequences separated by '|', indicating a choice, the whole being a possible substitution for the symbol on the left.
Concurrency operator.
In calculi of communicating processes (like pi-calculus), the vertical bar is used to indicate that processes execute in parallel.
APL.
The pipe in APL is the modulo or "residue" function between two operands and the absolute value function next to one operand.
List comprehensions.
The vertical bar is used for list comprehensions in some functional languages, e.g. Haskell and Erlang. Compare set-builder notation.
Text markup.
The vertical bar is used as a special character in lightweight markup languages, notably MediaWiki's Wikitext (in the templates and internal links).
In LaTeX text mode, the vertical bar produces an em dash (—). The codice_12 command can be used to produce a vertical bar.
Phonetics and orthography.
In the Khoisan languages and the International Phonetic Alphabet, the vertical bar is used to write the dental click (). A double vertical bar is used to write the alveolar lateral click (). Since these are technically letters, they have their own Unicode code points in the Latin Extended-B range: U+01C0 for the single bar and U+01C1 for the double bar.
Some Northwest and Northeast Caucasian languages written in the Cyrillic script have a vertical bar called palochka (Russian: , 'little stick'), indicating the preceding consonant is an ejective.
Longer single and double vertical bars are used to mark prosodic boundaries in the IPA.
Literature.
In medieval European manuscripts, a single vertical bar was a common variant of the virgula / used as a comma, or caesura mark.
In Sanskrit and other Indian languages, a single vertical mark, a danda, has a similar function as a period (full stop). Two bars || (a 'double danda') is the equivalent of a pilcrow in marking the end of a stanza, paragraph or section. The danda has its own Unicode code point, U+0964.
Poetry.
A double vertical bar ⟨ or ⟨ǁ⟩ is the standard caesura mark in English literary criticism and analysis. It marks the strong break or caesura common to many forms of poetry, particularly Old English verse. It is also traditionally used to mark the division between lines of verse printed as prose (the style preferred by Oxford University Press), though it is now often replaced by the forward slash.
Notation.
In the Geneva Bible and early printings of the King James Version, a double vertical bar is used to mark margin notes that contain an alternative translation from the original text. These margin notes always begin with the conjunction "Or". In later printings of the King James Version, the double vertical bar is irregularly used to mark any comment in the margins.
A double vertical bar symbol may be used to call out a footnote. (The traditional order of these symbols in English is *, †, ‡, §, ‖, ¶, so its use is very rare; in modern usage, numbers and letters are preferred for endnotes and footnotes.)
Music scoring.
In music, when writing chord sheets, single vertical bars associated with a colon (|: A / / / :|) represents the beginning and end of a section (e.g. Intro, Interlude, Verse, Chorus) of music. Single bars can also represent the beginning and end of measures (|: A / / / | D / / / | E / / / :|). A double vertical bar associated with a colon can represent the repeat of a given section (||: A / / / :|| - play twice).
Encoding.
Solid vertical bar versus broken bar.
Many early video terminals and dot-matrix printers rendered the vertical bar character as the allograph broken bar ¦. This may have been to distinguish the character from the lower-case 'L' and the upper-case 'I' on these limited-resolution devices, and to make a vertical line of them look more like a horizontal line of dashes. It was also (briefly) part of the ASCII standard.
An initial draft for a 7-bit character set that was published by the X3.2 subcommittee for Coded Character Sets and Data Format on June 8, 1961, was the first to include the vertical bar in a standard set. The bar was intended to be used as the representation for the logical OR symbol. A subsequent draft on May 12, 1966, places the vertical bar in column 7 alongside regional entry codepoints, and formed the basis for the original draft proposal used by the International Standards Organisation. This draft received opposition from the IBM user group SHARE, with its chairman, H. W. Nelson, writing a letter to the American Standards Association titled "The Proposed revised American Standard Code for Information Interchange does NOT meet the needs of computer programmers!"; in this letter, he argues that no characters within the international subset designated at columns 2-5 of the character set would be able to adequately represent logical OR and logical NOT in languages such as IBM's PL/I universally on all platforms. As a compromise, a requirement was introduced where the exclamation mark (!) and circumflex (^) would display as logical OR (|) and logical NOT (¬) respectively in use cases such as programming, while outside of these use cases they would represent their original typographic symbols:
The original vertical bar encoded at 0x7C in the original May 12, 1966 draft was then broken as ¦, so it could not be confused with the unbroken logical OR. In the 1967 revision of ASCII, along with the equivalent ISO 464 code published the same year, the code point was defined to be a broken vertical bar, and the exclamation mark character was allowed to be rendered as a solid vertical bar. However, the 1977 revision (ANSI X.3-1977) undid the changes made in the 1967 revision, enforcing that the circumflex could no longer be stylised as a logical NOT symbol, the exclamation mark likewise no longer allowing stylisation as a vertical bar, and defining the code point originally set to the broken bar as a solid vertical bar instead; the same changes were also reverted in ISO 646-1973 published four years prior.
Some variants of EBCDIC included both versions of the character as different code points. The broad implementation of the extended ASCII ISO/IEC 8859 series in the 1990s also made a distinction between the two forms. This was preserved in Unicode as a separate character at U+00A6 (the term "parted rule" is used sometimes in Unicode documentation). Some fonts draw the characters the same (both are solid vertical bars, or both are broken vertical bars).
Many keyboards with US, US-International, and German QWERTZ layout display the broken bar on a keycap even though the solid vertical bar character is produced. This is a legacy of keyboards manufactured during the 1980s and 1990s for IBM PC compatible computers, as the IBM PC continued to display the glyph for the broken bar at codepoint 7C on displays from MDA (1981) to VGA (1987) despite the changes made to ASCII in 1977. The UK/Ireland keyboard has both symbols engraved: the broken bar is given as an alternate graphic on the "grave" (backtick) key; the solid bar is on the backslash key.
The broken bar character can be typed (depending on the layout) as or or on Windows and on Linux. It can be inserted into HTML as
The broken bar does not appear to have any clearly identified uses distinct from those of the vertical bar. In non-computing use — for example in mathematics, physics and general typography — the broken bar is not an acceptable substitute for the vertical bar. In some dictionaries, the broken bar is used to mark stress that may be either primary or secondary: covers the pronunciations and .
Unicode code points.
These glyphs are encoded in Unicode as follows:
Notes.
<templatestyles src="Reflist/styles.css" />
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "|x|"
},
{
"math_id": 1,
"text": "|S|"
},
{
"math_id": 2,
"text": "|A|"
},
{
"math_id": 3,
"text": "\\begin{vmatrix} a & b \\\\ c & d\\end{vmatrix}"
},
{
"math_id": 4,
"text": "|G|"
},
{
"math_id": 5,
"text": "|g|"
},
{
"math_id": 6,
"text": "g \\in G"
},
{
"math_id": 7,
"text": "P(X|Y)"
},
{
"math_id": 8,
"text": "P|ab"
},
{
"math_id": 9,
"text": "P"
},
{
"math_id": 10,
"text": "ab"
},
{
"math_id": 11,
"text": "a \\mid b"
},
{
"math_id": 12,
"text": "f(x)|_{x=4}"
},
{
"math_id": 13,
"text": "f|_{A}"
},
{
"math_id": 14,
"text": "f"
},
{
"math_id": 15,
"text": "A"
},
{
"math_id": 16,
"text": "\\{x|x<2\\}"
},
{
"math_id": 17,
"text": "a|b"
},
{
"math_id": 18,
"text": "f(x) \\vert _a ^b"
},
{
"math_id": 19,
"text": "f(b) - f(a)"
},
{
"math_id": 20,
"text": "f(x|\\mu,\\sigma)"
},
{
"math_id": 21,
"text": "\\|"
},
{
"math_id": 22,
"text": "AB \\parallel CD"
},
{
"math_id": 23,
"text": "AB"
},
{
"math_id": 24,
"text": "CD"
},
{
"math_id": 25,
"text": "\\|A\\|"
},
{
"math_id": 26,
"text": "a : A"
},
{
"math_id": 27,
"text": "a"
},
{
"math_id": 28,
"text": " A"
},
{
"math_id": 29,
"text": "|a| : \\left\\| A \\right\\|"
},
{
"math_id": 30,
"text": "|a|"
},
{
"math_id": 31,
"text": "\\left\\| A \\right\\|"
},
{
"math_id": 32,
"text": "a | b \\| c"
},
{
"math_id": 33,
"text": "a \\mid b \\parallel c"
},
{
"math_id": 34,
"text": "|\\psi\\rangle"
},
{
"math_id": 35,
"text": "\\psi"
},
{
"math_id": 36,
"text": "\\langle\\psi|"
},
{
"math_id": 37,
"text": "\\langle\\psi|\\rho\\rangle"
},
{
"math_id": 38,
"text": "\\rho"
}
] |
https://en.wikipedia.org/wiki?curid=641073
|
64111989
|
1 Kings 22
|
1 Kings, chapter 22
1 Kings 22 is the 22nd (and the last) chapter of the First Book of Kings in the Old Testament of the Christian Bible or the first part of Books of Kings in the Hebrew Bible. The book is a compilation of various annals recording the acts of the kings of Israel and Judah by a Deuteronomic compiler in the seventh century BCE, with a supplement added in the sixth century BCE. This chapter belongs to the section comprising 1 Kings 16:15 to 2 Kings 8:29 which documents the period of the Omrides. The focus of this chapter is the reign of king Ahab and Ahaziah in the northern kingdom, as well as of king Jehoshaphat in the southern kingdom.
Text.
This chapter was originally written in the Hebrew language and since the 16th century is divided into 53 verses in Christian Bibles, but into 54 verses in the Hebrew Bible as in the verse numbering comparison table below.
Verse numbering.
This article generally follows the common numbering in Christian English Bible versions, with notes to the numbering in Hebrew Bible versions.
Textual witnesses.
Some early manuscripts containing the text of this chapter in Hebrew are of the Masoretic Text tradition, which includes the Codex Cairensis (895), Aleppo Codex (10th century), and Codex Leningradensis (1008). Fragments containing parts of this chapter in Hebrew were found among the Dead Sea Scrolls, that is, 6Q4 (6QpapKgs; 150–75 BCE) with extant verses 28–31.
There is also a translation into Koine Greek known as the Septuagint, made in the last few centuries BCE. Extant ancient manuscripts of the Septuagint version include Codex Vaticanus (B; formula_0B; 4th century) and Codex Alexandrinus (A; formula_0A; 5th century).
Death of Ahab (22:1–40).
Despite the announcement that his punishment for his crime against Naboth only befell his sons and he seemed to die of natural causes (1 Kings 22:40), Ahab was not left unreprimanded. The narrative of his death displays much life of Ahab into a single climactic story:
Three prophets, three warnings, three witnesses; these are the sign of Yahweh's continuing mercy to Ahab and Ahab cannot plead ignorance nor innocence: first warning, Ahab became sullen and angry; second warning, Ahab showed repentance; third warning: Ahab defiantly went to battle in disguise, but he got three chances, so it was strike three, in the third year (1 Kings 22:1), and he was removed.
The peace between Aram and Israel following the Battle of Aphek (1 Kings 20) lasted three years, Ahab decided to capture the strategic Transjordan trading hub, Ramoth Gilead, while he made use of the close ties between the kingdoms of Judah and Israel (remained until the rise of Jehu and Joash (2 Kings 9–11)).
Ahab did not hesitate to sacrifice Jehoshaphat (Ahab advised Jehoshaphat not to disguise) to the enemy in order to save himself who went in disguise (verses 29–30). However, the results were different (verses 31–36), as Jehoshaphat remained unhurt whereas a stray arrow hit Ahab and he could not leave the battlefield until evening (verse 38; related to Elijah's prophecy in 1 Kings 21:19).
The narrative also has an underlying theme of the battle between true and false prophecy (first initiated in 1 Kings 13). A fundamental problem regarding the prophets is the unaccountability of their own attitude towards God's messages (as in Jeremiah 28 and Micah 3:5–8). Micaiah ben Imlah states that the 'prophets' with opposing messages were possessed by an evil spirit who helped to drive Ahab to death, because he witnessed the discussions at a heavenly council (in a vision, cf. Isaiah 6). Just as Isaiah's warning to the people was ignored (Isaiah 6:9–10), Micaiah's message for Ahab to change course was not heard, so Ahab would meet his doom according to the true prophecy from YHWH.
"And they continued three years without war between Syria and Israel."
"And it came to pass in the third year, that Jehoshaphat the king of Judah came down to the king of Israel."
Jehoshaphat, the king of Judah (22:41–50).
Jehoshaphat was officially introduced, after the report that he was closely linked to Ahab, supporting the statement in the Annals that there was no war with Israel during his reign. The kingdom of Judah at this time controlled Edom and therefore had access to the Red Sea at the seaport of Ezion-geber, but they lacked the nautical skill to undertake trade projects and the big ships (the type which can sail to Tarshish) were wrecked at the harbor
"41 And Jehoshaphat the son of Asa began to reign over Judah in the fourth year of Ahab king of Israel."
"42 Jehoshaphat was thirty and five years old when he began to reign; and he reigned twenty and five years in Jerusalem. And his mother's name was Azubah the daughter of Shilhi."
"a And he walked in all the ways of Asa his father; he turned not aside from it, doing that which was right in the eyes of the Lord"
"b nevertheless the high places were not taken away; for the people offered and burnt incense yet in the high places."
Verse 43.
Verse 22:43b in the English Bible is numbered as 22:44 in the Hebrew text (BHS).
Ahaziah, the king of Israel (22:51–53).
Ahaziah, Ahab's son and successor, followed the footsteps of his father and his mother in his short reign, so did not change the punishment of Omri's dynasty which was only postponed.
"Ahaziah the son of Ahab began to reign over Israel in Samaria the seventeenth year of Jehoshaphat king of Judah, and he reigned two years over Israel."
"He did what the Lord considered evil. He followed the example of his father and mother and of Jeroboam (Nebat’s son) who led Israel to sin."
Notes.
<templatestyles src="Reflist/styles.css" />
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": " \\mathfrak{G}"
}
] |
https://en.wikipedia.org/wiki?curid=64111989
|
64111990
|
1 Chronicles 1
|
First Book of Chronicles, chapter 1
1 Chronicles 1 is the first chapter of the Books of Chronicles in the Hebrew Bible or the First Book of Chronicles in the Old Testament of the Christian Bible. The book is compiled from older sources by an unknown person or group, designated by modern scholars as "the Chronicler", and had the final shape established in late fifth or fourth century BCE. The content of this chapter is the genealogy list from Adam to Israel (=Jacob) in the following structure: Adam to Noah (verses 1–4); Noah's descendants from his three sons Shem, Ham, and Japheth: the Japhethites (verses 5–7), Hamites (verses 8–23), Semites (verses 24–27); the sons of Abraham (verses 28–34a); the sons of Isaac (34b–54; continued to 2:2 for Israel's sons). This chapter belongs to the section focusing on the list of genealogies from Adam to the lists of the people returning from exile in Babylon ( to ).
Text.
This chapter was originally written in the Hebrew language and is divided into 54 verses.
Textual witnesses.
Some early manuscripts containing the text of this chapter in Hebrew are of the Masoretic Text tradition, which includes the Aleppo Codex (10th century), and Codex Leningradensis (1008).
There is also a translation into Koine Greek known as the Septuagint, made in the last few centuries BCE. Extant ancient manuscripts of the Septuagint version include Codex Vaticanus (B; formula_0B; 4th century) and Codex Alexandrinus (A; formula_0A; 5th century).
From Adam to Abraham (1:1–27).
The list of names is taken exclusively from the Book of Genesis and reduced to a 'skeletal framework', with some omissions of those 'whose lines ended with their deaths', such as Cain's descendants and Abraham's brothers. It links the origin of Israel to the origin of all people – Abraham's ancestry in Adam and Noah's – and thus, within the whole human history. Verses 1–4 (from Adam to Noah) match closely to the genealogy in ; verses 5–12 (the genealogy of Noah's sons) match that in ; verses 13–27 (Shem's descendants until Abraham) parallel the genealogy in . Verse 27 contains "Abram, that is, Abraham" (the name first given by God in ), representing a jump from Genesis 11 to Genesis 17.
"Adam, Seth, Enosh;"
Verse 1.
Noah was the immediate descendant of Seth, so it is not necessary to mention Cain and Abel, or any of the other sons of Adam.
"Noah, Shem, Ham, and Japheth."
"And the sons of Gomer; Ashchenaz, and Riphath, and Togarmah."
The descendants of Abraham (1:28–54).
This section focuses on the offsprings of Abraham (but none of his brothers'). Verses 32–40 lists Abraham's sons other than Isaac and Ishmael with the direct connection to verse 28 and has been more extensively reworked than other genealogies in this chapter, whereas verses 43–54 contain an extensive reworking of Genesis 36 to list the descendants of Edom who are Judah's neighbors with 'the closest ties through the best and worst of times'.
"And the sons of Lotan; Hori, and Homam: and Timna was Lotan's sister."
Notes.
<templatestyles src="Reflist/styles.css" />
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": " \\mathfrak{G}"
}
] |
https://en.wikipedia.org/wiki?curid=64111990
|
641132
|
Edgeworth series
|
The Gram–Charlier A series (named in honor of Jørgen Pedersen Gram and Carl Charlier), and the Edgeworth series (named in honor of Francis Ysidro Edgeworth) are series that approximate a probability distribution in terms of its cumulants. The series are the same; but, the arrangement of terms (and thus the accuracy of truncating the series) differ. The key idea of these expansions is to write the characteristic function of the distribution whose probability density function f is to be approximated in terms of the characteristic function of a distribution with known and suitable properties, and to recover f through the inverse Fourier transform.
Gram–Charlier A series.
We examine a continuous random variable. Let formula_0 be the characteristic function of its distribution whose density function is f, and formula_1 its cumulants. We expand in terms of a known distribution with probability density function ψ, characteristic function formula_2, and cumulants formula_3. The density ψ is generally chosen to be that of the normal distribution, but other choices are possible as well. By the definition of the cumulants, we have (see Wallace, 1958)
formula_4 and
formula_5
which gives the following formal identity:
formula_6
By the properties of the Fourier transform, formula_7 is the Fourier transform of formula_8, where D is the differential operator with respect to x. Thus, after changing formula_9 with formula_10 on both sides of the equation, we find for f the formal expansion
formula_11
If ψ is chosen as the normal density
formula_12
with mean and variance as given by f, that is, mean formula_13 and variance formula_14, then the expansion becomes
formula_15
since formula_16 for all r > 2, as higher cumulants of the normal distribution are 0. By expanding the exponential and collecting terms according to the order of the derivatives, we arrive at the Gram–Charlier A series. Such an expansion can be written compactly in terms of Bell polynomials as
formula_17
Since the n-th derivative of the Gaussian function formula_18 is given in terms of Hermite polynomial as
formula_19
this gives us the final expression of the Gram–Charlier A series as
formula_20
Integrating the series gives us the cumulative distribution function
formula_21
where formula_22 is the CDF of the normal distribution.
If we include only the first two correction terms to the normal distribution, we obtain
formula_23
with formula_24 and formula_25.
Note that this expression is not guaranteed to be positive, and is therefore not a valid probability distribution. The Gram–Charlier A series diverges in many cases of interest—it converges only if formula_26 falls off faster than formula_27 at infinity (Cramér 1957). When it does not converge, the series is also not a true asymptotic expansion, because it is not possible to estimate the error of the expansion. For this reason, the Edgeworth series (see next section) is generally preferred over the Gram–Charlier A series.
The Edgeworth series.
Edgeworth developed a similar expansion as an improvement to the central limit theorem. The advantage of the Edgeworth series is that the error is controlled, so that it is a true asymptotic expansion.
Let formula_28 be a sequence of independent and identically distributed random variables with finite mean formula_29 and variance formula_30, and let formula_31 be their standardized sums:
formula_32
Let formula_33 denote the cumulative distribution functions of the variables formula_31. Then by the central limit theorem,
formula_34
for every formula_9, as long as the mean and variance are finite.
The standardization of formula_28 ensures that the first two cumulants of formula_31 are formula_35 and formula_36 Now assume that, in addition to having mean formula_29 and variance formula_30, the i.i.d. random variables formula_37 have higher cumulants formula_38. From the additivity and homogeneity properties of cumulants, the cumulants of formula_31 in terms of the cumulants of formula_37 are for formula_39,
formula_40
If we expand the formal expression of the characteristic function formula_41 of formula_33 in terms of the standard normal distribution, that is, if we set
formula_42
then the cumulant differences in the expansion are
formula_43
formula_44
formula_45
The Gram–Charlier A series for the density function of formula_31 is now
formula_46
The Edgeworth series is developed similarly to the Gram–Charlier A series, only that now terms are collected according to powers of formula_47. The coefficients of "n"−"m"/2 term can be obtained by collecting the monomials of the Bell polynomials corresponding to the integer partitions of "m". Thus, we have the characteristic function as
formula_48
where formula_49 is a polynomial of degree formula_50. Again, after inverse Fourier transform, the density function formula_51 follows as
formula_52
Likewise, integrating the series, we obtain the distribution function
formula_53
We can explicitly write the polynomial formula_54 as
formula_55
where the summation is over all the integer partitions of "m" such that formula_56 and formula_57 and formula_58
For example, if "m" = 3, then there are three ways to partition this number: 1 + 1 + 1 = 2 + 1 = 3. As such we need to examine three cases:
Thus, the required polynomial is
formula_59
The first five terms of the expansion are
formula_60
Here, φ("j")("x") is the "j"-th derivative of φ(·) at point "x". Remembering that the derivatives of the density of the normal distribution are related to the normal density by formula_61, (where formula_62 is the Hermite polynomial of order "n"), this explains the alternative representations in terms of the density function. Blinnikov and Moessner (1998) have given a simple algorithm to calculate higher-order terms of the expansion.
Note that in case of a lattice distributions (which have discrete values), the Edgeworth expansion must be adjusted to account for the discontinuous jumps between lattice points.
Illustration: density of the sample mean of three χ² distributions.
Take formula_63 and the sample mean formula_64.
We can use several distributions for formula_65:
References.
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|
[
{
"math_id": 0,
"text": "\\hat{f}"
},
{
"math_id": 1,
"text": "\\kappa_r"
},
{
"math_id": 2,
"text": "\\hat{\\psi}"
},
{
"math_id": 3,
"text": "\\gamma_r"
},
{
"math_id": 4,
"text": "\\hat{f}(t)= \\exp\\left[\\sum_{r=1}^\\infty\\kappa_r\\frac{(it)^r}{r!}\\right]"
},
{
"math_id": 5,
"text": " \\hat{\\psi}(t)=\\exp\\left[\\sum_{r=1}^\\infty\\gamma_r\\frac{(it)^r}{r!}\\right],"
},
{
"math_id": 6,
"text": "\\hat{f}(t)=\\exp\\left[\\sum_{r=1}^\\infty(\\kappa_r-\\gamma_r)\\frac{(it)^r}{r!}\\right]\\hat{\\psi}(t)\\,."
},
{
"math_id": 7,
"text": "(it)^r \\hat{\\psi}(t)"
},
{
"math_id": 8,
"text": "(-1)^r[D^r\\psi](-x)"
},
{
"math_id": 9,
"text": "x"
},
{
"math_id": 10,
"text": "-x"
},
{
"math_id": 11,
"text": "f(x) = \\exp\\left[\\sum_{r=1}^\\infty(\\kappa_r - \\gamma_r)\\frac{(-D)^r}{r!}\\right]\\psi(x)\\,."
},
{
"math_id": 12,
"text": "\\phi(x) = \\frac{1}{\\sqrt{2\\pi}\\sigma}\\exp\\left[-\\frac{(x-\\mu)^2}{2\\sigma^2}\\right]"
},
{
"math_id": 13,
"text": "\\mu = \\kappa_1"
},
{
"math_id": 14,
"text": "\\sigma^2 = \\kappa_2"
},
{
"math_id": 15,
"text": "f(x) = \\exp\\left[\\sum_{r=3}^\\infty\\kappa_r\\frac{(-D)^r}{r!}\\right] \\phi(x),"
},
{
"math_id": 16,
"text": " \\gamma_r=0"
},
{
"math_id": 17,
"text": "\\exp\\left[\\sum_{r=3}^\\infty\\kappa_r\\frac{(-D)^r}{r!}\\right] = \\sum_{n=0}^\\infty B_n(0,0,\\kappa_3,\\ldots,\\kappa_n)\\frac{(-D)^n}{n!}. "
},
{
"math_id": 18,
"text": "\\phi"
},
{
"math_id": 19,
"text": "\\phi^{(n)}(x) = \\frac{(-1)^n}{\\sigma^n} He_n \\left( \\frac{x-\\mu}{\\sigma} \\right) \\phi(x),"
},
{
"math_id": 20,
"text": " f(x) = \\phi(x) \\sum_{n=0}^\\infty \\frac{1}{n! \\sigma^n} B_n(0,0,\\kappa_3,\\ldots,\\kappa_n) He_n \\left( \\frac{x-\\mu}{\\sigma} \\right)."
},
{
"math_id": 21,
"text": " F(x) = \\int_{-\\infty}^x f(u) du = \\Phi(x) - \\phi(x) \\sum_{n=3}^\\infty \\frac{1}{n! \\sigma^{n-1}} B_n(0,0,\\kappa_3,\\ldots,\\kappa_n) He_{n-1} \\left( \\frac{x-\\mu}{\\sigma} \\right), "
},
{
"math_id": 22,
"text": "\\Phi"
},
{
"math_id": 23,
"text": " f(x) \\approx \\frac{1}{\\sqrt{2\\pi}\\sigma}\\exp\\left[-\\frac{(x-\\mu)^2}{2\\sigma^2}\\right]\\left[1+\\frac{\\kappa_3}{3!\\sigma^3}He_3\\left(\\frac{x-\\mu}{\\sigma}\\right)+\\frac{\\kappa_4}{4!\\sigma^4}He_4\\left(\\frac{x-\\mu}{\\sigma}\\right)\\right]\\,,"
},
{
"math_id": 24,
"text": "He_3(x)=x^3-3x"
},
{
"math_id": 25,
"text": "He_4(x)=x^4 - 6x^2 + 3"
},
{
"math_id": 26,
"text": "f(x)"
},
{
"math_id": 27,
"text": "\\exp(-(x^2)/4)"
},
{
"math_id": 28,
"text": "\\{Z_i\\}"
},
{
"math_id": 29,
"text": "\\mu"
},
{
"math_id": 30,
"text": "\\sigma^2"
},
{
"math_id": 31,
"text": "X_n"
},
{
"math_id": 32,
"text": "X_n = \\frac{1}{\\sqrt{n}} \\sum_{i=1}^n \\frac{Z_i - \\mu}{\\sigma}."
},
{
"math_id": 33,
"text": "F_n"
},
{
"math_id": 34,
"text": "\n \\lim_{n\\to\\infty} F_n(x) = \\Phi(x) \\equiv \\int_{-\\infty}^x \\tfrac{1}{\\sqrt{2\\pi}}e^{-\\frac{1}{2}q^2}dq\n "
},
{
"math_id": 35,
"text": "\\kappa_1^{F_n} = 0"
},
{
"math_id": 36,
"text": "\\kappa_2^{F_n} = 1."
},
{
"math_id": 37,
"text": "Z_i"
},
{
"math_id": 38,
"text": " \\kappa_r"
},
{
"math_id": 39,
"text": "r \\geq 2"
},
{
"math_id": 40,
"text": " \\kappa_r^{F_n} = \\frac{n \\kappa_r}{\\sigma^r n^{r/2}} = \\frac{\\lambda_r}{n^{r/2 - 1}} \\quad \\mathrm{where} \\quad \\lambda_r = \\frac{\\kappa_r}{\\sigma^r}. "
},
{
"math_id": 41,
"text": "\\hat{f}_n(t)"
},
{
"math_id": 42,
"text": "\\phi(x)=\\frac{1}{\\sqrt{2\\pi}}\\exp(-\\tfrac{1}{2}x^2),"
},
{
"math_id": 43,
"text": " \\kappa^{F_n}_1-\\gamma_1 = 0,"
},
{
"math_id": 44,
"text": " \\kappa^{F_n}_2-\\gamma_2 = 0,"
},
{
"math_id": 45,
"text": " \\kappa^{F_n}_r-\\gamma_r = \\frac{\\lambda_r}{n^{r/2-1}}; \\qquad r\\geq 3."
},
{
"math_id": 46,
"text": " f_n(x) = \\phi(x) \\sum_{r=0}^\\infty \\frac{1}{r!} B_r \\left(0,0,\\frac{\\lambda_3}{n^{1/2}},\\ldots,\\frac{\\lambda_r}{n^{r/2-1}}\\right) He_r(x)."
},
{
"math_id": 47,
"text": "n"
},
{
"math_id": 48,
"text": " \\hat{f}_n(t)=\\left[1+\\sum_{j=1}^\\infty \\frac{P_j(it)}{n^{j/2}}\\right] \\exp(-t^2/2)\\,,"
},
{
"math_id": 49,
"text": "P_j(x)"
},
{
"math_id": 50,
"text": "3j"
},
{
"math_id": 51,
"text": "f_n"
},
{
"math_id": 52,
"text": " f_n(x) = \\phi(x) + \\sum_{j=1}^\\infty \\frac{P_j(-D)}{n^{j/2}} \\phi(x)\\,."
},
{
"math_id": 53,
"text": " F_n(x) = \\Phi(x) + \\sum_{j=1}^\\infty \\frac{1}{n^{j/2}} \\frac{P_j(-D)}{D} \\phi(x)\\,. "
},
{
"math_id": 54,
"text": "P_m(-D)"
},
{
"math_id": 55,
"text": " P_m(-D) = \\sum \\prod_i \\frac{1}{k_i!} \\left(\\frac{\\lambda_{l_i}}{l_i!}\\right)^{k_i} (-D)^s,"
},
{
"math_id": 56,
"text": "\\sum_i i k_i = m"
},
{
"math_id": 57,
"text": "l_i = i+2"
},
{
"math_id": 58,
"text": "s = \\sum_i k_i l_i."
},
{
"math_id": 59,
"text": "\n\\begin{align}\nP_3(-D) &= \\frac{1}{3!} \\left(\\frac{\\lambda_3}{3!}\\right)^3 (-D)^9 + \\frac{1}{1! 1!} \\left(\\frac{\\lambda_3}{3!}\\right) \\left(\\frac{\\lambda_4}{4!}\\right) (-D)^7 + \\frac{1}{1!} \\left(\\frac{\\lambda_5}{5!}\\right) (-D)^5 \\\\\n&= \\frac{\\lambda_3^3}{1296} (-D)^9 + \\frac{\\lambda_3 \\lambda_4}{144} (-D)^7 + \\frac{\\lambda_5}{120} (-D)^5.\n\\end{align}\n"
},
{
"math_id": 60,
"text": "\\begin{align}\nf_n(x) &= \\phi(x) \\\\\n&\\quad -\\frac{1}{n^{\\frac{1}{2}}}\\left(\\tfrac{1}{6}\\lambda_3\\,\\phi^{(3)}(x) \\right) \\\\\n&\\quad +\\frac{1}{n}\\left(\\tfrac{1}{24}\\lambda_4\\,\\phi^{(4)}(x) + \\tfrac{1}{72}\\lambda_3^2\\,\\phi^{(6)}(x) \\right) \\\\\n&\\quad -\\frac{1}{n^{\\frac{3}{2}}}\\left(\\tfrac{1}{120}\\lambda_5\\,\\phi^{(5)}(x) + \\tfrac{1}{144}\\lambda_3\\lambda_4\\,\\phi^{(7)}(x) + \\tfrac{1}{1296}\\lambda_3^3\\,\\phi^{(9)}(x)\\right) \\\\\n&\\quad + \\frac{1}{n^2}\\left(\\tfrac{1}{720}\\lambda_6\\,\\phi^{(6)}(x) + \\left(\\tfrac{1}{1152}\\lambda_4^2 + \\tfrac{1}{720}\\lambda_3\\lambda_5\\right)\\phi^{(8)}(x) + \\tfrac{1}{1728}\\lambda_3^2\\lambda_4\\,\\phi^{(10)}(x) + \\tfrac{1}{31104}\\lambda_3^4\\,\\phi^{(12)}(x) \\right)\\\\\n&\\quad + O \\left (n^{-\\frac{5}{2}} \\right ).\n\\end{align}"
},
{
"math_id": 61,
"text": "\\phi^{(n)}(x) = (-1)^n He_n(x)\\phi(x)"
},
{
"math_id": 62,
"text": "He_n"
},
{
"math_id": 63,
"text": " X_i \\sim \\chi^2(k=2), \\, i=1, 2, 3 \\, (n=3)"
},
{
"math_id": 64,
"text": " \\bar X = \\frac{1}{3} \\sum_{i=1}^{3} X_i "
},
{
"math_id": 65,
"text": " \\bar X "
},
{
"math_id": 66,
"text": " \\bar X \\sim \\mathrm{Gamma}\\left(\\alpha=n\\cdot k /2, \\theta= 2/n \\right)=\\mathrm{Gamma}\\left(\\alpha=3, \\theta= 2/3 \\right)"
},
{
"math_id": 67,
"text": " \\bar X \\xrightarrow{n \\to \\infty} N(k, 2\\cdot k /n ) = N(2, 4/3 )"
},
{
"math_id": 68,
"text": "[0,1]"
}
] |
https://en.wikipedia.org/wiki?curid=641132
|
6412297
|
5
|
Integer number 5
Natural number
5 (five) is a number, numeral and digit. It is the natural number, and cardinal number, following 4 and preceding 6, and is a prime number.
Humans, and many other animals, have 5 digits on their limbs.
Mathematics.
Five is the second Fermat prime, the third Mersenne prime exponent, as well as a Fibonacci number. 5 is the first congruent number, as well as the length of the hypotenuse of the smallest integer-sided right triangle, making part of the smallest Pythagorean triple (3, 4, 5).
Geometry.
A shape with five sides is called a pentagon. The pentagon is the first regular polygon that does not tile the plane with copies of itself. It is the largest face any of the five regular three-dimensional regular Platonic solid can have.
A conic is determined using five points in the same way that two points are needed to determine a line.A pentagram, or five-pointed polygram, is a star polygon constructed by connecting some non-adjacent of a regular pentagon as self-intersecting edges.
5 is the first safe prime where formula_0 for a prime formula_1 is also prime (2), and the first good prime, since it is the first prime number whose square (25) is greater than the product of any two primes at the same number of positions before and after it in the sequence of primes (i.e., 3 × 7 = 21 and 11 × 2 = 22 are less than 25). 11, the fifth prime number, is the next good prime, that also forms the first pair of sexy primes with 5. 5 is the second Fermat prime of the form formula_2, of a total of five known Fermat primes.
The internal geometry of the pentagon and pentagram (represented by its Schläfli symbol {5/2}) appears prominently in Penrose tilings. They are facets inside Kepler–Poinsot star polyhedra and Schläfli–Hess star polychora.
There are five Platonic solids in three-dimensional space that are regular: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron.
The chromatic number of the plane is the minimum number of colors required to color the plane such that no pair of points at a distance of 1 has the same color. Five is a lower depending for the chromatic number of the plane, but this may depend on the choice of set-theoretical axioms:
The plane contains a total of five Bravais lattices, or arrays of points defined by discrete translation operations. Uniform tilings of the plane, are generated from combinations of only five regular polygons.
Higher Dimensional Geometry.
A hypertetrahedron, or 5-cell the 4 dimensional analogue of the tetrahedron, has five vertices. Its orthographic projection is homomorphic to the group "K"5.
There are five fundamental mirror symmetry point group families in 4-dimensions. There are also 5 compact hyperbolic Coxeter groups, or 4-prisms, of rank 5, each generating uniform honeycombs in hyperbolic 4-space as permutations of rings of the Coxeter diagrams.
Algebra.
5 is the value of the central cell of the first non-trivial normal magic square, called the "Luoshu" square. 5 is also the first of three known Wilson primes (5, 13, 563), where the square of a prime formula_3 divides formula_4 As a consequence of Fermat's little theorem and Euler's criterion, all squares are congruent to 0, 1, 4 (or −1) modulo 5. All integers formula_5 can be expressed as the sum of five non-zero squares. There are five countably infinite Ramsey classes of permutations.
Five is conjectured to be the only odd, untouchable number; if this is the case, then five will be the only odd prime number that is not the base of an aliquot tree.
Every odd number greater than five is conjectured to be expressible as the sum of three prime numbers; Helfgott has provided a proof of this (also known as the odd Goldbach conjecture) that is already widely acknowledged by mathematicians as it still undergoes peer-review. On the other hand, every odd number greater than one is the sum of at most five prime numbers (as a lower limit).<templatestyles src="Unsolved/styles.css" />
Unsolved problem in mathematics:
Is 5 the only odd, untouchable number?
Group Theory.
In graph theory, all graphs with four or fewer vertices are planar, however, there is a graph with five vertices that is not: "K"5, the complete graph with five vertices. By Kuratowski's theorem, a finite graph is planar if and only if it does not contain a subgraph that is a subdivision of , or "K"3,3, the utility graph.
There are five complex exceptional Lie algebras. The five Mathieu groups constitute the in the happy family of sporadic groups. These are also the first five sporadic groups to have been described. A centralizer of an element of order 5 inside the largest sporadic group formula_6 arises from the product between Harada–Norton sporadic group formula_7 and a group of order 5.
Evolution of the Arabic digit.
The evolution of the modern Western digit for the numeral for five is traced back to the Indian system of numerals, where on some earlier versions, the numeral bore resemblance to variations of the number four, rather than "5" (as it is represented today). The Kushana and Gupta empires in what is now India had among themselves several forms that bear no resemblance to the modern digit. Later on, Arabic traditions transformed the digit in several ways, producing forms that were still similar to the numeral for four, with similarities to the numeral for three; yet, still unlike the modern five. It was from those digits that Europeans finally came up with the modern 5 (represented in writings by Dürer, for example).
While the shape of the character for the digit 5 has an ascender in most modern typefaces, in typefaces with text figures the glyph usually has a descender, as, for example, in .
On the seven-segment display of a calculator and digital clock, it is represented by five segments at four successive turns from top to bottom, rotating counterclockwise first, then clockwise, and vice-versa. It is one of three numbers, along with 4 and 6, where the number of segments matches the number.
Other fields.
Astronomy.
There are five Lagrangian points in a two-body system.
Biology.
There are usually considered to be five senses (in general terms); the five basic tastes are sweet, salty, sour, bitter, and umami. Almost all amphibians, reptiles, and mammals which have fingers or toes have five of them on each extremity. Five is the number of appendages on most starfish, which exhibit pentamerism.
Computing.
5 is the ASCII code of the Enquiry character, which is abbreviated to ENQ.
Literature.
Poetry.
A pentameter is verse with five repeating feet per line; the iambic pentameter was the most prominent form used by William Shakespeare.
Music.
Modern musical notation uses a musical staff made of five horizontal lines. A scale with five notes per octave is called a pentatonic scale. A perfect fifth is the most consonant harmony, and is the basis for most western tuning systems. In harmonics, the fifth partial (or 4th overtone) of a fundamental has a frequency ratio of 5:1 to the frequency of that fundamental. This ratio corresponds to the interval of 2 octaves plus a pure major third. Thus, the interval of 5:4 is the interval of the pure third. A major triad chord when played in just intonation (most often the case in a cappella vocal ensemble singing), will contain such a pure major third.
Five is the lowest possible number that can be the top number of a time signature with an asymmetric meter.
Religion.
Judaism.
The Book of Numbers is one of five books in the Torah; the others being the books of Genesis, Exodus, Leviticus, and Deuteronomy. They are collectively called the Five Books of Moses, the Pentateuch (Greek for "five containers", referring to the scroll cases in which the books were kept), or Humash (, Hebrew for "fifth"). The Khamsa, an ancient symbol shaped like a hand with four fingers and one thumb, is used as a protective amulet by Jews; that same symbol is also very popular in Arabic culture, known to protect from envy and the evil eye.
Christianity.
There are traditionally five wounds of Jesus Christ in Christianity: the nail wounds in Christ's two hands, the nail wounds in Christ's two feet, and the Spear Wound of Christ (respectively at the four extremities of the body, and the head).
Islam.
The Five Pillars of Islam. The five-pointed simple star ☆ is one of the five used in Islamic Girih tiles.
Mysticism.
Gnosticism.
The number five was an important symbolic number in Manichaeism, with heavenly beings, concepts, and others often grouped in sets of five.
Alchemy.
According to ancient Greek philosophers such as Aristotle, the universe is made up of five classical elements: water, earth, air, fire, and ether. This concept was later adopted by medieval alchemists and more recently by practitioners of Neo-Pagan religions such as Wicca. There are five elements in the universe according to Hindu cosmology: (earth, fire, water, air and space, respectively). In East Asian tradition, there are five elements: water, fire, earth, wood, and metal. The Japanese names for the days of the week, Tuesday through Saturday, come from these elements via the identification of the elements with the five planets visible with the naked eye. Also, the traditional Japanese calendar has a five-day weekly cycle that can be still observed in printed mixed calendars combining Western, Chinese-Buddhist, and Japanese names for each weekday. There are also five elements in the traditional Chinese "Wuxing."
Quintessence, meaning "fifth element", refers to the elusive fifth element that completes the basic four elements (water, fire, air, and earth), as a union of these. The pentagram, or five-pointed star, bears mystic significance in various belief systems including Baháʼí, Christianity, Freemasonry, Satanism, Taoism, Thelema, and Wicca.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "(p - 1)/2"
},
{
"math_id": 1,
"text": "p"
},
{
"math_id": 2,
"text": "2^{2^{n}} + 1"
},
{
"math_id": 3,
"text": "p^2"
},
{
"math_id": 4,
"text": "(p-1)!+1."
},
{
"math_id": 5,
"text": "n \\ge 34"
},
{
"math_id": 6,
"text": "\\mathrm {F_1}"
},
{
"math_id": 7,
"text": "\\mathrm{HN}"
}
] |
https://en.wikipedia.org/wiki?curid=6412297
|
64123397
|
Charge-pump phase-locked loop
|
Charge-pump phase-locked loop (CP-PLL) is a modification of phase-locked loops with phase-frequency detectors and square waveform signals. A CP-PLL allows for a quick lock of the phase of the incoming signal, achieving low steady state phase error.
Phase-frequency detector (PFD).
Phase-frequency detector (PFD) is triggered by the trailing edges of the reference (Ref) and controlled (VCO) signals.
The output signal of PFD formula_0 can have only three states: 0, formula_1, and formula_2.
A trailing edge of the reference signal forces the PFD to switch to a higher state,
unless it is already in the state formula_1.
A trailing edge of the VCO signal forces the PFD to switch to a lower state,
unless it is already in the state formula_2.
If both trailing edges happen at the same time, then the PFD switches to zero.
Mathematical models of CP-PLL.
A first linear mathematical model of second-order CP-PLL was suggested by F. Gardner in 1980. A nonlinear model without the VCO overload was suggested by M. van Paemel in 1994
and then refined by N. Kuznetsov et al. in 2019. The closed form mathematical model of CP-PLL taking into account the VCO overload is derived in.
These mathematical models of CP-PLL allow to get analytical estimations of the hold-in range
(a maximum range of the input signal period
such that there exists a locked state at which the VCO is not overloaded)
and the pull-in range (a maximum range of the input signal period
within the hold-in range such that for any initial state the CP-PLL
acquires a locked state).
Continuous time linear model of the second order CP-PLL and Gardner's conjecture.
Gardner's analysis is based on the following approximation: time interval on which PFD has non-zero state on each period of reference signal is
formula_3
Then averaged output of charge-pump PFD is
formula_4
with corresponding transfer function
formula_5
Using filter transfer function formula_6 and VCO transfer function formula_7 one gets Gardner's linear approximated average model of second-order CP-PLL
formula_8
In 1980, F. Gardner, based on the above reasoning, conjectured that "transient response of practical charge-pump PLL's can be expected to be nearly the same as the response of the equivalent classical PLL" (Gardner's conjecture on CP-PLL).
Following Gardner's results, by analogy with the Egan conjecture on the pull-in range of type 2 APLL, Amr M. Fahim conjectured in his book that in order to have an infinite pull-in(capture) range, an active filter must be used for the loop filter in CP-PLL (Fahim-Egan's conjecture on the pull-in range of type II CP-PLL).
Continuous time nonlinear model of the second order CP-PLL.
Without loss of generality it is supposed that trailing edges of the VCO and Ref signals occur
when the corresponding phase reaches an integer number.
Let the time instance of the first trailing edge of the Ref signal is defined as formula_9.
The PFD state formula_10 is determined by the PFD initial state formula_11,
the initial phase shifts of the VCO formula_12
and Ref formula_13 signals.
The relationship between the input current formula_0 and the output voltage formula_14 for a
proportionally integrating (perfect PI) filter based on resistor and capacitor is as follows
formula_15
where formula_16 is a resistance, formula_17 is a capacitance,
and formula_18 is a capacitor charge.
The control signal formula_14 adjusts the VCO frequency:
formula_19
where formula_20 is the VCO free-running (quiescent) frequency
(i.e. for formula_21), formula_22
is the VCO gain (sensivity), and formula_23 is the VCO phase.
Finally, the continuous time nonlinear mathematical model of CP-PLL is as follows
formula_24
with the following discontinuous piece-wise constant nonlinearity
formula_25
and the initial conditions formula_26.
This model is a nonlinear, non-autonomous, discontinuous, switching system.
Discrete time nonlinear model of the second-order CP-PLL.
The reference signal frequency is assumed to be constant:
formula_27
where formula_28, formula_29 and formula_30
are a period, frequency and a phase of the reference signal.
Let formula_31.
Denote by formula_32 the first instant of time such that the PFD output becomes zero
and by formula_35 the first trailing edge of the VCO or Ref.
Further the corresponding increasing sequences formula_36 and formula_37 for formula_38 are defined.
Let formula_39.
Then for formula_40 the formula_41 is a non-zero constant (formula_42).
Denote by formula_43 the PFD pulse width (length of the time interval,
where the PFD output is a non-zero constant), multiplied by the sign of the PFD output:
i.e. formula_44 for formula_45 and
formula_46 for formula_47.
If the VCO trailing edge hits before the Ref trailing edge,
then formula_48 and in the opposite case we have formula_49, i.e. formula_43 shows how one signal lags behind another. Zero output of PFD formula_50 on the interval formula_51:
formula_52 for formula_53.
The transformation of variables formula_54 to
formula_55
allows to reduce the number of parameters to two:
formula_56
Here formula_57 is a normalized phase shift and formula_58 is a ratio of the VCO frequency formula_59 to the reference frequency formula_60.
Finally, the discrete-time model of second order CP-PLL without the VCO overload
formula_61
where
formula_62
This discrete-time model has the only one steady state at formula_63 and allows to estimate the hold-in and pull-in ranges.
If the VCO is overloaded, i.e. formula_64 is zero,
or what is the same:
formula_65 or
formula_66,
then the additional cases of the CP-PLL dynamics
have to be taken into account.
For any parameters the VCO overload may occur for sufficiently large frequency difference between the VCO and reference signals.
In practice the VCO overload should be avoided.
Nonlinear models of high-order CP-PLL.
Derivation of nonlinear mathematical models of high-order CP-PLL leads to transcendental phase equations that cannot be solved analytically and require numerical approaches like the classical fixed-point method or the Newton-Raphson approach.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "i(t)"
},
{
"math_id": 1,
"text": "+I_p"
},
{
"math_id": 2,
"text": "-I_p"
},
{
"math_id": 3,
"text": "t_p = |\\theta_e|/\\omega_{\\rm ref},\\ \\theta_e = \\theta_{\\rm ref} - \\theta_{\\rm vco}."
},
{
"math_id": 4,
"text": "i_d = I_p \\theta_e/2\\pi"
},
{
"math_id": 5,
"text": "I_d(s) = I_p\\theta_e(s)/2\\pi"
},
{
"math_id": 6,
"text": "F(s) = R + \\frac{1}{Cs}"
},
{
"math_id": 7,
"text": "\\theta_{\\rm vco}(s) = K_{\\rm vco}I_d(s)F(s)/s"
},
{
"math_id": 8,
"text": "\n\\frac{\\theta_e(s)}{\\theta_{\\rm ref}(s)} = \\frac{2\\pi s}{2\\pi s + K_{\\rm vco}I_p\\left(R + \\frac{1}{Cs}\\right)}.\n"
},
{
"math_id": 9,
"text": "t = 0"
},
{
"math_id": 10,
"text": "i(0)"
},
{
"math_id": 11,
"text": "i(0-)"
},
{
"math_id": 12,
"text": "\\theta_{vco}(0)"
},
{
"math_id": 13,
"text": "\\theta_{ref}(0)"
},
{
"math_id": 14,
"text": "v_F(t)"
},
{
"math_id": 15,
"text": "\n \\begin{align}\n v_F(t) = v_c(0) + Ri(t) + \\frac{1}{C}\\int\\limits_0^t i(\\tau)d\\tau\n \\end{align}\n"
},
{
"math_id": 16,
"text": "R>0"
},
{
"math_id": 17,
"text": "C>0"
},
{
"math_id": 18,
"text": "v_c(t)"
},
{
"math_id": 19,
"text": " \n \\begin{align} \n \\dot\\theta_{vco}(t) =\n \\omega_{vco}(t) =\n \\omega_{vco}^{\\text{free}} + K_{vco}v_F(t),\n \\end{align}\n"
},
{
"math_id": 20,
"text": "\\omega_{vco}^{\\text{free}}"
},
{
"math_id": 21,
"text": "v_F(t)\\equiv 0"
},
{
"math_id": 22,
"text": "K_{vco}"
},
{
"math_id": 23,
"text": "\\theta_{vco}(t)"
},
{
"math_id": 24,
"text": "\n\\begin{align}\n \\dot v_c(t) = \\tfrac{1}{C}i(t), \\quad\n \\dot\\theta_{vco}(t) =\n \\omega_{vco}^{\\text{free}} + K_{vco}\n (\n Ri(t)\n + v_c(t)\n )\n\\end{align}\n"
},
{
"math_id": 25,
"text": "\n i(t) = i\\big(i(t-), \\theta_{ref}(t), \\theta_{vco}(t)\\big)\n"
},
{
"math_id": 26,
"text": "\\big(v_c(0), \\theta_{vco}(0)\\big)"
},
{
"math_id": 27,
"text": "\n\\theta_{ref}(t) = \\omega_{ref}t = \\frac{t}{T_{ref}},\n"
},
{
"math_id": 28,
"text": "T_{ref}"
},
{
"math_id": 29,
"text": "\\omega_{ref}"
},
{
"math_id": 30,
"text": "\\theta_{ref}(t)"
},
{
"math_id": 31,
"text": "t_0 = 0"
},
{
"math_id": 32,
"text": "t_0^{\\rm middle}"
},
{
"math_id": 33,
"text": "i(0)=0"
},
{
"math_id": 34,
"text": "t_0^{\\rm middle}=0"
},
{
"math_id": 35,
"text": "t_1"
},
{
"math_id": 36,
"text": "\\{t_k\\}"
},
{
"math_id": 37,
"text": "\\{t_k^{\\rm middle}\\}"
},
{
"math_id": 38,
"text": "k=0,1,2..."
},
{
"math_id": 39,
"text": "t_k < t_k^{\\rm middle}"
},
{
"math_id": 40,
"text": "t \\in [t_k,t_k^{\\rm middle})"
},
{
"math_id": 41,
"text": "\\text{sign}(i(t))"
},
{
"math_id": 42,
"text": "\\pm1"
},
{
"math_id": 43,
"text": "\\tau_k"
},
{
"math_id": 44,
"text": " \\tau_k = (t_k^{\\rm middle} - t_k)\\text{sign}(i(t)) "
},
{
"math_id": 45,
"text": " t \\in [t_k,t_k^{\\rm middle}) "
},
{
"math_id": 46,
"text": " \\tau_k = 0 "
},
{
"math_id": 47,
"text": " t_k=t_k^{\\rm middle} "
},
{
"math_id": 48,
"text": "\\tau_k < 0"
},
{
"math_id": 49,
"text": "\\tau_k > 0"
},
{
"math_id": 50,
"text": "i(t) \\equiv 0"
},
{
"math_id": 51,
"text": "(t_k^{\\rm middle},t_{k+1})"
},
{
"math_id": 52,
"text": " v_F(t) \\equiv v_k "
},
{
"math_id": 53,
"text": " t \\in [t_k^{\\rm middle},t_{k+1}) "
},
{
"math_id": 54,
"text": "(\\tau_k,v_k)"
},
{
"math_id": 55,
"text": "\n p_k = \\frac{\\tau_k}{T_{\\rm ref}}, \n u_k=T_{\\rm ref}\n (\n \\omega_{\\rm vco}^{\\text{free}} + K_{\\rm vco}v_k\n ) - 1,\n"
},
{
"math_id": 56,
"text": "\n \\alpha = K_{\\rm vco}I_pT_{\\rm ref}R,\n \\beta = \\frac{K_{\\rm vco}I_pT_{\\rm ref}^2}{2C}.\n"
},
{
"math_id": 57,
"text": "p_k"
},
{
"math_id": 58,
"text": "u_k+1"
},
{
"math_id": 59,
"text": "\\omega_{\\rm vco}^{\\text{free}} + K_{\\rm vco}v_k"
},
{
"math_id": 60,
"text": "\\frac{1}{T_{\\rm ref}}"
},
{
"math_id": 61,
"text": "\n \\begin{align}\n & u_{k+1} = u_k +2\\beta p_{k+1},\\\\\n & p_{k+1} =\n \\begin{cases}\n \\frac{-(u_k + \\alpha + 1) + \\sqrt{(u_k + \\alpha + 1)^2 - 4\\beta c_k}}{2\\beta},\n \\quad \\text{ for } p_k \\geq 0, \\quad c_k \\leq 0,\n \\\\\n \\frac{1}{ u_k + 1} -1 + ( p_k \\text{ mod }1),\n \\quad \\text{ for } p_k \\geq 0, \\quad c_k > 0,\n \\\\\n l_k-1,\n \\quad \\text{ for } p_k < 0, \\quad l_k \\leq 1,\n \\\\\n \\frac{-(u_k + \\alpha + 1) + \\sqrt{(u_k + \\alpha + 1)^2 - 4\\beta d_k}}{2\\beta},\n \\quad \\text{ for } p_k < 0, \\quad l_k > 1,\n \\end{cases}\n \\end{align}\n"
},
{
"math_id": 62,
"text": "\n\\begin{align}\n c_k = (1 - ( p_k \\text{ mod }1))( u_k +1) - 1, \n S_{l_k} = -( u_k - \\alpha + 1 ) p_k + \\beta p_k^2, \n l_k = \\frac{1 - (S_{l_k} \\text{ mod }1)}{ u_k + 1},\n d_k = (S_{l_k} \\text{ mod }1) + u_k.\n\\end{align}\n"
},
{
"math_id": 63,
"text": "(u_k=0,p_k=0)"
},
{
"math_id": 64,
"text": " \\dot\\theta_{\\rm vco}(t)"
},
{
"math_id": 65,
"text": " (p_k>0, u_k<2\\beta p_k-1)"
},
{
"math_id": 66,
"text": "(p_k<0, u_k<\\alpha-1)"
}
] |
https://en.wikipedia.org/wiki?curid=64123397
|
64125867
|
Total subset
|
Subset T of a topological vector space X where the linear span of T is a dense subset of X
In mathematics, more specifically in functional analysis, a subset formula_0 of a topological vector space formula_1 is said to be a total subset of formula_1 if the linear span of formula_0 is a dense subset of formula_2
This condition arises frequently in many theorems of functional analysis.
Examples.
Unbounded self-adjoint operators on Hilbert spaces are defined on total subsets.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "T"
},
{
"math_id": 1,
"text": "X"
},
{
"math_id": 2,
"text": "X."
}
] |
https://en.wikipedia.org/wiki?curid=64125867
|
64129234
|
Mercier criterion
|
Plasma criteria
The Mercier criterion is a criterion for plasma used in the theoretical study of plasma instability. It was first proposed in 1954 by C. Mercier, who applied the perturbation method (in which formula_0 represents the frequency
and formula_1 is the z-direction of the unit vector) to the plasma mathematical model to compute calculations.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\omega =\\frac {\\partial } {\\partial t}"
},
{
"math_id": 1,
"text": "\\mathbf k = \\frac {\\partial} {\\partial z}"
}
] |
https://en.wikipedia.org/wiki?curid=64129234
|
641307
|
Harshad number
|
Integer divisible by sum of its digits
In mathematics, a harshad number (or Niven number) in a given number base is an integer that is divisible by the sum of its digits when written in that base.
Harshad numbers in base n are also known as n-harshad (or n-Niven) numbers.
Harshad numbers were defined by D. R. Kaprekar, a mathematician from India. The word "harshad" comes from the Sanskrit ' (joy) + ' (give), meaning joy-giver. The term "Niven number" arose from a paper delivered by Ivan M. Niven at a conference on number theory in 1977.
Definition.
Stated mathematically, let X be a positive integer with m digits when written in base n, and let the digits be formula_0 (formula_1). (It follows that formula_0 must be either zero or a positive integer up to &NoBreak;&NoBreak;.) X can be expressed as
formula_2
X is a harshad number in base n if:
formula_3
A number which is a harshad number in every number base is called an all-harshad number, or an all-Niven number. There are only four all-harshad numbers: 1, 2, 4, and 6. The number 12 is a harshad number in all bases except octal.
Properties.
Given the divisibility test for 9, one might be tempted to generalize that all numbers divisible by 9 are also harshad numbers. But for the purpose of determining the harshadness of n, the digits of n can only be added up once and n must be divisible by that sum; otherwise, it is not a harshad number. For example, 99 is not a harshad number, since 9 + 9 = 18, and 99 is not divisible by 18.
The base number (and furthermore, its powers) will always be a harshad number in its own base, since it will be represented as "10" and 1 + 0 = 1.
All numbers whose base "b" digit sum divides "b"−1 are harshad numbers in base "b".
For a prime number to also be a harshad number it must be less than or equal to the base number, otherwise the digits of the prime will add up to a number that is more than 1, but less than the prime, and will not be divisible. For example: 11 is not harshad in base 10 because the sum of its digits “11” is 1 + 1 = 2, and 11 is not divisible by 2; while in base 12 the number 11 may be represented as “B”, the sum of whose digits is also B. Since B is divisible by itself, it is harshad in base 12.
Although the sequence of factorials starts with harshad numbers in base 10, not all factorials are harshad numbers. 432! is the first that is not. (432! has digit sum 3897 = 32 × 433 in base 10, thus not dividing 432!)
The smallest k such that formula_4 is a harshad number are
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 9, 3, 2, 3, 6, 1, 6, 1, 1, 5, 9, 1, 2, 6, 1, 3, 9, 1, 12, 6, 4, 3, 2, 1, 3, 3, 3, 1, 10, 1, 12, 3, 1, 5, 9, 1, 8, 1, 2, 3, 18, 1, 2, 2, 2, 9, 9, 1, 12, 6, 1, 3, 3, 2, 3, 3, 3, 1, 18, 1, 7, 3, 2, 2, 4, 2, 9, 1, ... (sequence in the OEIS).
The smallest k such that formula_4 is not a harshad number are
11, 7, 5, 4, 3, 11, 2, 2, 11, 13, 1, 8, 1, 1, 1, 1, 1, 161, 1, 8, 5, 1, 1, 4, 1, 1, 7, 1, 1, 13, 1, 1, 1, 1, 1, 83, 1, 1, 1, 4, 1, 4, 1, 1, 11, 1, 1, 2, 1, 5, 1, 1, 1, 537, 1, 1, 1, 1, 1, 83, 1, 1, 3, 1, 1, 1, 1, 1, 1, 5, 1, 68, 1, 1, 1, 1, 1, 1, 1, 2, ... (sequence in the OEIS).
Other bases.
The harshad numbers in base 12 are:
1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, 10, 1A, 20, 29, 30, 38, 40, 47, 50, 56, 60, 65, 70, 74, 80, 83, 90, 92, A0, A1, B0, 100, 10A, 110, 115, 119, 120, 122, 128, 130, 134, 137, 146, 150, 153, 155, 164, 172, 173, 182, 191, 1A0, 1B0, 1BA, 200, ...
where A represents ten and B represents eleven.
Smallest k such that formula_4 is a base-12 harshad number are (written in base 10):
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 12, 6, 4, 3, 10, 2, 11, 3, 4, 1, 7, 1, 12, 6, 4, 3, 11, 2, 11, 3, 1, 5, 9, 1, 12, 11, 4, 3, 11, 2, 11, 1, 4, 4, 11, 1, 16, 6, 4, 3, 11, 2, 1, 3, 11, 11, 11, 1, 12, 11, 5, 7, 9, 1, 7, 3, 3, 9, 11, 1, ...
Smallest k such that formula_4 is not a base-12 harshad number are (written in base 10):
13, 7, 5, 4, 3, 3, 2, 2, 2, 2, 13, 16, 1, 1, 1, 1, 1, 1, 1, 1, 1, 157, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 13, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 157, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1885, 1, 1, 1, 1, 1, 3, ...
Similar to base 10, not all factorials are harshad numbers in base 12. After 7! (= 5040 = 2B00 in base 12, with digit sum 13 in base 12, and 13 does not divide 7!), 1276! is the next that is not. (1276! has digit sum 14201 = 11 × 1291 in base 12, thus does not divide 1276!)
Consecutive harshad numbers.
Maximal runs of consecutive harshad numbers.
Cooper and Kennedy proved in 1993 that no 21 consecutive integers are all harshad numbers in base 10. They also constructed infinitely many 20-tuples of consecutive integers that are all 10-harshad numbers, the smallest of which exceeds 1044363342786.
H. G. Grundman (1994) extended the Cooper and Kennedy result to show that there are 2"b" but not 2"b" + 1 consecutive "b"-harshad numbers for any base "b".
This result was strengthened to show that there are infinitely many runs of 2"b" consecutive "b"-harshad numbers for "b" = 2 or 3 by T. Cai (1996) and for arbitrary "b" by Brad Wilson in 1997.
In binary, there are thus infinitely many runs of four consecutive harshad numbers and in ternary infinitely many runs of six.
In general, such maximal sequences run from "N"·"bk" − "b" to "N"·"bk" + ("b" − 1), where "b" is the base, "k" is a relatively large power, and "N" is a constant.
Given one such suitably chosen sequence, we can convert it to a larger one as follows:
Thus our initial sequence yields an infinite set of solutions.
First runs of exactly n consecutive 10-harshad numbers.
The smallest naturals starting runs of "exactly" n consecutive 10-harshad numbers (i.e., the smallest x such that formula_5 are harshad numbers but formula_6 and formula_7 are not) are as follows (sequence in the OEIS):
<templatestyles src="alternating rows table/styles.css" />
By the previous section, no such x exists for formula_8
Estimating the density of harshad numbers.
If we let formula_9 denote the number of harshad numbers formula_10, then for any given formula_11
formula_12
as shown by Jean-Marie De Koninck and Nicolas Doyon; furthermore, De Koninck, Doyon and Kátai proved that
formula_13
where formula_14 and the formula_15 term uses Big O notation.
Sums of harshad numbers.
Every natural number not exceeding one billion is either a harshad number or the sum of two harshad numbers. Conditional to a technical hypothesis on the zeros of certain Dedekind zeta functions, Sanna proved that there exists a positive integer formula_16 such that every natural number is the sum of at most formula_16 harshad numbers, that is, the set of harshad numbers is an additive basis.
The number of ways that each natural number 1, 2, 3, ... can be written as sum of two harshad numbers is:
0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 5, 5, 5, 4, 4, 3, 3, 3, 3, 3, 4, 3, 4, 4, 4, 4, 5, 4, 5, 4, 4, 4, 3, 2, 4, 3, 3, 4, 3, 3, 5, 3, 4, 5, 4, 4, 7, 4, 5, 6, 5, 3, 7, 4, 4, 6, 4, 2, 7, 3, 4, 5, 4, 3, 7, 3, 4, 5, 4, 3, 8, 3, 4, 6, 3, 3, 6, 2, 5, 6, 5, 3, 8, 4, 4, 6, ... (sequence in the OEIS).
The smallest number that can be written in exactly 1, 2, 3, ... ways as the sum of two harshad numbers is:
2, 4, 6, 8, 10, 51, 48, 72, 108, 126, 90, 138, 144, 120, 198, 162, 210, 216, 315, 240, 234, 306, 252, 372, 270, 546, 360, 342, 444, 414, 468, 420, 642, 450, 522, 540, 924, 612, 600, 666, 630, 888, 930, 756, 840, 882, 936, 972, 1098, 1215, 1026, 1212, 1080, ... (sequence in the OEIS).
Nivenmorphic numbers.
A Nivenmorphic number or harshadmorphic number for a given number base is an integer t such that there exists some harshad number N whose digit sum is t, and t, written in that base, terminates N written in the same base.
For example, 18 is a Nivenmorphic number for base 10:
16218 is a harshad number
16218 has 18 as digit sum
18 terminates 16218
Sandro Boscaro determined that for base 10 all positive integers are Nivenmorphic numbers except 11. In fact, for an even integer "n" > 1, all positive integers except "n"+1 are Nivenmorphic numbers for base "n", and for an odd integer "n" > 1, all positive integers are Nivenmorphic numbers for base "n". e.g. the Nivenmorphic numbers in base 12 are OEIS: (all positive integers except 13).
The smallest number with base 10 digit sum "n" and terminates "n" written in base 10 are: (0 if no such number exists)
1, 2, 3, 4, 5, 6, 7, 8, 9, 910, 0, 912, 11713, 6314, 915, 3616, 15317, 918, 17119, 9920, 18921, 9922, 82823, 19824, 9925, 46826, 18927, 18928, 78329, 99930, 585931, 388832, 1098933, 198934, 289835, 99936, 99937, 478838, 198939, 1999840, 2988941, 2979942, 2979943, 999944, 999945, 4698946, 4779947, 2998848, 2998849, 9999950, ... (sequence in the OEIS)
Multiple harshad numbers.
defines a multiple harshad number as a harshad number that, when divided by the sum of its digits, produces another harshad number. He states that 6804 is "MHN-4" on the grounds that
formula_17
and went on to show that 2016502858579884466176 is MHN-12. The number 10080000000000 = 1008 × 1010, which is smaller, is also MHN-12. In general, 1008 × 10"n" is MHN-("n"+2).
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "a_i"
},
{
"math_id": 1,
"text": "i = 0, 1, \\ldots, m-1"
},
{
"math_id": 2,
"text": "X=\\sum_{i=0}^{m-1} a_i n^i."
},
{
"math_id": 3,
"text": "X \\equiv 0 \\bmod {\\sum_{i=0}^{m-1} a_i}."
},
{
"math_id": 4,
"text": "k \\cdot n"
},
{
"math_id": 5,
"text": "x, x+1, \\cdots, x+n-1"
},
{
"math_id": 6,
"text": "x-1"
},
{
"math_id": 7,
"text": "x+n"
},
{
"math_id": 8,
"text": "n > 20."
},
{
"math_id": 9,
"text": "N(x)"
},
{
"math_id": 10,
"text": "\\le x"
},
{
"math_id": 11,
"text": "\\varepsilon > 0,"
},
{
"math_id": 12,
"text": "x^{1-\\varepsilon} \\ll N(x) \\ll \\frac{x\\log\\log x}{\\log x}"
},
{
"math_id": 13,
"text": "N(x)=(c+o(1))\\frac{x}{\\log x},"
},
{
"math_id": 14,
"text": "c = (14/27) \\log 10 \\approx 1.1939"
},
{
"math_id": 15,
"text": "o(1)"
},
{
"math_id": 16,
"text": "k"
},
{
"math_id": 17,
"text": "\\begin{align}\n6804/18 &= 378\\\\\n378/18 &= 21\\\\\n21/3 &= 7\\\\\n7/7 &= 1\n\\end{align}"
},
{
"math_id": 18,
"text": "1/1=1"
}
] |
https://en.wikipedia.org/wiki?curid=641307
|
64133229
|
Order summable
|
In mathematics, specifically in order theory and functional analysis, a sequence of positive elements formula_0 in a preordered vector space formula_1 (that is, formula_2 for all formula_3) is called order summable if formula_4 exists in formula_1.
For any formula_5, we say that a sequence formula_0 of positive elements of formula_1 is of type formula_6 if there exists some formula_7 and some sequence formula_8 in formula_6 such that formula_9 for all formula_3.
The notion of order summable sequences is related to the completeness of the order topology.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\left(x_i\\right)_{i=1}^{\\infty}"
},
{
"math_id": 1,
"text": "X"
},
{
"math_id": 2,
"text": "x_i \\geq 0"
},
{
"math_id": 3,
"text": "i"
},
{
"math_id": 4,
"text": "\\sup_{n = 1, 2, \\ldots} \\sum_{i=1}^n x_i"
},
{
"math_id": 5,
"text": "1 \\leq p \\leq \\infty"
},
{
"math_id": 6,
"text": "\\ell^p"
},
{
"math_id": 7,
"text": "z \\in X"
},
{
"math_id": 8,
"text": "\\left(c_i\\right)_{i=1}^{\\infty}"
},
{
"math_id": 9,
"text": "0 \\leq x_i \\leq c_i z"
}
] |
https://en.wikipedia.org/wiki?curid=64133229
|
64162417
|
Order convergence
|
In mathematics, specifically in order theory and functional analysis, a filter formula_0 in an order complete vector lattice formula_1 is order convergent if it contains an order bounded subset (that is, a subset contained in an interval of the form formula_2) and if
formula_3
where formula_4 is the set of all order bounded subsets of "X", in which case this common value is called the order limit of formula_0 in formula_5
Order convergence plays an important role in the theory of vector lattices because the definition of order convergence does not depend on any topology.
Definition.
A net formula_6 in a vector lattice formula_1 is said to decrease to formula_7 if formula_8 implies formula_9 and formula_10 in formula_5
A net formula_6 in a vector lattice formula_1 is said to order-converge to formula_7 if there is a net formula_11 in formula_1 that decreases to formula_12 and satisfies formula_13 for all formula_14.
Order continuity.
A linear map formula_15 between vector lattices is said to be order continuous if whenever formula_6 is a net in formula_1 that order-converges to formula_16 in formula_17 then the net formula_18 order-converges to formula_19 in formula_20
formula_21 is said to be sequentially order continuous if whenever formula_22 is a sequence in formula_1 that order-converges to formula_16 in formula_17then the sequence formula_23 order-converges to formula_19 in formula_20
Related results.
In an order complete vector lattice formula_1 whose order is regular, formula_1 is of minimal type if and only if every order convergent filter in formula_1 converges when formula_1 is endowed with the order topology.
References.
<templatestyles src="Reflist/styles.css" />
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\mathcal{F}"
},
{
"math_id": 1,
"text": "X"
},
{
"math_id": 2,
"text": "[a, b] := \\{ x \\in X : a \\leq x \\text{ and } x \\leq b \\}"
},
{
"math_id": 3,
"text": "\\sup \\left\\{ \\inf S : S \\in \\operatorname{OBound}(X) \\cap \\mathcal{F} \\right\\} = \\inf \\left\\{ \\sup S : S \\in \\operatorname{OBound}(X) \\cap \\mathcal{F} \\right\\},"
},
{
"math_id": 4,
"text": "\\operatorname{OBound}(X)"
},
{
"math_id": 5,
"text": "X."
},
{
"math_id": 6,
"text": "\\left(x_{\\alpha}\\right)_{\\alpha \\in A}"
},
{
"math_id": 7,
"text": "x_0 \\in X"
},
{
"math_id": 8,
"text": "\\alpha \\leq \\beta"
},
{
"math_id": 9,
"text": "x_{\\beta} \\leq x_{\\alpha}"
},
{
"math_id": 10,
"text": "x_0 = inf \\left\\{ x_{\\alpha} : \\alpha \\in A \\right\\}"
},
{
"math_id": 11,
"text": "\\left(y_{\\alpha}\\right)_{\\alpha \\in A}"
},
{
"math_id": 12,
"text": "0"
},
{
"math_id": 13,
"text": "\\left|x_{\\alpha} - x_0\\right| \\leq y_{\\alpha}"
},
{
"math_id": 14,
"text": "\\alpha \\in A"
},
{
"math_id": 15,
"text": "T : X \\to Y"
},
{
"math_id": 16,
"text": "x_0"
},
{
"math_id": 17,
"text": "X,"
},
{
"math_id": 18,
"text": "\\left(T\\left(x_{\\alpha}\\right)\\right)_{\\alpha \\in A}"
},
{
"math_id": 19,
"text": "T\\left(x_0\\right)"
},
{
"math_id": 20,
"text": "Y."
},
{
"math_id": 21,
"text": "T"
},
{
"math_id": 22,
"text": "\\left(x_n\\right)_{n \\in \\N}"
},
{
"math_id": 23,
"text": "\\left(T\\left(x_n\\right)\\right)_{n \\in \\N}"
}
] |
https://en.wikipedia.org/wiki?curid=64162417
|
64164703
|
2 Kings 15
|
2 Kings, chapter 15
2 Kings 15 is the fifteenth chapter of the second part of the Books of Kings in the Hebrew Bible or the Second Book of Kings in the Old Testament of the Christian Bible. The book is a compilation of various annals recording the acts of the kings of Israel and Judah by a Deuteronomic compiler in the seventh century BCE, with a supplement added in the sixth century BCE. This chapter records the events during the reigns of Azariah (Uzziah) and his son, Jotham, the kings of Judah, as well as of Zechariah, Shallum, Menahem, Pekahiah and Pekah, the kings of Israel. Twelve first verses of the narrative belong to a major section 2 Kings 9:1–15:12 covering the period of Jehu's dynasty.
Text.
This chapter was originally written in the Hebrew language. It is divided into 38 verses.
Textual witnesses.
Some early manuscripts containing the text of this chapter in Hebrew are of the Masoretic Text tradition, which includes the Codex Cairensis (895), and Codex Leningradensis (1008).
There is also a translation into Koine Greek known as the Septuagint, made in the last few centuries BCE. Extant ancient manuscripts of the Septuagint version include Codex Vaticanus (B; formula_0B; 4th century) and Codex Alexandrinus (A; formula_0A; 5th century).
Structure.
This chapter can be divided into the following sections:
Analysis.
This chapter displays a contrast between the stability of the southern kingdom and the downward sliding of the northern kingdom, with two royal records of Judah bracketing the narrative of five Israel kings in quick succession. Each reign is judged using a standard formula, one for the kings of Judah (verses , ) and another for the kings of Israel (verses , , , ).
Azariah (Uzziah), king of Judah (15:1–7).
The regnal records of Azariah the son of Amaziah, the king of Judah, can be demarcated by the introductory form (verses 1–4) and the concluding form (verses 5–7). The main account is in verse 5 regarding the king's leprosy and the active role of his son, Jotham, in ruling the kingdom on his behalf, but the length of the co-regency is not explicitly recorded. The period of his reign coincides largely with the reign of Jeroboam, who ruled over a kingdom territory comparable to that of Solomon, so Azariah's kingdom was a vassal to the kingdom of Israel. provides a more detailed account of Azariah's reign, especially the reason God striking him with leprosy, his 'military actions against Philistia, the Arabs of Geur-Baal, and the Meunites', as well as 'his efforts to fortify Jerusalem and to secure the hold on the Shephelah.'
"In the twenty-seventh year of Jeroboam king of Israel, Azariah the son of Amaziah, king of Judah, began to reign."
"He was sixteen years old when he became king, and he reigned fifty-two years in Jerusalem. His mother’s name was Jecholiah of Jerusalem."
"And the Lord touched the king, so that he was a leper to the day of his death, and he lived in a separate house. And Jotham the king's son was over the household, governing the people of the land."
"And Azariah slept with his fathers, and they buried him with his fathers in the city of David, and Jotham his son reigned in his place."
Verse 7.
The time of Azariah's death coincides with the time Isaiah received his call to be a prophet ("in the year that King Uzziah died"; ).
E.L. Sukenik found an Aramaic inscription that reads, "Here were brought the bones of Uzziah, king of Judah. Do not open!" and once marked the tomb of Uzziah outside Jerusalem.
Zechariah, king of Israel (15:8–12).
Zechariah, the last ruler of the Jehu dynasty, only reigned for six months and his assassination ends a long period of stability in the kingdom of Israel. It is set in the frame of the divine guidance that God himself announced to the founder of the dynasty (2 Kings 10:30) and confirms the fulfillment of it in verse 12.
"In the thirty-eighth year of Azariah king of Judah, Zechariah the son of Jeroboam reigned over Israel in Samaria six months."
"And Shallum the son of Jabesh conspired against him, and smote him before the people, and slew him, and reigned in his stead"
"This was the word of the Lord which He spoke to Jehu, saying, “Your sons shall sit on the throne of Israel to the fourth generation.” And so it was."
Shallum, king of Israel (15:13–16).
After bringing an end to the Jehu dynasty (verse 10), Shallum could only reign for a month before he was slain by Menahem. The literary structure consists of an 'introductory regnal form' (verse ), the body of the account (verse 14) and the 'concluding regnal form' (verse ). Menahem's submission to Assyria (verses ) suggests that his action was to stop an attempt to revolt against the Assyrian by Shallum.
Menahem, king of Israel (15:17–22).
The 10-year reign of Menahem provides a 'rare period of stability' in the final years of the northern kingdom, which was the result of Menahem's tributary payment to the Assyrian king, Tiglath-Pileser III (also known as "Pul", cf. "ANET" 272). The tribute, along with those from other monarch, is listed with Menahem's name explicitly in the annals of the Assyria ("ANET" 283–284). To pay that tribute, Menahem instituted an oppressive tax, fifty shekels (about 1<templatestyles src="Fraction/styles.css" />1⁄4 pounds, or 575 grams) of silver per person from all the wealthy men in Israel (verse 20), which may contribute to the coup against his son after he died.
"In the thirty-ninth year of Azariah king of Judah, Menahem the son of Gadi began to reign over Israel, and he reigned ten years in Samaria."
"And Pul the king of Assyria came against the land: and Menahem gave Pul a thousand talents of silver, that his hand might be with him to confirm the kingdom in his hand."
Verse 19.
Tiglath-Pileser records the tribute from Menahem in one of his inscriptions ("ANET3 283").
Pekahiah, king of Israel (15:23–26).
The main regnal account of Pekahiah, the 17th king of Israel, only mentions his assassination by a group of 50 men from Gilead led by Pekah ben Remaliah, his own captain (verse 25).
"In the fiftieth year of Azariah king of Judah Pekahiah the son of Menahem began to reign over Israel in Samaria, and reigned two years."
Pekah, king of Israel (15:27–31).
The main record of Pekah's reign in this section focuses on the invasion of Tiglath-Pileser III into Israel in 734–732 BCE and his murder in a coup led by Hosea ben Elah, backed by the Assyrians, as noted in the annals of Assyria ("ANET" 284). Pekah's alliance with Rezin of Damascus in the Syro-Ephraimite War to resist the Assyrians and attack Judah, a vassal to the Assyrians, is recorded in multiple passages (verse 37, 2 Kings 16:5, 7–9; Isaiah 7:1–17; Isaiah 9:1) and also in the annals of the Assyrians ("ANET" 283–284).
"In the fifty-second year of Azariah king of Judah, Pekah the son of Remaliah became king over Israel in Samaria, and reigned twenty years."
Jotham, king of Judah (15:32–38).
Like his father (Azariah or Uzziah), Jotham was given a good assessment 'in the sight of the LORD' (verse 34; cf. verse 3), although both kings did not remove the 'high places', which was later done by Hezekiah () and Josiah (), nor perform notable political actions. Jotham's memorable achievement was the building of 'the upper gate of the house of the LORD' (verse ).
"In the second year of Pekah the son of Remaliah, king of Israel, Jotham the son of Uzziah, king of Judah, began to reign."
"He was twenty-five years old when he began to reign, and he reigned sixteen years in Jerusalem. His mother's name was Jerusha the daughter of Zadok."
Notes.
<templatestyles src="Reflist/styles.css" />
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": " \\mathfrak{G}"
}
] |
https://en.wikipedia.org/wiki?curid=64164703
|
64168728
|
Potassium telluride
|
<templatestyles src="Chembox/styles.css"/>
Chemical compound
Potassium telluride is an inorganic compound with a chemical formula K2Te. It is formed from potassium and tellurium, making it a telluride.<ref name="DOI10.1002/anie.197806841">Brigitte Eisenmann, Herbert Schäfer: "K2Te3 : The First Binary Alkali-Metal Polytelluride with Te2−3-Ions." In: "Angewandte Chemie International Edition in English." 17, 1978, S. 684, Error: Bad DOI specified!.</ref> Potassium telluride is a white powder. Like rubidium telluride and caesium telluride, it can be used as an ultraviolet detector in space. Its crystal structure is similar to other tellurides, which have an anti-fluorite structure.
Production.
Tellurium will react with melting potassium cyanide (KCN) producing potassium telluride. It can also be produced by direct combination of potassium and tellurium, usually in liquid ammonia solvent:
formula_0
Reactions.
Adding potassium telluride to water and letting the filtrate stand in air leads to an oxidation reaction that generates potassium hydroxide (KOH) and elemental tellurium:
formula_1
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\mathrm{2 \\ K + Te \\longrightarrow K_2Te}"
},
{
"math_id": 1,
"text": "\\mathrm{2 \\ K_2Te + 2 \\ H_2O + O_2 \\longrightarrow 4 \\ KOH + 2 \\ Te}"
}
] |
https://en.wikipedia.org/wiki?curid=64168728
|
64170595
|
Lomonosovite
|
Phosphate–silicate mineral
Lomonosovite is a phosphate–silicate mineral with the idealized formula Na10Ti4(Si2O7)2(PO4)2O4 early Na5Ti2(Si2O7)(PO4)O2 or Na2Ti2Si2O9*Na3PO4.
The main admixtures are niobium (up to 11.8% Nb2O5), manganese (up to 4.5 %MnO) and iron (up to 2.8%).
Discovery and name.
The mineral was discovered by V.I. Gerasimovskii in Lovozersky agpaitic massif. Named for Mikhail Lomonosov – famous Russian poet, chemist and philosopher, but the earlier – mining engineer.
Crystal structure.
According to X-ray data, lomonosovite structure was determined is triclinic unit cell with parameters: a = 5.44 Å, b = 7.163 Å, c = 14.83 Å, α = 99°, β = 106°, and γ = 90°, usually centrosymmetric (sp. gr. P-1), but acentric varieties (polytype) are also reported.
The crystal structure of lomonosovite is based on three-layer HOH packets consisting of a central octahedral O layer and two outer heteropolyhedral H layers. Ti- and Na centered octahedra are distinguished in the O layer, whereas the H layers are composed of Ti-centered octahedra and Si2O7 diorthogroups, (like in other heterophyllosilicates, for example lamprophyllite). The interpacket space includes Na+ cations and PO43- anions.
Properties.
Lomonosovite forms lamellar and tabular crystals with perfect cleavage. It is macroscopically brown, from cinnamon-brown to black. It is transparent in thin plates. The luster vitreous to adamantine.
Its pleochroism is strong from colorless to brown. The refractive index is formula_0= 1.654–1.670 formula_1 = 1.736 – 1.750 formula_2=1.764–1.778 2V=56–69.
Hardness 3–4 Density 3.12 – 3.15.
Origin.
Accessory mineral of peralkaline agpaitic nepheline syenites (like Khibina and Lovozero massif, Russia, Ilimaussaq intrusion, Greenland) important mineral of agpaitic pegmatites and peralkaline fenites.
|
[
{
"math_id": 0,
"text": "\\alpha"
},
{
"math_id": 1,
"text": "\\beta"
},
{
"math_id": 2,
"text": "\\gamma"
}
] |
https://en.wikipedia.org/wiki?curid=64170595
|
641715
|
Limaçon
|
Type of roulette curve
In geometry, a limaçon or limacon , also known as a limaçon of Pascal or Pascal's Snail, is defined as a roulette curve formed by the path of a point fixed to a circle when that circle rolls around the outside of a circle of equal radius. It can also be defined as the roulette formed when a circle rolls around a circle with half its radius so that the smaller circle is inside the larger circle. Thus, they belong to the family of curves called centered trochoids; more specifically, they are epitrochoids. The cardioid is the special case in which the point generating the roulette lies on the rolling circle; the resulting curve has a cusp.
Depending on the position of the point generating the curve, it may have inner and outer loops (giving the family its name), it may be heart-shaped, or it may be oval.
A limaçon is a bicircular rational plane algebraic curve of degree 4.
History.
The earliest formal research on limaçons is generally attributed to Étienne Pascal, father of Blaise Pascal. However, some insightful investigations regarding them had been undertaken earlier by the German Renaissance artist Albrecht Dürer. Dürer's "Underweysung der Messung (Instruction in Measurement)" contains specific geometric methods for producing limaçons. The curve was named by Gilles de Roberval when he used it as an example for finding tangent lines.
Equations.
The equation (up to translation and rotation) of a limaçon in polar coordinates has the form
formula_0
This can be converted to Cartesian coordinates by multiplying by "r" (thus introducing a point at the origin which in some cases is spurious), and substituting formula_1 and formula_2 to obtain
formula_3
Applying the parametric form of the polar to Cartesian conversion, we also have
formula_4
formula_5
while setting
formula_6
yields this parameterization as a curve in the complex plane:
formula_7
If we were to shift horizontally by formula_8, i.e.,
formula_9,
we would, by changing the location of the origin, convert to the usual form of the equation of a centered trochoid. Note the change of independent variable at this point to make it clear that we are no longer using the default polar coordinate parameterization formula_10.
Special cases.
In the special case formula_11, the polar equation is
formula_12
or
formula_13
making it a member of the sinusoidal spiral family of curves. This curve is the cardioid.
In the special case formula_14, the centered trochoid form of the equation becomes
formula_15
or, in polar coordinates,
formula_16
making it a member of the rose family of curves. This curve is a trisectrix, and is sometimes called the limaçon trisectrix.
Form.
When formula_17, the limaçon is a simple closed curve. However, the origin satisfies the Cartesian equation given above, so the graph of this equation has an acnode or isolated point.
When formula_18, the area bounded by the curve is convex, and when formula_19, the curve has an indentation bounded by two inflection points. At formula_20, the point formula_21 is a point of 0 curvature.
As formula_22 is decreased relative to formula_23, the indentation becomes more pronounced until, at formula_24, the curve becomes a cardioid, and the indentation becomes a cusp. For formula_25, the cusp expands to an inner loop, and the curve crosses itself at the origin. As formula_22 approaches 0, the loop fills up the outer curve and, in the limit, the limaçon becomes a circle traversed twice.
Measurement.
The area enclosed by the limaçon formula_26 is formula_27. When formula_28 this counts the area enclosed by the inner loop twice. In this case the curve crosses the origin at angles formula_29, the area enclosed by the inner loop is
formula_30
the area enclosed by the outer loop is
formula_31
and the area between the loops is
formula_32
The circumference of the limaçon is given by a complete elliptic integral of the second kind:
formula_33
formula_37
which is the equation of a conic section with eccentricity formula_38 and focus at the origin. Thus a limaçon can be defined as the inverse of a conic where the center of inversion is one of the foci. If the conic is a parabola then the inverse will be a cardioid, if the conic is a hyperbola then the corresponding limaçon will have an inner loop, and if the conic is an ellipse then the corresponding limaçon will have no loop.
|
[
{
"math_id": 0,
"text": "r = b + a \\cos \\theta."
},
{
"math_id": 1,
"text": "r^2 = x^2+y^2"
},
{
"math_id": 2,
"text": "r \\cos \\theta = x"
},
{
"math_id": 3,
"text": "\\left(x^2 + y^2 - ax\\right)^2 = b^2\\left(x^2 + y^2\\right)."
},
{
"math_id": 4,
"text": "x = (b + a\\cos \\theta)\\cos \\theta = {a\\over 2} + b \\cos \\theta + {a\\over 2} \\cos 2 \\theta,"
},
{
"math_id": 5,
"text": "y = (b + a\\cos \\theta)\\sin \\theta = b \\sin \\theta + {a\\over 2} \\sin 2 \\theta;"
},
{
"math_id": 6,
"text": "z = x + i y = ( b + a \\cos \\theta )( \\cos \\theta + i \\sin \\theta )"
},
{
"math_id": 7,
"text": "z = {a \\over 2} + b e^{i\\theta} + {a\\over 2} e^{2i\\theta}."
},
{
"math_id": 8,
"text": "-\\frac{1}{2}a"
},
{
"math_id": 9,
"text": "z = b e^{it} + {a \\over 2} e^{2it}"
},
{
"math_id": 10,
"text": "\\theta = \\arg z"
},
{
"math_id": 11,
"text": "a = b"
},
{
"math_id": 12,
"text": "r = b(1 + \\cos \\theta) = 2b\\cos^2 \\frac{\\theta}{2}"
},
{
"math_id": 13,
"text": "r^{1 \\over 2} = (2b)^{1 \\over 2} \\cos \\frac{\\theta}{2},"
},
{
"math_id": 14,
"text": "a = 2b"
},
{
"math_id": 15,
"text": "z = b \\left(e^{it} + e^{2it}\\right) = b e^{3it\\over 2} \\left(e^{it\\over 2} + e^{-it\\over 2}\\right) = 2b e^{3it\\over 2} \\cos{t \\over 2},"
},
{
"math_id": 16,
"text": "r = 2b\\cos{\\theta \\over 3}"
},
{
"math_id": 17,
"text": "b > a"
},
{
"math_id": 18,
"text": "b > 2a"
},
{
"math_id": 19,
"text": "a < b < 2a"
},
{
"math_id": 20,
"text": "b = 2a"
},
{
"math_id": 21,
"text": "(-a, 0)"
},
{
"math_id": 22,
"text": "b"
},
{
"math_id": 23,
"text": "a"
},
{
"math_id": 24,
"text": "b = a"
},
{
"math_id": 25,
"text": "0 < b < a"
},
{
"math_id": 26,
"text": "r = b + a \\cos \\theta"
},
{
"math_id": 27,
"text": "\\left(b^2 + {{a^2} \\over 2}\\right)\\pi"
},
{
"math_id": 28,
"text": "b < a"
},
{
"math_id": 29,
"text": "\\pi \\pm \\arccos {b \\over a}"
},
{
"math_id": 30,
"text": " \\left (b^2 + {{a^2}\\over 2} \\right )\\arccos {b \\over a} - {3\\over 2} b \\sqrt{a^2 - b^2},"
},
{
"math_id": 31,
"text": " \\left(b^2 + {{a^2}\\over 2} \\right ) \\left (\\pi - \\arccos {b \\over a} \\right ) + {3\\over 2} b \\sqrt{a^2 -b^2},"
},
{
"math_id": 32,
"text": " \\left (b^2 + {{a^2}\\over 2} \\right ) \\left (\\pi - 2\\arccos {b \\over a} \\right ) + 3b \\sqrt{a^2 -b^2}."
},
{
"math_id": 33,
"text": " 4(a + b)E\\left({{2\\sqrt{ab}} \\over a + b}\\right)."
},
{
"math_id": 34,
"text": "P"
},
{
"math_id": 35,
"text": "C"
},
{
"math_id": 36,
"text": "(a, 0)"
},
{
"math_id": 37,
"text": "r = {1 \\over {b + a \\cos \\theta}}"
},
{
"math_id": 38,
"text": "\\tfrac{a}{b}"
}
] |
https://en.wikipedia.org/wiki?curid=641715
|
64172078
|
Ordered algebra
|
In mathematics, an ordered algebra is an algebra over the real numbers formula_0 with unit "e" together with an associated order such that "e" is positive (i.e. "e" ≥ 0), the product of any two positive elements is again positive, and when "A" is considered as a vector space over formula_0 then it is an Archimedean ordered vector space.
Properties.
Let "A" be an ordered algebra with unit "e" and let "C"* denote the cone in "A"* (the algebraic dual of "A") of all positive linear forms on "A".
If "f" is a linear form on "A" such that "f"("e") = 1 and "f" generates an extreme ray of "C"* then "f" is a multiplicative homomorphism.
Results.
Stone's Algebra Theorem: Let "A" be an ordered algebra with unit "e" such that "e" is an order unit in "A", let "A"* denote the algebraic dual of "A", and let "K" be the formula_1-compact set of all multiplicative positive linear forms satisfying "f"("e") = 1. Then under the evaluation map, "A" is isomorphic to a dense subalgebra of formula_2. If in addition every positive sequence of type "l"1 in "A" is order summable then "A" together with the Minkowski functional "p""e" is isomorphic to the Banach algebra formula_2.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\mathbb{R}"
},
{
"math_id": 1,
"text": "\\sigma\\left( A^{*}, A \\right)"
},
{
"math_id": 2,
"text": "C_{\\mathbb{R}}(X)"
}
] |
https://en.wikipedia.org/wiki?curid=64172078
|
64175714
|
Kan-Thurston theorem
|
In mathematics, particularly algebraic topology, the Kan-Thurston theorem associates a discrete group formula_0 to every path-connected topological space formula_1 in such a way that the group cohomology of formula_0 is the same as the cohomology of the space formula_1. The group formula_0 might then be regarded as a good approximation to the space formula_1, and consequently the theorem is sometimes interpreted to mean that homotopy theory can be viewed as part of group theory.
More precisely, the theorem states that every path-connected topological space is homology-equivalent to the classifying space formula_2 of a discrete group formula_0, where homology-equivalent means there is a map formula_3 inducing an isomorphism on homology.
The theorem is attributed to Daniel Kan and William Thurston who published their result in 1976.
Statement of the Kan-Thurston theorem.
Let formula_1 be a path-connected topological space. Then, naturally associated to formula_1, there is a Serre fibration formula_4 where formula_5 is an aspherical space. Furthermore,
Notes.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "G"
},
{
"math_id": 1,
"text": "X"
},
{
"math_id": 2,
"text": "K(G,1)"
},
{
"math_id": 3,
"text": "K(G,1) \\rightarrow X"
},
{
"math_id": 4,
"text": "<Math>t_x \\colon T_X \\to X</Math>"
},
{
"math_id": 5,
"text": "T_X"
},
{
"math_id": 6,
"text": "<Math>\\pi_1(T_X) \\to \\pi_1(X)</math>"
},
{
"math_id": 7,
"text": "A"
},
{
"math_id": 8,
"text": "H_*(TX;A) \\to H_*(X;A)"
},
{
"math_id": 9,
"text": "H^*(TX;A) \\to H^*(X;A)"
},
{
"math_id": 10,
"text": "t_x"
}
] |
https://en.wikipedia.org/wiki?curid=64175714
|
64176408
|
2 Kings 17
|
2 Kings, chapter 17
2 Kings 17 is the seventeenth chapter of the second part of the Books of Kings in the Hebrew Bible or the Second Book of Kings in the Old Testament of the Christian Bible. The book is a compilation of various annals recording the acts of the kings of Israel and Judah by a Deuteronomic compiler in the seventh century BCE, with a supplement added in the sixth century BCE. This chapter records the events during the reigns of Hoshea the last king of Israel, the capture of Samaria and the deportation of the northern kingdom population by the Assyrians.
Text.
This chapter was originally written in the Hebrew language. It is divided into 41 verses.
Textual witnesses.
Some early manuscripts containing the text of this chapter in Hebrew are of the Masoretic Text tradition, which includes the Codex Cairensis (895), and Codex Leningradensis (1008).
There is also a translation into Koine Greek known as the Septuagint, made in the last few centuries BCE. Extant ancient manuscripts of the Septuagint version include Codex Vaticanus (B; formula_0B; 4th century) and Codex Alexandrinus (A; formula_0A; 5th century).
Structure.
This chapter can be divided into the following sections:
The skeletal narrative structure in this chapter is shaped by the actions of the king of Assyria, with the narrative followed by the commentary (twice):
Analysis.
This chapter provides a significant theological interpretation of Israel history connecting the long chronicles of the sin of the nation to the resulting divine punishment with the fall of the northern kingdom, as reflected by a 'dense concentration of Deuteronomistic language'. It also gives a glimpse to Judah's eventual fate, linking to other 'dense concentrations of Deuteronomistic judgment language' in ; ; . The northern prophets, Amos and Hosea, provide additional reflection on the reasons for the judgment.
Hoshea, king of Israel (17:1–6).
The regnal records of Hoshea, the last king of Israel, is evaluated less negatively than the previous kings of the northern kingdom, but his deeds are still 'evil in the sight of the Lord.' Hoshea's shift of allegiance from Assyria to Egypt has a disastrous consequence. Shalmaneser V, the king of Assyria, soon went up against Hoshea and laid siege on Samaria that last for three years, but Sargon II made the claim in his annals to have taken Samaria ("ANET" 284–285).
"In the twelfth year of Ahaz king of Judah reigned hath Hoshea son of Elah in Samaria, over Israel -- nine years."
"Against him came up Shalmaneser king of Assyria. And Hoshea became his vassal and paid him tribute."
"And the king of Assyria found conspiracy in Hoshea: for he had sent messengers to So king of Egypt, and brought no present to the king of Assyria, as he had done year by year: therefore the king of Assyria shut him up, and bound him in prison."
"In the ninth year of Hoshea, the king of Assyria captured Samaria, and he carried the Israelites away to Assyria and placed them in Halah, and on the Habor, the river of Gozan, and in the cities of the Medes."
Verse 6.
The deportees were displaced decentrally to various location in the north-east Syria, effectively destroying the races, so the exiled northern Israelite people left few traces in history and tradition (becoming "Ten Lost Tribes" of Israel), unlike the Jews (the people of Judah) who were later moved "en bloc" to Babylon.
Theological cause of the catastrophe (17:7–23).
The exposition in this section consists of two parts: about Israel (verses 7–18) and involving Judah (verses 19–23). The first part is marked by the term "because" of verse 7 to the "therefore" in the beginning of verse 18:
A general indictment (verses 7–8)
B specific crimes (verses (9–12)
C prophetic warning unheeded (verses 13–14)
A' general indictment (verses 15–16a)
B' specific crimes (verses 16b–17)
C' result (verse 18)
In the second part, the idolatry in kingdom of Judah is coordinated with that in the northern kingdom (verse 19; cf. verse 13), although the narrator at this point only hints the demise of Judah (as the punishment for its sins).
The immigrants from the east and their cults (17:24–41).
Following the principle of destroying races in the conquered territory, the Assyrians not only displaced the Israelites from their land, but also deported people from other lands into Israel. The places listed in verses 24, 29–41 are partly in Mesopotamia and partly in Syria. This mixing of ethnicity would avoid the development of large-scale resistance and 'paralyse the regions using the tension between people' of different origins. The Deuteronomistic narrative focuses on the religious impacts of this policy, that 'the religion (gods and ritual traditions) in the province of Samaria 'became mixed'. It is noted that the worship of YHWH still exists, but 'united syncretistically' with other religions (verses 32–34, 41), as explained using the episode recorded in verses 25–28.
Notes.
<templatestyles src="Reflist/styles.css" />
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": " \\mathfrak{G}"
}
] |
https://en.wikipedia.org/wiki?curid=64176408
|
64177800
|
Moduli of abelian varieties
|
Abelian varieties are a natural generalization of elliptic curves, including algebraic tori in higher dimensions. Just as elliptic curves have a natural moduli space formula_0 over characteristic 0 constructed as a quotient of the upper-half plane by the action of formula_1, there is an analogous construction for abelian varieties formula_2 using the Siegel upper half-space and the symplectic group formula_3.
Constructions over characteristic 0.
Principally polarized Abelian varieties.
Recall that the Siegel upper-half plane is given byformula_4which is an open subset in the formula_5 symmetric matrices (since formula_6 is an open subset of formula_7, and formula_8 is continuous). Notice if formula_9 this gives formula_10 matrices with positive imaginary part, hence this set is a generalization of the upper half plane. Then any point formula_11 gives a complex torus formula_12with a principal polarization formula_13 from the matrix formula_14page 34. It turns out all principally polarized Abelian varieties arise this way, giving formula_15 the structure of a parameter space for all principally polarized Abelian varieties. But, there exists an equivalence whereformula_16 for formula_17hence the moduli space of principally polarized abelian varieties is constructed from the stack quotientformula_18which gives a Deligne-Mumford stack over formula_19. If this is instead given by a GIT quotient, then it gives the coarse moduli space formula_20.
Principally polarized Abelian varieties with level "n"-structure.
In many cases, it is easier to work with the moduli space of principally polarized Abelian varieties with level "n"-structure because it creates a rigidification of the moduli problem which gives a moduli functor instead of a moduli stack. This means the functor is representable by an algebraic manifold, such as a variety or scheme, instead of a stack. A level "n"-structure is given by a fixed basis of
formula_21
where formula_22 is the lattice formula_23. Fixing such a basis removes the automorphisms of an abelian variety at a point in the moduli space, hence there exists a bona-fide algebraic manifold without a stabilizer structure. Denoteformula_24and defineformula_25as a quotient variety.
|
[
{
"math_id": 0,
"text": "\\mathcal{M}_{1,1}"
},
{
"math_id": 1,
"text": "SL_2(\\mathbb{Z})"
},
{
"math_id": 2,
"text": "\\mathcal{A}_g"
},
{
"math_id": 3,
"text": "\\operatorname{Sp}_{2g}(\\mathbb{Z})"
},
{
"math_id": 4,
"text": "H_g = \\{ \\Omega \\in \\operatorname{Mat}_{g,g}(\\mathbb{C}) : \\Omega^T =\\Omega, \\operatorname{Im}(\\Omega) > 0 \\} \\subseteq \\operatorname{Sym}_g(\\mathbb{C})"
},
{
"math_id": 5,
"text": "g\\times g"
},
{
"math_id": 6,
"text": "\\operatorname{Im}(\\Omega) > 0"
},
{
"math_id": 7,
"text": "\\mathbb{R}"
},
{
"math_id": 8,
"text": "\\operatorname{Im}"
},
{
"math_id": 9,
"text": "g=1"
},
{
"math_id": 10,
"text": "1\\times 1"
},
{
"math_id": 11,
"text": "\\Omega \\in H_g"
},
{
"math_id": 12,
"text": "X_\\Omega = \\mathbb{C}^g/(\\Omega\\mathbb{Z}^g + \\mathbb{Z}^g)"
},
{
"math_id": 13,
"text": "H_\\Omega"
},
{
"math_id": 14,
"text": "\\Omega^{-1}"
},
{
"math_id": 15,
"text": "H_g"
},
{
"math_id": 16,
"text": "X_\\Omega \\cong X_{\\Omega'} \\iff \\Omega = M\\Omega'"
},
{
"math_id": 17,
"text": "M \\in \\operatorname{Sp}_{2g}(\\mathbb{Z})"
},
{
"math_id": 18,
"text": "\\mathcal{A}_g = [\\operatorname{Sp}_{2g}(\\mathbb{Z})\\backslash H_g]"
},
{
"math_id": 19,
"text": "\\operatorname{Spec}(\\mathbb{C})"
},
{
"math_id": 20,
"text": "A_g"
},
{
"math_id": 21,
"text": "H_1(X_\\Omega, \\mathbb{Z}/n) \\cong \\frac{1}{n}\\cdot L/L \\cong n\\text{-torsion of } X_\\Omega"
},
{
"math_id": 22,
"text": "L"
},
{
"math_id": 23,
"text": "\\Omega\\mathbb{Z}^g + \\mathbb{Z}^g \\subset \\mathbb{C}^{2g}"
},
{
"math_id": 24,
"text": "\\Gamma(n) = \\ker [\\operatorname{Sp}_{2g}(\\mathbb{Z}) \\to \\operatorname{Sp}_{2g}(\\mathbb{Z})/n]"
},
{
"math_id": 25,
"text": "A_{g,n} = \\Gamma(n)\\backslash H_g"
}
] |
https://en.wikipedia.org/wiki?curid=64177800
|
6417882
|
Weight (strings)
|
The formula_0-weight of a string, for a letter formula_0, is the number of times that letter occurs in the string. More precisely, let formula_1 be a finite set (called the "alphabet"), formula_2 a "letter" of formula_1, and formula_3 a
"string" (where formula_4 is the free monoid generated by the elements of formula_1, equivalently the set of strings, including the empty string, whose letters are from formula_1). Then the formula_0-"weight" of formula_5, denoted by formula_6, is the number of times the generator formula_0 occurs in the unique expression for formula_5 as a product (concatenation) of letters in formula_1.
If formula_1 is an abelian group, the Hamming weight formula_7 of formula_5,
often simply referred to as "weight", is the number of nonzero letters in formula_5.
Examples.
"This article incorporates material from Weight (strings) on PlanetMath, which is licensed under the ."
|
[
{
"math_id": 0,
"text": "a"
},
{
"math_id": 1,
"text": "A"
},
{
"math_id": 2,
"text": "a\\in A"
},
{
"math_id": 3,
"text": "c\\in A^*"
},
{
"math_id": 4,
"text": "A^*"
},
{
"math_id": 5,
"text": "c"
},
{
"math_id": 6,
"text": "\\mathrm{wt}_a(c)"
},
{
"math_id": 7,
"text": "\\mathrm{wt}(c)"
},
{
"math_id": 8,
"text": "A=\\{x,y,z\\}"
},
{
"math_id": 9,
"text": "c=yxxzyyzxyzzyx"
},
{
"math_id": 10,
"text": "y"
},
{
"math_id": 11,
"text": "\\mathrm{wt}_y(c)=5"
},
{
"math_id": 12,
"text": "A=\\mathbf{Z}_3=\\{0,1,2\\}"
},
{
"math_id": 13,
"text": "c=002001200"
},
{
"math_id": 14,
"text": "\\mathrm{wt}_0(c)=6"
},
{
"math_id": 15,
"text": "\\mathrm{wt}_1(c)=1"
},
{
"math_id": 16,
"text": "\\mathrm{wt}_2(c)=2"
},
{
"math_id": 17,
"text": "\\mathrm{wt}(c)=\\mathrm{wt}_1(c)+\\mathrm{wt}_2(c)=3"
}
] |
https://en.wikipedia.org/wiki?curid=6417882
|
64189898
|
Lithium selenide
|
Chemical compound
<templatestyles src="Chembox/styles.css"/>
Chemical compound
Properties.
Lithium selenide is an inorganic compound that formed by selenium and lithium. It is a selenide with a chemical formula Li2Se. Lithium selenide has the same crystal form as other selenides, which is cubic, belonging to the anti-fluorite structure, the space group is formula_0, each unit cell has 4 units.
Synthesis.
Lithium Selenide can be synthesized via the reaction between 1.0 equivalents of grey elemental selenium and 2.1 equivalents of lithium trialkylborohydride. The reaction takes place in a solution of THF (tetrahydrofuran) under with stirring (minimum of 20 minutes) at room temperature according to the reaction below: To increase yields and harmful byproducts, naphthalene can be added to the reaction as a catalyst.
"Se + 2Li(C2H5)3BH → Li2Se + 2(C2H5)3B + H2"
Another method of synthesis involves the reduction of selenium with lithium in liquid ammonia. The "Li2Se" can be extracted after evaporation of the ammonia.
Uses.
One of the most contemporary uses of "Li2Se" compounds is in the creation of high-density capacitors and batteries. Lithium selenide can act as an excellent prelithiation agent, which helps to prevent the loss of capacity and efficiency during the formation of the solid electrolyte interphase (SEI). Additionally, the high relative conductivity and solubility of the products of lithium selenide decomposition makes it an ideal prelithiation agent. No harmful byproducts or gases are created during this decomposition of "Li2Se". One potential drawback to the use of "Li2Se" is the dissolution and shuttle problems inherent to the transition metals like selenide. To avoid this problem, evolving heterostructure materials can be used to inhibit the dissolution and shuttle effects of "Li2Se".
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "Fm\\bar{3}m"
}
] |
https://en.wikipedia.org/wiki?curid=64189898
|
6419365
|
Kolmogorov backward equations (diffusion)
|
The Kolmogorov backward equation (KBE) (diffusion) and its adjoint sometimes known as the Kolmogorov forward equation (diffusion) are partial differential equations (PDE) that arise in the theory of continuous-time continuous-state Markov processes. Both were published by Andrey Kolmogorov in 1931. Later it was realized that the forward equation was already known to physicists under the name Fokker–Planck equation; the KBE on the other hand was new.
Informally, the Kolmogorov forward equation addresses the following problem. We have information about the state "x" of the system at time "t" (namely a probability distribution formula_0); we want to know the probability distribution of the state at a later time formula_1. The adjective 'forward' refers to the fact that formula_0 serves as the initial condition and the PDE is integrated forward in time (in the common case where the initial state is known exactly, formula_0 is a Dirac delta function centered on the known initial state).
The Kolmogorov backward equation on the other hand is useful when we are interested at time "t" in whether at a future time "s" the system will be in a given subset of states "B", sometimes called the "target set". The target is described by a given function formula_2 which is equal to 1 if state "x" is in the target set at time "s", and zero otherwise. In other words, formula_3, the indicator function for the set "B". We want to know for every state "x" at time formula_4 what is the probability of ending up in the target set at time "s" (sometimes called the hit probability). In this case formula_2 serves as the final condition of the PDE, which is integrated backward in time, from "s" to "t".
Formulating the Kolmogorov backward equation.
Assume that the system state formula_5 evolves according to the stochastic differential equation
formula_6
then the Kolmogorov backward equation is
formula_7
for formula_8, subject to the final condition formula_9.
This can be derived using Itō's lemma on formula_10 and setting the formula_11 term equal to zero.
This equation can also be derived from the Feynman–Kac formula by noting that the hit probability is the same as the expected value of the indicator function formula_12 over all paths that originate from state formula_13 at time formula_14:
formula_15
Historically, the KBE was developed before the Feynman–Kac formula (1949).
Formulating the Kolmogorov forward equation.
With the same notation as before, the corresponding Kolmogorov forward equation is
formula_16
for formula_17, with initial condition formula_18. For more on this equation see Fokker–Planck equation.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "p_t(x)"
},
{
"math_id": 1,
"text": "s>t"
},
{
"math_id": 2,
"text": "u_s(x)"
},
{
"math_id": 3,
"text": "u_s(x) = 1_B "
},
{
"math_id": 4,
"text": "t,\\ (t<s)"
},
{
"math_id": 5,
"text": "X_t"
},
{
"math_id": 6,
"text": "dX_t = \\mu(X_t,t)\\,dt + \\sigma(X_t,t)\\,dW_t\\,,"
},
{
"math_id": 7,
"text": "\\frac{\\partial}{\\partial t}p(x,t)=-\\mu(x,t)\\frac{\\partial}{\\partial x}p(x,t) - \\frac{1}{2}\\sigma^2(x,t)\\frac{\\partial^2}{\\partial x^{2}}p(x,t),"
},
{
"math_id": 8,
"text": "t\\le s"
},
{
"math_id": 9,
"text": "p(x,s)=u_s(x)"
},
{
"math_id": 10,
"text": " p(x,t) "
},
{
"math_id": 11,
"text": "dt"
},
{
"math_id": 12,
"text": "u_s(x) = \\mathbf{1}_{B}(x)"
},
{
"math_id": 13,
"text": "x"
},
{
"math_id": 14,
"text": "t"
},
{
"math_id": 15,
"text": " \\Pr(X_s \\in B \\mid X_t = x) = E[\\mathbf{1}_B(X_s) \\mid X_t = x]."
},
{
"math_id": 16,
"text": "\\frac{\\partial}{\\partial s}p(x,s)=-\\frac{\\partial}{\\partial x}[\\mu(x,s)p(x,s)] + \\frac{1}{2}\\frac{\\partial^2}{\\partial x^2}[\\sigma^2(x,s)p(x,s)], "
},
{
"math_id": 17,
"text": "s \\ge t"
},
{
"math_id": 18,
"text": "p(x,t)=p_t(x)"
}
] |
https://en.wikipedia.org/wiki?curid=6419365
|
6419756
|
Petrophysics
|
Study of physical and chemical properties of rocks
Petrophysics (from the Greek πέτρα, "petra", "rock" and φύσις, "physis", "nature") is the study of physical and chemical rock properties and their interactions with fluids.
A major application of petrophysics is in studying reservoirs for the hydrocarbon industry. Petrophysicists work together with reservoir engineers and geoscientists to understand the porous media properties of the reservoir. Particularly how the pores are interconnected in the subsurface, controlling the accumulation and migration of hydrocarbons. Some fundamental petrophysical properties determined are lithology, porosity, water saturation, permeability, and capillary pressure.
The petrophysicists workflow measures and evaluates these petrophysical properties through well-log interpretation (i.e. in-situ reservoir conditions) and core analysis in the laboratory. During well perforation, different well-log tools are used to measure the petrophysical and mineralogical properties through radioactivity and seismic technologies in the borehole. In addition, core plugs are taken from the well as sidewall core or whole core samples. These studies are combined with geological, geophysical, and reservoir engineering studies to model the reservoir and determine its economic feasibility.
While most petrophysicists work in the hydrocarbon industry, some also work in the mining, water resources, geothermal energy, and carbon capture and storage industries. Petrophysics is part of the geosciences, and its studies are used by petroleum engineering, geology, geochemistry, exploration geophysics and others.
Fundamental petrophysical properties.
The following are the fundamental petrophysical properties used to characterize a reservoir:
Rock mechanical properties.
The rock's mechanical or geomechanical properties are also used within petrophysics to determine the reservoir strength, elastic properties, hardness, ultrasonic behaviour, index characteristics and in situ stresses.
Petrophysicists use acoustic and density measurements of rocks to compute their mechanical properties and strength. They measure the compressional (P) wave velocity of sound through the rock and the shear (S) wave velocity and use these with the density of the rock to compute the rock's "compressive strength", which is the compressive stress that causes a rock to fail, and the rocks' "flexibility", which is the relationship between stress and deformation for a rock. Converted-wave analysis is also determines the subsurface lithology and porosity.
Geomechanics measurements are useful for drillability assessment, wellbore and open-hole stability design, log strength and stress correlations, and formation and strength characterization. These measurements are also used to design dams, roads, foundations for buildings, and many other large construction projects. They can also help interpret seismic signals from the Earth, either manufactured seismic signals or those from earthquakes.
Methods of petrophysical analysis.
Core analysis.
As "core samples" are the only evidence of the reservoir's formation rock structure, the "Core analysis" is the "ground truth" data measured at laboratory to comprehend the key petrophysical features of the in-situ reservoir. In the petroleum industry, rock samples are retrieved from the subsurface and measured by oil or service companies' core laboratories. This process is time-consuming and expensive; thus, it can only be applied to some of the wells drilled in a field. Also, proper design, planning and supervision decrease data redundancy and uncertainty. Client and laboratory teams must work aligned to optimise the core analysis process.
Well-logging.
"Well Logging" is a relatively inexpensive method to obtain petrophysical properties downhole. Measurement tools are conveyed downhole using either wireline or LWD method.
An example of wireline logs is shown in Figure 1. The first “track” shows the natural gamma radiation level of the rock. The gamma radiation level “log” shows increasing radiation to the right and decreasing radiation to the left. The rocks emitting less radiation have more yellow shading. The detector is very sensitive, and the amount of radiation is very low. In clastic rock formations, rocks with smaller amounts of radiation are more likely to be coarser-grained and have more pore space, while rocks with higher amounts of radiation are more likely to have finer grains and less pore space.
The second track in the plot records the depth below the reference point, usually the Kelly bush or rotary table in feet, so these rock formations are 11,900 feet below the Earth's surface.
In the third track, the electrical resistivity of the rock is presented. The water in this rock is salty. The electrolytes flowing inside the pore space within the water conduct electricity resulting in lower resistivity of the rock. This also indicates an increased water saturation and decreased hydrocarbon saturation.
The fourth track shows the computed water saturation, both as “total” water (including the water bound to the rock) in magenta and the “effective water” or water that is free to flow in black. Both quantities are given as a fraction of the total pore space.
The fifth track shows the fraction of the total rock that is pore space filled with fluids (i.e. porosity). The display of the pore space is divided into green for oil and blue for movable water. The black line shows the fraction of the pore space, which contains either water or oil that can move or be "produced" (i.e. effective porosity). While the magenta line indicates the toral porosity, meaning that it includes the water that is permanently bound to the rock.
The last track represents the rock lithology divided into sandstone and shale portions. The yellow pattern represents the fraction of the rock (excluding fluids) composed of coarser-grained sandstone. The gray pattern represents the fraction of rock composed of finer-grained, i.e. "shale." The sandstone is the part of the rock that contains the producible hydrocarbons and water.
Modelling.
Reservoir models are built by reservoir engineering in specialised software with the petrophysical dataset elaborated by the petrophysicist to estimate the amount of hydrocarbon present in the reservoir, the rate at which that hydrocarbon can be produced to the Earth's surface through wellbores and the fluid flow in rocks. Similar models in the water resource industry compute how much water can be produced to the surface over long periods without depleting the aquifer.
Rock volumetric model for shaly sand formation.
Shaly sand is a term referred to as a mixture of shale or clay and sandstone. Hence, a significant portion of clay minerals and silt-size particles results in a fine-grained sandstone with higher density and rock complexity.
The shale/clay volume is an essential petrophysical parameter to estimate since it contributes to the rock bulk volume, and for correct porosity and water saturation, evaluation needs to be correctly defined. As shown in Figure 2, for modelling clastic rock formation, there are four components whose definitions are typical for shaly or clayey sands that assume: the rock matrix (grains), clay portion that surrounds the grains, water, and hydrocarbons. These two fluids are stored only in pore space in the rock matrix.
Due to the complex microstructure, for a water-wet rock, the following terms comprised a clastic reservoir formation:
"Vma" = volume of matrix grains.
"Vdcl" = volme of dry clay.
"Vcbw" = volume of clay bound water.
"Vcl" = volume of wet clay ("Vdcl" +"Vcbw").
"Vcap" = volume of capillary bound water.
"Vfw" = volume of free water.
"Vhyd" = volume of hydrocarbon.
"ΦT" = Total porosity (PHIT), which includes the connected and not connected pore throats.
"Φe" = Effective porosity which includes only the inter-connected pore throats.
"Vb" = bulk volume of the rock.
Key equations:
"Vma" + "Vcl" + "Vfw" + "Vhyd" = 1
Rock matrix volume + wet clay volume + water free volume + hydrocarbon volume = bulk rock volume
Scholarly societies.
The Society of Petrophysicists and Well Log Analysts (SPWLA) is an organisation whose mission is to increase the awareness of petrophysics, formation evaluation, and well logging best practices in the oil and gas industry and the scientific community at large.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\phi"
},
{
"math_id": 1,
"text": "S_w"
},
{
"math_id": 2,
"text": "k"
}
] |
https://en.wikipedia.org/wiki?curid=6419756
|
641995
|
Asymptotic analysis
|
Description of limiting behavior of a function
In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing limiting behavior.
As an illustration, suppose that we are interested in the properties of a function "f"&hairsp;("n") as n becomes very large. If "f"("n") = "n"2 + 3"n", then as n becomes very large, the term 3"n" becomes insignificant compared to "n"2. The function "f"("n") is said to be ""asymptotically equivalent" to "n"2, as "n" → ∞". This is often written symbolically as "f"&hairsp;("n") ~ "n"2, which is read as ""f"("n") is asymptotic to "n"2".
An example of an important asymptotic result is the prime number theorem. Let π("x") denote the prime-counting function (which is not directly related to the constant pi), i.e. π("x") is the number of prime numbers that are less than or equal to x. Then the theorem states that
formula_0
Asymptotic analysis is commonly used in computer science as part of the analysis of algorithms and is often expressed there in terms of big O notation.
Definition.
Formally, given functions "f"&hairsp;("x") and "g"("x"), we define a binary relation
formula_1
if and only if
formula_2
The symbol ~ is the tilde. The relation is an equivalence relation on the set of functions of x; the functions f and g are said to be "asymptotically equivalent". The domain of f and g can be any set for which the limit is defined: e.g. real numbers, complex numbers, positive integers.
The same notation is also used for other ways of passing to a limit: e.g. "x" → 0, "x" ↓ 0, . The way of passing to the limit is often not stated explicitly, if it is clear from the context.
Although the above definition is common in the literature, it is problematic if "g"("x") is zero infinitely often as x goes to the limiting value. For that reason, some authors use an alternative definition. The alternative definition, in little-o notation, is that "f" ~ "g" if and only if
formula_3
This definition is equivalent to the prior definition if "g"("x") is not zero in some neighbourhood of the limiting value.
Properties.
If formula_4 and formula_5, then, under some mild conditions, the following hold:
Such properties allow asymptotically equivalent functions to be freely exchanged in many algebraic expressions.
Asymptotic expansion.
An asymptotic expansion of a function "f"("x") is in practice an expression of that function in terms of a series, the partial sums of which do not necessarily converge, but such that taking any initial partial sum provides an asymptotic formula for f. The idea is that successive terms provide an increasingly accurate description of the order of growth of f.
In symbols, it means we have formula_15 but also formula_16 and formula_17 for each fixed "k". In view of the definition of the formula_18 symbol, the last equation means formula_19 in the little o notation, i.e., formula_20 is much smaller than formula_21
The relation formula_17 takes its full meaning if formula_22 for all "k", which means the formula_23 form an asymptotic scale. In that case, some authors may abusively write formula_24 to denote the statement formula_25 One should however be careful that this is not a standard use of the formula_18 symbol, and that it does not correspond to the definition given in .
In the present situation, this relation formula_26 actually follows from combining steps "k" and "k"−1; by subtracting formula_27 from formula_28 one gets formula_29 i.e. formula_30
In case the asymptotic expansion does not converge, for any particular value of the argument there will be a particular partial sum which provides the best approximation and adding additional terms will decrease the accuracy. This optimal partial sum will usually have more terms as the argument approaches the limit value.
Worked example.
Asymptotic expansions often occur when an ordinary series is used in a formal expression that forces the taking of values outside of its domain of convergence. For example, we might start with the ordinary series
formula_34
The expression on the left is valid on the entire complex plane formula_35, while the right hand side converges only for formula_36. Multiplying by formula_37 and integrating both sides yields
formula_38
The integral on the left hand side can be expressed in terms of the exponential integral. The integral on the right hand side, after the substitution formula_39, may be recognized as the gamma function. Evaluating both, one obtains the asymptotic expansion
formula_40
Here, the right hand side is clearly not convergent for any non-zero value of "t". However, by keeping "t" small, and truncating the series on the right to a finite number of terms, one may obtain a fairly good approximation to the value of formula_41. Substituting formula_42 and noting that formula_43 results in the asymptotic expansion given earlier in this article.
Asymptotic distribution.
In mathematical statistics, an asymptotic distribution is a hypothetical distribution that is in a sense the "limiting" distribution of a sequence of distributions. A distribution is an ordered set of random variables "Z""i" for "i" = 1, …, "n", for some positive integer "n". An asymptotic distribution allows "i" to range without bound, that is, "n" is infinite.
A special case of an asymptotic distribution is when the late entries go to zero—that is, the "Z""i" go to 0 as "i" goes to infinity. Some instances of "asymptotic distribution" refer only to this special case.
This is based on the notion of an asymptotic function which cleanly approaches a constant value (the "asymptote") as the independent variable goes to infinity; "clean" in this sense meaning that for any desired closeness epsilon there is some value of the independent variable after which the function never differs from the constant by more than epsilon.
An asymptote is a straight line that a curve approaches but never meets or crosses. Informally, one may speak of the curve meeting the asymptote "at infinity" although this is not a precise definition. In the equation formula_44 "y" becomes arbitrarily small in magnitude as "x" increases.
Applications.
Asymptotic analysis is used in several mathematical sciences. In statistics, asymptotic theory provides limiting approximations of the probability distribution of sample statistics, such as the likelihood ratio statistic and the expected value of the deviance. Asymptotic theory does not provide a method of evaluating the finite-sample distributions of sample statistics, however. Non-asymptotic bounds are provided by methods of approximation theory.
Examples of applications are the following.
Asymptotic analysis is a key tool for exploring the ordinary and partial differential equations which arise in the mathematical modelling of real-world phenomena. An illustrative example is the derivation of the boundary layer equations from the full Navier-Stokes equations governing fluid flow. In many cases, the asymptotic expansion is in power of a small parameter, ε: in the boundary layer case, this is the nondimensional ratio of the boundary layer thickness to a typical length scale of the problem. Indeed, applications of asymptotic analysis in mathematical modelling often center around a nondimensional parameter which has been shown, or assumed, to be small through a consideration of the scales of the problem at hand.
Asymptotic expansions typically arise in the approximation of certain integrals (Laplace's method, saddle-point method, method of steepest descent) or in the approximation of probability distributions (Edgeworth series). The Feynman graphs in quantum field theory are another example of asymptotic expansions which often do not converge.
Asymptotic versus Numerical Analysis.
Debruijn illustrates the use of asymptotics in the following dialog between Dr. N.A., a Numerical Analyst, and Dr. A.A., an Asymptotic Analyst:
N.A.: I want to evaluate my function formula_45 for large values of formula_46, with a relative error of at most 1%.
A.A.: formula_47.
N.A.: I am sorry, I don't understand.
A.A.: formula_48
N.A.: But my value of formula_46 is only 100.
A.A.: Why did you not say so? My evaluations giveformula_49
N.A.: This is no news to me. I know already that formula_50.
A.A.: I can gain a little on some of my estimates. Now I find thatformula_51
N.A.: I asked for 1%, not for 20%.
A.A.: It is almost the best thing I possibly can get. Why don't you take larger values of formula_46?
N.A.: !!! I think it's better to ask my electronic computing machine.
Machine: f(100) = 0.01137 42259 34008 67153
A.A.: Haven't I told you so? My estimate of 20% was not far off from the 14% of the real error.
N.A.: !!! . . . !
Some days later, Miss N.A. wants to know the value of f(1000), but her machine would take a month of computation to give the answer. She returns to her Asymptotic Colleague, and gets a fully satisfactory reply.
See also.
<templatestyles src="Div col/styles.css"/>
|
[
{
"math_id": 0,
"text": "\\pi(x)\\sim\\frac{x}{\\ln x}."
},
{
"math_id": 1,
"text": "f(x) \\sim g(x) \\quad (\\text{as } x\\to\\infty)"
},
{
"math_id": 2,
"text": "\\lim_{x \\to \\infty} \\frac{f(x)}{g(x)} = 1."
},
{
"math_id": 3,
"text": "f(x)=g(x)(1+o(1))."
},
{
"math_id": 4,
"text": "f \\sim g"
},
{
"math_id": 5,
"text": "a \\sim b"
},
{
"math_id": 6,
"text": "f^r \\sim g^r"
},
{
"math_id": 7,
"text": "\\log(f) \\sim \\log(g)"
},
{
"math_id": 8,
"text": "\\lim g \\neq 1 "
},
{
"math_id": 9,
"text": "f\\times a \\sim g\\times b"
},
{
"math_id": 10,
"text": "f / a \\sim g / b"
},
{
"math_id": 11,
"text": "n! \\sim \\sqrt{2\\pi n}\\left(\\frac{n}{e}\\right)^n"
},
{
"math_id": 12,
"text": "p(n)\\sim \\frac{1}{4n\\sqrt{3}} e^{\\pi\\sqrt{\\frac{2n}{3}}}"
},
{
"math_id": 13,
"text": "\\operatorname{Ai}(x) \\sim \\frac{e^{-\\frac{2}{3} x^\\frac{3}{2}}}{2\\sqrt{\\pi} x^{1/4}}"
},
{
"math_id": 14,
"text": "\\begin{align}\n H_\\alpha^{(1)}(z) &\\sim \\sqrt{\\frac{2}{\\pi z}} e^{ i\\left(z - \\frac{2\\pi\\alpha - \\pi}{4}\\right)} \\\\\n H_\\alpha^{(2)}(z) &\\sim \\sqrt{\\frac{2}{\\pi z}} e^{-i\\left(z - \\frac{2\\pi\\alpha - \\pi}{4}\\right)}\n\\end{align}"
},
{
"math_id": 15,
"text": "f \\sim g_1,"
},
{
"math_id": 16,
"text": "f - g_1 \\sim g_2"
},
{
"math_id": 17,
"text": "f - g_1 - \\cdots - g_{k-1} \\sim g_{k}"
},
{
"math_id": 18,
"text": "\\sim"
},
{
"math_id": 19,
"text": "f - (g_1 + \\cdots + g_k) = o(g_k)"
},
{
"math_id": 20,
"text": "f - (g_1 + \\cdots + g_k)"
},
{
"math_id": 21,
"text": "g_k."
},
{
"math_id": 22,
"text": "g_{k+1} = o(g_k)"
},
{
"math_id": 23,
"text": "g_k"
},
{
"math_id": 24,
"text": "f \\sim g_1 + \\cdots + g_k"
},
{
"math_id": 25,
"text": "f - (g_1 + \\cdots + g_k) = o(g_k)."
},
{
"math_id": 26,
"text": "g_{k} = o(g_{k-1})"
},
{
"math_id": 27,
"text": "f - g_1 - \\cdots - g_{k-2} = g_{k-1} + o(g_{k-1})"
},
{
"math_id": 28,
"text": "f - g_1 - \\cdots - g_{k-2} - g_{k-1} = g_{k} + o(g_{k}),"
},
{
"math_id": 29,
"text": "g_{k} + o(g_{k})=o(g_{k-1}),"
},
{
"math_id": 30,
"text": "g_{k} = o(g_{k-1})."
},
{
"math_id": 31,
"text": "\\frac{e^x}{x^x \\sqrt{2\\pi x}} \\Gamma(x+1) \\sim 1+\\frac{1}{12x}+\\frac{1}{288x^2}-\\frac{139}{51840x^3}-\\cdots\n \\ (x \\to \\infty)"
},
{
"math_id": 32,
"text": "xe^xE_1(x) \\sim \\sum_{n=0}^\\infty \\frac{(-1)^nn!}{x^n} \\ (x \\to \\infty) "
},
{
"math_id": 33,
"text": " \\sqrt{\\pi}x e^{x^2}\\operatorname{erfc}(x) \\sim 1+\\sum_{n=1}^\\infty (-1)^n \\frac{(2n-1)!!}{n!(2x^2)^n} \\ (x \\to \\infty)"
},
{
"math_id": 34,
"text": "\\frac{1}{1-w}=\\sum_{n=0}^\\infty w^n"
},
{
"math_id": 35,
"text": "w \\ne 1"
},
{
"math_id": 36,
"text": "|w|< 1"
},
{
"math_id": 37,
"text": "e^{-w/t}"
},
{
"math_id": 38,
"text": " \\int_0^\\infty \\frac{e^{-\\frac{w}{t}}}{1 - w} \\, dw = \\sum_{n=0}^\\infty t^{n+1} \\int_0^\\infty e^{-u} u^n \\, du"
},
{
"math_id": 39,
"text": "u=w/t"
},
{
"math_id": 40,
"text": "e^{-\\frac{1}{t}} \\operatorname{Ei}\\left(\\frac{1}{t}\\right) = \\sum _{n=0}^\\infty n! \\; t^{n+1} "
},
{
"math_id": 41,
"text": "\\operatorname{Ei}(1/t)"
},
{
"math_id": 42,
"text": "x = -1/t"
},
{
"math_id": 43,
"text": "\\operatorname{Ei}(x) = -E_1(-x)"
},
{
"math_id": 44,
"text": "y = \\frac{1}{x},"
},
{
"math_id": 45,
"text": "f(x)"
},
{
"math_id": 46,
"text": "x"
},
{
"math_id": 47,
"text": "f(x)=x^{-1}+\\mathrm O(x^{-2}) \\qquad (x\\to\\infty)"
},
{
"math_id": 48,
"text": "|f(x)-x^{-1}|<8x^{-2} \\qquad (x>10^4)."
},
{
"math_id": 49,
"text": "|f(x)-x^{-1}|<57000x^{-2} \\qquad (x>100)."
},
{
"math_id": 50,
"text": "0<f(100)<1"
},
{
"math_id": 51,
"text": "|f(x)-x^{-1}|<20x^{-2} \\qquad (x>100)."
}
] |
https://en.wikipedia.org/wiki?curid=641995
|
642006
|
Laplace's method
|
Method for approximate evaluation of integrals
In mathematics, Laplace's method, named after Pierre-Simon Laplace, is a technique used to approximate integrals of the form
formula_0
where formula_1 is a twice-differentiable function, formula_2 is a large number, and the endpoints formula_3 and formula_4 could be infinite. This technique was originally presented in the book by .
In Bayesian statistics, Laplace's approximation can refer to either approximating the posterior normalizing constant with Laplace's method or approximating the posterior distribution with a Gaussian centered at the maximum a posteriori estimate. Laplace approximations are used in the integrated nested Laplace approximations method for fast approximations of Bayesian inference.
Concept.
Let the function formula_6 have a unique global maximum at formula_7. formula_8 is a constant here. The following two functions are considered:
formula_9
It must be noted that formula_7 will be the global maximum of formula_10 and formula_11 as well. Hence, it is observed:
formula_12
As "M" increases, the ratio for formula_11 will grow exponentially, while the ratio for formula_10 does not change. Thus, significant contributions to the integral of this function will come only from points formula_13 in a neighborhood of formula_7, which can then be estimated.
General theory.
To state and motivate the method, one must make several assumptions. It is assumed that formula_7 is not an endpoint of the interval of integration and that the values formula_6 cannot be very close to formula_14 unless formula_13 is close to formula_7.
formula_6 can be expanded around "x"0 by Taylor's theorem,
formula_15
where formula_16 (see: big O notation).
Since formula_1 has a global maximum at formula_7, and formula_7 is not an endpoint, it is a stationary point, i.e. formula_17. Therefore, the second-order Taylor polynomial approximating formula_6 is
formula_18
Then, just one more step is needed to get a Gaussian distribution. Since formula_7 is a global maximum of the function formula_1 it can be stated, by definition of the second derivative, that formula_19, thus giving the relation
formula_20
for formula_13 close to formula_7. The integral can then be approximated with:
formula_21
If formula_22 this latter integral becomes a Gaussian integral if we replace the limits of integration by formula_23 and formula_24; when formula_2 is large this creates only a small error because the exponential decays very fast away from formula_7. Computing this Gaussian integral we obtain:
formula_25
A generalization of this method and extension to arbitrary precision is provided by the book .
Formal statement and proof.
Suppose formula_6 is a twice continuously differentiable function on formula_26 and there exists a unique point formula_27 such that:
formula_28
Then:
formula_29
<templatestyles src="Template:Hidden begin/styles.css"/>Proof
Lower bound: Let formula_30. Since formula_31 is continuous there exists formula_32 such that if formula_33 then formula_34 By Taylor's Theorem, for any formula_35
formula_36
Then we have the following lower bound:
formula_37
where the last equality was obtained by a change of variables
formula_38
Remember formula_39 so we can take the square root of its negation.
If we divide both sides of the above inequality by
formula_40
and take the limit we get:
formula_41
since this is true for arbitrary formula_42 we get the lower bound:
formula_43
Note that this proof works also when formula_44 or formula_45 (or both).
Upper bound: The proof is similar to that of the lower bound but there are a few inconveniences. Again we start by picking an formula_46 but in order for the proof to work we need formula_42 small enough so that formula_47 Then, as above, by continuity of formula_31 and Taylor's Theorem we can find formula_48 so that if formula_49, then
formula_50
Lastly, by our assumptions (assuming formula_51 are finite) there exists an formula_52 such that if formula_53, then formula_54.
Then we can calculate the following upper bound:
formula_55
If we divide both sides of the above inequality by
formula_56
and take the limit we get:
formula_57
Since formula_42 is arbitrary we get the upper bound:
formula_58
And combining this with the lower bound gives the result.
Note that the above proof obviously fails when formula_44 or formula_59 (or both). To deal with these cases, we need some extra assumptions. A sufficient (not necessary) assumption is that for formula_60
formula_61
and that the number formula_62 as above exists (note that this must be an assumption in the case when the interval formula_63 is infinite). The proof proceeds otherwise as above, but with a slightly different approximation of integrals:
formula_64
When we divide by
formula_65
we get for this term
formula_66
whose limit as formula_67 is formula_68. The rest of the proof (the analysis of the interesting term) proceeds as above.
The given condition in the infinite interval case is, as said above, sufficient but not necessary. However, the condition is fulfilled in many, if not in most, applications: the condition simply says that the integral we are studying must be well-defined (not infinite) and that the maximum of the function at formula_7 must be a "true" maximum (the number formula_69 must exist). There is no need to demand that the integral is finite for formula_70 but it is enough to demand that the integral is finite for some formula_71
This method relies on 4 basic concepts such as
<templatestyles src="Template:Hidden begin/styles.css"/>Concepts
1. Relative error
The “approximation” in this method is related to the relative error and not the absolute error. Therefore, if we set
formula_72
the integral can be written as
formula_73
where formula_74 is a small number when formula_2 is a large number obviously and the relative error will be
formula_75
Now, let us separate this integral into two parts: formula_76 region and the rest.
2. formula_77 around the stationary point when formula_2 is large enough
Let’s look at the Taylor expansion of formula_78 around "x"0 and translate "x" to "y" because we do the comparison in y-space, we will get
formula_79
Note that formula_17 because formula_7 is a stationary point. From this equation you will find that the terms higher than second derivative in this Taylor expansion is suppressed as the order of formula_80 so that formula_81 will get closer to the Gaussian function as shown in figure. Besides,
formula_82
3. The larger formula_2 is, the smaller range of formula_13 is related
Because we do the comparison in y-space, formula_83 is fixed in formula_76 which will cause formula_84; however, formula_74 is inversely proportional to formula_85, the chosen region of formula_13 will be smaller when formula_2 is increased.
4. If the integral in Laplace's method converges, the contribution of the region which is not around the stationary point of the integration of its relative error will tend to zero as formula_2 grows.
Relying on the 3rd concept, even if we choose a very large "Dy", "sDy" will finally be a very small number when formula_2 is increased to a huge number. Then, how can we guarantee the integral of the rest will tend to 0 when formula_2 is large enough?
The basic idea is to find a function formula_86 such that formula_87 and the integral of formula_88 will tend to zero when formula_2 grows. Because the exponential function of formula_89 will be always larger than zero as long as formula_86 is a real number, and this exponential function is proportional to formula_90 the integral of formula_5 will tend to zero. For simplicity, choose formula_86 as a tangent through the point formula_91 as shown in the figure:
If the interval of the integration of this method is finite, we will find that no matter formula_6 is continue in the rest region, it will be always smaller than formula_86 shown above when formula_2 is large enough. By the way, it will be proved later that the integral of formula_88 will tend to zero when formula_2 is large enough.
If the interval of the integration of this method is infinite, formula_86 and formula_6 might always cross to each other. If so, we cannot guarantee that the integral of formula_5 will tend to zero finally. For example, in the case of formula_92 formula_93 will always diverge. Therefore, we need to require that formula_94 can converge for the infinite interval case. If so, this integral will tend to zero when formula_95 is large enough and we can choose this formula_95 as the cross of formula_86 and formula_96
You might ask why not choose formula_97 as a convergent integral? Let me use an example to show you the reason. Suppose the rest part of formula_6 is formula_98 then formula_99 and its integral will diverge; however, when formula_100 the integral of formula_101 converges. So, the integral of some functions will diverge when formula_2 is not a large number, but they will converge when formula_2 is large enough.
Based on these four concepts, we can derive the relative error of this method.
Other formulations.
Laplace's approximation is sometimes written as
formula_102
where formula_11 is positive.
Importantly, the accuracy of the approximation depends on the variable of integration, that is, on what stays in formula_103 and what goes into formula_104
<templatestyles src="Template:Hidden begin/styles.css"/>The derivation of its relative error
First, use formula_105 to denote the global maximum, which will simplify this derivation. We are interested in the relative error, written as formula_106,
formula_107
where
formula_108
So, if we let
formula_109
and formula_110, we can get
formula_111
since formula_112.
For the upper bound, note that formula_113 thus we can separate this integration into 5 parts with 3 different types (a), (b) and (c), respectively. Therefore,
formula_114
where formula_115 and formula_116 are similar, let us just calculate formula_115 and formula_117 and formula_118 are similar, too, I’ll just calculate formula_117.
For formula_115, after the translation of formula_119, we can get
formula_120
This means that as long as formula_121 is large enough, it will tend to zero.
For formula_117, we can get
formula_122
where
formula_123
and formula_124 should have the same sign of formula_125 during this region. Let us choose formula_86 as the tangent across the point at formula_91 , i.e. formula_126 which is shown in the figure
From this figure you can find that when formula_74 or formula_121 gets smaller, the region satisfies the above inequality will get larger. Therefore, if we want to find a suitable formula_86 to cover the whole formula_6 during the interval of formula_117, formula_121 will have an upper limit. Besides, because the integration of formula_127 is simple, let me use it to estimate the relative error contributed by this formula_117.
Based on Taylor expansion, we can get
formula_128
and
formula_129
and then substitute them back into the calculation of formula_117; however, you can find that the remainders of these two expansions are both inversely proportional to the square root of formula_2, let me drop them out to beautify the calculation. Keeping them is better, but it will make the formula uglier.
formula_130
Therefore, it will tend to zero when formula_121 gets larger, but don't forget that the upper bound of formula_121 should be considered during this calculation.
About the integration near formula_131, we can also use Taylor's Theorem to calculate it. When formula_132
formula_133
and you can find that it is inversely proportional to the square root of formula_2. In fact, formula_134 will have the same behave when formula_124 is a constant.
Conclusively, the integral near the stationary point will get smaller as formula_85 gets larger, and the rest parts will tend to zero as long as formula_121 is large enough; however, we need to remember that formula_121 has an upper limit which is decided by whether the function formula_86 is always larger than formula_135 in the rest region. However, as long as we can find one formula_86 satisfying this condition, the upper bound of formula_121 can be chosen as directly proportional to formula_85 since formula_86 is a tangent across the point of formula_135 at formula_91. So, the bigger formula_2 is, the bigger formula_121 can be.
In the multivariate case, where formula_136 is a formula_95-dimensional vector and formula_137 is a scalar function of formula_136, Laplace's approximation is usually written as:
formula_138
where formula_139 is the Hessian matrix of formula_1 evaluated at formula_140 and where formula_141 denotes matrix determinant. Analogously to the univariate case, the Hessian is required to be negative-definite.
By the way, although formula_136 denotes a formula_95-dimensional vector, the term formula_142 denotes an infinitesimal volume here, i.e. formula_143.
Steepest descent extension.
In extensions of Laplace's method, complex analysis, and in particular Cauchy's integral formula, is used to find a contour "of steepest descent" for an (asymptotically with large "M") equivalent integral, expressed as a line integral. In particular, if no point "x"0 where the derivative of formula_1 vanishes exists on the real line, it may be necessary to deform the integration contour to an optimal one, where the above analysis will be possible. Again, the main idea is to reduce, at least asymptotically, the calculation of the given integral to that of a simpler integral that can be explicitly evaluated. See the book of Erdelyi (1956) for a simple discussion (where the method is termed "steepest descents").
The appropriate formulation for the complex "z"-plane is
formula_144
for a path passing through the saddle point at "z"0. Note the explicit appearance of a minus sign to indicate the direction of the second derivative: one must "not" take the modulus. Also note that if the integrand is meromorphic, one may have to add residues corresponding to poles traversed while deforming the contour (see for example section 3 of Okounkov's paper "Symmetric functions and random partitions").
Further generalizations.
An extension of the "steepest descent method" is the so-called "nonlinear stationary phase/steepest descent method". Here, instead of integrals, one needs to evaluate asymptotically solutions of Riemann–Hilbert factorization problems.
Given a contour "C" in the complex sphere, a function formula_1 defined on that contour and a special point, such as infinity, a holomorphic function "M" is sought away from "C", with prescribed jump across "C", and with a given normalization at infinity. If formula_1 and hence "M" are matrices rather than scalars this is a problem that in general does not admit an explicit solution.
An asymptotic evaluation is then possible along the lines of the linear stationary phase/steepest descent method. The idea is to reduce asymptotically the solution of the given Riemann–Hilbert problem to that of a simpler, explicitly solvable, Riemann–Hilbert problem. Cauchy's theorem is used to justify deformations of the jump contour.
The nonlinear stationary phase was introduced by Deift and Zhou in 1993, based on earlier work of Its. A (properly speaking) nonlinear steepest descent method was introduced by Kamvissis, K. McLaughlin and P. Miller in 2003, based on previous work of Lax, Levermore, Deift, Venakides and Zhou. As in the linear case, "steepest descent contours" solve a min-max problem. In the nonlinear case they turn out to be "S-curves" (defined in a different context back in the 80s by Stahl, Gonchar and Rakhmanov).
The nonlinear stationary phase/steepest descent method has applications to the theory of soliton equations and integrable models, random matrices and combinatorics.
Median-point approximation generalization.
In the generalization, evaluation of the integral is considered equivalent to finding the norm of the distribution with density
formula_145
Denoting the cumulative distribution formula_146, if there is a diffeomorphic Gaussian distribution with density
formula_147
the norm is given by
formula_148
and the corresponding diffeomorphism is
formula_149
where formula_150 denotes cumulative standard normal distribution function.
In general, any distribution diffeomorphic to the Gaussian distribution has density
formula_151
and the median-point is mapped to the median of the Gaussian distribution. Matching the logarithm of the density functions and their derivatives at the median point up to a given order yields a system of equations that determine the approximate values of formula_152 and formula_10.
The approximation was introduced in 2019 by D. Makogon and C. Morais Smith primarily in the context of partition function evaluation for a system of interacting fermions.
Complex integrals.
For complex integrals in the form:
formula_153
with formula_154 we make the substitution "t" = "iu" and the change of variable formula_155 to get the bilateral Laplace transform:
formula_156
We then split "g"("c" + "ix") in its real and complex part, after which we recover "u" = "t"/"i". This is useful for inverse Laplace transforms, the Perron formula and complex integration.
Example: Stirling's approximation.
Laplace's method can be used to derive Stirling's approximation
formula_157
for a large integer "N".
From the definition of the Gamma function, we have
formula_158
Now we change variables, letting formula_159 so that formula_160 Plug these values back in to obtain
formula_161
This integral has the form necessary for Laplace's method with
formula_162
which is twice-differentiable:
formula_163
formula_164
The maximum of formula_165 lies at "z"0 = 1, and the second derivative of formula_165 has the value −1 at this point. Therefore, we obtain
formula_166
Notes.
<templatestyles src="Reflist/styles.css" />
References.
<templatestyles src="Refbegin/styles.css" />
"This article incorporates material from saddle point approximation on PlanetMath, which is licensed under the ."
|
[
{
"math_id": 0,
"text": "\\int_a^b e^{Mf(x)} \\, dx,"
},
{
"math_id": 1,
"text": "f"
},
{
"math_id": 2,
"text": "M"
},
{
"math_id": 3,
"text": "a"
},
{
"math_id": 4,
"text": "b"
},
{
"math_id": 5,
"text": "e^{Mf(x)}"
},
{
"math_id": 6,
"text": "f(x)"
},
{
"math_id": 7,
"text": "x_0"
},
{
"math_id": 8,
"text": "M>0"
},
{
"math_id": 9,
"text": "\\begin{align}\ng(x) &= Mf(x), \\\\\nh(x) &= e^{Mf(x)}.\n\\end{align}"
},
{
"math_id": 10,
"text": "g"
},
{
"math_id": 11,
"text": "h"
},
{
"math_id": 12,
"text": "\\begin{align}\n\\frac{g(x_0)}{g(x)} &= \\frac{M f(x_0)}{M f(x)} = \\frac{f(x_0)}{f(x)}, \\\\[4pt]\n\\frac{h(x_0)}{h(x)} &= \\frac{e^{M f(x_0)}}{e^{M f(x)}} = e^{M(f(x_0) - f(x))}.\n\\end{align}"
},
{
"math_id": 13,
"text": "x"
},
{
"math_id": 14,
"text": "f(x_0)"
},
{
"math_id": 15,
"text": "f(x) = f(x_0) + f'(x_0)(x-x_0) + \\frac{1}{2} f''(x_0)(x-x_0)^2 + R"
},
{
"math_id": 16,
"text": "R = O\\left((x-x_0)^3\\right)"
},
{
"math_id": 17,
"text": "f'(x_0)=0"
},
{
"math_id": 18,
"text": "f(x) \\approx f(x_0) + \\frac{1}{2} f''(x_0) (x-x_0)^2."
},
{
"math_id": 19,
"text": "f''(x_0) \\le 0"
},
{
"math_id": 20,
"text": "f(x) \\approx f(x_0) - \\frac{1}{2} |f''(x_0)| (x-x_0)^2"
},
{
"math_id": 21,
"text": "\\int_a^b e^{M f(x)}\\, dx\\approx e^{M f(x_0)}\\int_a^b e^{-\\frac{1}{2} M|f''(x_0)| (x-x_0)^2} \\, dx"
},
{
"math_id": 22,
"text": "f''(x_0) < 0"
},
{
"math_id": 23,
"text": "-\\infty"
},
{
"math_id": 24,
"text": "+\\infty"
},
{
"math_id": 25,
"text": "\\int_a^b e^{M f(x)}\\, dx\\approx \\sqrt{\\frac{2\\pi}{M|f''(x_0)|}}e^{M f(x_0)} \\text { as } M\\to\\infty."
},
{
"math_id": 26,
"text": "[a,b],"
},
{
"math_id": 27,
"text": "x_0 \\in (a,b)"
},
{
"math_id": 28,
"text": "f(x_0) = \\max_{x \\in [a,b]} f(x) \\quad \\text{and} \\quad f''(x_0)<0."
},
{
"math_id": 29,
"text": "\\lim_{n\\to\\infty} \\frac{\\int_a^b e^{nf(x)} \\, dx}{e^{nf(x_0)} \\sqrt{\\frac{2\\pi}{n\\left(-f''(x_0)\\right)}}}= 1. "
},
{
"math_id": 30,
"text": "\\varepsilon > 0"
},
{
"math_id": 31,
"text": "f''"
},
{
"math_id": 32,
"text": "\\delta >0"
},
{
"math_id": 33,
"text": "|x_0-c|< \\delta"
},
{
"math_id": 34,
"text": "f''(c) \\ge f''(x_0) - \\varepsilon."
},
{
"math_id": 35,
"text": "x \\in (x_0 - \\delta, x_0 + \\delta),"
},
{
"math_id": 36,
"text": "f(x) \\ge f(x_0) + \\frac{1}{2}(f''(x_0) - \\varepsilon)(x-x_0)^2."
},
{
"math_id": 37,
"text": "\\begin{align}\n\\int_a^b e^{nf(x)} \\, dx &\\ge \\int_{x_0 - \\delta}^{x_0 + \\delta} e^{nf(x)} \\, dx \\\\\n &\\ge e^{nf(x_0)} \\int_{x_0 - \\delta}^{x_0 + \\delta} e^{\\frac{n}{2}(f''(x_0) - \\varepsilon)(x-x_0)^2} \\, dx \\\\\n &= e^{nf(x_0)} \\sqrt{\\frac{1}{n(-f''(x_0) + \\varepsilon)}} \\int_{-\\delta \\sqrt{n(-f''(x_0) + \\varepsilon)} }^{\\delta \\sqrt{n(-f''(x_0) + \\varepsilon)} } e^{-\\frac{1}{2}y^2} \\, dy\n\\end{align}"
},
{
"math_id": 38,
"text": "y= \\sqrt{n(-f''(x_0) + \\varepsilon)} (x-x_0)."
},
{
"math_id": 39,
"text": "f''(x_0)<0"
},
{
"math_id": 40,
"text": "e^{nf(x_0)}\\sqrt{\\frac{2 \\pi}{n(-f''(x_0))}}"
},
{
"math_id": 41,
"text": "\\lim_{n \\to \\infty} \\frac{\\int_a^b e^{nf(x)} \\,dx}{e^{nf(x_0)}\\sqrt{\\frac{2\\pi}{n(-f''(x_0))}}} \\ge \\lim_{n \\to \\infty} \\frac{1}{\\sqrt{2\\pi}} \\int_{-\\delta\\sqrt{n(-f''(x_0) + \\varepsilon)} }^{\\delta \\sqrt{n(-f''(x_0) + \\varepsilon)}} e^{-\\frac{1}{2}y^2} \\, dy \\, \\cdot \\sqrt{\\frac{-f''(x_0)}{-f''(x_0) + \\varepsilon}} = \\sqrt{\\frac{-f''(x_0)}{-f''(x_0) + \\varepsilon}}"
},
{
"math_id": 42,
"text": "\\varepsilon"
},
{
"math_id": 43,
"text": "\\lim_{n \\to \\infty} \\frac{\\int_a^b e^{nf(x)} \\, dx}{ e^{nf(x_0)}\\sqrt{\\frac{2\\pi}{n(-f''(x_0))}}} \\ge 1"
},
{
"math_id": 44,
"text": "a = -\\infty"
},
{
"math_id": 45,
"text": "b= \\infty"
},
{
"math_id": 46,
"text": "\\varepsilon >0"
},
{
"math_id": 47,
"text": "f''(x_0) + \\varepsilon < 0."
},
{
"math_id": 48,
"text": "\\delta>0"
},
{
"math_id": 49,
"text": "|x-x_0| < \\delta"
},
{
"math_id": 50,
"text": "f(x) \\le f(x_0) + \\frac{1}{2} (f''(x_0) + \\varepsilon)(x-x_0)^2."
},
{
"math_id": 51,
"text": "a,b"
},
{
"math_id": 52,
"text": "\\eta >0"
},
{
"math_id": 53,
"text": "|x-x_0|\\ge \\delta"
},
{
"math_id": 54,
"text": "f(x) \\le f(x_0) - \\eta"
},
{
"math_id": 55,
"text": "\\begin{align}\n\\int_a^b e^{nf(x)} \\, dx &\\le \\int_a^{x_0-\\delta} e^{nf(x)} \\, dx + \\int_{x_0-\\delta}^{x_0 + \\delta} e^{nf(x)} \\, dx + \\int_{x_0 + \\delta}^b e^{nf(x)} \\, dx \\\\\n&\\le (b-a)e^{n(f(x_0)-\\eta)} + \\int_{x_0-\\delta}^{x_0 + \\delta} e^{n f(x) } \\, dx \\\\\n&\\le (b-a)e^{n(f(x_0)-\\eta)} + e^{nf(x_0)} \\int_{x_0-\\delta}^{x_0 + \\delta} e^{\\frac{n}{2}(f''(x_0)+\\varepsilon)(x-x_0)^2} \\, dx\\\\\n&\\le (b-a)e^{n(f(x_0)-\\eta)} + e^{nf(x_0)} \\int_{-\\infty}^{+\\infty} e^{\\frac{n}{2}(f''(x_0)+\\varepsilon)(x-x_0)^2} \\, dx \\\\\n&\\le (b-a)e^{n(f(x_0)-\\eta)} + e^{nf(x_0)} \\sqrt{\\frac{2 \\pi}{n (-f''(x_0) - \\varepsilon)}}\n\\end{align}"
},
{
"math_id": 56,
"text": "e^{nf(x_0)}\\sqrt{\\frac{2 \\pi}{n (-f''(x_0))}}"
},
{
"math_id": 57,
"text": "\\lim_{n \\to \\infty} \\frac{\\int_a^b e^{nf(x)} \\, dx}{e^{nf(x_0)}\\sqrt{\\frac{2\\pi}{n(-f''(x_0))}}} \\le \\lim_{n \\to \\infty} (b-a) e^{-\\eta n} \\sqrt{\\frac{n(-f''(x_0))}{2\\pi}} + \\sqrt{\\frac{-f''(x_0)}{-f''(x_0) - \\varepsilon}} = \\sqrt{\\frac{-f''(x_0)}{-f''(x_0) - \\varepsilon}}"
},
{
"math_id": 58,
"text": "\\lim_{n \\to \\infty} \\frac{\\int_a^b e^{nf(x)} \\, dx}{e^{nf(x_0)}\\sqrt{\\frac{2 \\pi}{n (-f''(x_0))}}} \\le 1"
},
{
"math_id": 59,
"text": "b = \\infty"
},
{
"math_id": 60,
"text": "n = 1,"
},
{
"math_id": 61,
"text": "\\int_a^b e^{nf(x)} \\, dx < \\infty,"
},
{
"math_id": 62,
"text": "\\eta"
},
{
"math_id": 63,
"text": "[a,b]"
},
{
"math_id": 64,
"text": "\\int_a^{x_0-\\delta} e^{nf(x)} \\, dx + \\int_{x_0 + \\delta}^b e^{nf(x)} \\, dx \\le \\int_a^b e^{f(x)}e^{(n-1)(f(x_0) - \\eta)} \\, dx = e^{(n-1)(f(x_0) - \\eta)} \\int_a^b e^{f(x)} \\, dx."
},
{
"math_id": 65,
"text": "e^{nf(x_0)}\\sqrt{\\frac{2\\pi}{n(-f''(x_0))}},"
},
{
"math_id": 66,
"text": "\\frac{e^{(n-1)(f(x_0) - \\eta)} \\int_a^b e^{f(x)} \\, dx}{e^{nf(x_0)}\\sqrt{\\frac{2\\pi}{n(-f''(x_0))}}} = e^{-(n-1)\\eta} \\sqrt{n} e^{-f(x_0)} \\int_a^b e^{f(x)} \\, dx \\sqrt{\\frac{-f''(x_0)}{2\\pi}}"
},
{
"math_id": 67,
"text": "n \\to \\infty"
},
{
"math_id": 68,
"text": "0"
},
{
"math_id": 69,
"text": "\\eta > 0"
},
{
"math_id": 70,
"text": "n=1"
},
{
"math_id": 71,
"text": "n=N."
},
{
"math_id": 72,
"text": "s = \\sqrt{\\frac{2\\pi}{M\\left|f''(x_0)\\right|}},"
},
{
"math_id": 73,
"text": "\\begin{align} \n\\int_a^b e^{M f(x)} \\, dx &= se^{Mf(x_0)} \\frac{1}{s}\\int_a^b e^{M(f(x)-f(x_0))}\\, dx \\\\ \n& = se^{Mf(x_0)} \\int_{\\frac{a-x_0}{s}}^{\\frac{b-x_0}{s}} e^{M(f(sy+x_0)-f(x_0))}\\,dy \n\\end{align}"
},
{
"math_id": 74,
"text": "s"
},
{
"math_id": 75,
"text": "\\left| \\int_{\\frac{a-x_0}{s}}^{\\frac{b-x_0}{s}} e^{M(f(sy+x_0)-f(x_0))} dy-1 \\right|."
},
{
"math_id": 76,
"text": "y\\in[-D_y,D_y]"
},
{
"math_id": 77,
"text": "e^{M (f(sy+x_0)-f(x_0))} \\to e^{-\\pi y^2}"
},
{
"math_id": 78,
"text": "M(f(x)-f(x_0))"
},
{
"math_id": 79,
"text": "M(f(x)-f(x_0)) = \\frac{Mf''(x_0)}{2}s^2y^2 +\\frac{Mf'''(x_0)}{6}s^3y^3+ \\cdots = -\\pi y^2 +O\\left(\\frac{1}{\\sqrt{M}}\\right)."
},
{
"math_id": 80,
"text": "\\tfrac{1}{\\sqrt{M}}"
},
{
"math_id": 81,
"text": "\\exp(M(f(x)-f(x_0)))"
},
{
"math_id": 82,
"text": "\\int_{-\\infty}^{\\infty}e^{-\\pi y^2} dy =1."
},
{
"math_id": 83,
"text": "y"
},
{
"math_id": 84,
"text": "x\\in[-sD_y, sD_y]"
},
{
"math_id": 85,
"text": "\\sqrt{M}"
},
{
"math_id": 86,
"text": "m(x)"
},
{
"math_id": 87,
"text": "m(x)\\ge f(x)"
},
{
"math_id": 88,
"text": "e^{Mm(x)}"
},
{
"math_id": 89,
"text": "Mm(x)"
},
{
"math_id": 90,
"text": "m(x),"
},
{
"math_id": 91,
"text": "x=sD_y"
},
{
"math_id": 92,
"text": "f(x)=\\tfrac{\\sin(x)}{x},"
},
{
"math_id": 93,
"text": "\\int^{\\infty}_{0}e^{Mf(x)} dx"
},
{
"math_id": 94,
"text": "\\int^{\\infty}_{d}e^{Mf(x)} dx"
},
{
"math_id": 95,
"text": "d"
},
{
"math_id": 96,
"text": "f(x)."
},
{
"math_id": 97,
"text": "\\int^{\\infty}_{d}e^{f(x)} dx"
},
{
"math_id": 98,
"text": "-\\ln x,"
},
{
"math_id": 99,
"text": "e^{f(x)}=\\tfrac{1}{x}"
},
{
"math_id": 100,
"text": "M=2,"
},
{
"math_id": 101,
"text": "e^{Mf(x)}=\\tfrac{1}{x^2}"
},
{
"math_id": 102,
"text": "\\int_a^b h(x) e^{M g(x)}\\, dx \\approx \\sqrt{\\frac{2\\pi}{M|g''(x_0)|}} h(x_0) e^{M g(x_0)} \\ \\text { as } M\\to\\infty"
},
{
"math_id": 103,
"text": "g(x)"
},
{
"math_id": 104,
"text": "h(x)."
},
{
"math_id": 105,
"text": "x_0=0"
},
{
"math_id": 106,
"text": "|R|"
},
{
"math_id": 107,
"text": "\\int_a^b h(x) e^{M g(x)}\\, dx = h(0)e^{Mg(0)}s \\underbrace{\\int_{a/s}^{b/s}\\frac{h(x)}{h(0)}e^{M\\left[ g(sy)-g(0) \\right]} dy}_{1+R},"
},
{
"math_id": 108,
"text": "s\\equiv\\sqrt{\\frac{2\\pi}{M\\left| g'' (0) \\right|}}."
},
{
"math_id": 109,
"text": "A\\equiv \\frac{h(sy)}{h(0)}e^{M\\left[g(sy)-g(0) \\right]}"
},
{
"math_id": 110,
"text": "A_0\\equiv e^{-\\pi y^2}"
},
{
"math_id": 111,
"text": "\\left| R\\right| = \\left| \\int_{a/s}^{b/s}A\\,dy -\\int_{-\\infty}^{\\infty}A_0\\,dy \\right|"
},
{
"math_id": 112,
"text": "\\int_{-\\infty}^{\\infty}A_0\\,dy =1"
},
{
"math_id": 113,
"text": "|A+B| \\le |A|+|B|,"
},
{
"math_id": 114,
"text": "|R| < \\underbrace{\\left| \\int_{D_y}^{\\infty}A_0 dy \\right|}_{(a_1)} + \\underbrace{\\left| \\int_{D_y}^{b/s}A dy \\right|}_{(b_1)}+ \\underbrace{\\left| \\int_{-D_y}^{D_y}\\left(A-A_0\\right) dy \\right|}_{(c)} + \\underbrace{\\left| \\int_{a/s}^{-D_y}A dy \\right|}_{(b_2)} + \\underbrace{\\left| \\int_{-\\infty}^{-D_y}A_0 dy \\right|}_{(a_2)}"
},
{
"math_id": 115,
"text": "(a_1)"
},
{
"math_id": 116,
"text": "(a_2)"
},
{
"math_id": 117,
"text": "(b_1)"
},
{
"math_id": 118,
"text": "(b_2)"
},
{
"math_id": 119,
"text": "z\\equiv\\pi y^2"
},
{
"math_id": 120,
"text": "(a_1) = \\left| \\frac{1}{2\\sqrt{\\pi}}\\int_{\\pi D_y^2}^{\\infty} e^{-z}z^{-1/2} dz\\right| <\\frac{e^{-\\pi D_y^2}}{2\\pi D_y}."
},
{
"math_id": 121,
"text": "D_y"
},
{
"math_id": 122,
"text": "(b_1)\\le\\left| \\int_{D_y}^{b/s}\\left[\\frac{h(sy)}{h(0)}\\right]_{\\text{max}} e^{Mm(sy)}dy \\right|"
},
{
"math_id": 123,
"text": "m(x) \\ge g(x)-g(0) \\text{as} x\\in [sD_y,b]"
},
{
"math_id": 124,
"text": "h(x)"
},
{
"math_id": 125,
"text": "h(0)"
},
{
"math_id": 126,
"text": "m(sy)= g(sD_y)-g(0) +g'(sD_y)\\left( sy-sD_y \\right)"
},
{
"math_id": 127,
"text": "e^{-\\alpha x}"
},
{
"math_id": 128,
"text": "\\begin{align}\nM\\left[g(sD_y)-g(0)\\right] &= M\\left[ \\frac{g''(0)}{2}s^2D_y^2 +\\frac{g'''(\\xi)}{6}s^3D_y^3 \\right] && \\text{as } \\xi\\in[0,sD_y] \\\\\n& = -\\pi D_y^2 +\\frac{(2\\pi)^{3/2}g'''(\\xi)D_y^3}{6\\sqrt{M}|g''(0)|^{\\frac{3}{2}}},\n\\end{align}"
},
{
"math_id": 129,
"text": "\\begin{align}\nMsg'(sD_y) &= Ms\\left(g''(0)sD_y +\\frac{g'''(\\zeta)}{2}s^2D_y^2\\right) && \\text{as } \\zeta\\in[0,sD_y] \\\\\n &= -2\\pi D_y +\\sqrt{\\frac{2}{M}}\\left( \\frac{\\pi}{|g''(0)|} \\right)^{\\frac{3}{2}}g'''(\\zeta)D_y^2,\n\\end{align}"
},
{
"math_id": 130,
"text": "\\begin{align}\n(b_1) &\\le \\left|\\left[ \\frac{h(sy)}{h(0)} \\right]_{\\max} e^{-\\pi D_y^2}\\int_0^{b/s-D_y}e^{-2\\pi D_y y} dy \\right| \\\\\n &\\le \\left|\\left[ \\frac{h(sy)}{h(0)} \\right]_{\\max} e^{-\\pi D_y^2}\\frac{1}{2\\pi D_y} \\right|.\n\\end{align}"
},
{
"math_id": 131,
"text": "x=0"
},
{
"math_id": 132,
"text": "h'(0) \\ne 0"
},
{
"math_id": 133,
"text": "\\begin{align}\n(c) &\\le \\int_{-D_y}^{D_y} e^{-\\pi y^2} \\left| \\frac{sh'(\\xi)}{h(0)}y \\right|\\, dy \\\\\n &< \\sqrt{\\frac{2}{\\pi M |g''(0)|}} \\left| \\frac{h'(\\xi)}{h(0)} \\right|_\\max \\left( 1-e^{-\\pi D_y^2} \\right)\n\\end{align}"
},
{
"math_id": 134,
"text": "(c)"
},
{
"math_id": 135,
"text": "g(x)-g(0)"
},
{
"math_id": 136,
"text": "\\mathbf{x}"
},
{
"math_id": 137,
"text": "f(\\mathbf{x})"
},
{
"math_id": 138,
"text": "\\int h(\\mathbf{x})e^{M f(\\mathbf{x})}\\, d\\mathbf{x} \\approx \\left(\\frac{2\\pi}{M}\\right)^{d/2} \\frac{h(\\mathbf{x}_0)e^{M f(\\mathbf{x}_0)}}{\\left|-H(f)(\\mathbf{x}_0)\\right|^{1/2}} \\text { as } M\\to\\infty"
},
{
"math_id": 139,
"text": "H(f)(\\mathbf{x}_0)"
},
{
"math_id": 140,
"text": "\\mathbf{x}_0"
},
{
"math_id": 141,
"text": "|\\cdot|"
},
{
"math_id": 142,
"text": "d\\mathbf{x}"
},
{
"math_id": 143,
"text": "d\\mathbf{x} := dx_1dx_2\\cdots dx_d"
},
{
"math_id": 144,
"text": "\\int_a^b e^{M f(z)}\\, dz \\approx \\sqrt{\\frac{2\\pi}{-Mf''(z_0)}}e^{M f(z_0)} \\text{ as } M\\to\\infty."
},
{
"math_id": 145,
"text": "e^{M f(x)}."
},
{
"math_id": 146,
"text": "F(x)"
},
{
"math_id": 147,
"text": "e^{-g -\\frac{\\gamma}{2}y^2}"
},
{
"math_id": 148,
"text": "\\sqrt{2\\pi\\gamma^{-1}}e^{-g}"
},
{
"math_id": 149,
"text": "y(x)=\\frac{1}{\\sqrt{\\gamma}}\\Phi^{-1}{\\left(\\frac{F(x)}{F(\\infty)}\\right)},"
},
{
"math_id": 150,
"text": "\\Phi"
},
{
"math_id": 151,
"text": "e^{-g -\\frac{\\gamma}{2}y^2(x)}y'(x)"
},
{
"math_id": 152,
"text": "\\gamma"
},
{
"math_id": 153,
"text": "\\frac{1}{2\\pi i}\\int_{c-i\\infty}^{c+i\\infty} g(s)e^{st} \\,ds"
},
{
"math_id": 154,
"text": "t \\gg 1,"
},
{
"math_id": 155,
"text": "s=c+ix"
},
{
"math_id": 156,
"text": "\\frac{1}{2 \\pi}\\int_{-\\infty}^\\infty g(c+ix)e^{-ux}e^{icu} \\, dx."
},
{
"math_id": 157,
"text": "N!\\approx \\sqrt{2\\pi N} (\\frac{N}{e})^N \\,"
},
{
"math_id": 158,
"text": "N! = \\Gamma(N+1)=\\int_0^\\infty e^{-x} x^N \\, dx."
},
{
"math_id": 159,
"text": "x=Nz"
},
{
"math_id": 160,
"text": "dx = Ndz."
},
{
"math_id": 161,
"text": "\\begin{align}\nN! &= \\int_0^\\infty e^{-Nz} (Nz)^N N \\, dz \\\\\n &= N^{N+1} \\int_0^\\infty e^{-Nz} z^N \\, dz \\\\\n &= N^{N+1} \\int_0^\\infty e^{-Nz} e^{N\\ln z} \\, dz \\\\\n &= N^{N+1} \\int_0^\\infty e^{N(\\ln z-z)} \\, dz.\n\\end{align}"
},
{
"math_id": 162,
"text": "f(z) = \\ln{z}-z"
},
{
"math_id": 163,
"text": "f'(z) = \\frac{1}{z}-1,"
},
{
"math_id": 164,
"text": "f''(z) = -\\frac{1}{z^2}."
},
{
"math_id": 165,
"text": "f(z)"
},
{
"math_id": 166,
"text": "N! \\approx N^{N+1}\\sqrt{\\frac{2\\pi}{N}} e^{-N}=\\sqrt{2\\pi N} N^N e^{-N}."
}
] |
https://en.wikipedia.org/wiki?curid=642006
|
64202283
|
Higuchi dimension
|
Fractal geometry concept
In fractal geometry, the Higuchi dimension (or Higuchi fractal dimension (HFD)) is an approximate value for the box-counting dimension of the graph of a real-valued function or time series. This value is obtained via an algorithmic approximation so one also talks about the Higuchi method. It has many applications in science and engineering and has been applied to subjects like characterizing primary waves in seismograms, clinical neurophysiology and analyzing changes in the electroencephalogram in Alzheimer's disease.
Formulation of the method.
The original formulation of the method is due to T. Higuchi. Given a time series formula_0 consisting of formula_1 data points and a parameter formula_2 the Higuchi Fractal dimension (HFD) of formula_3 is calculated in the following way: For each formula_4 and formula_5 define the length formula_6 by
formula_7
The length formula_8 is defined by the average value of the formula_9 lengths formula_10,
formula_11
The slope of the best-fitting linear function through the data points formula_12 is defined to be the Higuchi fractal dimension of the time-series formula_3.
Application to functions.
For a real-valued function formula_13 one can partition the unit interval formula_14 into formula_1 equidistantly intervals formula_15 and apply the Higuchi algorithm to the times series formula_16. This results into the Higuchi fractal dimension of the function formula_17. It was shown that in this case the Higuchi method yields an approximation for the box-counting dimension of the graph of formula_17 as it follows a geometrical approach (see Liehr & Massopust 2020).
Robustness and stability.
Applications to fractional Brownian functions and the Weierstrass function reveal that the Higuchi fractal dimension can be close to the box-dimension. On the other hand, the method can be unstable in the case where the data formula_18 are periodic or if subsets of it lie on a horizontal line (see Liehr & Massopust 2020).
|
[
{
"math_id": 0,
"text": "X:\\{1, \\dots, N \\} \\to \\mathbb{R}"
},
{
"math_id": 1,
"text": "N"
},
{
"math_id": 2,
"text": "k_{\\mathrm{max}} \\geq 2"
},
{
"math_id": 3,
"text": "X"
},
{
"math_id": 4,
"text": "k \\in \\{ 1, \\dots, k_{\\mathrm{max}} }\\"
},
{
"math_id": 5,
"text": "m \\in \\{1, \\dots, k}\\"
},
{
"math_id": 6,
"text": "L_m(k)"
},
{
"math_id": 7,
"text": "L_m(k) = \\frac{N-1}{\\lfloor \\frac{N-m}{k} \\rfloor k^2} \\sum_{i=1}^{\\lfloor \\frac{N-m}{k} \\rfloor} |X_N(m+ik)-X_N(m+(i-1)k)|."
},
{
"math_id": 8,
"text": "L(k)"
},
{
"math_id": 9,
"text": "k"
},
{
"math_id": 10,
"text": "L_1(k), \\dots, L_k(k)"
},
{
"math_id": 11,
"text": "L(k) = \\frac{1}{k} \\sum_{m=1}^k L_m(k)."
},
{
"math_id": 12,
"text": "\\left \\{ \\left ( \\log \\frac{1}{k} ,\\log L(k) \\right ) \\right \\}"
},
{
"math_id": 13,
"text": "f:[0,1] \\to \\mathbb{R}"
},
{
"math_id": 14,
"text": "[0,1]"
},
{
"math_id": 15,
"text": "[t_j,t_{j+1})"
},
{
"math_id": 16,
"text": "X(j) = f(t_j)"
},
{
"math_id": 17,
"text": "f"
},
{
"math_id": 18,
"text": "X(1), \\dots, X(N)"
}
] |
https://en.wikipedia.org/wiki?curid=64202283
|
6420246
|
Sinusoidal plane-wave solutions of the electromagnetic wave equation
|
Particular solutions to the electromagnetic wave equation
Sinusoidal plane-wave solutions are particular solutions to the wave equation.
The general solution of the electromagnetic wave equation in homogeneous, linear, time-independent media can be written as a linear superposition of plane-waves of different frequencies and polarizations.
The treatment in this article is classical but, because of the generality of Maxwell's equations for electrodynamics, the treatment can be converted into the quantum mechanical treatment with only a reinterpretation of classical quantities (aside from the quantum mechanical treatment needed for charge and current densities).
The reinterpretation is based on the theories of Max Planck and the interpretations by Albert Einstein of those theories and of other experiments. The quantum generalization of the classical treatment can be found in the articles on photon polarization and photon dynamics in the double-slit experiment.
Explanation.
Experimentally, every light signal can be decomposed into a spectrum of frequencies and wavelengths associated with sinusoidal solutions of the wave equation. Polarizing filters can be used to decompose light into its various polarization components. The polarization components can be linear, circular or elliptical.
Plane waves.
The plane sinusoidal solution for an electromagnetic wave traveling in the z direction is
formula_0
for the electric field and
formula_1
for the magnetic field, where k is the wavenumber,
formula_2
formula_3 is the angular frequency of the wave, and formula_4 is the speed of light. The hats on the vectors indicate unit vectors in the x, y, and z directions. r = ("x", "y", "z") is the position vector (in meters).
The plane wave is parameterized by the amplitudes
formula_5
and phases
formula_6
where
formula_7
and
formula_8
Polarization state vector.
Jones vector.
All the polarization information can be reduced to a single vector, called the Jones vector, in the x-y plane. This vector, while arising from a purely classical treatment of polarization, can be interpreted as a quantum state vector. The connection with quantum mechanics is made in the article on photon polarization.
The vector emerges from the plane-wave solution. The electric field solution can be rewritten in complex notation as
formula_9
where
formula_10
is the Jones vector in the x-y plane. The notation for this vector is the bra–ket notation of Dirac, which is normally used in a quantum context. The quantum notation is used here in anticipation of the interpretation of the Jones vector as a quantum state vector.
Dual Jones vector.
The Jones vector has a dual given by
formula_11
Normalization of the Jones vector.
A Jones vector represents a specific wave with a specific phase, amplitude and state of polarization. When one is using a Jones vector simply to indicate a state of polarization, then it is customary for it to be normalized. That requires that the inner product of the vector with itself to be unity:
formula_12
An arbitrary Jones vector can simply be scaled to achieve this property. All normalized Jones vectors represent a wave of the same intensity (within a particular isotropic medium). Even given a normalized Jones vector, multiplication by a pure phase factor will result in a different normalized Jones vector representing the same state of polarization.
Polarization states.
Linear polarization.
In general, the wave is linearly polarized when the phase angles formula_13 are equal,
formula_14
This represents a wave polarized at an angle formula_15 with respect to the x axis. In that case the Jones vector can be written
formula_16
Elliptical and circular polarization.
The general case in which the electric field is not confined to one direction but rotates in the "x"-"y" plane is called elliptical polarization. The state vector is given by
formula_17
In the special case of formula_18, this reduces to linear polarization.
Circular polarization corresponds to the special cases of formula_19 with formula_20. The two circular polarization states are thus given by the Jones vectors:
formula_21
|
[
{
"math_id": 0,
"text": " \\begin{align}\n\\mathbf{E} ( \\mathbf{r} , t )\n&= \\begin{pmatrix}\n E_{0,x} \\cos \\left ( kz-\\omega t + \\alpha_x \\right ) \\\\\n E_{0,y} \\cos \\left ( kz-\\omega t + \\alpha_y \\right ) \\\\\n 0\n\\end{pmatrix} \\\\[1ex]\n&= E_{0,x} \\cos \\left ( kz-\\omega t + \\alpha_x \\right ) \\, \\hat {\\mathbf{x}} \\; + \\; E_{0,y} \\cos \\left ( kz-\\omega t + \\alpha_y \\right ) \\, \\hat {\\mathbf{y}}\n\\end{align} "
},
{
"math_id": 1,
"text": " \\begin{align}\nc \\, \\mathbf{B} ( \\mathbf{r} , t )\n&= \\hat { \\mathbf{z} } \\times \\mathbf{E} ( \\mathbf{r} , t )\n\\\\[1ex]\n&= \\begin{pmatrix}\n -E_{0,y} \\cos \\left ( kz-\\omega t + \\alpha_y \\right ) \\\\\n \\hphantom{-}E_{0,x} \\cos \\left ( kz-\\omega t + \\alpha_x \\right ) \\\\\n 0\n\\end{pmatrix}\n\\\\[1ex]\n&= -E_{0,y} \\cos \\left ( kz-\\omega t + \\alpha_y \\right ) \\hat {\\mathbf{x}} \\; + \\; E_{0,x} \\cos \\left ( kz-\\omega t + \\alpha_x \\right ) \\hat {\\mathbf{y}}\n\\end{align} "
},
{
"math_id": 2,
"text": " \\omega = c k"
},
{
"math_id": 3,
"text": " \\omega"
},
{
"math_id": 4,
"text": " c "
},
{
"math_id": 5,
"text": "\\begin{align}\n E_{0,x} &= \\left| \\mathbf{E} \\right| \\cos \\theta \\\\[1.56ex]\n E_{0,y} &= \\left| \\mathbf{E} \\right| \\sin \\theta\n\\end{align} "
},
{
"math_id": 6,
"text": " \\alpha_x , \\alpha_y "
},
{
"math_id": 7,
"text": " \\theta \\ \\stackrel{\\mathrm{def}}{=}\\ \\tan^{-1} \\left ( \\frac{E_{0,y}}{E_{0, x}} \\right ) ."
},
{
"math_id": 8,
"text": " \\left| \\mathbf{E} \\right|^2 \\ \\stackrel{\\mathrm{def}}{=}\\ \\left ( E_{0,x} \\right )^2 + \\left ( E_{0,y} \\right )^2 ."
},
{
"math_id": 9,
"text": " \\mathbf{E} ( \\mathbf{r} , t ) = |\\mathbf{E}| \\, \\operatorname\\mathcal{R_e}\\left[ |\\psi\\rangle e^{ i ( kz - \\omega t ) } \\right] "
},
{
"math_id": 10,
"text": " |\\psi\\rangle \\ \\stackrel{\\mathrm{def}}{=}\\ \\begin{pmatrix} \\psi_x \\\\ \\psi_y \\end{pmatrix} = \\begin{pmatrix} \\cos(\\theta) e^{i \\alpha_x} \\\\ \\sin(\\theta) e^{i \\alpha_y} \\end{pmatrix} "
},
{
"math_id": 11,
"text": " \\langle \\psi | \\ \\stackrel{\\mathrm{def}}{=}\\ \\begin{pmatrix} \\psi_x^* & \\psi_y^* \\end{pmatrix} = \\begin{pmatrix} \\cos(\\theta) e^{-i \\alpha_x} & \\sin(\\theta) e^{-i \\alpha_y} \\end{pmatrix}."
},
{
"math_id": 12,
"text": " \\langle \\psi | \\psi \\rangle = \\begin{pmatrix} \\psi_x^* & \\psi_y^* \\end{pmatrix} \\begin{pmatrix} \\psi_x \\\\ \\psi_y \\end{pmatrix} = 1 . "
},
{
"math_id": 13,
"text": " \\alpha_x , \\alpha_y "
},
{
"math_id": 14,
"text": " \\alpha_x = \\alpha_y \\ \\stackrel{\\mathrm{def}}{=}\\ \\alpha ."
},
{
"math_id": 15,
"text": " \\theta "
},
{
"math_id": 16,
"text": " |\\psi\\rangle = \\begin{pmatrix} \\cos\\theta \\\\ \\sin\\theta \\end{pmatrix} e^{i \\alpha} ."
},
{
"math_id": 17,
"text": " |\\psi\\rangle = \\begin{pmatrix} \\psi_x \\\\ \\psi_y \\end{pmatrix} = \\begin{pmatrix} \\cos(\\theta) e^{i \\alpha_x} \\\\ \\sin(\\theta) e^{i \\alpha_y} \\end{pmatrix} = e^{i \\alpha} \\begin{pmatrix} \\cos(\\theta) \\\\ \\sin(\\theta) e^{i \\Delta \\alpha} \\end{pmatrix} ."
},
{
"math_id": 18,
"text": "\\Delta\\alpha = 0"
},
{
"math_id": 19,
"text": "\\theta=\\pm\\pi/4"
},
{
"math_id": 20,
"text": "\\Delta\\alpha=\\pi/2"
},
{
"math_id": 21,
"text": " |\\psi\\rangle = \\begin{pmatrix} \\psi_x \\\\ \\psi_y \\end{pmatrix} =\ne^{i \\alpha} \\frac 1 \\sqrt{2}\n \\begin{pmatrix} 1 \\\\ \\pm i \\end{pmatrix}. "
}
] |
https://en.wikipedia.org/wiki?curid=6420246
|
64203241
|
Chlorophyllide a reductase
|
Enzyme
Chlorophyllide "a" reductase (EC 1.3.7.15), also known as COR, is an enzyme with systematic name "bacteriochlorophyllide-a:ferredoxin 7,8-oxidoreductase". It catalyses the following chemical reaction
chlorophyllide "a" + 2 reduced ferredoxin + ATP + H2O + 2 H+ formula_0 3-deacetyl 3-vinylbacteriochlorophyllide "a" + 2 oxidized ferredoxin + ADP + phosphate
This reduction (with trans stereochemistry) of the pyrrole ring B, gives the characteristic 18-electron aromatic system that distinguishes bacteriochlorophylls from chlorophylls, which retain the chlorin system of Chlorophyllide "a".
This enzyme is present in purple bacteria such as "Rhodobacter capsulatus" and "Rhodobacter sphaeroides", and Pseudomonadota. It is a component of the biosynthetic pathway to bacteriochlorophylls.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=64203241
|
64204
|
Kinetic theory of gases
|
Understanding of gas properties in terms of molecular motion
The kinetic theory of gases is a simple classical model of the thermodynamic behavior of gases. It treats a gas as composed of numerous particles, too small to see with a microscope, which are constantly in random motion. Their collisions with each other and with the walls of their container are used to explain physical properties of the gas—for example, the relationship between its temperature, pressure, and volume. The particles are now known to be the atoms or molecules of the gas.
The basic version of the model describes an ideal gas. It treats the collisions as perfectly elastic and as the only interaction between the particles, which are additionally assumed to be much smaller than their average distance apart.
The theory's introduction allowed many principal concepts of thermodynamics to be established. It explains the macroscopic properties of gases, such as volume, pressure, and temperature, as well as transport properties such as viscosity, thermal conductivity and mass diffusivity. Due to the time reversibility of microscopic dynamics (microscopic reversibility), the kinetic theory is also connected to the principle of detailed balance, in terms of the fluctuation-dissipation theorem (for Brownian motion) and the Onsager reciprocal relations.
The theory was historically significant as the first explicit exercise of the ideas of statistical mechanics.
History.
In about 50 BCE, the Roman philosopher Lucretius proposed that apparently static macroscopic bodies were composed on a small scale of rapidly moving atoms all bouncing off each other. This Epicurean atomistic point of view was rarely considered in the subsequent centuries, when Aristotlean ideas were dominant.
In 1738 Daniel Bernoulli published "Hydrodynamica", which laid the basis for the kinetic theory of gases. In this work, Bernoulli posited the argument, that gases consist of great numbers of molecules moving in all directions, that their impact on a surface causes the pressure of the gas, and that their average kinetic energy determines the temperature of the gas. The theory was not immediately accepted, in part because conservation of energy had not yet been established, and it was not obvious to physicists how the collisions between molecules could be perfectly elastic.
Other pioneers of the kinetic theory, whose work was also largely neglected by their contemporaries, were Mikhail Lomonosov (1747), Georges-Louis Le Sage (ca. 1780, published 1818), John Herapath (1816) and John James Waterston (1843), which connected their research with the development of mechanical explanations of gravitation.
In 1856 August Krönig created a simple gas-kinetic model, which only considered the translational motion of the particles. In 1857 Rudolf Clausius developed a similar, but more sophisticated version of the theory, which included translational and, contrary to Krönig, also rotational and vibrational molecular motions. In this same work he introduced the concept of mean free path of a particle. In 1859, after reading a paper about the diffusion of molecules by Clausius, Scottish physicist James Clerk Maxwell formulated the Maxwell distribution of molecular velocities, which gave the proportion of molecules having a certain velocity in a specific range. This was the first-ever statistical law in physics. Maxwell also gave the first mechanical argument that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium. In his 1873 thirteen page article 'Molecules', Maxwell states: "we are told that an 'atom' is a material point, invested and surrounded by 'potential forces' and that when 'flying molecules' strike against a solid body in constant succession it causes what is called pressure of air and other gases."
In 1871, Ludwig Boltzmann generalized Maxwell's achievement and formulated the Maxwell–Boltzmann distribution. The logarithmic connection between entropy and probability was also first stated by Boltzmann.
At the beginning of the 20th century, atoms were considered by many physicists to be purely hypothetical constructs, rather than real objects. An important turning point was Albert Einstein's (1905) and Marian Smoluchowski's (1906) papers on Brownian motion, which succeeded in making certain accurate quantitative predictions based on the kinetic theory.
Following the development of the Boltzmann equation, a framework for its use in developing transport equations was developed independently by David Enskog and Sydney Chapman in 1917 and 1916. The framework provided a route to prediction of the transport properties of dilute gases, and became known as Chapman–Enskog theory. The framework was gradually expanded throughout the following century, eventually becoming a route to prediction of transport properties in real, dense gases.
Assumptions.
The application of kinetic theory to ideal gases makes the following assumptions:
Thus, the dynamics of particle motion can be treated classically, and the equations of motion are time-reversible.
As a simplifying assumption, the particles are usually assumed to have the same mass as one another; however, the theory can be generalized to a mass distribution, with each mass type contributing to the gas properties independently of one another in agreement with Dalton's Law of partial pressures. Many of the model's predictions are the same whether or not collisions between particles are included, so they are often neglected as a simplifying assumption in derivations (see below).
More modern developments, such as Revised Enskog Theory and the Extended BGK model, relax one or more of the above assumptions. These can accurately describe the properties of dense gases, and gases with internal degrees of freedom, because they include the volume of the particles as well as contributions from intermolecular and intramolecular forces as well as quantized molecular rotations, quantum rotational-vibrational symmetry effects, and electronic excitation. While theories relaxing the assumptions that the gas particles occupy negligible volume and that collisions are strictly elastic have been successful, it has been shown that relaxing the requirement of interactions being binary and uncorrelated will eventually lead to divergent results.
Equilibrium properties.
Pressure and kinetic energy.
In the kinetic theory of gases, the pressure is assumed to be equal to the force (per unit area) exerted by the individual gas atoms or molecules hitting and rebounding from the gas container's surface.
Consider a gas particle traveling at velocity, formula_0, along the formula_1-direction in an enclosed volume with characteristic length, formula_2, cross-sectional area, formula_3, and volume, formula_4. The gas particle encounters a boundary after characteristic time
formula_5
The momentum of the gas particle can then be described as
formula_6
We combine the above with Newton's second law, which states that the force experienced by a particle is related to the time rate of change of its momentum, such that
formula_7
Now consider a large number, "N", of gas particles with random orientation in a three-dimensional volume. Because the orientation is random, the average particle speed, formula_8, in every direction is identical
formula_9
Further, assume that the volume is symmetrical about its three dimensions, formula_10, such that
formula_11
formula_12
formula_13
The total surface area on which the gas particles act is therefore
formula_14
The pressure exerted by the collisions of the "N" gas particles with the surface can then be found by adding the force contribution of every particle and dividing by the interior surface area of the volume,
formula_15
formula_16
The total translational kinetic energy formula_17 of the gas is defined as
formula_18
providing the result
formula_19
This is an important, non-trivial result of the kinetic theory because it relates pressure, a macroscopic property, to the translational kinetic energy of the molecules, which is a microscopic property.
Temperature and kinetic energy.
Rewriting the above result for the pressure as formula_20, we may combine it with the ideal gas law
where formula_21 is the Boltzmann constant and formula_22 the absolute temperature defined by the ideal gas law, to obtain
formula_23
which leads to a simplified expression of the average translational kinetic energy per molecule,
formula_24
The translational kinetic energy of the system is formula_25 times that of a molecule, namely formula_26. The temperature, formula_22 is related to the translational kinetic energy by the description above, resulting in
which becomes
Equation (3) is one important result of the kinetic theory:
"The average molecular kinetic energy is proportional to the ideal gas law's absolute temperature".
From equations (1) and (3), we have
Thus, the product of pressure and volume per mole is proportional to the average
translational molecular kinetic energy.
Equations (1) and (4) are called the "classical results", which could also be derived from statistical mechanics;
for more details, see:
The equipartition theorem requires that kinetic energy is partitioned equally between all kinetic degrees of freedom, "D". A monotatomic gas is axially symmetric about each spatial axis, so that "D" = 3 comprising translational motion along each axis. A diatomic gas is axially symmetric about only one axis, so that "D" = 5, comprising translational motion along three axes and rotational motion along two axes. A polyatomic gas, like water, is not radially symmetric about any axis, resulting in "D" = 6, comprising 3 translational and 3 rotational degrees of freedom.
Because the equipartition theorem requires that kinetic energy is partitioned equally, the total kinetic energy is
formula_27
Thus, the energy added to the system per gas particle kinetic degree of freedom is
formula_28
Therefore, the kinetic energy per kelvin of one mole of monatomic ideal gas ("D" = 3) is
formula_29
where formula_30 is the Avogadro constant, and "R" is the ideal gas constant.
Thus, the kinetic energy per unit kelvin of an ideal monoatomic gas can be calculated easily:
At standard temperature (273.15 K), the kinetic energy can also be obtained:
At higher temperatures (typically thousands of kelvins), vibrational modes become active to provide additional degrees of freedom, creating a temperature-dependence on "D" and the total molecular energy. Quantum statistical mechanics is needed to accurately compute these contributions.
Collisions with container wall.
For an ideal gas in equilibrium, the rate of collisions with the container wall and velocity distribution of particles hitting the container wall can be calculated based on naive kinetic theory, and the results can be used for analyzing effusive flow rates, which is useful in applications such as the gaseous diffusion method for isotope separation.
Assume that in the container, the number density (number per unit volume) is formula_31 and that the particles obey Maxwell's velocity distribution:
formula_32
Then for a small area formula_33 on the container wall, a particle with speed formula_34 at angle formula_35 from the normal of the area formula_33, will collide with the area within time interval formula_36, if it is within the distance formula_37 from the area formula_33. Therefore, all the particles with speed formula_34 at angle formula_35 from the normal that can reach area formula_33 within time interval formula_36 are contained in the tilted pipe with a height of formula_38 and a volume of formula_39.
The total number of particles that reach area formula_33 within time interval formula_36 also depends on the velocity distribution; All in all, it calculates to be:formula_40
Integrating this over all appropriate velocities within the constraint formula_41 yields the number of atomic or molecular collisions with a wall of a container per unit area per unit time:
formula_42
This quantity is also known as the "impingement rate" in vacuum physics. Note that to calculate the average speed formula_43 of the Maxwell's velocity distribution, one has to integrate overformula_44.
The momentum transfer to the container wall from particles hitting the area formula_33 with speed formula_34 at angle formula_35 from the normal, in time interval formula_36 is:formula_45Integrating this over all appropriate velocities within the constraint formula_41 yields the pressure (consistent with Ideal gas law):formula_46If this small area formula_47 is punched to become a small hole, the effusive flow rate will be: formula_48
Combined with the ideal gas law, this yields
formula_49
The above expression is consistent with Graham's law.
To calculate the velocity distribution of particles hitting this small area, we must take into account that all the particles with formula_50 that hit the area formula_33 within the time interval formula_36 are contained in the tilted pipe with a height of formula_38 and a volume of formula_39; Therefore, compared to the Maxwell distribution, the velocity distribution will have an extra factor of formula_51:
formula_52
with the constraint formula_53. The constant formula_54 can be determined by the normalization condition formula_55 to be formula_56, and overall:formula_57
Speed of molecules.
From the kinetic energy formula it can be shown that
formula_58
formula_59
formula_60
where "v" is in m/s, "T" is in kelvin, and "m" is the mass of one molecule of gas in kg. The most probable (or mode) speed formula_61 is 81.6% of the root-mean-square speed formula_62, and the mean (arithmetic mean, or average) speed formula_43 is 92.1% of the rms speed (isotropic distribution of speeds).
See:
Mean free path.
In kinetic theory of gases, the mean free path is the average distance traveled by a molecule, or a number of molecules per volume, before they make their first collision. Let formula_63 be the collision cross section of one molecule colliding with another. As in the previous section, the number density formula_64 is defined as the number of molecules per (extensive) volume, or formula_65. The collision cross section per volume or collision cross section density is formula_66, and it is related to the mean free path formula_67 byformula_68
Notice that the unit of the collision cross section per volume formula_66 is reciprocal of length.
Transport properties.
The kinetic theory of gases deals not only with gases in thermodynamic equilibrium, but also very importantly with gases not in thermodynamic equilibrium. This means using Kinetic Theory to consider what are known as "transport properties", such as viscosity, thermal conductivity, mass diffusivity and thermal diffusion.
In its most basic form, Kinetic gas theory is only applicable to dilute gases. The extension of Kinetic gas theory to dense gas mixtures, Revised Enskog Theory, was developed in 1983-1987 by E. G. D. Cohen, J. M. Kincaid and M. Lòpez de Haro, building on work by H. van Beijeren and M. H. Ernst.
Viscosity and kinetic momentum.
In books on elementary kinetic theory one can find results for dilute gas modeling that are used in many fields. Derivation of the kinetic model for shear viscosity usually starts by considering a Couette flow where two parallel plates are separated by a gas layer. The upper plate is moving at a constant velocity to the right due to a force "F". The lower plate is stationary, and an equal and opposite force must therefore be acting on it to keep it at rest. The molecules in the gas layer have a forward velocity component formula_69 which increase uniformly with distance formula_70 above the lower plate. The non-equilibrium flow is superimposed on a Maxwell-Boltzmann equilibrium distribution of molecular motions.
Inside a dilute gas in a Couette flow setup, let formula_71 be the forward velocity of the gas at a horizontal flat layer (labeled as formula_72); formula_71 is along the horizontal direction. The number of molecules arriving at the area formula_33 on one side of the gas layer, with speed formula_34 at angle formula_35 from the normal, in time interval formula_36 is
formula_73
These molecules made their last collision at formula_74, where formula_67 is the mean free path. Each molecule will contribute a forward momentum of
formula_75
where plus sign applies to molecules from above, and minus sign below. Note that the forward velocity gradient formula_76 can be considered to be constant over a distance of mean free path.
Integrating over all appropriate velocities within the constraint
formula_77
yields the forward momentum transfer per unit time per unit area (also known as shear stress):
formula_78
The net rate of momentum per unit area that is transported across the imaginary surface is thus
formula_79
Combining the above kinetic equation with Newton's law of viscosity
formula_80
gives the equation for shear viscosity, which is usually denoted formula_81 when it is a dilute gas:
formula_82
Combining this equation with the equation for mean free path gives
formula_83
Maxwell-Boltzmann distribution gives the average (equilibrium) molecular speed as
formula_84
where formula_85 is the most probable speed. We note that
formula_86
and insert the velocity in the viscosity equation above. This gives the well known equation (with formula_87 subsequently estimated below) for shear viscosity for dilute gases:
formula_88
and formula_89 is the molar mass. The equation above presupposes that the gas density is low (i.e. the pressure is low). This implies that the transport of momentum through the gas due to the translational motion of molecules is much larger than the transport due to momentum being transferred between molecules during collisions. The transfer of momentum between molecules is explicitly accounted for in Revised Enskog theory, which relaxes the requirement of a gas being dilute. The viscosity equation further presupposes that there is only one type of gas molecules, and that the gas molecules are perfect elastic and hard core particles of spherical shape. This assumption of elastic, hard core spherical molecules, like billiard balls, implies that the collision cross section of one molecule can be estimated by
formula_90
The radius formula_91 is called collision cross section radius or kinetic radius, and the diameter formula_92 is called collision cross section diameter or kinetic diameter of a molecule in a monomolecular gas. There are no simple general relation between the collision cross section and the hard core size of the (fairly spherical) molecule. The relation depends on shape of the potential energy of the molecule. For a real spherical molecule (i.e. a noble gas atom or a reasonably spherical molecule) the interaction potential is more like the Lennard-Jones potential or Morse potential which have a negative part that attracts the other molecule from distances longer than the hard core radius. The radius for zero Lennard-Jones potential may then be used as a rough estimate for the kinetic radius. However, using this estimate will typically lead to an erroneous temperature dependency of the viscosity. For such interaction potentials, significantly more accurate results are obtained by numerical evaluation of the required collision integrals.
The expression for viscosity obtained from Revised Enskog Theory reduces to the above expression in the limit of infinite dilution, and can be written as
formula_93
where formula_94 is a term that tends to zero in the limit of infinite dilution that accounts for excluded volume, and formula_95 is a term accounting for the transfer of momentum over a non-zero distance between particles during a collision.
Thermal conductivity and heat flux.
Following a similar logic as above, one can derive the kinetic model for thermal conductivity of a dilute gas:
Consider two parallel plates separated by a gas layer. Both plates have uniform temperatures, and are so massive compared to the gas layer that they can be treated as thermal reservoirs. The upper plate has a higher temperature than the lower plate. The molecules in the gas layer have a molecular kinetic energy formula_96 which increases uniformly with distance formula_70 above the lower plate. The non-equilibrium energy flow is superimposed on a Maxwell-Boltzmann equilibrium distribution of molecular motions.
Let formula_97 be the molecular kinetic energy of the gas at an imaginary horizontal surface inside the gas layer. The number of molecules arriving at an area formula_33 on one side of the gas layer, with speed formula_34 at angle formula_35 from the normal, in time interval formula_36 is
formula_98
These molecules made their last collision at a distance formula_99 above and below the gas layer, and each will contribute a molecular kinetic energy of
formula_100
where formula_101 is the specific heat capacity. Again, plus sign applies to molecules from above, and minus sign below. Note that the temperature gradient formula_102 can be considered to be constant over a distance of mean free path.
Integrating over all appropriate velocities within the constraint
formula_77
yields the energy transfer per unit time per unit area (also known as heat flux):
formula_103
Note that the energy transfer from above is in the formula_104 direction, and therefore the overall minus sign in the equation. The net heat flux across the imaginary surface is thus
formula_105
Combining the above kinetic equation with Fourier's law
formula_106
gives the equation for thermal conductivity, which is usually denoted formula_107 when it is a dilute gas:
formula_108
Similarly to viscosity, Revised Enskog Theory yields an expression for thermal conductivity that reduces to the above expression in the limit of infinite dilution, and which can be written as
formula_109
where formula_110 is a term that tends to unity in the limit of infinite dilution, accounting for excluded volume, and formula_111 is a term accounting for the transfer of energy across a non-zero distance between particles during a collision.
Diffusion Coefficient and diffusion flux.
Following a similar logic as above, one can derive the kinetic model for mass diffusivity of a dilute gas:
Consider a steady diffusion between two regions of the same gas with perfectly flat and parallel boundaries separated by a layer of the same gas. Both regions have uniform number densities, but the upper region has a higher number density than the lower region. In the steady state, the number density at any point is constant (that is, independent of time). However, the number density formula_112 in the layer increases uniformly with distance formula_70 above the lower plate. The non-equilibrium molecular flow is superimposed on a Maxwell-Boltzmann equilibrium distribution of molecular motions.
Let formula_113 be the number density of the gas at an imaginary horizontal surface inside the layer. The number of molecules arriving at an area formula_33 on one side of the gas layer, with speed formula_34 at angle formula_35 from the normal, in time interval formula_36 is
formula_114
These molecules made their last collision at a distance formula_99 above and below the gas layer, where the local number density is
formula_115
Again, plus sign applies to molecules from above, and minus sign below. Note that the number density gradient formula_116 can be considered to be constant over a distance of mean free path.
Integrating over all appropriate velocities within the constraint
formula_77
yields the molecular transfer per unit time per unit area (also known as diffusion flux):
formula_117
Note that the molecular transfer from above is in the formula_104 direction, and therefore the overall minus sign in the equation. The net diffusion flux across the imaginary surface is thus
formula_118
Combining the above kinetic equation with Fick's first law of diffusion
formula_119
gives the equation for mass diffusivity, which is usually denoted formula_120 when it is a dilute gas:
formula_121
The corresponding expression obtained from Revised Enskog Theory may be written as
formula_122
where formula_123 is a factor that tends to unity in the limit of infinite dilution, which accounts for excluded volume and the variation chemical potentials with density.
Detailed balance.
Fluctuation and dissipation.
The kinetic theory of gases entails that due to the microscopic reversibility of the gas particles' detailed dynamics, the system must obey the principle of detailed balance. Specifically, the fluctuation-dissipation theorem applies to the Brownian motion (or diffusion) and the drag force, which leads to the Einstein–Smoluchowski equation:formula_124where
Note that the mobility "μ" = "v"d/"F" can be calculated based on the viscosity of the gas; Therefore, the Einstein–Smoluchowski equation also provides a relation between the mass diffusivity and the viscosity of the gas.
Onsager reciprocal relations.
The mathematical similarities between the expressions for shear viscocity, thermal conductivity and diffusion coefficient of the ideal (dilute) gas is not a coincidence; It is a direct result of the Onsager reciprocal relations (i.e. the detailed balance of the reversible dynamics of the particles), when applied to the convection (matter flow due to temperature gradient, and heat flow due to pressure gradient) and advection (matter flow due to the velocity of particles, and momentum transfer due to pressure gradient) of the ideal (dilute) gas.
Notes.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "v_i"
},
{
"math_id": 1,
"text": "\\hat{i}"
},
{
"math_id": 2,
"text": "L_i"
},
{
"math_id": 3,
"text": "A_i"
},
{
"math_id": 4,
"text": "V = A_i L_i"
},
{
"math_id": 5,
"text": " t = L_i / v_i."
},
{
"math_id": 6,
"text": " p_i = m v_i = m L_i / t ."
},
{
"math_id": 7,
"text": "F_i = \\frac{\\mathrm{d}p_i}{\\mathrm{d}t} = \\frac{m L_i}{t^2}=\\frac{m v_i^2}{L_i}."
},
{
"math_id": 8,
"text": " v "
},
{
"math_id": 9,
"text": "v^2 = {\\vec{v}_x^2} = {\\vec{v}_y^2} = {\\vec{v}_z^2}."
},
{
"math_id": 10,
"text": "\\hat{i}, \\hat{j}, \\hat{k}"
},
{
"math_id": 11,
"text": "v = v_i=v_j=v_k,"
},
{
"math_id": 12,
"text": "F = F_i = F_j = F_k,"
},
{
"math_id": 13,
"text": "A_i=A_j=A_k."
},
{
"math_id": 14,
"text": "A = 3 * A_i."
},
{
"math_id": 15,
"text": "P = \\frac{N \\overline{F}}{A}=\\frac{NLF}{V} "
},
{
"math_id": 16,
"text": " \\Rightarrow PV=NLF = \\frac{N}{3} m v^2."
},
{
"math_id": 17,
"text": "K_t "
},
{
"math_id": 18,
"text": "K_t = \\frac{N}{2} m v^2 ,"
},
{
"math_id": 19,
"text": "PV = \\frac{2}{3} K_t ."
},
{
"math_id": 20,
"text": "PV = \\frac{Nmv^2}{3} "
},
{
"math_id": 21,
"text": " k_\\mathrm{B}"
},
{
"math_id": 22,
"text": " T"
},
{
"math_id": 23,
"text": "k_\\mathrm{B} T = {m v^2\\over 3}, "
},
{
"math_id": 24,
"text": " \\frac{1}{2} m v^2 = \\frac{3}{2} k_\\mathrm{B} T."
},
{
"math_id": 25,
"text": "N"
},
{
"math_id": 26,
"text": " K_t = \\frac{1}{2} N m v^2 "
},
{
"math_id": 27,
"text": " K =D K_t = \\frac{D}{2} N m v^2. "
},
{
"math_id": 28,
"text": " \\frac{K}{ND} = \\frac{1}{2} k_B T . "
},
{
"math_id": 29,
"text": " K = \\frac{D}{2} k_B N_A T = \\frac{3}{2} R, "
},
{
"math_id": 30,
"text": "N_A"
},
{
"math_id": 31,
"text": "n=N/V"
},
{
"math_id": 32,
"text": "f_\\text{Maxwell}(v_x,v_y,v_z) \\, dv_x \\, dv_y \\, dv_z = \\left(\\frac{m}{2 \\pi k_B T}\\right)^{3/2} e^{- \\frac{mv^2}{2k_BT}} \\, dv_x \\, dv_y \\, dv_z"
},
{
"math_id": 33,
"text": "dA"
},
{
"math_id": 34,
"text": "v"
},
{
"math_id": 35,
"text": "\\theta"
},
{
"math_id": 36,
"text": "dt"
},
{
"math_id": 37,
"text": "vdt"
},
{
"math_id": 38,
"text": "v\\cos (\\theta) dt"
},
{
"math_id": 39,
"text": "v\\cos (\\theta) dAdt"
},
{
"math_id": 40,
"text": "n v \\cos(\\theta) \\, dA\\, dt \\times\\left(\\frac{m}{2 \\pi k_BT}\\right)^{3/2} e^{- \\frac{mv^2}{2k_BT}} \\left( v^2 \\sin(\\theta) \\, dv \\, d\\theta \\, d\\phi \\right)."
},
{
"math_id": 41,
"text": "v>0,0<\\theta<\\pi/2,0<\\phi<2\\pi"
},
{
"math_id": 42,
"text": "J_\\text{collision} =\\frac{\\int_0^{\\pi/2}\\cos \\theta \\sin \\theta d\\theta}{\\int_0^{\\pi}\\sin \\theta d\\theta}\\times n \\bar v= \\frac{1}{4}n \\bar v = \\frac{n}{4} \\sqrt{\\frac{8 k_\\mathrm{B} T}{\\pi m}}. "
},
{
"math_id": 43,
"text": "\\bar v"
},
{
"math_id": 44,
"text": "v>0,0<\\theta<\\pi,0<\\phi<2\\pi"
},
{
"math_id": 45,
"text": "[2mv \\cos(\\theta)]\\times n v \\cos(\\theta) \\, dA\\, dt \\times\\left(\\frac{m}{2 \\pi k_BT}\\right)^{3/2} e^{- \\frac{mv^2}{2k_BT}} \\left( v^2 \\sin(\\theta) \\, dv \\, d\\theta \\, d\\phi \\right)."
},
{
"math_id": 46,
"text": "P=\\frac{2\\int_0^{\\pi/2}\\cos^2 \\theta \\sin \\theta d\\theta}{\\int_0^{\\pi}\\sin \\theta d\\theta}\\times n mv_\\text{rms}^2=\\frac{1}{3}n mv_\\text{rms}^2=\\frac{2}{3}n\\langle E_{kin}\\rangle=nk_\\mathrm{B} T "
},
{
"math_id": 47,
"text": "A"
},
{
"math_id": 48,
"text": "\\Phi_\\text{effusion} = J_\\text{collision} A= n A \\sqrt{\\frac{k_\\mathrm{B} T}{2 \\pi m}}. "
},
{
"math_id": 49,
"text": "\\Phi_\\text{effusion} = \\frac{P A}{\\sqrt{2 \\pi m k_\\mathrm{B} T}}. "
},
{
"math_id": 50,
"text": "(v,\\theta,\\phi)"
},
{
"math_id": 51,
"text": "v\\cos \\theta"
},
{
"math_id": 52,
"text": "\\begin{align}\nf(v,\\theta,\\phi) \\, dv \\, d\\theta \\, d\\phi\n&=\\lambda v\\cos{\\theta}{\\times} \\left(\\frac{m}{2 \\pi k T}\\right)^{3/2}e^{- \\frac{mv^2}{2k_\\mathrm{B} T}}(v^2\\sin{\\theta} \\, dv \\, d\\theta \\, d\\phi) \\\\\n\\end{align}"
},
{
"math_id": 53,
"text": "v>0,\\, 0<\\theta<\\frac \\pi 2,\\, 0<\\phi<2\\pi"
},
{
"math_id": 54,
"text": "\\lambda"
},
{
"math_id": 55,
"text": "\\int f(v,\\theta,\\phi) \\, dv \\, d\\theta \\, d\\phi=1"
},
{
"math_id": 56,
"text": "{4}/{\\bar v} "
},
{
"math_id": 57,
"text": "\\begin{align}\nf(v,\\theta,\\phi) \\, dv \\, d\\theta \\, d\\phi\n&=\\frac{1}{2\\pi} \\left(\\frac{m}{k_\\mathrm{B} T}\\right)^2e^{- \\frac{mv^2}{2k_\\mathrm{B} T}}\n(v^3\\sin{\\theta}\\cos{\\theta} \\, dv \\, d\\theta \\, d\\phi) \\\\\n\\end{align};\\quad v>0,\\, 0<\\theta<\\frac \\pi 2,\\, 0<\\phi<2\\pi"
},
{
"math_id": 58,
"text": "v_\\text{p} = \\sqrt{2 \\cdot \\frac{k_\\mathrm{B} T}{m}},"
},
{
"math_id": 59,
"text": " \\bar v = \\frac {2}{\\sqrt{\\pi}} v_p = \\sqrt{\\frac {8}{\\pi} \\cdot \\frac{k_\\mathrm{B} T}{m}},"
},
{
"math_id": 60,
"text": "v_\\text{rms} = \\sqrt{\\frac{3}{2}} v_p = \\sqrt{{3} \\cdot \\frac {k_\\mathrm{B} T}{m}},"
},
{
"math_id": 61,
"text": "v_\\text{p}"
},
{
"math_id": 62,
"text": "v_\\text{rms}"
},
{
"math_id": 63,
"text": " \\sigma "
},
{
"math_id": 64,
"text": " n "
},
{
"math_id": 65,
"text": " n = N/V "
},
{
"math_id": 66,
"text": " n \\sigma "
},
{
"math_id": 67,
"text": "l"
},
{
"math_id": 68,
"text": "l = \\frac {1} {n \\sigma \\sqrt{2}} "
},
{
"math_id": 69,
"text": "u"
},
{
"math_id": 70,
"text": "y"
},
{
"math_id": 71,
"text": " u_{0} "
},
{
"math_id": 72,
"text": "y=0"
},
{
"math_id": 73,
"text": "nv\\cos({\\theta})\\, dA \\, dt \\times \\left(\\frac{m}{2 \\pi k_\\mathrm{B} T}\\right)^{3/2} \\, e^{- \\frac{mv^2}{2 k_\\mathrm{B} T}} (v^2\\sin{\\theta} \\, dv \\, d\\theta \\, d\\phi)"
},
{
"math_id": 74,
"text": "y=\\pm l\\cos \\theta"
},
{
"math_id": 75,
"text": "p_{x}^{\\pm} = m \\left( u_{0} \\pm l \\cos \\theta \\,{d u \\over dy} \\right), "
},
{
"math_id": 76,
"text": "du/dy"
},
{
"math_id": 77,
"text": "\\begin{cases}\nv>0\\\\\n0<\\theta<\\pi/2\\\\\n0<\\phi<2\\pi\n\\end{cases}"
},
{
"math_id": 78,
"text": "\\tau^{\\pm} = \\frac {1}{4} \\bar v n \\cdot m \\left( u_{0} \\pm \\frac {2}{3} l \\,{d u \\over dy} \\right) "
},
{
"math_id": 79,
"text": "\\tau = \\tau^{+} - \\tau^{-} = \\frac {1}{3} \\bar v n m \\cdot l \\,{d u \\over dy} "
},
{
"math_id": 80,
"text": "\\tau = \\eta \\,{d u \\over dy} "
},
{
"math_id": 81,
"text": " \\eta_{0} "
},
{
"math_id": 82,
"text": "\\eta_{0} = \\frac {1} {3} \\bar v n m l "
},
{
"math_id": 83,
"text": "\\eta_{0} = \\frac {1} {3 \\sqrt{2} } \\frac {m \\cdot \\bar v} {\\sigma}"
},
{
"math_id": 84,
"text": "\\bar v = \\frac{2}{\\sqrt{\\pi}} v_{p} = 2 \\sqrt{\\frac{2}{\\pi} \\cdot \\frac {k_\\mathrm{B}T}{m}} "
},
{
"math_id": 85,
"text": "v_{p}"
},
{
"math_id": 86,
"text": "k_{B} \\cdot N_{A} = R \\quad \\text{and} \\quad M = m \\cdot N_{A} "
},
{
"math_id": 87,
"text": "\\sigma"
},
{
"math_id": 88,
"text": "\\eta_{0}\n= \\frac {2} {3 \\sqrt{\\pi} } \\cdot \\frac {\\sqrt{m k_\\mathrm{B} T}} { \\sigma } \n= \\frac {2} {3 \\sqrt{\\pi} } \\cdot \\frac {\\sqrt{MRT}} { \\sigma \\cdot N_{A} } "
},
{
"math_id": 89,
"text": " M "
},
{
"math_id": 90,
"text": "\\sigma = \\pi \\left( 2 r \\right)^2 = \\pi d^2 "
},
{
"math_id": 91,
"text": "r"
},
{
"math_id": 92,
"text": "d"
},
{
"math_id": 93,
"text": "\\eta = (1 + \\alpha_\\eta)\\eta_0 + \\eta_c "
},
{
"math_id": 94,
"text": "\\alpha_\\eta"
},
{
"math_id": 95,
"text": "\\eta_c"
},
{
"math_id": 96,
"text": "\\varepsilon"
},
{
"math_id": 97,
"text": " \\varepsilon_{0} "
},
{
"math_id": 98,
"text": " nv \\cos(\\theta)\\, dA \\, dt \\times \\left(\\frac{m}{2 \\pi k_\\mathrm{B}T}\\right)^{3 / 2} e^{- \\frac{mv^2}{2k_BT}} (v^2 \\sin(\\theta) \\, dv \\, d\\theta \\, d\\phi)"
},
{
"math_id": 99,
"text": "l\\cos \\theta"
},
{
"math_id": 100,
"text": " \\varepsilon^{\\pm} = \\left( \\varepsilon_{0} \\pm m c_v l \\cos \\theta \\, {d T \\over dy} \\right), "
},
{
"math_id": 101,
"text": "c_v"
},
{
"math_id": 102,
"text": "dT/dy"
},
{
"math_id": 103,
"text": " q_y^{\\pm} = -\\frac {1}{4} \\bar v n \\cdot \\left( \\varepsilon_{0} \\pm \\frac {2}{3} m c_v l \\,{d T \\over dy} \\right) "
},
{
"math_id": 104,
"text": "-y"
},
{
"math_id": 105,
"text": " q = q_y^{+} - q_y^{-} = -\\frac {1}{3} \\bar v n m c_v l \\,{d T \\over dy} "
},
{
"math_id": 106,
"text": " q = -\\kappa \\,{d T \\over dy} "
},
{
"math_id": 107,
"text": " \\kappa_{0} "
},
{
"math_id": 108,
"text": " \\kappa_{0} = \\frac {1} {3} \\bar v n m c_v l "
},
{
"math_id": 109,
"text": " \\kappa = \\alpha_\\kappa \\kappa_0 + \\kappa_c "
},
{
"math_id": 110,
"text": " \\alpha_\\kappa "
},
{
"math_id": 111,
"text": " \\kappa_c "
},
{
"math_id": 112,
"text": "n"
},
{
"math_id": 113,
"text": " n_{0} "
},
{
"math_id": 114,
"text": " nv\\cos(\\theta) \\, dA \\, dt \\times \\left(\\frac{m}{2 \\pi k_\\mathrm{B}T}\\right)^{3 / 2} e^{- \\frac{mv^2}{2k_BT}} (v^2\\sin(\\theta) \\, dv\\, d\\theta \\, d\\phi)"
},
{
"math_id": 115,
"text": " n^{\\pm} = \\left( n_{0} \\pm l \\cos \\theta \\, {d n \\over dy} \\right) "
},
{
"math_id": 116,
"text": "dn/dy"
},
{
"math_id": 117,
"text": " J_y^{\\pm} = -\\frac {1}{4} \\bar v \\cdot \\left( n_{0} \\pm \\frac {2}{3} l \\, {d n \\over dy} \\right) "
},
{
"math_id": 118,
"text": " J = J_y^{+} - J_y^{-} = -\\frac {1}{3} \\bar v l {d n \\over dy} "
},
{
"math_id": 119,
"text": " J = -D {d n \\over dy} "
},
{
"math_id": 120,
"text": " D_{0} "
},
{
"math_id": 121,
"text": " D_{0} = \\frac {1} {3} \\bar v l "
},
{
"math_id": 122,
"text": " D = \\alpha_D D_0 "
},
{
"math_id": 123,
"text": " \\alpha_D "
},
{
"math_id": 124,
"text": " D = \\mu \\, k_\\text{B} T, "
}
] |
https://en.wikipedia.org/wiki?curid=64204
|
6420610
|
Ovoid (polar space)
|
In mathematics, an ovoid "O" of a (finite) polar space of rank "r" is a set of points, such that every subspace of rank formula_0 intersects "O" in exactly one point.
Cases.
Symplectic polar space.
An ovoid of formula_1 (a symplectic polar space of rank "n") would contain formula_2 points.
However it only has an ovoid if and only formula_3 and "q" is even. In that case, when the polar space is embedded into formula_4 the classical way, it is also an ovoid in the projective geometry sense.
Hermitian polar space.
Ovoids of formula_5 and formula_6 would contain formula_7 points.
Hyperbolic quadrics.
An ovoid of a hyperbolic quadricformula_8would contain formula_9 points.
Parabolic quadrics.
An ovoid of a parabolic quadric formula_10 would contain formula_2 points. For formula_3, it is easy to see to obtain an ovoid by cutting the parabolic quadric with a hyperplane, such that the intersection is an elliptic quadric. The intersection is an ovoid.
If "q" is even, formula_11 is isomorphic (as polar space) with formula_1, and thus due to the above, it has no ovoid for formula_12.
Elliptic quadrics.
An ovoid of an elliptic quadric formula_13would contain formula_14 points.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "r-1"
},
{
"math_id": 1,
"text": "W_{2 n-1}(q)"
},
{
"math_id": 2,
"text": "q^n+1"
},
{
"math_id": 3,
"text": "n=2"
},
{
"math_id": 4,
"text": "PG(3,q)"
},
{
"math_id": 5,
"text": "H(2n,q^2)(n\\geq 2)"
},
{
"math_id": 6,
"text": "H(2n+1,q^2)(n\\geq 1)"
},
{
"math_id": 7,
"text": "q^{2n+1}+1"
},
{
"math_id": 8,
"text": " Q^{+}(2n-1,q)(n\\geq 2)"
},
{
"math_id": 9,
"text": "q^{n-1}+1"
},
{
"math_id": 10,
"text": "Q(2 n,q)(n\\geq 2)"
},
{
"math_id": 11,
"text": "Q(2n,q)"
},
{
"math_id": 12,
"text": "n\\geq 3"
},
{
"math_id": 13,
"text": "Q^{-}(2n+1,q)(n\\geq 2)"
},
{
"math_id": 14,
"text": "q^{n}+1"
}
] |
https://en.wikipedia.org/wiki?curid=6420610
|
642090
|
Asymptotic expansion
|
Series of functions in mathematics
In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to a given function as the argument of the function tends towards a particular, often infinite, point. Investigations by revealed that the divergent part of an asymptotic expansion is latently meaningful, i.e. contains information about the exact value of the expanded function.
The theory of asymptotic series was created by Poincaré (and independently by Stieltjes) in 1886.
The most common type of asymptotic expansion is a power series in either positive or negative powers. Methods of generating such expansions include the Euler–Maclaurin summation formula and integral transforms such as the Laplace and Mellin transforms. Repeated integration by parts will often lead to an asymptotic expansion.
Since a "convergent" Taylor series fits the definition of asymptotic expansion as well, the phrase "asymptotic series" usually implies a "non-convergent" series. Despite non-convergence, the asymptotic expansion is useful when truncated to a finite number of terms. The approximation may provide benefits by being more mathematically tractable than the function being expanded, or by an increase in the speed of computation of the expanded function. Typically, the best approximation is given when the series is truncated at the smallest term. This way of optimally truncating an asymptotic expansion is known as superasymptotics. The error is then typically of the form ~ exp(−"c"/ε) where ε is the expansion parameter. The error is thus beyond all orders in the expansion parameter. It is possible to improve on the superasymptotic error, e.g. by employing resummation methods such as Borel resummation to the divergent tail. Such methods are often referred to as hyperasymptotic approximations.
See asymptotic analysis and big O notation for the notation used in this article.
Formal definition.
First we define an asymptotic scale, and then give the formal definition of an asymptotic expansion.
If formula_0 is a sequence of continuous functions on some domain, and if formula_1 is a limit point of the domain, then the sequence constitutes an asymptotic scale if for every n,
formula_2
(formula_1 may be taken to be infinity.) In other words, a sequence of functions is an asymptotic scale if each function in the sequence grows strictly slower (in the limit formula_3) than the preceding function.
If formula_4 is a continuous function on the domain of the asymptotic scale, then f has an asymptotic expansion of order formula_5 with respect to the scale as a formal series
formula_6
if
formula_7
or the weaker condition
formula_8
is satisfied. Here, formula_9 is the little o notation. If one or the other holds for all formula_5, then we write
formula_10
In contrast to a convergent series for formula_4, wherein the series converges for any "fixed" formula_11 in the limit formula_12, one can think of the asymptotic series as converging for "fixed" formula_5 in the limit formula_3 (with formula_1 possibly infinite).
Worked example.
Asymptotic expansions often occur when an ordinary series is used in a formal expression that forces the taking of values outside of its domain of convergence. Thus, for example, one may start with the ordinary series
formula_21
The expression on the left is valid on the entire complex plane formula_22, while the right hand side converges only for formula_23. Multiplying by formula_24 and integrating both sides yields
formula_25
after the substitution formula_26 on the right hand side. The integral on the left hand side, understood as a Cauchy principal value, can be expressed in terms of the exponential integral. The integral on the right hand side may be recognized as the gamma function. Evaluating both, one obtains the asymptotic expansion
formula_27
Here, the right hand side is clearly not convergent for any non-zero value of "t". However, by truncating the series on the right to a finite number of terms, one may obtain a fairly good approximation to the value of formula_28 for sufficiently small "t". Substituting formula_29 and noting that formula_30 results in the asymptotic expansion given earlier in this article.
Integration by parts.
Using integration by parts, we can obtain an explicit formulaformula_31For any fixed formula_32, the absolute value of the error term formula_33 decreases, then increases. The minimum occurs at formula_34, at which point formula_35. This bound is said to be "asymptotics beyond all orders".
Properties.
Uniqueness for a given asymptotic scale.
For a given asymptotic scale formula_36 the asymptotic expansion of function formula_37 is unique. That is the coefficients formula_38 are uniquely determined in the following way:
formula_39
where formula_40 is the limit point of this asymptotic expansion (may be formula_41).
Non-uniqueness for a given function.
A given function formula_37 may have many asymptotic expansions (each with a different asymptotic scale).
Subdominance.
An asymptotic expansion may be an asymptotic expansion to more than one function.
|
[
{
"math_id": 0,
"text": "\\ \\varphi_n\\ "
},
{
"math_id": 1,
"text": "\\ L\\ "
},
{
"math_id": 2,
"text": "\\varphi_{n+1}(x) = o(\\varphi_n(x)) \\quad (x \\to L)\\ ."
},
{
"math_id": 3,
"text": "\\ x \\to L\\ "
},
{
"math_id": 4,
"text": "\\ f\\ "
},
{
"math_id": 5,
"text": "\\ N\\ "
},
{
"math_id": 6,
"text": " \\sum_{n=0}^N a_n \\varphi_{n}(x) "
},
{
"math_id": 7,
"text": " f(x) - \\sum_{n=0}^{N-1} a_n \\varphi_{n}(x) = O(\\varphi_{N}(x)) \\quad (x \\to L) "
},
{
"math_id": 8,
"text": " f(x) - \\sum_{n=0}^{N-1} a_n \\varphi_{n}(x) = o(\\varphi_{N-1}(x)) \\quad (x \\to L)\\ "
},
{
"math_id": 9,
"text": "o"
},
{
"math_id": 10,
"text": " f(x) \\sim \\sum_{n=0}^\\infty a_n \\varphi_n(x) \\quad (x \\to L)\\ ."
},
{
"math_id": 11,
"text": "\\ x\\ "
},
{
"math_id": 12,
"text": "N \\to \\infty"
},
{
"math_id": 13,
"text": " \\frac{e^x}{x^x\\sqrt{2\\pi x}} \\Gamma(x+1) \\sim 1+\\frac{1}{12x}+\\frac{1}{288x^2}-\\frac{139}{51840x^3}-\\cdots\\ (x \\to \\infty)"
},
{
"math_id": 14,
"text": "x e^x E_1(x) \\sim \\sum_{n=0}^\\infty \\frac{(-1)^nn!}{x^n} \\ (x \\to \\infty) "
},
{
"math_id": 15,
"text": "\\operatorname{li}(x) \\sim \\frac{x}{\\ln x} \\sum_{k=0}^{\\infty} \\frac{k!}{(\\ln x)^k}"
},
{
"math_id": 16,
"text": "\\zeta(s) \\sim \\sum_{n=1}^{N}n^{-s} + \\frac{N^{1-s}}{s-1} - \\frac{N^{-s}}{2} + N^{-s} \\sum_{m=1}^\\infty \\frac{B_{2m} s^{\\overline{2m-1}}}{(2m)! N^{2m-1}}"
},
{
"math_id": 17,
"text": "B_{2m}"
},
{
"math_id": 18,
"text": "s^{\\overline{2m-1}}"
},
{
"math_id": 19,
"text": "N > |s|"
},
{
"math_id": 20,
"text": " \\sqrt{\\pi}x e^{x^2}{\\rm erfc}(x) \\sim 1+\\sum_{n=1}^\\infty (-1)^n \\frac{(2n-1)!!}{(2x^2)^n} \\ (x \\to \\infty)"
},
{
"math_id": 21,
"text": "\\frac{1}{1-w}=\\sum_{n=0}^\\infty w^n."
},
{
"math_id": 22,
"text": "w\\ne 1"
},
{
"math_id": 23,
"text": "|w|< 1"
},
{
"math_id": 24,
"text": "e^{-w/t}"
},
{
"math_id": 25,
"text": "\\int_0^\\infty \\frac{e^{-\\frac{w}{t}}}{1-w}\\, dw = \\sum_{n=0}^\\infty t^{n+1} \\int_0^\\infty e^{-u} u^n\\, du,"
},
{
"math_id": 26,
"text": "u=w/t"
},
{
"math_id": 27,
"text": "e^{-\\frac{1}{t}} \\operatorname{Ei}\\left(\\frac{1}{t}\\right) = \\sum_{n=0}^\\infty n! t^{n+1}. "
},
{
"math_id": 28,
"text": "\\operatorname{Ei} \\left (\\tfrac{1}{t} \\right )"
},
{
"math_id": 29,
"text": "x=-\\tfrac{1}{t}"
},
{
"math_id": 30,
"text": "\\operatorname{Ei}(x)=-E_1(-x)"
},
{
"math_id": 31,
"text": "\\operatorname{Ei}(z) = \\frac{e^{z}} {z} \\left (\\sum _{k=0}^{n} \\frac{k!} {z^{k}} + e_{n}(z)\\right), \\quad e_{n}(z) \\equiv (n + 1)!\\ ze^{-z}\\int _{ -\\infty }^{z} \\frac{e^{t}} {t^{n+2}}\\,dt"
},
{
"math_id": 32,
"text": "z"
},
{
"math_id": 33,
"text": "|e_n(z)|"
},
{
"math_id": 34,
"text": "n\\sim |z|"
},
{
"math_id": 35,
"text": "\\vert e_{n}(z)\\vert \\leq \\sqrt{\\frac{2\\pi } {\\vert z\\vert }}e^{-\\vert z\\vert }"
},
{
"math_id": 36,
"text": "\\{\\varphi_n(x)\\}"
},
{
"math_id": 37,
"text": "f(x)"
},
{
"math_id": 38,
"text": "\\{a_n\\}"
},
{
"math_id": 39,
"text": "\\begin{align}\na_0 &= \\lim_{x \\to L} \\frac{f(x)}{\\varphi_0(x)} \\\\\na_1 &= \\lim_{x \\to L} \\frac{f(x) - a_0 \\varphi_0(x)} {\\varphi_1(x)} \\\\\n& \\;\\;\\vdots \\\\\na_N &= \\lim_{x \\to L} \\frac {f(x) - \\sum_{n=0}^{N-1} a_n \\varphi_n(x)} {\\varphi_N(x)}\n\\end{align}"
},
{
"math_id": 40,
"text": "L"
},
{
"math_id": 41,
"text": "\\pm \\infty"
}
] |
https://en.wikipedia.org/wiki?curid=642090
|
642101
|
Cardioid
|
Type of plane curve
In geometry, a cardioid (from el " "καρδιά" (kardiá)" 'heart') is a plane curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius. It can also be defined as an epicycloid having a single cusp. It is also a type of sinusoidal spiral, and an inverse curve of the parabola with the focus as the center of inversion. A cardioid can also be defined as the set of points of reflections of a fixed point on a circle through all tangents to the circle.
The name was coined by Giovanni Salvemini in 1741 but the cardioid had been the subject of study decades beforehand. Although named for its heart-like form, it is shaped more like the outline of the cross-section of a round apple without the stalk.
A cardioid microphone exhibits an acoustic pickup pattern that, when graphed in two dimensions, resembles a cardioid (any 2d plane containing the 3d straight line of the microphone body). In three dimensions, the cardioid is shaped like an apple centred around the microphone which is the "stalk" of the apple.
Equations.
Let formula_0 be the common radius of the two generating circles with midpoints formula_1, formula_2 the rolling angle and the origin the starting point (see picture). One gets the
Proof for the parametric representation.
A proof can be established using complex numbers and their common description as the complex plane. The rolling movement of the black circle on the blue one can be split into two rotations. In the complex plane a rotation around point formula_8 (the origin) by an angle formula_2 can be performed by multiplying a point formula_9 (complex number) by formula_10. Hence
the rotation formula_11 around point formula_0 isformula_12,
the rotation formula_13 around point formula_14 is: formula_15.
A point formula_16 of the cardioid is generated by rotating the origin around point formula_0 and subsequently rotating around formula_14 by the same angle formula_2:
formula_17
From here one gets the parametric representation above:
formula_18
Metric properties.
For the cardioid as defined above the following formulas hold:
The proofs of these statements use in both cases the polar representation of the cardioid. For suitable formulas see polar coordinate system (arc length) and polar coordinate system (area)
<templatestyles src="Math_proof/styles.css" />Proof of the area formula
formula_25
<templatestyles src="Math_proof/styles.css" />Proof of the arc length formula
formula_26
<templatestyles src="Math_proof/styles.css" />Proof for the radius of curvature
The radius of curvature formula_27 of a curve in polar coordinates with equation formula_28 is (s. )
formula_29
For the cardioid formula_30 one gets
formula_31
Properties.
Chords through the cusp.
Proof of C1.
The points formula_33 are on a chord through the cusp (=origin). Hence formula_34
Proof for C2.
For the proof the representation in the complex plane (see above) is used. For the points formula_35 and formula_36
the midpoint of the chord formula_37 is formula_38 which lies on the perimeter of the circle with midpoint formula_14 and radius formula_0 (see picture).
A cardioid is the inverse curve of a parabola with its focus at the center of inversion (see graph)
Cardioid as inverse curve of a parabola.
For the example shown in the graph the generator circles have radius formula_39. Hence the cardioid has the polar representation
formula_40
and its inverse curve
formula_41
which is a parabola (s. parabola in polar coordinates) with the equation formula_42 in Cartesian coordinates.
"Remark: "Not every inverse curve of a parabola is a cardioid. For example, if a parabola is inverted across a circle whose center lies at the "vertex" of the parabola, then the result is a cissoid of Diocles.
Cardioid as envelope of a pencil of circles.
In the previous section if one inverts additionally the tangents of the parabola one gets a pencil of circles through the center of inversion (origin). A detailed consideration shows: The midpoints of the circles lie on the perimeter of the fixed generator circle. (The generator circle is the inverse curve of the parabola's directrix.)
This property gives rise to the following simple method to "draw" a cardioid:
<templatestyles src="Math_proof/styles.css" />Proof with envelope condition
The envelope of the pencil of implicitly given curves formula_45 with parameter formula_46 consists of such points formula_47 which are solutions of the non-linear system
formula_48 which is the envelope condition. Note that formula_49 means the partial derivative for parameter formula_46.
Let formula_43 be the circle with midpoint formula_50 and radius formula_51. Then formula_43 has parametric representation formula_52. The pencil of circles with centers on formula_43 containing point formula_53 can be represented implicitly by
formula_54
which is equivalent to
formula_55
The second envelope condition is
formula_56
One easily checks that the points of the cardioid with the parametric representation
formula_57
fulfill the non-linear system above. The parameter formula_46 is identical to the angle parameter of the cardioid.
Cardioid as envelope of a pencil of lines.
A similar and simple method to draw a cardioid uses a pencil of "lines". It is due to L. Cremona:
Proof.
The following consideration uses trigonometric formulae for formula_60, formula_61, formula_62, formula_63, and formula_64.
In order to keep the calculations simple, the proof is given for the cardioid with polar representation
formula_65 ("§ Cardioids in different positions").
2(1 + cos 𝜑)=====
From the parametric representation
formula_66
one gets the normal vector formula_67. The equation of the tangent
formula_68 is:
formula_69
With help of trigonometric formulae and subsequent division by formula_70, the equation of the tangent can be rewritten as:
formula_71
"Equation of the chord" of the "circle" with midpoint and radius 3.
For the equation of the secant line passing the two points formula_72 one gets:
formula_73
With help of trigonometric formulae and the subsequent division by formula_74 the equation of the secant line can be rewritten by:
formula_75
Conclusion.
Despite the two angles formula_76 have different meanings (s. picture) one gets for formula_77 the same line. Hence any secant line of the circle, defined above, is a tangent of the cardioid, too:
"The cardioid is the envelope of the chords of a circle."
"Remark:"
The proof can be performed with help of the "envelope conditions" (see previous section) of an implicit pencil of curves:
formula_78
is the pencil of secant lines of a circle (s. above) and
formula_79
For fixed parameter t both the equations represent lines. Their intersection point is
formula_80
which is a point of the cardioid with polar equation formula_81
Cardioid as caustic of a circle.
The considerations made in the previous section give a proof that the caustic of a circle with light source on the perimeter of the circle is a cardioid.
If in the plane there is a light source at a point formula_82 on the perimeter of a circle which is reflecting any ray, then the reflected rays within the circle are tangents of a cardioid.
<templatestyles src="Math_proof/styles.css" />Proof
As in the previous section the circle may have midpoint formula_83 and radius formula_84. Its parametric representation is
formula_85
The tangent at circle point formula_86 has normal vector formula_87. Hence the reflected ray has the normal vector formula_88 (see graph) and contains point formula_89. The reflected ray is part of the line with equation (see previous section)
formula_90
which is tangent of the cardioid with polar equation
formula_91
from the previous section.
"Remark:" For such considerations usually multiple reflections at the circle are neglected.
Cardioid as pedal curve of a circle.
The Cremona generation of a cardioid should not be confused with the following generation:
Let be formula_92 a circle and formula_44 a point on the perimeter of this circle. The following is true:
The foots of perpendiculars from point formula_44 on the tangents of circle formula_92 are points of a cardioid.
Hence a cardioid is a special pedal curve of a circle.
Proof.
In a Cartesian coordinate system circle formula_92 may have midpoint formula_93 and radius formula_94. The tangent at circle point formula_95 has the equation
formula_96
The foot of the perpendicular from point formula_44 on the tangent is point formula_97 with the still unknown distance formula_98 to the origin formula_44. Inserting the point into the equation of the tangent yields
formula_99
which is the polar equation of a cardioid.
"Remark:" If point formula_44 is not on the perimeter of the circle formula_92, one gets a limaçon of Pascal.
The evolute of a cardioid.
The evolute of a curve is the locus of centers of curvature. In detail: For a curve formula_100 with radius of curvature formula_101 the evolute has the representation
formula_102
with formula_103 the suitably oriented unit normal.
For a cardioid one gets:
The "evolute" of a cardioid is another cardioid, one third as large, and facing the opposite direction (s. picture).
Proof.
For the cardioid with parametric representation
formula_104
formula_105
the unit normal is
formula_106
and the radius of curvature
formula_107
Hence the parametric equations of the evolute are
formula_108
formula_109
These equations describe a cardioid a third as large, rotated 180 degrees and shifted along the x-axis by formula_110.
Orthogonal trajectories.
An orthogonal trajectory of a pencil of curves is a curve which intersects any curve of the pencil orthogonally. For cardioids the following is true:
<templatestyles src="Block indent/styles.css"/>The orthogonal trajectories of the pencil of cardioids with equations formula_112 are the cardioids with equations formula_113
Proof.
For a curve given in polar coordinates by a function formula_114 the following connection to Cartesian coordinates hold:
formula_115
and for the derivatives
formula_116
Dividing the second equation by the first yields the Cartesian slope of the tangent line to the curve at the point formula_117:
formula_118
For the cardioids with the equations formula_119 and formula_120 respectively one gets:
formula_121 and formula_122
Hence
formula_124
That means: Any curve of the first pencil intersects any curve of the second pencil orthogonally.
In different positions.
Choosing other positions of the cardioid within the coordinate system results in different equations. The picture shows the 4 most common positions of a cardioid and their polar equations.
In complex analysis.
In complex analysis, the image of any circle through the origin under the map formula_125 is a cardioid. One application of this result is that the boundary of the central period-1 component of the Mandelbrot set is a cardioid given by the equation
formula_126
The Mandelbrot set contains an infinite number of slightly distorted copies of itself and the central bulb of any of these smaller copies is an approximate cardioid.
Caustics.
Certain caustics can take the shape of cardioids. The catacaustic of a circle with respect to a point on the circumference is a cardioid. Also, the catacaustic of a cone with respect to rays parallel to a generating line is a surface whose cross section is a cardioid. This can be seen, as in the photograph to the right, in a conical cup partially filled with liquid when a light is shining from a distance and at an angle equal to the angle of the cone. The shape of the curve at the bottom of a cylindrical cup is half of a nephroid, which looks quite similar.
Notes.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "a"
},
{
"math_id": 1,
"text": "(-a,0), (a,0)"
},
{
"math_id": 2,
"text": "\\varphi"
},
{
"math_id": 3,
"text": "\\begin{align}\n x(\\varphi) &= 2a (1 - \\cos\\varphi)\\cdot\\cos\\varphi \\ , \\\\\n y(\\varphi) &= 2a (1 - \\cos\\varphi)\\cdot\\sin\\varphi \\ , \\qquad 0\\le \\varphi < 2\\pi\n\\end{align}"
},
{
"math_id": 4,
"text": "r(\\varphi) = 2a (1 - \\cos\\varphi)."
},
{
"math_id": 5,
"text": "\\cos\\varphi = x/r"
},
{
"math_id": 6,
"text": "r = \\sqrt{x^2 + y^2}"
},
{
"math_id": 7,
"text": "\\left(x^2 + y^2\\right)^2 + 4 a x \\left(x^2 + y^2\\right) - 4a^2 y^2 = 0."
},
{
"math_id": 8,
"text": "0"
},
{
"math_id": 9,
"text": "z"
},
{
"math_id": 10,
"text": " e^{i\\varphi}"
},
{
"math_id": 11,
"text": "\\Phi_+"
},
{
"math_id": 12,
"text": ":z \\mapsto a + (z - a)e^{i\\varphi}"
},
{
"math_id": 13,
"text": "\\Phi_-"
},
{
"math_id": 14,
"text": "-a"
},
{
"math_id": 15,
"text": "z \\mapsto -a + (z + a)e^{i\\varphi}"
},
{
"math_id": 16,
"text": "p(\\varphi)"
},
{
"math_id": 17,
"text": "p(\\varphi) = \\Phi_ - (\\Phi_+(0)) = \\Phi_-\\left(a - ae^{i\\varphi}\\right) = -a + \\left( a - ae^{i\\varphi} + a\\right)e^{i\\varphi} = a\\;\\left(-e^{i2\\varphi} + 2e^{i\\varphi} - 1\\right)."
},
{
"math_id": 18,
"text": "\\begin{array}{cclcccc}\n x(\\varphi) &=& a\\;(-\\cos(2\\varphi) + 2\\cos\\varphi - 1) &=& 2a(1 - \\cos\\varphi)\\cdot\\cos\\varphi & & \\\\\n y(\\varphi) &=& a\\;(-\\sin(2\\varphi) + 2\\sin\\varphi) &=& 2a(1 - \\cos\\varphi)\\cdot\\sin\\varphi &.&\n\\end{array}"
},
{
"math_id": 19,
"text": "e^{i\\varphi} = \\cos\\varphi + i\\sin\\varphi, \\ (\\cos\\varphi)^2 + (\\sin\\varphi)^2 = 1,"
},
{
"math_id": 20,
"text": "\\cos(2\\varphi) = (\\cos\\varphi)^2 - (\\sin\\varphi)^2, "
},
{
"math_id": 21,
"text": "\\sin (2\\varphi) = 2\\sin\\varphi\\cos\\varphi"
},
{
"math_id": 22,
"text": "A = 6\\pi a^2"
},
{
"math_id": 23,
"text": "L = 16 a"
},
{
"math_id": 24,
"text": "\\rho(\\varphi) = \\tfrac{8}{3}a\\sin\\tfrac{\\varphi}{2} \\, . "
},
{
"math_id": 25,
"text": "A = 2 \\cdot \\tfrac{1}{2}\\int_0^\\pi{(r(\\varphi))^2}\\; d\\varphi = \\int_0^\\pi{4a^2(1 - \\cos\\varphi)^2}\\; d\\varphi = \\cdots = 4a^2 \\cdot \\tfrac{3}{2}\\pi = 6\\pi a^2."
},
{
"math_id": 26,
"text": "L = 2\\int_0^\\pi{\\sqrt{r(\\varphi)^2 + (r'(\\varphi))^2}} \\; d\\varphi = \\cdots = 8a\\int_0^\\pi\\sqrt{\\tfrac{1}{2}(1 - \\cos\\varphi)}\\; d\\varphi = 8a\\int_0^\\pi\\sin\\left(\\tfrac{\\varphi}{2}\\right) d\\varphi = 16a."
},
{
"math_id": 27,
"text": " \\rho "
},
{
"math_id": 28,
"text": "r=r(\\varphi)"
},
{
"math_id": 29,
"text": "\\rho(\\varphi) = \\frac{\\left[r(\\varphi)^2 + \\dot r(\\varphi)^2\\right]^{3/2}}\n{r(\\varphi)^2 + 2 \\dot r(\\varphi)^2 - r(\\varphi) \\ddot r(\\varphi)} \\ ."
},
{
"math_id": 30,
"text": "r(\\varphi) = 2a (1 - \\cos\\varphi) = 4a \\sin^2\\left(\\tfrac{\\varphi}{2}\\right)"
},
{
"math_id": 31,
"text": "\\rho(\\varphi) = \\cdots = \\frac{\\left[16a^2\\sin^2\\frac{\\varphi}{2}\\right]^\\frac{3}{2}} {24a^2 \\sin^2\\frac{\\varphi}{2}} = \\frac{8}{3}a\\sin\\frac{\\varphi}{2} \\ . "
},
{
"math_id": 32,
"text": "4a"
},
{
"math_id": 33,
"text": "P: p(\\varphi),\\; Q: p(\\varphi + \\pi)"
},
{
"math_id": 34,
"text": "\\begin{align}\n |PQ| &= r(\\varphi) + r(\\varphi + \\pi) \\\\\n &= 2a (1 - \\cos\\varphi) + 2a (1 - \\cos(\\varphi + \\pi)) = \\cdots = 4a\n\\end{align}."
},
{
"math_id": 35,
"text": "P:\\ p(\\varphi) = a\\,\\left(-e^{i2\\varphi} + 2e^{i\\varphi} - 1\\right)"
},
{
"math_id": 36,
"text": "Q:\\ p(\\varphi + \\pi) = a\\,\\left(-e^{i2(\\varphi + \\pi)} + 2e^{i(\\varphi + \\pi)} - 1\\right) = a\\,\\left(-e^{i2\\varphi} - 2e^{i\\varphi} - 1\\right),"
},
{
"math_id": 37,
"text": "PQ"
},
{
"math_id": 38,
"text": "M:\\ \\tfrac{1}{2}(p(\\varphi) + p(\\varphi + \\pi)) = \\cdots = -a - ae^{i2\\varphi}"
},
{
"math_id": 39,
"text": "a = \\frac{1}{2}"
},
{
"math_id": 40,
"text": "r(\\varphi) = 1 - \\cos\\varphi"
},
{
"math_id": 41,
"text": "r(\\varphi) = \\frac{1}{1 - \\cos\\varphi},"
},
{
"math_id": 42,
"text": "x = \\tfrac{1}{2}\\left(y^2 - 1\\right)"
},
{
"math_id": 43,
"text": "c"
},
{
"math_id": 44,
"text": "O"
},
{
"math_id": 45,
"text": "F(x,y,t) = 0"
},
{
"math_id": 46,
"text": "t"
},
{
"math_id": 47,
"text": "(x,y)"
},
{
"math_id": 48,
"text": "F(x,y,t) = 0, \\quad F_t(x,y,t) = 0, "
},
{
"math_id": 49,
"text": "F_t"
},
{
"math_id": 50,
"text": "(-1,0)"
},
{
"math_id": 51,
"text": "1"
},
{
"math_id": 52,
"text": "(-1 + \\cos t, \\sin t)"
},
{
"math_id": 53,
"text": "O = (0,0)"
},
{
"math_id": 54,
"text": "F(x,y,t) = (x + 1 - \\cos t)^2 + (y - \\sin t)^2 - (2 - 2\\cos t) = 0,"
},
{
"math_id": 55,
"text": "F(x,y,t) = x^2 + y^2 + 2x\\; (1 - \\cos t) - 2 y\\; \\sin t = 0\\; ."
},
{
"math_id": 56,
"text": "F_t(x,y,t) = 2x\\; \\sin t - 2y\\; \\cos t = 0 ."
},
{
"math_id": 57,
"text": "x(t) = 2(1 - \\cos t)\\cos t,\\quad y(t) = 2(1 - \\cos t)\\sin t"
},
{
"math_id": 58,
"text": "2N"
},
{
"math_id": 59,
"text": "(1,2), (2,4), \\dots, (n,2n), \\dots, (N,2N), (N+1,2), (N+2,4), \\dots "
},
{
"math_id": 60,
"text": "\\cos\\alpha + \\cos\\beta"
},
{
"math_id": 61,
"text": "\\sin\\alpha + \\sin\\beta"
},
{
"math_id": 62,
"text": "1 + \\cos 2\\alpha "
},
{
"math_id": 63,
"text": "\\cos 2\\alpha"
},
{
"math_id": 64,
"text": "\\sin 2\\alpha"
},
{
"math_id": 65,
"text": "r = 2(1 \\mathbin{\\color{red}+} \\cos\\varphi)"
},
{
"math_id": 66,
"text": "\\begin{align}\n x(\\varphi) &= 2(1 + \\cos\\varphi) \\cos \\varphi, \\\\\n y(\\varphi) &= 2(1 + \\cos\\varphi) \\sin \\varphi\n\\end{align}"
},
{
"math_id": 67,
"text": "\\vec n = \\left(\\dot y , -\\dot x\\right)^\\mathsf{T}"
},
{
"math_id": 68,
"text": "\\dot y(\\varphi) \\cdot (x - x(\\varphi)) - \\dot x(\\varphi) \\cdot (y - y(\\varphi)) = 0"
},
{
"math_id": 69,
"text": "(\\cos2\\varphi + \\cos \\varphi)\\cdot x + (\\sin 2\\varphi + \\sin \\varphi)\\cdot y = 2(1 + \\cos \\varphi)^2 \\, ."
},
{
"math_id": 70,
"text": "\\cos\\frac{1}{2}\\varphi"
},
{
"math_id": 71,
"text": "\\cos(\\tfrac{3}{2}\\varphi) \\cdot x + \\sin\\left(\\tfrac{3}{2}\\varphi\\right) \\cdot y = 4 \\left(\\cos\\tfrac{1}{2}\\varphi\\right)^3 \\quad 0 < \\varphi < 2\\pi,\\ \\varphi \\ne \\pi ."
},
{
"math_id": 72,
"text": "(1 + 3\\cos\\theta, 3\\sin\\theta),\\ (1 + 3\\cos{\\color{red}2}\\theta, 3\\sin{\\color{red}2}\\theta))"
},
{
"math_id": 73,
"text": "(\\sin\\theta - \\sin 2\\theta) x + (\\cos 2\\theta - \\sin \\theta) y = -2\\cos \\theta - \\sin(2\\theta) \\, ."
},
{
"math_id": 74,
"text": "\\sin\\frac{1}{2}\\theta"
},
{
"math_id": 75,
"text": "\\cos\\left(\\tfrac{3}{2}\\theta\\right) \\cdot x + \\sin\\left(\\tfrac{3}{2}\\theta\\right) \\cdot y = 4 \\left(\\cos\\tfrac{1}{2}\\theta\\right)^3 \\quad 0 < \\theta < 2\\pi ."
},
{
"math_id": 76,
"text": "\\varphi, \\theta"
},
{
"math_id": 77,
"text": "\\varphi = \\theta "
},
{
"math_id": 78,
"text": "F(x, y, t) = \\cos\\left(\\tfrac{3}{2}t\\right) x + \\sin\\left(\\tfrac{3}{2}t\\right) y - 4 \\left(\\cos\\tfrac{1}{2}t\\right)^3 = 0 "
},
{
"math_id": 79,
"text": "F_t(x, y, t) = - \\tfrac{3}{2}\\sin\\left(\\tfrac{3}{2}t\\right) x + \\tfrac{3}{2}\\cos \\left(\\tfrac{3}{2}t\\right) y + 3\\cos\\left(\\tfrac{1}{2}t\\right) \\sin t = 0\\, ."
},
{
"math_id": 80,
"text": "x(t) = 2(1 + \\cos t)\\cos t,\\quad y(t) = 2(1 + \\cos t)\\sin t,"
},
{
"math_id": 81,
"text": "r = 2(1 + \\cos t)."
},
{
"math_id": 82,
"text": "Z"
},
{
"math_id": 83,
"text": "(1,0) "
},
{
"math_id": 84,
"text": "3"
},
{
"math_id": 85,
"text": "c(\\varphi) = (1 + 3\\cos\\varphi, 3\\sin\\varphi) \\ ."
},
{
"math_id": 86,
"text": "C:\\ k(\\varphi)"
},
{
"math_id": 87,
"text": "\\vec n_t = (\\cos\\varphi,\\sin\\varphi)^\\mathsf{T}"
},
{
"math_id": 88,
"text": "\\vec n_r = \\left(\\cos{\\color{red}\\tfrac{3}{2}} \\varphi, \\sin{\\color{red}\\tfrac{3}{2}} \\varphi\\right)^\\mathsf{T}"
},
{
"math_id": 89,
"text": "C:\\ (1 + 3\\cos\\varphi, 3\\sin\\varphi) "
},
{
"math_id": 90,
"text": "\\cos\\left(\\tfrac{3}{2}\\varphi\\right) x + \\sin \\left(\\tfrac{3}{2}\\varphi\\right) y = 4 \\left(\\cos\\tfrac{1}{2}\\varphi\\right)^3 \\, ,"
},
{
"math_id": 91,
"text": "r = 2(1 + \\cos\\varphi)"
},
{
"math_id": 92,
"text": "k"
},
{
"math_id": 93,
"text": "(2a,0)"
},
{
"math_id": 94,
"text": "2a"
},
{
"math_id": 95,
"text": "(2a + 2a\\cos\\varphi, 2a\\sin \\varphi)"
},
{
"math_id": 96,
"text": "(x - 2a) \\cdot \\cos\\varphi + y\\cdot\\sin\\varphi = 2a\\, ."
},
{
"math_id": 97,
"text": "(r\\cos \\varphi, r\\sin \\varphi)"
},
{
"math_id": 98,
"text": "r"
},
{
"math_id": 99,
"text": "(r\\cos\\varphi - 2a)\\cos\\varphi + r\\sin^2\\varphi = 2a \\quad \\rightarrow \\quad r = 2a(1 + \\cos \\varphi) "
},
{
"math_id": 100,
"text": "\\vec x(s) = \\vec c(s)"
},
{
"math_id": 101,
"text": "\\rho(s)"
},
{
"math_id": 102,
"text": "\\vec X(s) = \\vec c(s) + \\rho(s)\\vec n(s)."
},
{
"math_id": 103,
"text": "\\vec n(s)"
},
{
"math_id": 104,
"text": "x(\\varphi) = 2a (1 - \\cos\\varphi)\\cos\\varphi = 4a \\sin^2\\tfrac{\\varphi}{2}\\cos\\varphi\\, ,"
},
{
"math_id": 105,
"text": "y(\\varphi) = 2a (1 - \\cos\\varphi)\\sin\\varphi = 4a \\sin^2\\tfrac{\\varphi}{2}\\sin\\varphi"
},
{
"math_id": 106,
"text": "\\vec n(\\varphi) = (-\\sin\\tfrac{3}{2}\\varphi, \\cos\\tfrac{3}{2}\\varphi)"
},
{
"math_id": 107,
"text": "\\rho(\\varphi) = \\tfrac{8}{3}a\\sin\\tfrac{\\varphi}{2} \\, . "
},
{
"math_id": 108,
"text": "X(\\varphi) = 4a \\sin^2\\tfrac{\\varphi}{2}\\cos\\varphi-\\tfrac{8}{3}a\\sin\\tfrac{\\varphi}{2}\\cdot \\sin\\tfrac{3}{2} \\varphi = \\cdots = \\tfrac{4}{3}a\\cos^2\\tfrac{\\varphi}{2}\\cos\\varphi - \\tfrac{4}{3}a \\, , "
},
{
"math_id": 109,
"text": "Y(\\varphi) = 4a \\sin^2\\tfrac{\\varphi}{2} \\sin\\varphi + \\tfrac{8}{3}a \\sin\\tfrac{\\varphi}{2} \\cdot\\cos\\tfrac{3}{2} \\varphi = \\cdots = \\tfrac{4}{3}a \\cos^2\\tfrac{\\varphi}{2} \\sin\\varphi \\, . "
},
{
"math_id": 110,
"text": "-\\tfrac{4}{3} a"
},
{
"math_id": 111,
"text": "\n \\sin\\tfrac{3}{2}\\varphi = \\sin\\tfrac{\\varphi}{2}\\cos\\varphi + \\cos\\tfrac{\\varphi}{2}\\sin\\varphi\\ ,\\ \\cos\\tfrac{3}{2}\\varphi = \\cdots, \\ \n \\sin\\varphi = 2\\sin\\tfrac{\\varphi}{2}\\cos\\tfrac{\\varphi}{2}, \\ \n \\cos\\varphi= \\cdots \\ .\n"
},
{
"math_id": 112,
"text": "r=2a(1-\\cos\\varphi)\\ , \\; a>0 \\ , \\ "
},
{
"math_id": 113,
"text": "r=2b(1+\\cos\\varphi)\\ , \\; b>0 \\ . "
},
{
"math_id": 114,
"text": "r(\\varphi)"
},
{
"math_id": 115,
"text": "\\begin{align}\n x(\\varphi) &= r(\\varphi)\\cos\\varphi\\, ,\\\\\n y(\\varphi) &= r(\\varphi)\\sin\\varphi\n\\end{align} "
},
{
"math_id": 116,
"text": "\\begin{align}\n\\frac{dx}{d\\varphi} &= r'(\\varphi)\\cos\\varphi - r(\\varphi)\\sin\\varphi\\, ,\\\\\n\\frac{dy}{d\\varphi} &= r'(\\varphi)\\sin\\varphi + r(\\varphi)\\cos\\varphi\\, .\n\\end{align}"
},
{
"math_id": 117,
"text": "(r(\\varphi), \\varphi)"
},
{
"math_id": 118,
"text": "\\frac{dy}{dx} = \\frac{r'(\\varphi)\\sin\\varphi + r(\\varphi)\\cos\\varphi}{r'(\\varphi)\\cos\\varphi - r(\\varphi)\\sin\\varphi}."
},
{
"math_id": 119,
"text": "r=2a(1-\\cos\\varphi) \\; "
},
{
"math_id": 120,
"text": "r = 2b(1 + \\cos\\varphi)\\ "
},
{
"math_id": 121,
"text": "\\frac{dy_a}{dx} = \\frac{\\cos(\\varphi) - \\cos(2\\varphi)}{\\sin(2\\varphi) - \\sin(\\varphi)} "
},
{
"math_id": 122,
"text": " \\frac{dy_b}{dx} = -\\frac{\\cos(\\varphi) + \\cos(2\\varphi)}{\\sin(2\\varphi) + \\sin(\\varphi)}\\ ."
},
{
"math_id": 123,
"text": "b"
},
{
"math_id": 124,
"text": "\\frac{dy_a}{dx}\\cdot \\frac{dy_b}{dx} = \\cdots = -\\frac{\\cos^2\\varphi-\\cos^2 (2\\varphi)}{\\sin^2 (2\\varphi)-\\sin^2\\varphi} = -\\frac{-1 + \\cos^2\\varphi + 1 - \\cos^2 2\\varphi}{\\sin^2 (2\\varphi) - \\sin^2(\\varphi)} = -1\\, ."
},
{
"math_id": 125,
"text": "z \\to z^2"
},
{
"math_id": 126,
"text": " c \\,=\\, \\frac{1 - \\left(e^{it} - 1\\right)^2}{4}."
}
] |
https://en.wikipedia.org/wiki?curid=642101
|
64215003
|
Diósi–Penrose model
|
The Diósi–Penrose model was introduced as a possible solution to the measurement problem, where the wave function collapse is related to gravity. The model was first suggested by Lajos Diósi when studying how possible gravitational fluctuations may affect the dynamics of quantum systems. Later, following a different line of reasoning, Roger Penrose arrived at an estimation for the collapse time of a superposition due to gravitational effects, which is the same (within an unimportant numerical factor) as that found by Diósi, hence the name Diósi–Penrose model. However, it should be pointed out that while Diósi gave a precise dynamical equation for the collapse, Penrose took a more conservative approach, estimating only the collapse time of a superposition.
The Diósi model.
In the Diósi model, the wave-function collapse is induced by the interaction of the system with a classical noise field, where the spatial correlation function of this noise is related to the Newtonian potential. The evolution of the state vector formula_0 deviates from the Schrödinger equation and has the typical structure of the collapse models equations:
where
is the mass density function, with formula_1, formula_2 and formula_3 respectively the mass, the position operator and the mass density function of the formula_4-th particle of the system. formula_5 is a parameter introduced to smear the mass density function, required since taking a point-like mass distribution
formula_6
would lead to divergences in the predictions of the model, e.g. an infinite collapse rate or increase of energy. Typically, two different distributions for the mass density formula_7 have been considered in the literature: a spherical or a Gaussian mass density profile, given respectively by
formula_8
and
formula_9
Choosing one or another distribution formula_7 does not affect significantly the predictions of the model, as long as the same value for formula_5 is considered. The noise field formula_10 in Eq. (1) has zero average and correlation given by
where “formula_11” denotes the average over the noise. One can then understand from Eq. (1) and (3) in which sense the model is gravity-related: the coupling constant between the system and the noise is proportional to the gravitational constant formula_12, and the spatial correlation of the noise field formula_13 has the typical form of a Newtonian potential. Similarly to other collapse models, the Diósi–Penrose model shares the following two features:
In order to show these features, it is convenient to write the master equation for the statistical operator formula_14 corresponding to Eq. (1):
It is interesting to point out that this master equation has more recently been re-derived by L. Diósi using an hybrid approach where quantized massive particles interact with classical gravitational fields.
If one considers the master equation in the position basis, introducing formula_15 with formula_16, where formula_17 is a position eigenstate of the formula_4-th particle, neglecting the free evolution, one finds
with
where
formula_18
is the mass density when the particles of the system are centered at the points formula_19, ..., formula_20. Eq. (5) can be solved exactly, and one gets
where
As expected, for the diagonal terms of the density matrix, when formula_21, one has formula_22, i.e. the time of decay goes to infinity, implying that states with well-localized position are not affected by the collapse. On the contrary, the off-diagonal terms formula_23, which are different from zero when a spatial superposition is involved, will decay with a time of decay given by Eq. (8).
To get an idea of the scale at which the gravitationally induced collapse becomes relevant, one can compute the time of decay in Eq. (8) for the case of a sphere with radius formula_5 and mass formula_24 in a spatial superposition at a distance formula_25. Then the time of decay can be computed) using Eq. (8) with
where formula_26. To give some examples, if one considers a proton, for which formula_27 kg and formula_28 m, in a superposition with formula_29, one gets formula_30 years. On the contrary, for a dust grain with formula_31 kg and formula_32 m, one gets one gets formula_33 s. Therefore, contrary to what might be expected considering the weaknesses of gravitational force, the effects of the gravity-related collapse become relevant already at the mesoscopic scale.
Recently, the model have been generalized by including dissipative and non-Markovian effects.
Penrose's proposal.
It is well known that general relativity and quantum mechanics, our most fundamental theories for describing the universe, are not compatible, and the unification of the two is still missing. The standard approach to overcome this situation is to try to modify general relativity by quantizing gravity. Penrose suggests an opposite approach, what he calls “gravitization of quantum mechanics”, where quantum mechanics gets modified when gravitational effects become relevant. The reasoning underlying this approach is the following one: take a massive system of well-localized states in space. In this case, the state being well-localized, the induced space–time curvature is well defined. According to quantum mechanics, because of the superposition principle, the system can be placed (at least in principle) in a superposition of two well-localized states, which would lead to a superposition of two different space–times. The key idea is that since space–time metric should be well defined, nature “dislikes” these space–time superpositions and suppresses them by collapsing the wave function to one of the two localized states.
To set these ideas on a more quantitative ground, Penrose suggested that a way for measuring the difference between two space–times, in the Newtonian limit, is
where formula_34 is the Newtonian gravitational acceleration at the point where the system is localized around formula_35. The acceleration formula_34 can be written in terms of the corresponding gravitational potentials formula_36, i.e. formula_37. Using this relation in Eq. (9), together with the Poisson equation formula_38, with formula_39 giving the mass density when the state is localized around formula_35, and its solution, one arrives at
The corresponding decay time can be obtained by the Heisenberg time–energy uncertainty:
which, apart for a factor formula_40 simply due to the use of different conventions, is exactly the same as the time decay formula_41 derived by Diósi's model. This is the reason why the two proposals are named together as the Diósi–Penrose model.
More recently, Penrose suggested a new and quite elegant way to justify the need for a gravity-induced collapse, based on avoiding tensions between the superposition principle and the equivalence principle, the cornerstones of quantum mechanics and general relativity. In order to explain it, let us start by comparing the evolution of a generic state in the presence of uniform gravitational acceleration formula_42. One way to perform the calculation, what Penrose calls “Newtonian perspective”, consists in working in an inertial frame, with space–time coordinates formula_43 and solve the Schrödinger equation in presence of the potential formula_44 (typically, one chooses the coordinates in such a way that the acceleration formula_42 is directed along the formula_45 axis, in which case formula_46). Alternatively, because of the equivalence principle, one can choose to go in the free-fall reference frame, with coordinates formula_47 related to formula_43 by formula_48 and formula_49, solve the free Schrödinger equation in that reference frame, and then write the results in terms of the inertial coordinates formula_43. This is what Penrose calls “Einsteinian perspective”. The solution formula_50 obtained in the Einsteinian perspective and the one formula_51 obtained in the Newtonian perspective are related to each other by
Since the two wave functions are equivalent apart from an overall phase, they lead to the same physical predictions, which implies that there are no problems in this situation where the gravitational field always has a well-defined value. However, if the space–time metric is not well defined, then we will be in a situation where there is a superposition of a gravitational field corresponding to the acceleration formula_52 and one corresponding to the acceleration formula_53. This does not create problems as long as one sticks to the Newtonian perspective. However, when using the Einstenian perspective, it will imply a phase difference between the two branches of the superposition given by formula_54. While the term in the exponent linear in the time formula_55 does not lead to any conceptual difficulty, the first term, proportional to formula_56, is problematic, since it is a non-relativistic residue of the so-called Unruh effect: in other words, the two terms in the superposition belong to different Hilbert spaces and, strictly speaking, cannot be superposed. Here is where the gravity-induced collapse plays a role, collapsing the superposition when the first term of the phase formula_57 becomes too large.
Further information on Penrose's idea for the gravity-induced collapse can be also found in the Penrose interpretation.
Experimental tests and theoretical bounds.
Since the Diósi–Penrose model predicts deviations from standard quantum mechanics, the model can be tested. The only free parameter of the model is the size of the mass density distribution, given by formula_5. All bounds present in the literature are based on an indirect effect of the gravitational-related collapse: a Brownian-like diffusion induced by the collapse on the motion of the particles. This Brownian-like diffusion is a common feature of all objective-collapse theories and, typically, allows to set the strongest bounds on the parameters of these models. The first bound on formula_5 was set by Ghirardi et al., where it was shown that formula_58 m to avoid unrealistic heating due to this Brownian-like induced diffusion. Then the bound has been further restricted to formula_59 m by the analysis of the data from gravitational wave detectors. and later to formula_60 m by studying the heating of neutron stars.
Regarding direct interferometric tests of the model, where a system is prepared in a spatial superposition, there are two proposals currently considered: an optomechanical setup with a mesoscopic mirror to be placed in a superposition by a laser, and experiments involving superpositions of Bose–Einstein condensates.
See also.
<templatestyles src="Div col/styles.css"/>
|
[
{
"math_id": 0,
"text": "|\\psi_{t}\\rangle"
},
{
"math_id": 1,
"text": "m_j"
},
{
"math_id": 2,
"text": "\\hat{\\mathbf{x}}_j"
},
{
"math_id": 3,
"text": "\\mu_{R_0}(\\mathbf{x})"
},
{
"math_id": 4,
"text": "j"
},
{
"math_id": 5,
"text": "R_0"
},
{
"math_id": 6,
"text": "\\mathcal{M}_\\text{point}(\\mathbf{x}) = \\sum_{j=1}^N m_j \\delta(\\mathbf{x} - \\hat{\\mathbf{x}}_j)"
},
{
"math_id": 7,
"text": "\\mu_{R_0}(\\mathbf{x} - \\hat{\\mathbf{x}}_j)"
},
{
"math_id": 8,
"text": "\\mu_{R_0}^\\text{s}(\\mathbf{x} - \\hat{\\mathbf{x}}_j) = \\frac{3}{4\\pi R_0^3} \\theta\\big(|\\mathbf{x} - \\hat{\\mathbf{x}}_j| - R_0\\big)"
},
{
"math_id": 9,
"text": "\\mu_{R_{0}}^\\text{g}(\\mathbf{x} - \\hat{\\mathbf{x}}_j) = \\frac{1}{(2\\pi R_0^2)^{3/2}}\\,\\exp\\left(-\\frac{(\\mathbf{x} - \\hat{\\mathbf{x}}_j)^2}{2R_0^2}\\right)."
},
{
"math_id": 10,
"text": "w(\\mathbf{x}, t) := \\frac{dW(\\mathbf{x}, t)}{dt}"
},
{
"math_id": 11,
"text": "\\mathbb{E}"
},
{
"math_id": 12,
"text": "G"
},
{
"math_id": 13,
"text": "w(\\mathbf{x}, t)"
},
{
"math_id": 14,
"text": "\\rho(t) = \\mathbb{E}\\big[|\\psi_t\\rangle \\langle\\psi_t|\\big]"
},
{
"math_id": 15,
"text": "\\rho(\\vec{\\boldsymbol{a}}, \\vec{\\boldsymbol{b}}, t) := \\langle\\vec{\\boldsymbol{a}}|\\rho(t)|\\vec{\\boldsymbol{b}}\\rangle"
},
{
"math_id": 16,
"text": "|\\vec{\\boldsymbol{a}}\\rangle := |\\boldsymbol{a}_1\\rangle \\otimes \\dots \\otimes |\\boldsymbol{a}_N\\rangle"
},
{
"math_id": 17,
"text": "|\\boldsymbol{a}_j\\rangle"
},
{
"math_id": 18,
"text": "\\mathcal{M}(\\mathbf{x}, \\vec{\\boldsymbol{a}}) := \\sum_j m_j \\mu_{R_0}(\\mathbf{x} - \\boldsymbol{a}_j)"
},
{
"math_id": 19,
"text": "\\boldsymbol{a}_1"
},
{
"math_id": 20,
"text": "\\boldsymbol{a}_N"
},
{
"math_id": 21,
"text": "\\vec{\\boldsymbol{a}} = \\vec{\\boldsymbol{b}}"
},
{
"math_id": 22,
"text": "\\Lambda(\\vec{\\boldsymbol{a}}, \\vec{\\boldsymbol{a}}) = 0"
},
{
"math_id": 23,
"text": "\\vec{\\boldsymbol{a}} \\neq \\vec{\\boldsymbol{b}}"
},
{
"math_id": 24,
"text": "m"
},
{
"math_id": 25,
"text": "d: = |\\boldsymbol{a} - \\boldsymbol{b}|"
},
{
"math_id": 26,
"text": "\\lambda = d/(2R_0)"
},
{
"math_id": 27,
"text": "m \\simeq 1.67 \\times 10^{-27}"
},
{
"math_id": 28,
"text": "R_0 \\simeq 10^{-15}"
},
{
"math_id": 29,
"text": "d \\gg R_0"
},
{
"math_id": 30,
"text": "\\tau_\\text{DP} \\simeq 10^6"
},
{
"math_id": 31,
"text": "m \\simeq 6 \\times 10^{-12}"
},
{
"math_id": 32,
"text": "R_0 \\simeq 10^{-5}"
},
{
"math_id": 33,
"text": "\\tau_\\text{DP} \\simeq 10^{-8}"
},
{
"math_id": 34,
"text": "g_i(\\boldsymbol{r})"
},
{
"math_id": 35,
"text": "i"
},
{
"math_id": 36,
"text": "\\Phi_i(\\boldsymbol{r})"
},
{
"math_id": 37,
"text": "g_i(\\boldsymbol{r}) = -\\nabla\\Phi_i(\\boldsymbol{r})"
},
{
"math_id": 38,
"text": "\\nabla^2\\Phi_i(\\boldsymbol{r}) = 4\\pi G\\mu_i(\\boldsymbol{r})"
},
{
"math_id": 39,
"text": "\\mu_i(\\boldsymbol{r})"
},
{
"math_id": 40,
"text": "8\\pi"
},
{
"math_id": 41,
"text": "\\tau_\\text{DP}"
},
{
"math_id": 42,
"text": "\\boldsymbol{g}"
},
{
"math_id": 43,
"text": "(\\boldsymbol{r}, t)"
},
{
"math_id": 44,
"text": "V(\\boldsymbol{x}) = m\\boldsymbol{g} \\cdot \\boldsymbol{x}"
},
{
"math_id": 45,
"text": "z"
},
{
"math_id": 46,
"text": "V(z) = mgz"
},
{
"math_id": 47,
"text": "(\\boldsymbol{R}, T)"
},
{
"math_id": 48,
"text": "\\boldsymbol{R} = \\boldsymbol{r} + \\frac{1}{2} \\boldsymbol{g} t^2"
},
{
"math_id": 49,
"text": "T = t"
},
{
"math_id": 50,
"text": "\\Psi(\\boldsymbol{r}, t)"
},
{
"math_id": 51,
"text": "\\psi(\\boldsymbol{r}, t)"
},
{
"math_id": 52,
"text": "\\boldsymbol{g}_a"
},
{
"math_id": 53,
"text": "\\boldsymbol{g}_b"
},
{
"math_id": 54,
"text": "e^{i\\frac{m}{\\hbar} \\left(\\frac{1}{6} (\\boldsymbol{g}_a - \\boldsymbol{g}_b)^2 t^3 + (\\boldsymbol{g}_a - \\boldsymbol{g}_b) \\cdot \\boldsymbol{r}\\,t\\right)}"
},
{
"math_id": 55,
"text": "t"
},
{
"math_id": 56,
"text": "t^3"
},
{
"math_id": 57,
"text": "\\frac{1}{6} (g_a - g_b)^2 t^3"
},
{
"math_id": 58,
"text": "R_0 > 10^{-15}"
},
{
"math_id": 59,
"text": "R_0 > 4 \\times 10^{-14}"
},
{
"math_id": 60,
"text": "R_0 \\gtrsim 10^{-13}"
}
] |
https://en.wikipedia.org/wiki?curid=64215003
|
64219
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Bernoulli's principle
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Principle relating to fluid dynamics
Bernoulli's principle is a key concept in fluid dynamics that relates pressure, speed and height. Bernoulli's principle states that an increase in the speed of a parcel of fluid occurs simultaneously with a decrease in either the pressure or the height above a datum.Ch.3§ 3.5 The principle is named after the Swiss mathematician and physicist Daniel Bernoulli, who published it in his book "Hydrodynamica" in 1738. Although Bernoulli deduced that pressure decreases when the flow speed increases, it was Leonhard Euler in 1752 who derived Bernoulli's equation in its usual form.
Bernoulli's principle can be derived from the principle of conservation of energy. This states that, in a steady flow, the sum of all forms of energy in a fluid is the same at all points that are free of viscous forces. This requires that the sum of kinetic energy, potential energy and internal energy remains constant.§ 3.5 Thus an increase in the speed of the fluid—implying an increase in its kinetic energy—occurs with a simultaneous decrease in (the sum of) its potential energy (including the static pressure) and internal energy. If the fluid is flowing out of a reservoir, the sum of all forms of energy is the same because in a reservoir the energy per unit volume (the sum of pressure and gravitational potential "ρ" "g" "h") is the same everywhere.Example 3.5 and p.116
Bernoulli's principle can also be derived directly from Isaac Newton's second Law of Motion. If a small volume of fluid is flowing horizontally from a region of high pressure to a region of low pressure, then there is more pressure behind than in front. This gives a net force on the volume, accelerating it along the streamline.
Fluid particles are subject only to pressure and their own weight. If a fluid is flowing horizontally and along a section of a streamline, where the speed increases it can only be because the fluid on that section has moved from a region of higher pressure to a region of lower pressure; and if its speed decreases, it can only be because it has moved from a region of lower pressure to a region of higher pressure. Consequently, within a fluid flowing horizontally, the highest speed occurs where the pressure is lowest, and the lowest speed occurs where the pressure is highest.
Bernoulli's principle is only applicable for isentropic flows: when the effects of irreversible processes (like turbulence) and non-adiabatic processes (e.g. thermal radiation) are small and can be neglected. However, the principle can be applied to various types of flow within these bounds, resulting in various forms of Bernoulli's equation. The simple form of Bernoulli's equation is valid for incompressible flows (e.g. most liquid flows and gases moving at low Mach number). More advanced forms may be applied to compressible flows at higher Mach numbers.
Incompressible flow equation.
In most flows of liquids, and of gases at low Mach number, the density of a fluid parcel can be considered to be constant, regardless of pressure variations in the flow. Therefore, the fluid can be considered to be incompressible, and these flows are called incompressible flows. Bernoulli performed his experiments on liquids, so his equation in its original form is valid only for incompressible flow.
A common form of Bernoulli's equation is:
where:
Bernoulli's equation and the Bernoulli constant are applicable throughout any region of flow where the energy per unit mass is uniform. Because the energy per unit mass of liquid in a well-mixed reservoir is uniform throughout, Bernoulli's equation can be used to analyze the fluid flow everywhere in that reservoir (including pipes or flow fields that the reservoir feeds) "except" where viscous forces dominate and erode the energy per unit mass.Example 3.5 and p.116
The following assumptions must be met for this Bernoulli equation to apply:265
For conservative force fields (not limited to the gravitational field), Bernoulli's equation can be generalized as:265
formula_5
where Ψ is the force potential at the point considered. For example, for the Earth's gravity Ψ = "gz".
By multiplying with the fluid density ρ, equation (A) can be rewritten as:
formula_6
or:
formula_7
where
The constant in the Bernoulli equation can be normalized. A common approach is in terms of total head or energy head H:
formula_8
The above equations suggest there is a flow speed at which pressure is zero, and at even higher speeds the pressure is negative. Most often, gases and liquids are not capable of negative absolute pressure, or even zero pressure, so clearly Bernoulli's equation ceases to be valid before zero pressure is reached. In liquids—when the pressure becomes too low—cavitation occurs. The above equations use a linear relationship between flow speed squared and pressure. At higher flow speeds in gases, or for sound waves in liquid, the changes in mass density become significant so that the assumption of constant density is invalid.
Simplified form.
In many applications of Bernoulli's equation, the change in the ρgz term is so small compared with the other terms that it can be ignored. For example, in the case of aircraft in flight, the change in height z is so small the ρgz term can be omitted. This allows the above equation to be presented in the following simplified form:
formula_9
where "p"0 is called total pressure, and q is " dynamic pressure". Many authors refer to the pressure p as static pressure to distinguish it from total pressure "p"0 and dynamic pressure q. In "Aerodynamics", L.J. Clancy writes: "To distinguish it from the total and dynamic pressures, the actual pressure of the fluid, which is associated not with its motion but with its state, is often referred to as the static pressure, but where the term pressure alone is used it refers to this static pressure."§ 3.5
The simplified form of Bernoulli's equation can be summarized in the following memorable word equation:§ 3.5
<templatestyles src="Block indent/styles.css"/>static pressure + dynamic pressure = total pressure
Every point in a steadily flowing fluid, regardless of the fluid speed at that point, has its own unique static pressure p and dynamic pressure q. Their sum "p" + "q" is defined to be the total pressure "p"0. The significance of Bernoulli's principle can now be summarized as "total pressure is constant in any region free of viscous forces". If the fluid flow is brought to rest at some point, this point is called a stagnation point, and at this point the static pressure is equal to the stagnation pressure.
If the fluid flow is irrotational, the total pressure is uniform and Bernoulli's principle can be summarized as "total pressure is constant everywhere in the fluid flow".Equation 3.12 It is reasonable to assume that irrotational flow exists in any situation where a large body of fluid is flowing past a solid body. Examples are aircraft in flight and ships moving in open bodies of water. However, Bernoulli's principle importantly does not apply in the boundary layer such as in flow through long pipes.
Unsteady potential flow.
The Bernoulli equation for unsteady potential flow is used in the theory of ocean surface waves and acoustics. For an irrotational flow, the flow velocity can be described as the gradient ∇"φ" of a velocity potential φ. In that case, and for a constant density ρ, the momentum equations of the Euler equations can be integrated to:383formula_10
which is a Bernoulli equation valid also for unsteady—or time dependent—flows. Here denotes the partial derivative of the velocity potential φ with respect to time t, and "v" = |∇"φ"| is the flow speed. The function "f"("t") depends only on time and not on position in the fluid. As a result, the Bernoulli equation at some moment t applies in the whole fluid domain. This is also true for the special case of a steady irrotational flow, in which case f and are constants so equation (A) can be applied in every point of the fluid domain.383 Further "f"("t") can be made equal to zero by incorporating it into the velocity potential using the transformation:formula_11
resulting in:
formula_12
Note that the relation of the potential to the flow velocity is unaffected by this transformation: ∇Φ = ∇"φ".
The Bernoulli equation for unsteady potential flow also appears to play a central role in Luke's variational principle, a variational description of free-surface flows using the Lagrangian mechanics.
Compressible flow equation.
Bernoulli developed his principle from observations on liquids, and Bernoulli's equation is valid for ideal fluids: those that are incompressible, irrotational, inviscid, and subjected to conservative forces. It is sometimes valid for the flow of gases: provided that there is no transfer of kinetic or potential energy from the gas flow to the compression or expansion of the gas. If both the gas pressure and volume change simultaneously, then work will be done on or by the gas. In this case, Bernoulli's equation—in its incompressible flow form—cannot be assumed to be valid. However, if the gas process is entirely isobaric, or isochoric, then no work is done on or by the gas (so the simple energy balance is not upset). According to the gas law, an isobaric or isochoric process is ordinarily the only way to ensure constant density in a gas. Also the gas density will be proportional to the ratio of pressure and absolute temperature; however, this ratio will vary upon compression or expansion, no matter what non-zero quantity of heat is added or removed. The only exception is if the net heat transfer is zero, as in a complete thermodynamic cycle or in an individual isentropic (frictionless adiabatic) process, and even then this reversible process must be reversed, to restore the gas to the original pressure and specific volume, and thus density. Only then is the original, unmodified Bernoulli equation applicable. In this case the equation can be used if the flow speed of the gas is sufficiently below the speed of sound, such that the variation in density of the gas (due to this effect) along each streamline can be ignored. Adiabatic flow at less than Mach 0.3 is generally considered to be slow enough.
It is possible to use the fundamental principles of physics to develop similar equations applicable to compressible fluids. There are numerous equations, each tailored for a particular application, but all are analogous to Bernoulli's equation and all rely on nothing more than the fundamental principles of physics such as Newton's laws of motion or the first law of thermodynamics.
Compressible flow in fluid dynamics.
For a compressible fluid, with a barotropic equation of state, and under the action of conservative forces,
formula_13
where:
In engineering situations, elevations are generally small compared to the size of the Earth, and the time scales of fluid flow are small enough to consider the equation of state as adiabatic. In this case, the above equation for an ideal gas becomes:§ 3.11
formula_14
where, in addition to the terms listed above:
In many applications of compressible flow, changes in elevation are negligible compared to the other terms, so the term gz can be omitted. A very useful form of the equation is then:
formula_15
where:
Compressible flow in thermodynamics.
The most general form of the equation, suitable for use in thermodynamics in case of (quasi) steady flow, is:§ 3.5§ 5§ 5.9
formula_16
Here w is the enthalpy per unit mass (also known as specific enthalpy), which is also often written as h (not to be confused with "head" or "height").
Note that
formula_17
where e is the thermodynamic energy per unit mass, also known as the specific internal energy. So, for constant internal energy formula_18 the equation reduces to the incompressible-flow form.
The constant on the right-hand side is often called the Bernoulli constant and denoted b. For steady inviscid adiabatic flow with no additional sources or sinks of energy, b is constant along any given streamline. More generally, when b may vary along streamlines, it still proves a useful parameter, related to the "head" of the fluid (see below).
When the change in Ψ can be ignored, a very useful form of this equation is:
formula_19
where "w"0 is total enthalpy. For a calorically perfect gas such as an ideal gas, the enthalpy is directly proportional to the temperature, and this leads to the concept of the total (or stagnation) temperature.
When shock waves are present, in a reference frame in which the shock is stationary and the flow is steady, many of the parameters in the Bernoulli equation suffer abrupt changes in passing through the shock. The Bernoulli parameter remains unaffected. An exception to this rule is radiative shocks, which violate the assumptions leading to the Bernoulli equation, namely the lack of additional sinks or sources of energy.
Unsteady potential flow.
For a compressible fluid, with a barotropic equation of state, the unsteady momentum conservation equation
formula_20
With the irrotational assumption, namely, the flow velocity can be described as the gradient ∇"φ" of a velocity potential "φ". The unsteady momentum conservation equation becomes
formula_21
which leads to
formula_22
In this case, the above equation for isentropic flow becomes:
formula_23
Derivations.
<templatestyles src="Math_proof/styles.css" />Bernoulli equation for incompressible fluids
The Bernoulli equation for incompressible fluids can be derived by either integrating Newton's second law of motion or by applying the law of conservation of energy, ignoring viscosity, compressibility, and thermal effects.
The simplest derivation is to first ignore gravity and consider constrictions and expansions in pipes that are otherwise straight, as seen in Venturi effect. Let the x axis be directed down the axis of the pipe.
Define a parcel of fluid moving through a pipe with cross-sectional area A, the length of the parcel is d"x", and the volume of the parcel "A" d"x". If mass density is ρ, the mass of the parcel is density multiplied by its volume "m" = "ρA" d"x". The change in pressure over distance d"x" is d"p" and flow velocity "v" =.
Apply Newton's second law of motion (force = mass × acceleration) and recognizing that the effective force on the parcel of fluid is −"A" d"p". If the pressure decreases along the length of the pipe, d"p" is negative but the force resulting in flow is positive along the x axis.
formula_24
In steady flow the velocity field is constant with respect to time, "v" = "v"("x") = "v"("x"("t")), so v itself is not directly a function of time t. It is only when the parcel moves through x that the cross sectional area changes: v depends on t only through the cross-sectional position "x"("t").
formula_25
With density ρ constant, the equation of motion can be written as
formula_26
by integrating with respect to x
formula_27
where C is a constant, sometimes referred to as the Bernoulli constant. It is not a universal constant, but rather a constant of a particular fluid system. The deduction is: where the speed is large, pressure is low and vice versa.
In the above derivation, no external work–energy principle is invoked. Rather, Bernoulli's principle was derived by a simple manipulation of Newton's second law.
Another way to derive Bernoulli's principle for an incompressible flow is by applying conservation of energy. In the form of the work-energy theorem, stating that
<templatestyles src="Block indent/styles.css"/>"the change in the kinetic energy ""E"kin" of the system equals the net work W done on the system"; formula_28
Therefore,
<templatestyles src="Block indent/styles.css"/> "the work done by the forces in the fluid equals increase in kinetic energy."
The system consists of the volume of fluid, initially between the cross-sections "A"1 and "A"2. In the time interval Δ"t" fluid elements initially at the inflow cross-section "A"1 move over a distance "s"1 = "v"1 Δ"t", while at the outflow cross-section the fluid moves away from cross-section "A"2 over a distance "s"2 = "v"2 Δ"t". The displaced fluid volumes at the inflow and outflow are respectively "A"1"s"1 and "A"2"s"2. The associated displaced fluid masses are – when ρ is the fluid's mass density – equal to density times volume, so "ρA"1"s"1 and "ρA"2"s"2. By mass conservation, these two masses displaced in the time interval Δ"t" have to be equal, and this displaced mass is denoted by Δ"m":
formula_29
The work done by the forces consists of two parts:
formula_31
Now, the work by the force of gravity is opposite to the change in potential energy, "W"gravity = −"ΔE"pot,gravity: while the force of gravity is in the negative z-direction, the work—gravity force times change in elevation—will be negative for a positive elevation change Δ"z" = "z"2 − "z"1, while the corresponding potential energy change is positive.14–4,&hairsp;§14–3 So: formula_32
And therefore the total work done in this time interval Δ"t" is
formula_33
The "increase in kinetic energy" is
formula_34
Putting these together, the work-kinetic energy theorem "W" = Δ"E"kin gives:
formula_35
or
formula_36
After dividing by the mass Δ"m" = "ρA"1"v"1 Δ"t" = "ρA"2"v"2 Δ"t" the result is:
formula_37
or, as stated in the first paragraph:
Further division by g produces the following equation. Note that each term can be described in the length dimension (such as meters). This is the head equation derived from Bernoulli's principle:
The middle term, z, represents the potential energy of the fluid due to its elevation with respect to a reference plane. Now, z is called the elevation head and given the designation "z"elevation.
A free falling mass from an elevation "z" > 0 (in a vacuum) will reach a speed
formula_38
when arriving at elevation "z" = 0. Or when rearranged as "head":
formula_39
The term is called the "velocity head", expressed as a length measurement. It represents the internal energy of the fluid due to its motion.
The hydrostatic pressure "p" is defined as
formula_40
with "p"0 some reference pressure, or when rearranged as "head":
formula_41
The term is also called the "pressure head", expressed as a length measurement. It represents the internal energy of the fluid due to the pressure exerted on the container. The head due to the flow speed and the head due to static pressure combined with the elevation above a reference plane, a simple relationship useful for incompressible fluids using the velocity head, elevation head, and pressure head is obtained.
If Eqn. 1 is multiplied by the density of the fluid, an equation with three pressure terms is obtained:
Note that the pressure of the system is constant in this form of the Bernoulli equation. If the static pressure of the system (the third term) increases, and if the pressure due to elevation (the middle term) is constant, then the dynamic pressure (the first term) must have decreased. In other words, if the speed of a fluid decreases and it is not due to an elevation difference, it must be due to an increase in the static pressure that is resisting the flow.
All three equations are merely simplified versions of an energy balance on a system.
<templatestyles src="Math_proof/styles.css" />Bernoulli equation for compressible fluids
The derivation for compressible fluids is similar. Again, the derivation depends upon (1) conservation of mass, and (2) conservation of energy. Conservation of mass implies that in the above figure, in the interval of time Δ"t", the amount of mass passing through the boundary defined by the area "A"1 is equal to the amount of mass passing outwards through the boundary defined by the area "A"2:
formula_42
Conservation of energy is applied in a similar manner: It is assumed that the change in energy of the volume
of the streamtube bounded by "A"1 and "A"2 is due entirely to energy entering or leaving through one or the other of these two boundaries. Clearly, in a more complicated situation such as a fluid flow coupled with radiation, such conditions are not met. Nevertheless, assuming this to be the case and assuming the flow is steady so that the net change in the energy is zero,
formula_43
where Δ"E"1 and Δ"E"2 are the energy entering through "A"1 and leaving through "A"2, respectively. The energy entering through "A"1 is the sum of the kinetic energy entering, the energy entering in the form of potential gravitational energy of the fluid, the fluid thermodynamic internal energy per unit of mass ("ε"1) entering, and the energy entering in the form of mechanical "p" d"V" work:
formula_44
where "Ψ" = "gz" is a force potential due to the Earth's gravity, g is acceleration due to gravity, and z is elevation above a reference plane. A similar expression for Δ"E"2 may easily be constructed.
So now setting 0 = Δ"E"1 − Δ"E"2:
formula_45
which can be rewritten as:
formula_46
Now, using the previously-obtained result from conservation of mass, this may be simplified to obtain
formula_47
which is the Bernoulli equation for compressible flow.
An equivalent expression can be written in terms of fluid enthalpy (h):
formula_48
Applications.
In modern everyday life there are many observations that can be successfully explained by application of Bernoulli's principle, even though no real fluid is entirely inviscid, and a small viscosity often has a large effect on the flow.
Misconceptions.
Airfoil lift.
One of the most common erroneous explanations of aerodynamic lift asserts that the air must traverse the upper and lower surfaces of a wing in the same amount of time, implying that since the upper surface presents a longer path the air must be moving over the top of the wing faster than over the bottom. Bernoulli's principle is then cited to conclude that the pressure on top of the wing must be lower than on the bottom.
Equal transit time applies to the flow around a body generating no lift, but there is no physical principle that requires equal transit time in cases of bodies generating lift. In fact, theory predicts – and experiments confirm – that the air traverses the top surface of a body experiencing lift in a "shorter" time than it traverses the bottom surface; the explanation based on equal transit time is false. While the equal-time explanation is false, it is not the Bernoulli principle that is false, because this principle is well established; Bernoulli's equation is used correctly in common mathematical treatments of aerodynamic lift.
Common classroom demonstrations.
There are several common classroom demonstrations that are sometimes incorrectly explained using Bernoulli's principle. One involves holding a piece of paper horizontally so that it droops downward and then blowing over the top of it. As the demonstrator blows over the paper, the paper rises. It is then asserted that this is because "faster moving air has lower pressure".
One problem with this explanation can be seen by blowing along the bottom of the paper: if the deflection was caused by faster moving air, then the paper should deflect downward; but the paper deflects upward regardless of whether the faster moving air is on the top or the bottom. Another problem is that when the air leaves the demonstrator's mouth it has the "same" pressure as the surrounding air; the air does not have lower pressure just because it is moving; in the demonstration, the static pressure of the air leaving the demonstrator's mouth is "equal" to the pressure of the surrounding air. A third problem is that it is false to make a connection between the flow on the two sides of the paper using Bernoulli's equation since the air above and below are "different" flow fields and Bernoulli's principle only applies within a flow field.
As the wording of the principle can change its implications, stating the principle correctly is important. What Bernoulli's principle actually says is that within a flow of constant energy, when fluid flows through a region of lower pressure it speeds up and vice versa. Thus, Bernoulli's principle concerns itself with "changes" in speed and "changes" in pressure "within" a flow field. It cannot be used to compare different flow fields.
A correct explanation of why the paper rises would observe that the plume follows the curve of the paper and that a curved streamline will develop a pressure gradient perpendicular to the direction of flow, with the lower pressure on the inside of the curve. Bernoulli's principle predicts that the decrease in pressure is associated with an increase in speed; in other words, as the air passes over the paper, it speeds up and moves faster than it was moving when it left the demonstrator's mouth. But this is not apparent from the demonstration.
Other common classroom demonstrations, such as blowing between two suspended spheres, inflating a large bag, or suspending a ball in an airstream are sometimes explained in a similarly misleading manner by saying "faster moving air has lower pressure".
Notes.
<templatestyles src="Reflist/styles.css" />
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "v"
},
{
"math_id": 1,
"text": "g"
},
{
"math_id": 2,
"text": "z"
},
{
"math_id": 3,
"text": "p"
},
{
"math_id": 4,
"text": "\\rho"
},
{
"math_id": 5,
"text": "\\frac{v^2}{2} + \\Psi + \\frac{p}{\\rho} = \\text{constant}"
},
{
"math_id": 6,
"text": "\\tfrac{1}{2} \\rho v^2 + \\rho g z + p = \\text{constant}"
},
{
"math_id": 7,
"text": "q + \\rho g h = p_0 + \\rho g z = \\text{constant}"
},
{
"math_id": 8,
"text": "H = z + \\frac{p}{\\rho g} + \\frac{v^2}{2g} = h + \\frac{v^2}{2g},"
},
{
"math_id": 9,
"text": "p + q = p_0"
},
{
"math_id": 10,
"text": "\\frac{\\partial \\varphi}{\\partial t} + \\tfrac12 v^2 + \\frac{p}{\\rho} + gz = f(t),"
},
{
"math_id": 11,
"text": "\\Phi = \\varphi - \\int_{t_0}^t f(\\tau)\\, \\mathrm{d}\\tau,"
},
{
"math_id": 12,
"text": "\\frac{\\partial \\Phi}{\\partial t} + \\tfrac12 v^2 + \\frac{p}{\\rho} + gz = 0."
},
{
"math_id": 13,
"text": "\\frac {v^2}{2}+ \\int_{p_1}^p \\frac {\\mathrm{d}\\tilde{p}}{\\rho\\left(\\tilde{p}\\right)} + \\Psi = \\text{constant (along a streamline)}"
},
{
"math_id": 14,
"text": "\\frac {v^2}{2}+ gz + \\left(\\frac {\\gamma}{\\gamma-1}\\right) \\frac {p}{\\rho} = \\text{constant (along a streamline)}"
},
{
"math_id": 15,
"text": "\\frac {v^2}{2}+\\left( \\frac {\\gamma}{\\gamma-1}\\right)\\frac {p}{\\rho} = \\left(\\frac {\\gamma}{\\gamma-1}\\right)\\frac {p_0}{\\rho_0}"
},
{
"math_id": 16,
"text": "\\frac{v^2}{2} + \\Psi + w = \\text{constant}."
},
{
"math_id": 17,
"text": "w =e + \\frac{p}{\\rho} ~~~\\left(= \\frac{\\gamma}{\\gamma-1} \\frac{p}{\\rho}\\right)"
},
{
"math_id": 18,
"text": "e"
},
{
"math_id": 19,
"text": "\\frac{v^2}{2} + w = w_0"
},
{
"math_id": 20,
"text": "\\frac{\\partial \\vec{v}}{\\partial t} + \\left(\\vec{v}\\cdot \\nabla\\right)\\vec{v} = -\\vec{g} - \\frac{\\nabla p}{\\rho}"
},
{
"math_id": 21,
"text": "\\frac{\\partial \\nabla \\phi}{\\partial t} + \\nabla \\left(\\frac{\\nabla \\phi \\cdot \\nabla \\phi}{2}\\right) = -\\nabla \\Psi - \\nabla \\int_{p_1}^{p}\\frac{d \\tilde{p}}{\\rho(\\tilde{p})}"
},
{
"math_id": 22,
"text": "\\frac{\\partial \\phi}{\\partial t} + \\frac{\\nabla \\phi \\cdot \\nabla \\phi}{2} + \\Psi + \\int_{p_1}^{p}\\frac{d \\tilde{p}}{\\rho(\\tilde{p})} = \\text{constant}"
},
{
"math_id": 23,
"text": "\\frac{\\partial \\phi}{\\partial t} + \\frac{\\nabla \\phi \\cdot \\nabla \\phi}{2} + \\Psi + \\frac{\\gamma}{\\gamma-1}\\frac{p}{\\rho} = \\text{constant}"
},
{
"math_id": 24,
"text": "\\begin{align}\nm \\frac{\\mathrm{d}v}{\\mathrm{d}t}&= F \\\\\n\\rho A \\mathrm{d}x \\frac{\\mathrm{d}v}{\\mathrm{d}t} &= -A \\mathrm{d}p \\\\\n\\rho \\frac{\\mathrm{d}v}{\\mathrm{d}t} &= -\\frac{\\mathrm{d}p}{\\mathrm{d}x} \n\\end{align}"
},
{
"math_id": 25,
"text": "\\frac{\\mathrm{d}v}{\\mathrm{d}t}\n= \\frac{\\mathrm{d}v}{\\mathrm{d}x}\\frac{\\mathrm{d}x}{\\mathrm{d}t}\n= \\frac{\\mathrm{d}v}{\\mathrm{d}x}v\n= \\frac{\\mathrm{d}}{\\mathrm{d}x} \\left( \\frac{v^2}{2} \\right)."
},
{
"math_id": 26,
"text": "\\frac{\\mathrm{d}}{\\mathrm{d}x} \\left( \\rho \\frac{v^2}{2} + p \\right) =0"
},
{
"math_id": 27,
"text": " \\frac{v^2}{2} + \\frac{p}{\\rho}= C"
},
{
"math_id": 28,
"text": "W = \\Delta E_\\text{kin}."
},
{
"math_id": 29,
"text": "\\begin{align}\n\\rho A_1 s_1 &= \\rho A_1 v_1 \\Delta t = \\Delta m, \\\\\n\\rho A_2 s_2 &= \\rho A_2 v_2 \\Delta t = \\Delta m.\n\\end{align}"
},
{
"math_id": 30,
"text": "W_\\text{pressure}=F_{1,\\text{pressure}} s_{1} - F_{2,\\text{pressure}} s_2 =p_1 A_1 s_1 - p_2 A_2 s_2 = \\Delta m \\frac{p_1}{\\rho} - \\Delta m \\frac{p_2}{\\rho}."
},
{
"math_id": 31,
"text": "\\Delta E_\\text{pot,gravity} = \\Delta m\\, g z_2 - \\Delta m\\, g z_1. "
},
{
"math_id": 32,
"text": "W_\\text{gravity} = -\\Delta E_\\text{pot,gravity} = \\Delta m\\, g z_1 - \\Delta m\\, g z_2."
},
{
"math_id": 33,
"text": "W = W_\\text{pressure} + W_\\text{gravity}."
},
{
"math_id": 34,
"text": "\\Delta E_\\text{kin} = \\tfrac12 \\Delta m\\, v_2^2-\\tfrac12 \\Delta m\\, v_1^2."
},
{
"math_id": 35,
"text": "\\Delta m \\frac{p_1}{\\rho} - \\Delta m \\frac{p_2}{\\rho} + \\Delta m\\, g z_1 - \\Delta m\\, g z_2 = \\tfrac12 \\Delta m\\, v_2^2 - \\tfrac12 \\Delta m\\, v_1^2"
},
{
"math_id": 36,
"text": "\\tfrac12 \\Delta m\\, v_1^2 + \\Delta m\\, g z_1 + \\Delta m \\frac{p_1}{\\rho} = \\tfrac12 \\Delta m\\, v_2^2 + \\Delta m\\, g z_2 + \\Delta m \\frac{p_2}{\\rho}."
},
{
"math_id": 37,
"text": "\\tfrac12 v_1^2 +g z_1 + \\frac{p_1}{\\rho}=\\tfrac12 v_2^2 +g z_2 + \\frac{p_2}{\\rho}"
},
{
"math_id": 38,
"text": "v = \\sqrt{ {2 g}{z} },"
},
{
"math_id": 39,
"text": "h_v =\\frac{v^2}{2 g}"
},
{
"math_id": 40,
"text": "p = p_0 - \\rho g z ,"
},
{
"math_id": 41,
"text": "\\psi = \\frac{p}{\\rho g}."
},
{
"math_id": 42,
"text": "0= \\Delta M_1 - \\Delta M_2 = \\rho_1 A_1 v_1 \\, \\Delta t - \\rho_2 A_2 v_2 \\, \\Delta t."
},
{
"math_id": 43,
"text": "\\Delta E_1 - \\Delta E_2 = 0"
},
{
"math_id": 44,
"text": "\\Delta E_1 = \\left(\\tfrac12 \\rho_1 v_1^2 + \\Psi_1 \\rho_1 + \\varepsilon_1 \\rho_1 + p_1 \\right) A_1 v_1 \\, \\Delta t"
},
{
"math_id": 45,
"text": "0 = \\left(\\tfrac12 \\rho_1 v_1^2+ \\Psi_1 \\rho_1 + \\varepsilon_1 \\rho_1 + p_1 \\right) A_1 v_1 \\, \\Delta t - \\left(\\tfrac12 \\rho_2 v_2^2 + \\Psi_2 \\rho_2 + \\varepsilon_2 \\rho_2 + p_2 \\right) A_2 v_2 \\, \\Delta t"
},
{
"math_id": 46,
"text": " 0 = \\left(\\tfrac12 v_1^2 + \\Psi_1 + \\varepsilon_1 + \\frac{p_1}{\\rho_1} \\right) \\rho_1 A_1 v_1 \\, \\Delta t - \\left(\\tfrac12 v_2^2 + \\Psi_2 + \\varepsilon_2 + \\frac{p_2}{\\rho_2} \\right) \\rho_2 A_2 v_2 \\, \\Delta t "
},
{
"math_id": 47,
"text": " \\tfrac12 v^2 + \\Psi + \\varepsilon + \\frac{p}{\\rho} = \\text{constant} \\equiv b "
},
{
"math_id": 48,
"text": " \\tfrac{1}{2} v^2 + \\Psi + h = \\text{constant} \\equiv b "
}
] |
https://en.wikipedia.org/wiki?curid=64219
|
64221
|
Biorhythm (pseudoscience)
|
The biorhythm theory is the pseudoscientific idea that peoples' daily lives are significantly affected by rhythmic cycles with periods of exactly 23, 28 and 33 days, typically a 23-day physical cycle, a 28-day emotional cycle, and a 33-day intellectual cycle. The idea was developed by Wilhelm Fliess in the late 19th century, and was popularized in the United States in the late 1970s. The proposal has been independently tested and, consistently, no validity for it has been found.
According to the notion of biorhythms, a person's life is influenced by rhythmic biological cycles that affect his or her ability in various domains, such as mental, physical, and emotional activity. These cycles begin at birth and oscillate in a steady (sine wave) fashion throughout life, and by modeling them mathematically, it is suggested that a person's level of ability in each of these domains can be predicted from day to day. It is built on the idea that the biofeedback chemical and hormonal secretion functions within the body could show a sinusoidal behavior over time.
Most biorhythm models use three cycles: a 23-day physical cycle, a 28-day emotional cycle, and a 33-day intellectual cycle. Although the 28-day cycle is the same length as the average woman's menstrual cycle and was originally described as a "female" cycle (see below), the two are not necessarily in synchronization. Each of these cycles varies between high and low extremes sinusoidally, with days where the cycle crosses the zero line described as "critical days" of greater risk or uncertainty.
The numbers from +100% (maximum) to -100% (minimum) indicate where on each cycle the rhythms are on a particular day. In general, a rhythm at 0% is crossing the midpoint and is thought to have no real impact on one's life, whereas a rhythm at +100% (at the peak of that cycle) would give one an edge in that area, and a rhythm at -100% (at the bottom of that cycle) would make life more difficult in that area. There is no particular meaning to a day on which one's rhythms are all high or all low, except the obvious benefits or hindrances that these rare extremes are thought to have on one's life.
In addition to the three popular cycles, various other cycles have been proposed, based on linear combination of the three, or on longer or shorter rhythms.
Calculation.
Theories published state the equations for the cycles as:
where formula_3 indicates the number of days since birth. Basic arithmetic shows that the combination of the simpler 23- and 28-day cycles repeats every 644 days (or 1<templatestyles src="Fraction/styles.css" />3⁄4 years), while the triple combination of 23-, 28-, and 33-day cycles repeats every 21,252 days (or 58.18+ years).
History.
The 23- and 28-day rhythms used by biorhythmists were first devised in the late 19th century by Wilhelm Fliess, a Berlin physician and friend of Sigmund Freud. Fliess believed that he observed regularities at 23- and 28-day intervals in a number of phenomena, including births and deaths. He labeled the 23-day rhythm "male" and the 28-day rhythm "female", matching the menstrual cycle.
In 1904, Viennese psychology professor Hermann Swoboda came to similar conclusions. Alfred Teltscher, professor of engineering at the University of Innsbruck, developed Swoboda's work and suggested that his students' good and bad days followed a rhythmic pattern; he believed that the brain's ability to absorb, mental ability, and alertness ran in 33-day cycles. One of the first academic researchers of biorhythms was Estonian-born Nikolai Pärna, who published a book in German called "Rhythm, Life and Creation" in 1923.
The practice of consulting biorhythms was popularized in the 1970s by a series of books by Bernard Gittelson, including "Biorhythm—A Personal Science", "Biorhythm Charts of the Famous and Infamous", and "Biorhythm Sports Forecasting". Gittelson's company, Biorhythm Computers, Inc., made a business selling personal biorhythm charts and calculators, but his ability to predict sporting events was not substantiated.
Charting biorhythms for personal use was popular in the United States during the 1970s; many places (especially video arcades and amusement areas) had a biorhythm machine that provided charts upon entry of date of birth. Biorhythm programs were a common application on personal computers; and in the late 1970s, there were also handheld biorhythm calculators on the market, the "Kosmos 1" and the Casio "Biolator".
Critical views.
There have been some three dozen published studies of biorhythm theory, but according to a study by Terence Hines, all of those either supported the null hypothesis that there is no correlation of human experience and the supposed biorhythms beyond what can be explained by coincidence, or, in cases where authors claimed to have evidence for biorhythm theory, methodological and statistical errors invalidated their conclusions. Hines therefore concluded that the theory is not valid.
Supporters continued to defend the theory in spite of the lack of corroborating scientific evidence, leading to the charge that it had become a kind of pseudoscience due to its proponents' rejection of empirical testing:
<templatestyles src="Template:Blockquote/styles.css" />An examination of some 134 biorhythm studies found that the theory is not valid (Hines, 1998). It is empirically testable and has been shown to be false. Terence Hines believes that this fact implies that biorhythm theory 'can not be properly termed a pseudoscientific theory'. However, when the advocates of an empirically testable theory refuse to give up the theory in the face of overwhelming evidence against it, it seems reasonable to call the theory pseudoscientific. For, in fact, the adherents to such a theory have declared by their behaviour that there is nothing that could falsify it, yet they continue to claim the theory is scientific. (from Carroll's "The Skeptic's Dictionary")
The physiologist Gordon Stein in the book "Encyclopedia of Hoaxes" (1993) wrote:Both the theoretical underpinning and the practical scientific verification of biorhythm theory are lacking. Without those, biorhythms became just another pseudoscientific claim that people are willing to accept without required evidence. Those pushing biorhythm calculators and books on a gullible public are guilty of making fraudulent claims. They are hoaxers of the public if they know what they are saying has no factual justification.A 1978 study of the incidence of industrial accidents found neither empirical nor theoretical support for the biorhythm model.
In Underwood Dudley's book, "Numerology: Or What Pythagoras Wrought", he provides an example of a situation in which a magician provides a woman her biorhythm chart that supposedly included the next two years of her life. The women sent letters to the magician describing how accurate the chart was. The magician purposely sent her a biorhythm chart based on a different birthdate. After he explained that he sent the wrong chart to her, he sent her another chart, also having the wrong birthdate. She then said that this new chart was even more accurate than the previous one. This kind of willful credulous belief in vague or inaccurate prognostication derives from motivated reasoning backed up by fallacious acceptance of confirmation bias, post hoc rationalization, and suggestibility.
Wilhelm Fliess "was able to impose his number patterns on virtually everything" and worked to convince others that cycles happen within men and women every 23 and 28 days. Mathematically, Fliess's equation, n = 23x +28y is unconstrained as there are infinitely many solutions for x and y, meaning that Fliess and Sigmund Freud (who adopted this idea in the early 1890s) could predict anything they wanted with the combination.
The skeptical evaluations of the various biorhythm proposals led to a number of critiques lambasting the subject published in the 1970s and 1980s. Biorhythm advocates who objected to the takedowns claimed that because circadian rhythms had been empirically verified in many organisms' sleep cycles, biorhythms were just as plausible. However, unlike biorhythms, which are claimed to have precise and unaltering periods, circadian rhythms are found by observing the cycle itself and the periods are found to vary in length based on biological and environmental factors. Assuming such factors were relevant to biorhythms would result in chaotic cycle combinations that remove any "predictive" features.
Additional studies.
Several controlled, experimental studies found no correlation between the 23, 28 and 33 day cycles and academic performance. These studies include:
James (1984).
James hypothesized that if biorhythms were rooted in science, then each proposed biorhythm cycle would contribute to task performance. Further, he predicted that each type of biorhythm cycle (i.e., intellectual, physical, and emotional) would be most influential on tasks associated with the corresponding cycle type. For example, he postulated that intellectual biorhythm cycles would be most influential on academic testing performance. In order to test his hypotheses, James observed 368 participants, noting their performance on tasks associated with intellectual, physical, and emotional functioning. Based on data collected from his experimental research, James concluded that there was no relation between subjects' biorhythmic status (on any of the three cycle types), and their performance on the associated practical tests.
Peveto (1980).
Peveto examined the proposed relationship between biorhythms and academic performance, specifically in terms of reading ability. Through examination of the data collected, Peveto concluded that there were no significant differences in the academic performance of the students, in regards to reading, during the high, low, or critical positions of neither the physical biorhythm cycle, the emotional biorhythm cycle, nor the intellectual biorhythm cycle. As a result, it was concluded that biorhythm cycles have no effect on the academic performance of students, when academic performance was measured using reading ability.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\sin(2\\pi t/23)"
},
{
"math_id": 1,
"text": "\\sin(2\\pi t/28)"
},
{
"math_id": 2,
"text": "\\sin(2\\pi t/33)"
},
{
"math_id": 3,
"text": "t"
}
] |
https://en.wikipedia.org/wiki?curid=64221
|
642330
|
Newtonian dynamics
|
Formulation of physics
In physics, Newtonian dynamics (also known as Newtonian mechanics) is the study of the dynamics of a particle or a small body according to Newton's laws of motion.
Mathematical generalizations.
Typically, the Newtonian dynamics occurs in a three-dimensional Euclidean space, which is flat. However, in mathematics Newton's laws of motion can be generalized to multidimensional and curved spaces. Often the term Newtonian dynamics is narrowed to Newton's second law formula_0.
Newton's second law in a multidimensional space.
Consider formula_1 particles with masses formula_2 in the regular three-dimensional Euclidean space. Let formula_3 be their radius-vectors in some inertial coordinate system. Then the motion of these particles is governed by Newton's second law applied to each of them
The three-dimensional radius-vectors formula_4 can be built into a single formula_5-dimensional radius-vector. Similarly, three-dimensional velocity vectors formula_6 can be built into a single formula_5-dimensional velocity vector:
In terms of the multidimensional vectors (2) the equations (1) are written as
i.e. they take the form of Newton's second law applied to a single particle with the unit mass formula_7.
Definition. The equations (3) are called the
equations of a Newtonian dynamical system in a flat multidimensional Euclidean space, which is called the configuration space of this system. Its points are marked by the radius-vector
formula_8. The space whose points are marked by the pair of vectors formula_9 is called the phase space of the dynamical system (3).
Euclidean structure.
The configuration space and the phase space of the dynamical system (3) both are Euclidean spaces, i. e. they are equipped with a Euclidean structure. The Euclidean structure of them is defined so that the kinetic energy of the single multidimensional particle with the unit mass formula_7 is equal to the sum of kinetic energies of the three-dimensional particles with the masses formula_2:
Constraints and internal coordinates.
In some cases the motion of the particles with the masses formula_2 can be constrained. Typical constraints look like scalar equations of the form
Constraints of the form (5) are called holonomic and scleronomic. In terms of the radius-vector formula_8 of the Newtonian dynamical system (3) they are written as
Each such constraint reduces by one the number of degrees of freedom of the Newtonian dynamical system (3). Therefore, the constrained system has formula_10 degrees of freedom.
Definition. The constraint equations (6) define an formula_11-dimensional manifold formula_12 within the configuration space of the Newtonian dynamical system (3). This manifold formula_12 is called the configuration space of the constrained system. Its tangent bundle formula_13 is called the phase space of the constrained system.
Let formula_14 be the internal coordinates of a point of formula_12. Their usage is typical for the Lagrangian mechanics. The radius-vector formula_8 is expressed as some definite function of formula_14:
The vector-function (7) resolves the constraint equations (6) in the sense that upon substituting (7) into (6) the equations (6) are fulfilled identically in formula_14.
Internal presentation of the velocity vector.
The velocity vector of the constrained Newtonian dynamical system is expressed in terms of the partial derivatives of the vector-function
(7):
The quantities formula_15 are called internal components of the velocity vector. Sometimes they are denoted with the use of a separate symbol
and then treated as independent variables. The quantities
are used as internal coordinates of a point of the phase space formula_13 of the constrained Newtonian dynamical system.
Embedding and the induced Riemannian metric.
Geometrically, the vector-function (7) implements an embedding of the configuration space formula_12 of the constrained Newtonian dynamical system into the formula_16-dimensional flat configuration space of the unconstrained
Newtonian dynamical system (3). Due to this embedding the Euclidean structure of the ambient space induces the Riemannian metric onto the manifold formula_12. The components of the metric tensor of this induced metric are given by the formula
where formula_17 is the scalar product associated with the Euclidean structure (4).
Kinetic energy of a constrained Newtonian dynamical system.
Since the Euclidean structure of an unconstrained system of formula_1 particles is introduced through their kinetic energy, the induced Riemannian structure on the configuration space formula_1 of a constrained system preserves this relation to the kinetic energy:
The formula (12) is derived by substituting (8) into (4) and taking into account (11).
Constraint forces.
For a constrained Newtonian dynamical system the constraints described by the equations (6) are usually implemented by some mechanical framework. This framework produces some auxiliary forces including the force that maintains the system within its configuration manifold formula_12. Such a maintaining force is perpendicular to formula_12. It is called the normal force. The force formula_18 from (6) is subdivided into two components
The first component in (13) is tangent to the configuration manifold formula_12. The second component is perpendicular to formula_12. In coincides with the normal force formula_19.<br>
Like the velocity vector (8), the tangent force
formula_20 has its internal presentation
The quantities formula_21 in (14) are called the internal components of the force vector.
Newton's second law in a curved space.
The Newtonian dynamical system (3) constrained to the configuration manifold formula_12 by the constraint equations (6) is described by the differential equations
where formula_22 are Christoffel symbols of the metric connection produced by the Riemannian metric (11).
Relation to Lagrange equations.
Mechanical systems with constraints are usually described by Lagrange equations:
where formula_23 is the kinetic energy the constrained dynamical system given by the formula (12). The quantities formula_24 in
(16) are the inner covariant components of the tangent force vector formula_25 (see (13) and (14)). They are produced from the inner contravariant components formula_21 of the vector formula_25 by means of the standard index lowering procedure using the metric (11):
The equations (16) are equivalent to the equations (15). However, the metric (11) and
other geometric features of the configuration manifold formula_12 are not explicit in (16). The metric (11) can be recovered from the kinetic energy formula_26 by means of the formula
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\displaystyle m\\,\\mathbf a=\\mathbf F"
},
{
"math_id": 1,
"text": "\\displaystyle N"
},
{
"math_id": 2,
"text": "\\displaystyle m_1,\\,\\ldots,\\,m_N"
},
{
"math_id": 3,
"text": "\\displaystyle \\mathbf r_1,\\,\\ldots,\\,\\mathbf r_N"
},
{
"math_id": 4,
"text": "\\displaystyle\\mathbf r_1,\\,\\ldots,\\,\\mathbf r_N"
},
{
"math_id": 5,
"text": "\\displaystyle n=3N"
},
{
"math_id": 6,
"text": "\\displaystyle\\mathbf v_1,\\,\\ldots,\\,\\mathbf v_N"
},
{
"math_id": 7,
"text": "\\displaystyle m=1"
},
{
"math_id": 8,
"text": "\\displaystyle\\mathbf r"
},
{
"math_id": 9,
"text": "\\displaystyle(\\mathbf r,\\mathbf v)"
},
{
"math_id": 10,
"text": "\\displaystyle n=3\\,N-K"
},
{
"math_id": 11,
"text": "\\displaystyle n"
},
{
"math_id": 12,
"text": "\\displaystyle M"
},
{
"math_id": 13,
"text": "\\displaystyle TM"
},
{
"math_id": 14,
"text": "\\displaystyle q^1,\\,\\ldots,\\,q^n"
},
{
"math_id": 15,
"text": "\\displaystyle\\dot q^1,\\,\\ldots,\\,\\dot q^n"
},
{
"math_id": 16,
"text": "\\displaystyle 3\\,N"
},
{
"math_id": 17,
"text": "\\displaystyle(\\ ,\\ )"
},
{
"math_id": 18,
"text": "\\displaystyle\\mathbf F"
},
{
"math_id": 19,
"text": "\\displaystyle\\mathbf N"
},
{
"math_id": 20,
"text": "\\displaystyle\\mathbf F_\\parallel"
},
{
"math_id": 21,
"text": "F^1,\\,\\ldots,\\,F^n"
},
{
"math_id": 22,
"text": "\\Gamma^s_{ij}"
},
{
"math_id": 23,
"text": "T=T(q^1,\\ldots,q^n,w^1,\\ldots,w^n)"
},
{
"math_id": 24,
"text": "Q_1,\\,\\ldots,\\,Q_n"
},
{
"math_id": 25,
"text": "\\mathbf F_\\parallel"
},
{
"math_id": 26,
"text": "\\displaystyle T"
}
] |
https://en.wikipedia.org/wiki?curid=642330
|
64237234
|
Weyl expansion
|
In physics, the Weyl expansion, also known as the Weyl identity or angular spectrum expansion, expresses an outgoing spherical wave as a linear combination of plane waves. In a Cartesian coordinate system, it can be denoted as
Outgoing spherical wave as a linear combination of plane waves
formula_0,
where formula_1, formula_2 and formula_3 are the wavenumbers in their respective coordinate axes:
formula_4.
The expansion is named after Hermann Weyl, who published it in 1919. The Weyl identity is largely used to characterize the reflection and transmission of spherical waves at planar interfaces; it is often used to derive the Green's functions for Helmholtz equation in layered media. The expansion also covers evanescent wave components. It is often preferred to the Sommerfeld identity when the field representation is needed to be in Cartesian coordinates.
The resulting Weyl integral is commonly encountered in microwave integrated circuit analysis and electromagnetic radiation over a stratified medium; as in the case for Sommerfeld integral, it is numerically evaluated. As a result, it is used in calculation of Green's functions for method of moments for such geometries. Other uses include the descriptions of dipolar emissions near surfaces in nanophotonics, holographic inverse scattering problems, Green's functions in quantum electrodynamics and acoustic or seismic waves.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\frac{e^{-j k_0 r}}{r}=\\frac{1}{j 2\\pi} \\int_{-\\infty}^{\\infty} \\int_{-\\infty}^{\\infty} dk_x dk_y e^{-j(k_x x + k_y y)} \\frac{e^{-jk_z |z|}}{k_z}"
},
{
"math_id": 1,
"text": "k_x"
},
{
"math_id": 2,
"text": "k_y"
},
{
"math_id": 3,
"text": "k_z"
},
{
"math_id": 4,
"text": "k_0=\\sqrt{k_x^2+k_y^2+k_z^2}"
}
] |
https://en.wikipedia.org/wiki?curid=64237234
|
64237800
|
Fundamental theorem of Hilbert spaces
|
In mathematics, specifically in functional analysis and Hilbert space theory, the fundamental theorem of Hilbert spaces gives a necessarily and sufficient condition for a Hausdorff pre-Hilbert space to be a Hilbert space in terms of the canonical isometry of a pre-Hilbert space into its anti-dual.
Preliminaries.
Antilinear functionals and the anti-dual.
Suppose that H is a topological vector space (TVS).
A function "f" : "H" → formula_0 is called semilinear or antilinear if for all "x", "y" ∈ "H" and all scalars c ,
The vector space of all continuous antilinear functions on H is called the anti-dual space or complex conjugate dual space of H and is denoted by formula_1 (in contrast, the continuous dual space of H is denoted by formula_2), which we make into a normed space by endowing it with the canonical norm (defined in the same way as the canonical norm on the continuous dual space of H).
Pre-Hilbert spaces and sesquilinear forms.
A sesquilinear form is a map "B" : "H" × "H" → formula_0 such that for all "y" ∈ "H", the map defined by "x" ↦ "B"("x", "y") is linear, and for all "x" ∈ "H", the map defined by "y" ↦ "B"("x", "y") is antilinear.
Note that in Physics, the convention is that a sesquilinear form is linear in its "second" coordinate and antilinear in its first coordinate.
A sesquilinear form on H is called positive definite if "B"("x", "x") > 0 for all non-0 "x" ∈ "H"; it is called non-negative if "B"("x", "x") ≥ 0 for all "x" ∈ "H".
A sesquilinear form B on H is called a Hermitian form if in addition it has the property that formula_3 for all "x", "y" ∈ "H".
Pre-Hilbert and Hilbert spaces.
A pre-Hilbert space is a pair consisting of a vector space H and a non-negative sesquilinear form B on H;
if in addition this sesquilinear form B is positive definite then ("H", "B") is called a Hausdorff pre-Hilbert space.
If B is non-negative then it induces a canonical seminorm on H, denoted by formula_4, defined by "x" ↦ "B"("x", "x")1/2, where if B is also positive definite then this map is a norm.
This canonical semi-norm makes every pre-Hilbert space into a seminormed space and every Hausdorff pre-Hilbert space into a normed space.
The sesquilinear form "B" : "H" × "H" → formula_0 is separately uniformly continuous in each of its two arguments and hence can be extended to a separately continuous sesquilinear form on the completion of H; if H is Hausdorff then this completion is a Hilbert space.
A Hausdorff pre-Hilbert space that is complete is called a Hilbert space.
Canonical map into the anti-dual.
Suppose ("H", "B") is a pre-Hilbert space. If "h" ∈ "H", we define the canonical maps:
"B"("h", •) : "H" → formula_0 where "y" ↦ "B"("h", "y"), and
"B"(•, "h") : "H" → formula_0 where "x" ↦ "B"("x", "h")
The canonical map from H into its anti-dual formula_1 is the map
formula_5 defined by "x" ↦ "B"("x", •).
If ("H", "B") is a pre-Hilbert space then this canonical map is linear and continuous;
this map is an isometry onto a vector subspace of the anti-dual if and only if ("H", "B") is a Hausdorff pre-Hilbert.
There is of course a canonical antilinear surjective isometry formula_6 that sends a continuous linear functional f on H to the continuous antilinear functional denoted by and defined by "x" ↦ "f" ("x").
Fundamental theorem of Hilbert spaces: Suppose that ("H", "B") is a Hausdorff pre-Hilbert space where "B" : "H" × "H" → formula_0 is a sesquilinear form that is linear in its first coordinate and antilinear in its second coordinate. Then the canonical linear mapping from H into the anti-dual space of H is surjective if and only if ("H", "B") is a Hilbert space, in which case the canonical map is a surjective isometry of H onto its anti-dual.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\mathbb{C}"
},
{
"math_id": 1,
"text": "\\overline{H}^{\\prime}"
},
{
"math_id": 2,
"text": "H^{\\prime}"
},
{
"math_id": 3,
"text": "B(x, y) = \\overline{B(y, x)}"
},
{
"math_id": 4,
"text": "\\| \\cdot \\|"
},
{
"math_id": 5,
"text": "H \\to \\overline{H}^{\\prime}"
},
{
"math_id": 6,
"text": "H^{\\prime} \\to \\overline{H}^{\\prime}"
}
] |
https://en.wikipedia.org/wiki?curid=64237800
|
64239792
|
2 Kings 5
|
2 Kings, chapter 5
2 Kings 5 is the fifth chapter of the second part of the Books of Kings in the Hebrew Bible or the Second Book of Kings in the Old Testament of the Christian Bible. The book is a compilation of various annals recording the acts of the kings of Israel and Judah by a Deuteronomic compiler in the seventh century BCE, with a supplement added in the sixth century BCE. This chapter records an astonishing healing of Naaman, an Aramean general, by the prophet Elisha.
Text.
This chapter was originally written in the Hebrew language. It is divided into 27 verses.
Textual witnesses.
Some early manuscripts containing the text of this chapter in Hebrew are of the Masoretic Text tradition, which includes the Codex Cairensis (895), Aleppo Codex (10th century), and Codex Leningradensis (1008). A fragment containing a part of this chapter in Hebrew was found among the Dead Sea Scrolls, that is, 6Q4 (6QpapKgs; 150–75 BCE) with the extant verse 26.
There is also a translation into Koine Greek known as the Septuagint, made in the last few centuries BCE. Extant ancient manuscripts of the Septuagint version include Codex Vaticanus (B; formula_0B; 4th century) and Codex Alexandrinus (A; formula_0A; 5th century).
The healing of Naaman (5:1–19).
This story of Elisha healing neighboring Aram's highest-ranking military officer, Naaman, of an uncurable illness happened in a period of significant Aramean control over Israel (verse 2; Aram could give Israel orders, verses 6–7), perhaps during the time of Ben-Hadad II and Jehoram, or during the time of Hazael of Aram (reigned 842–796 BCE) Jehu (reigned 841–814 BCE), Jehoahaz (reigned 814–798 BCE) or Joash of Israel (reigned 798–782 BCE; cf. 2 Kings 8:11–12; 10:32–33; 13:22). Elisha's reputation as a miracle-worker spread to Aram through a young female Israelite prisoner-of-war (verse 3), whose information not only helped her master, but also her people in the service of her God. In helping the Aramean general, Elisha simultaneously helped the Israelite king. The Aramean king sent a lot of money and ordered his vassal in Samaria to do impossible task: to immediately produce the necessary miracle to heal Naaman (verse 6–7), but Elisha somehow knew about the letter ("seper"; literally "scroll") from Aram and sent his own letter to the Israelite king asking Naaman to be directed to the prophet for treatment. Naaman who expected respectful conventional behavior of miracle-healing was understandably unhappy that Elisha did not meet him personally and only prescribed instructions to ritually bathe in the Jordan (verses 9–12), yet after advised by his more sensible soldiers (verse 13), Naaman complied and immediately experienced complete healing (verse 14). Naaman quickly returned to his benefactor, wishing to ensure the future proximity of YHWH whose power had convinced him. Since this God resides only in Israel, he took two mule-loads of Israelite earth to Damascus in order to be able to sacrifice to YHWH there (verses 15a, 17; a sincere 'earthbound understanding of God') with the blessing of Elisha who parted from Naaman in peace (verse 19).
"Now Naaman, commander of the army of the king of Syria, was a great and honorable man in the eyes of his master, because by him the Lord had given victory to Syria. He was also a mighty man of valor, but a leper."
2 "And the Syrians had gone out on raids, and had brought back captive a young girl from the land of Israel. She waited on Naaman's wife."
3 "Then she said to her mistress, "If only my master were with the prophet who is in Samaria! For he would heal him of his leprosy.""
Verses 2–3.
The young girl had much reason to doubt the power of YHWH because of her abduction, but nonetheless showed her confidence in YHWH when informing her mistress about Elisha, in contrast to Naaman who was responsible to subjugate Israel and take away slaves but powerless about his disease.
""Are not the Abanah and the Pharpar, the rivers of Damascus, better than all the waters of Israel? Could I not wash in them and be clean?" So he turned and went away in a rage."
Gehazi's greed and punishment (5:20–27).
This passage is an appendix to the main story, the healing of Naaman, with the same purpose of hailing the glory of God and Elisha, but here in the teaching of disciples: what can a prophet accept as recompense for services to God and at what point is it considered selling one's soul? In verses 15b,16, Elisha showed a good example: in a case like this, a prophet accepts nothing, clarifying that great power and wealth cannot force or buy the support of prophets and God, nor must prophets allowed themselves be used as tools for any interest groups (cf. Micah 3:5). Gehazi, Elisha's servant (also mentioned in 2 Kings 4:27–37; 8:4–5) became the complementary negative example: cunningly accepting the presents brought by Naaman for himself, but then receiving condemnation by his master for the act and afflicted by Naaman's former sickness.
26 "Then he said to him, “Did not my heart go with you when the man turned back from his chariot to meet you? Is it time to receive money and to receive clothing, olive groves and vineyards, sheep and oxen, male and female servants?"
27 "Therefore the leprosy of Naaman shall cling to you and your descendants forever.” And he went out from his presence leprous, as white as snow."
Verses 26–27.
Elisha has been directing Naaman's thoughts to YHWH alone as the healer of the disease, so the prophet was out of sight until Naaman was fully cured and steadfastly refused any present to remove any indication that he was in any way instrumental in the healing. Naaman must have been very impressed with the act and pledged to worship YHWH. However, Gehazi's actions possibly obliterated the impression. In listing of all the Gehazi's plan to purchase using the ill-gotten talents Elisha showed Gehazi that he has been reading all his thoughts.
Notes.
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References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": " \\mathfrak{G}"
}
] |
https://en.wikipedia.org/wiki?curid=64239792
|
64239793
|
2 Kings 6
|
2 Kings, chapter 6
2 Kings 6 is the sixth chapter of the second part of the Books of Kings in the Hebrew Bible or the Second Book of Kings in the Old Testament of the Christian Bible. The book is a compilation of various annals recording the acts of the kings of Israel and Judah by a Deuteronomic compiler in the seventh century BC, with a supplement added in the sixth century BCE. This chapter records some miraculous deeds of the prophet Elisha.
Text.
This chapter was originally written in the Hebrew language. It is divided into 33 verses.
Textual witnesses.
Some early manuscripts containing the text of this chapter in Hebrew are of the Masoretic Text tradition, which includes the Codex Cairensis (895), Aleppo Codex (10th century), and Codex Leningradensis (1008). A fragment containing a part of this chapter in Hebrew was found among the Dead Sea Scrolls, that is, 6Q4 (6QpapKgs; 150–75 BCE) with the extant verse 32.
There is also a translation into Koine Greek known as the Septuagint, made in the last few centuries BCE. Extant ancient manuscripts of the Septuagint version include Codex Vaticanus (B; formula_0B; 4th century) and Codex Alexandrinus (A; formula_0A; 5th century).
The axe head recovered (6:1–7).
The passage shows how Elisha helped his disciples, even for something seemingly trivial. Elisha's followers lost a borrowed axe in the water (hence an obligation to repay its owner for the loss; cf. Exodus 22:13–14), and the prophet came to help by using "a kind of analogical magic" on the last spot of the axe, before letting the disciple picked it up out of the water. This episode is tied syntactically to the earlier passage by an 'initial "waw"-consecutive verb' (in "and they said") and thematically by similar emphasis of Elisha's 'divinely granted powers' as well as in its relation to Jordan River.
"Then the man of God said, “Where did it fall?” When he showed him the place, he cut off a stick and threw it in there and made the iron float."
Elisha captures Arameans and subsequently ensures their release (6:8–23).
The scene moves to a larger political world, where Aramean troops attacked Israelite territory unhindered, but with the help from the prophet, the Israelite army could avoid falling into their hands several times. The Aramean king (likely Ben-Hadad II; 2 Kings 6:24) could only presume he had been betrayed (verse 11), until he found out that the Israelite king (verse 12, Jehoram) was guided by a 'clairvoyant prophet', so he sent an 'army regiment with horses and chariots' to Dothan (about 15 km. north of Samaria) to arrest Elisha. The prophet's servant saw in despair that the city was completely surrounded, yet Elisha could see a heavenly host with horses and chariots of fire (verse 17) guarding him (referring to Elisha's archaic title: 'chariots of Israel and its horsemen [better: horses]', 2 Kings 13:14). There was no battle with the Arameans, because God 'struck them with blindness' (verse 18), so that Elisha can mock them that the one they seek (which is the prophet himself) was not there, and he led them into his trap, right into the middle of the strongly fortified royal city of Samaria, where the Arameans were now completely surrounded with no escape (verses 19–23). However, Elisha prevented the king (who respectfully called the prophet 'father'; cf. 2 Kings 13:14) from simply killing the helpless prisoners, and instead to feed and release the Arameans (verses 21–22); a humane act which might help to reduce tensions and enmities at the time (verse 23). Initially the narrative refers Elisha as "the man of God" and only later employs his name, emphasizing that the prophet is indeed the man of God.
"And Elisha prayed, and said, "Lord, I pray, open his eyes that he may see." Then the Lord opened the eyes of the young man, and he saw. And behold, the mountain was full of horses and chariots of fire all around Elisha."
Ben-Hadad besieges Samaria(6:24–33).
Despite the kind gesture of 2 Kings 6:23, the Arameans who no longer made plundering raids through the country, now directly besieged the capital, Samaria. Such a siege in ancient times could last for months, even years, in order literally to starve out the people in the city (cf. 2 Kings 17:5; 25:1–2). The attacker is identified as Ben-Hadad II whom Ahab foolishly released in the time of Elijah (1 Kings 20), then later caused Ahab's death (1 Kings 22) and now threatened Ahab's son, Jehoram (as this report appeared within his regnal report in 2 Kings 3:1–8:15).
The narrative displays the increasingly desperate situation: poor-quality food and fuel became extremely expensive (verse 25), ravenous hunger drove people to cannibalism (verses 26–29, cf. also Lamentations 2:20; 4:10), the king was completely powerless and deeply dejected (verses 27, 30). At last, the prophet Elisha was mentioned—not as a possible helper, but as the king's enemy (verses 31–32), because apparently the prophet had encouraged resistance to the enemy and trust in YHWH, and now the king's patience had come to an end and sent messengers to arrest Elisha (verse 33). The location of Elisha's house was presumably in Samaria, not in Dothan, since Samaria was under siege. The presence of some elders in Elisha's house indicates a consultation session regarding oracles from YHWH (cf. Ezekiel 8:1). Elisha called the king as "this son of a murderer", likely recalling the acts of Ahab, the father of the present king Jehoram, in murdering the sons of the prophets (1 Kings 18).
"And there was a great famine in Samaria, as they besieged it, until a donkey's head was sold for eighty shekels of silver, and the fourth part of a kab of dove's dung for five shekels of silver."
Notes.
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References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": " \\mathfrak{G}"
}
] |
https://en.wikipedia.org/wiki?curid=64239793
|
64239794
|
2 Kings 7
|
2 Kings, chapter 7
2 Kings 7 is the seventh chapter of the second part of the Books of Kings in the Hebrew Bible or the Second Book of Kings in the Old Testament of the Christian Bible. The book is a compilation of various annals recording the acts of the kings of Israel and Judah by a Deuteronomic compiler in the seventh century BCE, with a supplement added in the sixth century BCE. This chapter records the fulfillment of Elisha's prophecy during the siege of Arameans on Samaria.
Text.
This chapter was originally written in the Hebrew language. It is divided into 20 verses.
Textual witnesses.
Some early manuscripts containing the text of this chapter in Hebrew are of the Masoretic Text tradition, which includes the Codex Cairensis (895), Aleppo Codex (10th century), and Codex Leningradensis (1008). Fragments containing parts of this chapter in Hebrew were found among the Dead Sea Scrolls, that is, 6Q4 (6QpapKgs; 150–75 BCE) with extant verses 8–10, 20.
There is also a translation into Koine Greek known as the Septuagint, made in the last few centuries BCE. Extant ancient manuscripts of the Septuagint version include Codex Vaticanus (B; formula_0B; 4th century) and Codex Alexandrinus (A; formula_0A; 5th century).
Elisha’s prophecy of plenty (7:1–2).
Facing the death threat from the Israelite king (2 Kings 6), Elisha attacked back using a prophecy from God that good-quality food would be available at normal prices within one day (verse 1). When the king's adviser showed doubts over the hardly imaginable salvation under the circumstances, Elisha even proclaimed a woeful prophecy against him (verse 2). The king's silence seems to indicate that he was ready to give Elisha one final chance.
"Then Elisha said, "Hear the word of the Lord. Thus says the Lord: 'Tomorrow about this time a seah of fine flour shall be sold for a shekel, and two seahs of barley for a shekel, at the gate of Samaria.""
The Syrians flee (7:3–15).
The narrative's dramatic climax starts with four lepers, who stood daily at the city gates, rejected and avoided by other city inhabitants, going to the Aramean encampment and becoming the first to witness the sudden retreat of the big army, but instead of taking personal advantage of the situation they decided to announce the news to state officials (verses 3–11; a wonderful precursor to Jesus' recognition that God loves making the last first; cf. Mark 10:31ff). An information was supplied (what the lepers did not know) that God brought hallucinations to the Arameans, convincing them that great Egyptian and Hittite armies advanced to attack, thus forcing them to break off the siege immediately (verses 6–7). The Israelite king suspected a trick (verse 12; cf. a very similar scene in ), but finally sent people to investigate the situation and found the Arameans' eastward retreat toward the Jordan leaving their weapons and goods in panic (verses 13–15).
Elisha’s prophecy fulfilled (7:16–20).
The report about Arameans' retreat triggered the people to enter the camp close to the city and take possession of their provisions, causing food prices to sink to the level forecast by Elisha (verse 16). The story ended with the fate of the doubting adviser who saw the prophecy fulfilled but was trampled to death before he could enjoy the victory (verses 17–20). Verse 19 quotes the words of the officer and the prophet Elisha to clarify the fulfillment of the prophecy.
Notes.
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References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": " \\mathfrak{G}"
}
] |
https://en.wikipedia.org/wiki?curid=64239794
|
64239801
|
2 Kings 11
|
2 Kings, chapter 11
2 Kings 11 is the eleventh chapter of the second part of the Books of Kings in the Hebrew Bible or the Second Book of Kings in the Old Testament of the Christian Bible. The book is a compilation of various annals recording the acts of the kings of Israel and Judah by a Deuteronomic compiler in the seventh century BCE, with a supplement added in the sixth century BCE. This chapter records the reign of Athaliah and Joash as the rulers of Judah.
Text.
This chapter was originally written in the Hebrew language. It is divided into 21 verses in Christian Bibles, but into 20 verses in the Hebrew Bible as in the verse numbering comparison table below.
Verse numbering.
This article generally follows the common numbering in Christian English Bible versions, with notes to the numbering in Hebrew Bible versions.
Textual witnesses.
Some early manuscripts containing the text of this chapter in Hebrew are of the Masoretic Text tradition, which includes the Codex Cairensis (895), Aleppo Codex (10th century), and Codex Leningradensis (1008).
There is also a translation into Koine Greek known as the Septuagint, made in the last few centuries BCE. Extant ancient manuscripts of the Septuagint version include Codex Vaticanus (B; formula_0B; 4th century) and Codex Alexandrinus (A; formula_0A; 5th century).
Analysis.
A parallel pattern of sequence is observed in the final sections of 2 Kings between 2 Kings 11–20 and 2 Kings 21–25, as follows:
A. Athaliah, daughter of Ahab, kills royal seed ()
B. Joash reigns (2 Kings 11–12)
C. Quick sequence of kings of Israel and Judah (2 Kings 13–16)
D. Fall of Samaria (2 Kings 17)
E. Revival of Judah under Hezekiah (2 Kings 18–20)
A'. Manasseh, a king like Ahab, promotes idolatry and kills the innocence (2 Kings 21)
B'. Josiah reigns (2 Kings 22–23)
C'. Quick succession of kings of Judah (2 Kings 24)
D'. Fall of Jerusalem (2 Kings 25)
E'. Elevation of Jehoiachin (2 Kings 25:27–30)
Athaliah's accession to power and Joash's rescue (11:1–3).
The record of Athaliah's reign in Judah was treated structurally as an appendix of the regnal account of Ahaziah ben Jehoram, the king of Judah (2 Kings 8:25–11:20), or as a revolt of a usurper (cf. northern tribes against Rehoboah in 1 Kings 12; Jehu's revolt against Jehoram in 2 Kings 9–10), so it lacks the usual formal structure of regnal accounts. Athaliah was Omri's 'granddaughter' (2 Kings 8:26), who married to Joram of the Davidic royal family and became the queen mother of Ahaziah ben Joram (2 Kings 8:18). When Jehu's coup left her with no male relatives in either Samaria or Jerusalem, she reacted brutally as a mass murderer of David's house (of what remained after Jehu's slaughter in 2 Kings 10:12–14) and—despite being a woman and an Omride—became the ruler of Judah, effectively personifying the Omridic politics that was violently cut away from (northern) Israel, for a further six years in Judah.
"And when Athaliah the mother of Ahaziah saw that her son was dead, she arose and destroyed all the seed royal."
"But Jehosheba, the daughter of king Joram, sister of Ahaziah, took Joash the son of Ahaziah, and stole him from among the king's sons which were slain; and they hid him, even him and his nurse, in the bedchamber from Athaliah, so that he was not slain."
Joash's enthronement and Athaliah's death (11:4–21).
The priest Jehoiada played a significant role in deposing Athaliah and putting the 7-year-old Joash on the throne after keeping the future king hidden for six years (). Jehoiada built up a 'subversive organization in the temple with a good infrastructure, sufficient weaponry', and a close relationship with the 'people of the land' (verses 14, 18, 20). The final sentence of verse 20 (contrasting the land/Judah and the city/Jerusalem) gives indication on the political constellation: Athaliah, like all Omrides, enjoyed the support of the urban and aristocratic circles of the capital city, whereas the opposition (such as also Jehu) received the support from the provincial farming population. The religious factors also played a role in the overthrow in Judah, as Jehoiada was a priest of the temple of Jerusalem, where since the time of Solomon, there had been syncretistic and strictly YHWH-worshipping tendencies there (cf. e.g. ; , 22), so the revolt might include anti-Baal sentiment (verse 18a). This chapter is a Judean counterpart to Jehu's revolt (2 Kings 9–10), which also eliminated a queen (Jezebel) and the Baal worship in (northern) Israel six years earlier.
"And he brought out the king’s son, put the crown on him, and gave him the Testimony; they made him king and anointed him, and they clapped their hands and said, "Long live the king!""
"And when she looked, behold, the king stood by a pillar, as the manner was, and the princes and the trumpeters by the king, and all the people of the land rejoiced, and blew with trumpets: and Athaliah rent her clothes, and cried, Treason, Treason."
"And all the people of the land went to the temple of Baal, and tore it down. They thoroughly broke in pieces its altars and images, and killed Mattan the priest of Baal before the altars. And the priest appointed officers over the house of the Lord."
"Jehoash was seven years old when he began to reign."
Notes.
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References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": " \\mathfrak{G}"
}
] |
https://en.wikipedia.org/wiki?curid=64239801
|
64239805
|
2 Kings 12
|
2 Kings, chapter 12
2 Kings 12 is the twelfth chapter of the second part of the Books of Kings in the Hebrew Bible or the Second Book of Kings in the Old Testament of the Christian Bible. The book is a compilation of various annals recording the acts of the kings of Israel and Judah by a Deuteronomic compiler in the seventh century BCE, with a supplement added in the sixth century BCE. This chapter records the reign of Joash as the king of Judah.
Text.
This chapter was originally written in the Hebrew language. It is divided into 21 verses in Christian Bibles, but into 22 verses in the Hebrew Bible as in the verse numbering comparison table below.
Verse numbering.
This article generally follows the common numbering in Christian English Bible versions, with notes to the numbering in Hebrew Bible versions.
Textual witnesses.
Some early manuscripts containing the text of this chapter in Hebrew are of the Masoretic Text tradition, which includes the Codex Cairensis (895), Aleppo Codex (10th century), and Codex Leningradensis (1008).
There is also a translation into Koine Greek known as the Septuagint, made in the last few centuries BCE. Extant ancient manuscripts of the Septuagint version include Codex Vaticanus (B; formula_0B; 4th century) and Codex Alexandrinus (A; formula_0A; 5th century).
Analysis.
A parallel pattern of sequence is observed in the final sections of 2 Kings between 2 Kings 11–20 and 2 Kings 21–25, as follows:
A. Athaliah, daughter of Ahab, kills royal seed ()
B. Joash reigns (2 Kings 11–12)
C. Quick sequence of kings of Israel and Judah (2 Kings 13–16)
D. Fall of Samaria (2 Kings 17)
E. Revival of Judah under Hezekiah (2 Kings 18–20)
A'. Manasseh, a king like Ahab, promotes idolatry and kills the innocence (2 Kings 21)
B'. Josiah reigns (2 Kings 22–23)
C'. Quick succession of kings of Judah (2 Kings 24)
D'. Fall of Jerusalem (2 Kings 25)
E'. Elevation of Jehoiachin (2 Kings 25:27–30)
This chapter consists of three parts:
The Temple renovation during the reign of Joash (12:1–16).
Joash (or Jehoash) is given a relatively positive rating in the books of Kings, first because of his succession to replace the Omride queen Athaliah, and secondly due to his care of the temple of YHWH (the Chronicler notes that Jehoash became corrupt after the death of Jehoiada; 2 Chronicles 24:15–22). Joash arranged that temple renovation was no longer solely directed by the priests, but was decreed by the palace, and that donations for this project were placed in a collection box, to be counted communally at intervals, then given to a building administration (verses 6–12, 15). As animal and vegetable sacrifices were reserved for God and his priests (verse 17), others could be made by paying in silver ("shekel"), so a group of lower caste 'priests who guarded the threshold' was assigned to deposit these in a designated chest (according to 2 Chronices 24:10, by the
time of exile, the believers threw their money into the collection box themselves).
"In the seventh year of Jehu Jehoash began to reign; and forty years reigned he in Jerusalem. And his mother's name was Zibiah of Beersheba."
"Now it was so, by the twenty-third year of King Jehoash, that the priests had not repaired the damages of the temple."
Joash's reign (12:17–21).
During the later parts of Joash's reign, Hazael, the king of Aram in Damascus (cf. 1 Kings 19:15–17; 2 Kings 8:7–15), placed both the northern kingdom of Jehu (cf. 2 Kings 10:32–33) and the kingdom of Judah under heavy burden of tributes. The threat of Hazael to Jerusalem indicates a continuous concern for the Aramean invasion to the land of Israel since the time of Omri's dynasty to the early parts of Jehu's dynasty until king Jehoash ben Jehoahaz of Israel (the third in Jehu's line of kings) defeated the Arameans following the death of prophet Elisha (2 Kings 13:14–21). The payment of tribute to Hazael may mean that all the funds for temple repairs collected by Jehoash (and his predecessors, such as Jehoshaphat, Jehoram and Ahaziah) were lost to the Arameans.
Jehoash's assassination could be explained from historiographical perspectives, beginning with Jehoshaphat giving his son Jehoram in marriage to Athaliah, so the house of David thereafter descended from the house of Omri, and the next three kings of Judah (three generations) were assassinated as the consequences of Elijah's prophecy that every male of Ahab in Israel would be cut off (2 Kings 21:21) until the reign of Uzziah ben Amaziah of Judah which coincides the time king Jeroboam ben Jehoash of Israel restored the borders of Israel.
"And the rest of the acts of Joash, and all that he did, are they not written in the book of the chronicles of the kings of Judah?"
"For Jozachar the son of Shimeath and Jehozabad the son of Shomer, his servants, struck him. So he died, and they buried him with his fathers in the City of David. Then Amaziah his son reigned in his place."
Notes.
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References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": " \\mathfrak{G}"
}
] |
https://en.wikipedia.org/wiki?curid=64239805
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64239807
|
2 Kings 21
|
2 Kings, chapter 21
2 Kings 21 is the twenty-first chapter of the second part of the Books of Kings in the Hebrew Bible or the Second Book of Kings in the Old Testament of the Christian Bible. The book is a compilation of various annals recording the acts of the kings of Israel and Judah by a Deuteronomic compiler in the seventh century BCE, with a supplement added in the sixth century BCE. This chapter records the events during the reign of Manasseh and Amon, the kings of Judah.
Text.
This chapter was originally written in the Hebrew language. It is divided into 26 verses.
Textual witnesses.
Some early manuscripts containing the text of this chapter in Hebrew are of the Masoretic Text tradition, which includes the Codex Cairensis (895), Aleppo Codex (10th century), and Codex Leningradensis (1008).
There is also a translation into Koine Greek known as the Septuagint, made in the last few centuries BCE. Extant ancient manuscripts of the Septuagint version include Codex Vaticanus (B; formula_0B; 4th century) and Codex Alexandrinus (A; formula_0A; 5th century).
Analysis.
A parallel pattern of sequence is observed in the final sections of 2 Kings between 2 Kings 11–20 and 2 Kings 21–25, as follows:
A. Athaliah, daughter of Ahab, kills royal seed ()
B. Joash reigns (2 Kings 11–12)
C. Quick sequence of kings of Israel and Judah (2 Kings 13–16)
D. Fall of Samaria (2 Kings 17)
E. Revival of Judah under Hezekiah (2 Kings 18–20)
A'. Manasseh, a king like Ahab, promotes idolatry and kills the innocence (2 Kings 21)
B'. Josiah reigns (2 Kings 22–23)
C'. Quick succession of kings of Judah (2 Kings 24)
D'. Fall of Jerusalem (2 Kings 25)
E'. Elevation of Jehoiachin (2 Kings 25:27–30)
Manasseh king of Judah (21:1–18).
The passage recording the reign of Manasseh consists of the 'introductory regnal form' (verses 1–3), the body/regnal account (verses 4–16; with major subunits in verses 4–5, 6–8, 9–15 and 16, each in the "waw"-consecutive narrative form) and the 'concluding regnal form' (verses 17–18). Manasseh's 55-year reign is the longest of all the kings of Judah, but in the Books of Kings he is considered the worst king of the southern kingdom. Manasseh behaved like Ahab, the king of Israel in Samaria:
Later, his grandson, king Josiah, must abolish all the deities reintroduced by Manasseh (cf. 2 Kings 23).
Manasseh was Assyria's vassal, that Assyrian sources mention as 'a bringer of tribute and as a military follower', without the slightest indication of resistance. This might be the reason for the length of his reign.
"Manasseh was twelve years old when he began to reign, and he reigned fifty-five years in Jerusalem. His mother's name was Hephzibah."
Verse 1.
Two seals appeared on the antiquities market in Jerusalem (first reported in 1963), both bearing the inscription, "Belonging to Manasseh, son of the king." As the term "son of the king" refers to royal princes, whether they eventually ascended the throne or not, the seal is considered to be Manasseh's during his co-regency with his father. It bears the same iconography of the Egyptian winged scarab as the seals attributed to King Hezekiah, recalling the alliance between Hezekiah and Egypt against the Assyrians (; ), and may symbolize 'a desire to permanently unite the northern and southern kingdoms together with God's divine blessing'.
Jar handles bearing a stamp with a winged-beetle and the phrase LMLK ("to the king"), along with the name of a city, have been unearthed throughout ancient Judah as well as in a large administrative complex discovered outside of the old city of Jerusalem and used to hold olive oil, food, wine, etc. – goods that were paid as taxes to the king, dated to the reigns of Hezekiah (cf. "Hezekiah's storehouses"; ) and Manasseh. These artifacts provide the evidence of 'a complex and highly organized tax system in Judah' from the time of Hezekiah extending into the time of Manasseh, among others to pay the tribute to the Assyrians.
Amon, king of Judah (21:19–26).
Amon, Manasseh's son and successor, is recorded to have 'walked in the way
which his father walked' (verse 21), but, unlike his father, he had a short period of reign. Then, 'the people of the land'—the same political group
who brought down the 'evil' queen Athaliah, enabling the 'good' king Joash to seize the throne (2 Kings 11:18, 20)—intervene to 'punish the king's murderers' and place Josiah, Amon's son, on the throne.
"Amon was twenty-two years old when he began to reign, and he reigned two years in Jerusalem. His mother's name was Meshullemeth the daughter of Haruz of Jotbah."
Archaeology.
Manasseh and the kingdom of Judah are only mentioned in the list of subservient kings/states in Assyrian inscriptions of Esarhaddon and Ashurbanipal. Manasseh is mentioned in the Esarhaddon Prism (dates to 673–672 BCE), discovered by archaeologist Reginald Campbell Thompson during the 1927–28 excavation season at the ancient Assyrian capital of Nineveh. The 493 lines of cuneiform inscribed on the sides of the prism describe the history of King Esarhaddon's reign and an account of the reconstruction of the Assyrian palace in Babylon, which reads "Together 22 kings of Hatti [this land includes Israel], the seashore and the islands. All these I sent out and made them transport under terrible difficulties"; one of these 22 kings was King Manasseh of Judah ("Menasii šar [âlu]Iaudi").
A record by Esarhaddon's son and successor, Ashurbanipal, mentions "Manasseh, king of Judah" who contributed to the invasion force against Egypt. This was recorded on the "Rassam cylinder" (or "Rassam Prism", now in the British Museum), named after Hormuzd Rassam, who discovered it in the North Palace of Nineveh in 1854. The ten-faced, cuneiform cylinder contains a record of Ashurbanipal's campaigns against Egypt and the Levant, that involved 22 kings "from the seashore, the islands and the mainland", who are called "servants who belong to me," clearly denoting them as Assyrian vassals. Manasseh was one of the kings who 'brought tribute to Ashurbanipal and kissed his feet'.
In rabbinic literature on "Isaiah" and Christian pseudepigrapha "Ascension of Isaiah", Manasseh is accused of executing the prophet Isaiah, who was identified as the maternal grandfather of Manasseh.
Manasseh is mentioned in chapter 21 of 1 Meqabyan, a book considered canonical in the Ethiopian Orthodox Tewahedo Church, where he is used as an example of ungodly king.
Notes.
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References.
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|
[
{
"math_id": 0,
"text": " \\mathfrak{G}"
}
] |
https://en.wikipedia.org/wiki?curid=64239807
|
64239808
|
2 Kings 22
|
2 Kings, chapter 22
2 Kings 22 is the twenty-second chapter of the second part of the Books of Kings in the Hebrew Bible or the Second Book of Kings in the Old Testament of the Christian Bible. The book is a compilation of various annals recording the acts of the kings of Israel and Judah by a Deuteronomic compiler in the seventh century BCE, with a supplement added in the sixth century BCE. This chapter records the events during the reign of Josiah, the king of Judah, especially the discovery of the Book of the Law (Torah) during the renovation of the Temple in Jerusalem.
Text.
This chapter was originally written in the Hebrew language. It is divided into 20 verses.
Textual witnesses.
Some early manuscripts containing the text of this chapter in Biblical Hebrew are of the Masoretic Text tradition, which includes the Aleppo Codex (10th century), Codex Leningradensis (1008), and the Codex Cairensis (11th century).
There is also a translation into Koine Greek known as the Septuagint, made in the last few centuries BCE. Extant ancient manuscripts of the Septuagint version include Codex Vaticanus (B; formula_0B; 4th century) and Codex Alexandrinus (A; formula_0A; 5th century).
Analysis.
A parallel pattern of sequence is observed in the final sections of 2 Kings between 2 Kings 11–20 and 2 Kings 21–25, as follows:
A. Athaliah, daughter of Ahab, kills royal seed ()
B. Joash reigns (2 Kings 11–12)
C. Quick sequence of kings of Israel and Judah (2 Kings 13–16)
D. Fall of Samaria (2 Kings 17)
E. Revival of Judah under Hezekiah (2 Kings 18–20)
A'. Manasseh, a king like Ahab, promotes idolatry and kills the innocence (2 Kings 21)
B'. Josiah reigns (2 Kings 22–23)
C'. Quick succession of kings of Judah (2 Kings 24)
D'. Fall of Jerusalem (2 Kings 25)
E'. Elevation of Jehoiachin (2 Kings 25:27–30)
2 Kings 22–23:30 mainly contains the story of Josiah's actions of his eighteenth year (22:3; 23:23) and the discovery of the book of the law (22:8-10; 23:24) as grouped based on five royal initiatives (using distinct verbs "send" and "command"):
Josiah king of Judah (22:1–7).
The account of Josiah ben Amon as the king of Judah is bracketed by the introductory regnal form in 2 Kings 22:1–2 and the concluding regnal form in 2 Kings 23:28–30, as the body in 2 Kings 22:3–23:27 highlights the religious reform and national restoration. The life of Josiah shows some similarities to the life of Joash, king of Judah, in that:
In 625 BCE Babylon achieved independence under Nabopolassar and in 612 BCE took the Assyrian capital Nineveh. This situation enables the kingdom of Judah, not under the threat of the Assyrians anymore, could make internal changes, including religious reforms.
records the instruction of Josiah, through the scribe Shaphan ben Azaliah ben Meshullam, to the high priest Hilkiah to lead the renovation of the Temple in Jerusalem.
" And it came to pass in the eighteenth year of king Josiah, that the king sent Shaphan the son of Azaliah, the son of Meshullam, the scribe, to the house of the LORD, saying,"
The Book of the Law was discovered (22:8–13).
Hilkiah reported to Shaphan about the discovery of a book of Torah in the temple during the renovations.(verse 8; cf. 2 Kings 12). Critical studies suggest that the discovered book was Deuteronomy or its core (Deuteronomy 6ab–28), which contains the speech made by Moses shortly before his death and might include some older materials as well. The closing admonitions (Deuteronomy 28), the strict demand for the exclusive worship of YHWH () and the cultic veneration of YHWH alone in the central holy site of Jerusalem () would impress Josiah, and rules such as the social laws of Deuteronomy (e.g. ) would become state law during his reign. Shaphan's report to King Josiah concerning the discovery of the Torah scroll and read the document (), causing Josiah's distress on hearing the words and his command to a delegation including Hilkiah the priest, Shaphan the scribe, and others to make an inquiry of YHWH to determine the significance of this discovery (), which led them to the home of the prophetess, Huldah, wife of Shallum ben Harhas, the keeper of garments.
"And Hilkiah the high priest said unto Shaphan the scribe, I have found the book of the law in the house of the Lord. And Hilkiah gave the book to Shaphan, and he read it."
"And the king commanded Hilkiah the priest, and Ahikam the son of Shaphan, and Achbor the son of Michaiah, and Shaphan the scribe, and Asahiah a servant of the king's, saying,"
Huldah's prophecy (22:14–20).
The prophetess Huldah pointed out the inevitability that the kingdom of Judah would suffer destruction because of the people's apostasy, although she showed supports for Josiah's reforms and indicated that Josiah's righteousness would earn him a peaceful death before the catastrophe struck.
""Surely, therefore, I will gather you to your fathers, and you shall be gathered to your grave in peace; and your eyes shall not see all the calamity which I will bring on this place." So they brought back word to the king."
Archaeology.
Two ostraca were found in 1997 ("Shlomo Moussaieff" #1 and #2) that seems to strengthen the evidence for a temple renovation during the reign of Josiah (see Bordreuil, Israel, and Pardee 1996 and 1998), but these artifacts did not come from regular excavations, so there is a suspicion of modern forgery. The first ostracon has a five-line inscription that records a royal contribution of three shekel of silver by a king ʾAshyahu to the temple of Yahweh to be made through a royal functionary named Zakaryahu, dated by palaeography to the time of Josiah. The name "Ashyahu" is determined as a short form of "Yo’shiyahu" ("Josiah"). The second ostracon contains a widow's plea about an inheritance which mentions Josiah's name and a short quote from .
Notes.
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References.
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|
[
{
"math_id": 0,
"text": " \\mathfrak{G}"
}
] |
https://en.wikipedia.org/wiki?curid=64239808
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64244772
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Stable principal bundle
|
In mathematics, and especially differential geometry and algebraic geometry, a stable principal bundle is a generalisation of the notion of a stable vector bundle to the setting of principal bundles. The concept of stability for principal bundles was introduced by Annamalai Ramanathan for the purpose of defining the moduli space of G-principal bundles over a Riemann surface, a generalisation of earlier work by David Mumford and others on the moduli spaces of vector bundles.
Many statements about the stability of vector bundles can be translated into the language of stable principal bundles. For example, the analogue of the Kobayashi–Hitchin correspondence for principal bundles, that a holomorphic principal bundle over a compact Kähler manifold admits a Hermite–Einstein connection if and only if it is polystable, was shown to be true in the case of projective manifolds by Subramanian and Ramanathan, and for arbitrary compact Kähler manifolds by Anchouche and Biswas.
Definition.
The essential definition of stability for principal bundles was made by Ramanathan, but applies only to the case of Riemann surfaces. In this section we state the definition as appearing in the work of Anchouche and Biswas which is valid over any Kähler manifold, and indeed makes sense more generally for algebraic varieties. This reduces to Ramanathan's definition in the case the manifold is a Riemann surface.
Let formula_0 be a connected reductive algebraic group over the complex numbers formula_1. Let formula_2 be a compact Kähler manifold of complex dimension formula_3. Suppose formula_4 is a holomorphic principal formula_0-bundle over formula_5. Holomorphic here means that the transition functions for formula_6 vary holomorphically, which makes sense as the structure group is a complex Lie group. The principal bundle formula_6 is called stable (resp. semi-stable) if for every reduction of structure group formula_7 for formula_8 a maximal parabolic subgroup where formula_9 is some open subset with the codimension formula_10, we have
formula_11
Here formula_12 is the relative tangent bundle of the fibre bundle formula_13 otherwise known as the vertical bundle of formula_14. Recall that the degree of a vector bundle (or coherent sheaf) formula_15 is defined to be
formula_16
where formula_17 is the first Chern class of formula_18. In the above setting the degree is computed for a bundle defined over formula_19 inside formula_5, but since the codimension of the complement of formula_19 is bigger than two, the value of the integral will agree with that over all of formula_5.
Notice that in the case where formula_20, that is where formula_5 is a Riemann surface, by assumption on the codimension of formula_19 we must have that formula_21, so it is enough to consider reductions of structure group over the entirety of formula_5, formula_22.
Relation to stability of vector bundles.
Given a principal formula_0-bundle for a complex Lie group formula_0 there are several natural vector bundles one may associate to it.
Firstly if formula_23, the general linear group, then the standard representation of formula_24 on formula_25 allows one to construct the associated bundle formula_26. This is a holomorphic vector bundle over formula_5, and the above definition of stability of the principal bundle is equivalent to slope stability of formula_27. The essential point is that a maximal parabolic subgroup formula_28 corresponds to a choice of flag formula_29, where formula_30 is invariant under the subgroup formula_31. Since the structure group of formula_6 has been reduced to formula_31, and formula_31 preserves the vector subspace formula_32, one may take the associated bundle formula_33, which is a sub-bundle of formula_27 over the subset formula_9 on which the reduction of structure group is defined, and therefore a subsheaf of formula_27 over all of formula_5. It can then be computed that
formula_34
where formula_35 denotes the slope of the vector bundles.
When the structure group is not formula_36 there is still a natural associated vector bundle to formula_6, the adjoint bundle formula_37, with fibre given by the Lie algebra formula_38 of formula_0. The principal bundle formula_6 is semistable if and only if the adjoint bundle formula_37 is slope semistable, and furthermore if formula_6 is stable, then formula_37 is slope polystable. Again the key point here is that for a parabolic subgroup formula_8, one obtains a parabolic subalgebra formula_39 and can take the associated subbundle. In this case more care must be taken because the adjoint representation of formula_0 on formula_38 is not always faithful or irreducible, the latter condition hinting at why stability of the principal bundle only leads to "polystability" of the adjoint bundle (because a representation that splits as a direct sum would lead to the associated bundle splitting as a direct sum).
Generalisations.
Just as one can generalise a vector bundle to the notion of a Higgs bundle, it is possible to formulate a definition of a principal formula_0-Higgs bundle. The above definition of stability for principal bundles generalises to these objects by requiring the reductions of structure group are compatible with the Higgs field of the principal Higgs bundle. It was shown by Anchouche and Biswas that the analogue of the nonabelian Hodge correspondence for Higgs vector bundles is true for principal formula_0-Higgs bundles in the case where the base manifold formula_2 is a complex projective variety.
References.
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[
{
"math_id": 0,
"text": "G"
},
{
"math_id": 1,
"text": "\\mathbb{C}"
},
{
"math_id": 2,
"text": "(X,\\omega)"
},
{
"math_id": 3,
"text": "n"
},
{
"math_id": 4,
"text": "P\\to X"
},
{
"math_id": 5,
"text": "X"
},
{
"math_id": 6,
"text": "P"
},
{
"math_id": 7,
"text": "\\sigma: U \\to P/Q"
},
{
"math_id": 8,
"text": "Q\\subset G"
},
{
"math_id": 9,
"text": "U\\subset X"
},
{
"math_id": 10,
"text": "\\operatorname{codim}(X\\backslash U) \\ge 2"
},
{
"math_id": 11,
"text": " \\deg \\sigma^* T_{\\operatorname{rel}} P/Q > 0 \\quad (\\text{resp. }\\ge 0)."
},
{
"math_id": 12,
"text": "T_{\\operatorname{rel}} P/Q"
},
{
"math_id": 13,
"text": "\\left.P/Q\\right|_U \\to U"
},
{
"math_id": 14,
"text": "T (\\left.P/Q\\right|_U)"
},
{
"math_id": 15,
"text": "F\\to X"
},
{
"math_id": 16,
"text": " \\operatorname{deg}(F) := \\int_X c_1(F) \\wedge \\omega^{n-1},"
},
{
"math_id": 17,
"text": "c_1(F)"
},
{
"math_id": 18,
"text": "F"
},
{
"math_id": 19,
"text": "U"
},
{
"math_id": 20,
"text": "\\dim X = 1"
},
{
"math_id": 21,
"text": "U=X"
},
{
"math_id": 22,
"text": "\\sigma: X \\to P/Q"
},
{
"math_id": 23,
"text": "G=\\operatorname{GL}(n,\\mathbb{C})"
},
{
"math_id": 24,
"text": "\\operatorname{GL}(n,\\mathbb{C})"
},
{
"math_id": 25,
"text": "\\mathbb{C}^n"
},
{
"math_id": 26,
"text": "E = P \\times_{\\operatorname{GL}(n,\\mathbb{C})} \\mathbb{C}^n"
},
{
"math_id": 27,
"text": "E"
},
{
"math_id": 28,
"text": "Q\\subset \\operatorname{GL}(n,\\mathbb{C})"
},
{
"math_id": 29,
"text": "0 \\subset W \\subset \\mathbb{C}^n"
},
{
"math_id": 30,
"text": "W"
},
{
"math_id": 31,
"text": "Q"
},
{
"math_id": 32,
"text": "W\\subset \\mathbb{C}^n"
},
{
"math_id": 33,
"text": "F = P\\times_Q W"
},
{
"math_id": 34,
"text": " \\deg \\sigma^* T_{\\operatorname{rel}} P/Q = \\mu (E) - \\mu(F)"
},
{
"math_id": 35,
"text": "\\mu"
},
{
"math_id": 36,
"text": "G=\\operatorname{GL}(n, \\mathbb{C})"
},
{
"math_id": 37,
"text": "\\operatorname{ad} P"
},
{
"math_id": 38,
"text": "\\mathfrak{g}"
},
{
"math_id": 39,
"text": "\\mathfrak{q} \\subset \\mathfrak{g}"
}
] |
https://en.wikipedia.org/wiki?curid=64244772
|
64247707
|
Krein–Smulian theorem
|
In mathematics, particularly in functional analysis, the Krein-Smulian theorem can refer to two theorems relating the closed convex hull and compactness in the weak topology. They are named after Mark Krein and Vitold Shmulyan, who published them in 1940.
Statement.
Both of the following theorems are referred to as the Krein-Smulian Theorem.
<templatestyles src="Math_theorem/styles.css" />
Krein-Smulian Theorem: —
Let formula_0 be a Banach space and formula_1 a weakly compact subset of formula_0 (that is, formula_1 is compact when formula_0 is endowed with the weak topology). Then the closed convex hull of formula_1 in formula_0 is weakly compact.
<templatestyles src="Math_theorem/styles.css" />
Krein-Smulian Theorem —
Let formula_0 be a Banach space and formula_2 a convex subset of the continuous dual space formula_3 of formula_0. If for all formula_4 formula_5 is weak-* closed in formula_3 then formula_2 is weak-* closed.
References.
<templatestyles src="Reflist/styles.css" />
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "X"
},
{
"math_id": 1,
"text": "K"
},
{
"math_id": 2,
"text": "A"
},
{
"math_id": 3,
"text": "X^{\\prime}"
},
{
"math_id": 4,
"text": "r > 0,"
},
{
"math_id": 5,
"text": "A \\cap \\left\\{x^{\\prime} \\in X^{\\prime} : \\left\\| x^{\\prime} \\right\\| \\leq r\\right\\}"
}
] |
https://en.wikipedia.org/wiki?curid=64247707
|
64250099
|
Continuous Bernoulli distribution
|
Probability distribution
In probability theory, statistics, and machine learning, the continuous Bernoulli distribution is a family of continuous probability distributions parameterized by a single shape parameter formula_1, defined on the unit interval formula_0, by:
formula_2
The continuous Bernoulli distribution arises in deep learning and computer vision, specifically in the context of variational autoencoders, for modeling the pixel intensities of natural images. As such, it defines a proper probabilistic counterpart for the commonly used binary cross entropy loss, which is often applied to continuous, formula_3-valued data. This practice amounts to ignoring the normalizing constant of the continuous Bernoulli distribution, since the binary cross entropy loss only defines a true log-likelihood for discrete, formula_4-valued data.
The continuous Bernoulli also defines an exponential family of distributions. Writing formula_5 for the natural parameter, the density can be rewritten in canonical form:
formula_6.
Related distributions.
Bernoulli distribution.
The continuous Bernoulli can be thought of as a continuous relaxation of the Bernoulli distribution, which is defined on the discrete set formula_7 by the probability mass function:
formula_8
where formula_9 is a scalar parameter between 0 and 1. Applying this same functional form on the continuous interval formula_10 results in the continuous Bernoulli probability density function, up to a normalizing constant.
Beta distribution.
The Beta distribution has the density function:
formula_11
which can be re-written as:
formula_12
where formula_13 are positive scalar parameters, and formula_14 represents an arbitrary point inside the 1-simplex, formula_15. Switching the role of the parameter and the argument in this density function, we obtain:
formula_16
This family is only identifiable up to the linear constraint formula_17, whence we obtain:
formula_18
corresponding exactly to the continuous Bernoulli density.
Exponential distribution.
An exponential distribution restricted to the unit interval is equivalent to a continuous Bernoulli distribution with appropriate parameter.
Continuous categorical distribution.
The multivariate generalization of the continuous Bernoulli is called the continuous-categorical.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "x \\in [0, 1]"
},
{
"math_id": 1,
"text": "\\lambda \\in (0, 1)"
},
{
"math_id": 2,
"text": " p(x | \\lambda) \\propto \\lambda^x (1-\\lambda)^{1-x}. "
},
{
"math_id": 3,
"text": "[0,1]"
},
{
"math_id": 4,
"text": "\\{0,1\\}"
},
{
"math_id": 5,
"text": "\\eta = \\log\\left(\\lambda/(1-\\lambda)\\right)"
},
{
"math_id": 6,
"text": " p(x | \\eta) \\propto \\exp (\\eta x) "
},
{
"math_id": 7,
"text": " \\{0,1\\} "
},
{
"math_id": 8,
"text": " p(x) = p^x (1-p)^{1-x}, "
},
{
"math_id": 9,
"text": " p "
},
{
"math_id": 10,
"text": " [0,1] "
},
{
"math_id": 11,
"text": " p(x) \\propto x^{\\alpha - 1} (1-x)^{\\beta - 1}, "
},
{
"math_id": 12,
"text": " p(x) \\propto x_1^{\\alpha_1 - 1} x_2^{\\alpha_2 - 1}, "
},
{
"math_id": 13,
"text": " \\alpha_1, \\alpha_2 "
},
{
"math_id": 14,
"text": "(x_1, x_2)"
},
{
"math_id": 15,
"text": " \\Delta^{1} = \\{ (x_1, x_2): x_1 > 0, x_2 > 0, x_1 + x_2 = 1 \\} "
},
{
"math_id": 16,
"text": " p(x) \\propto \\alpha_1^{x_1} \\alpha_2^{x_2}. "
},
{
"math_id": 17,
"text": " \\alpha_1 + \\alpha_2 = 1 "
},
{
"math_id": 18,
"text": " p(x) \\propto \\lambda^{x_1} (1-\\lambda)^{x_2}, "
}
] |
https://en.wikipedia.org/wiki?curid=64250099
|
64254647
|
Tetrastix
|
In geometry, it is possible to fill 3/4 of the volume of three-dimensional Euclidean space by three sets of infinitely-long square prisms aligned with the three coordinate axes, leaving cubical voids; John Horton Conway, Heidi Burgiel and Chaim Goodman-Strauss have named this structure tetrastix.
Applications.
The motivation for some of the early studies of this structure was for its applications in the crystallography of crystal structures formed by rod-shaped molecules.
Shrinking the square cross-sections of the prisms slightly causes the remaining space, consisting of the cubical voids, to become linked up into a single polyhedral set, bounded by axis-parallel faces. Polyhedra constructed in this way from finitely many prisms provide examples of axis-parallel polyhedra with formula_0 vertices and faces that require formula_1 pieces when subdivided into convex pieces; they have been called Thurston polyhedra, after William Thurston, who suggested using these shapes for this lower bound application. Like the Schönhardt polyhedron, these polyhedra have no triangulation into tetrahedra unless additional vertices are introduced.
Anduriel Widmark has used the tetrastix and hexastix structures as the basis for artworks made from glass rods, fused to form tangled knots.
Related structures.
The space occupied by the union of the prisms can be divided into the prisms of the tetrastix structure in two different ways. If the prisms are divided into unit cubes, offset by half a unit from the integer grid aligned with the prism sides, then these cubes together with the unit cube voids of the tetrastix structure form a tiling of space by cubes, combinatorially equivalent to the Weaire–Phelan structure for tiling space with unit volumes of low surface area. The tetrastix and Weaire–Phelan structures have the same group of symmetries. Although this cube tiling includes some cubes (the ones filling the voids of the tetrastix) that do not meet face-to-face with any other cube, results of Oskar Perron on Keller's conjecture prove that (like the cubes within each prism of the tetrastix) every tiling of three-dimensional space by unit cubes must include an infinite column of cubes that all meet face-to-face.
Similar constructions to the tetrastix are possible with triangular and hexagonal prisms, in four directions, called by Conway et al. "tristix" and hexastix.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "n"
},
{
"math_id": 1,
"text": "\\Omega(n^{3/2})"
}
] |
https://en.wikipedia.org/wiki?curid=64254647
|
64256623
|
BitFunnel
|
Search engine indexing algorithm for Bing
BitFunnel is the search engine indexing algorithm and a set of components used in the Bing search engine, which were made open source in 2016. BitFunnel uses bit-sliced signatures instead of an inverted index in an attempt to reduce operations cost.
History.
Progress on the implementation of BitFunnel was made public in early 2016, with the expectation that there would be a usable implementation later that year. In September 2016, the source code was made available via GitHub. A paper discussing the BitFunnel algorithm and implementation was released as through the Special Interest Group on Information Retrieval of the Association for Computing Machinery in 2017 and won the Best Paper Award.
Components.
BitFunnel consists of three major components:
Algorithm.
Initial problem and solution overview.
The BitFunnel paper describes the "matching problem", which occurs when an algorithm must identify documents through the usage of keywords. The goal of the problem is to identify a set of matches given a corpus to search and a query of keyword terms to match against. This problem is commonly solved through inverted indexes, where each searchable item is maintained with a map of keywords.
In contrast, BitFunnel represents each searchable item through a signature. A signature is a sequence of bits which describe a Bloom filter of the searchable terms in a given searchable item. The bloom filter is constructed through hashing through several bit positions.
Theoretical implementation of bit-string signatures.
The signature of a document (D) can be described as the logical-or of its term signatures:
formula_0
Similarly, a query for a document (Q) can be defined as a union:
formula_1
Additionally, a document D is a member of the set "M"' when the following condition is satisfied:
formula_2
This knowledge is then combined to produce a formula where "M"' is identified by documents which match the query signature:
formula_3
These steps and their proofs are discussed in the 2017 paper.
Pseudocode for bit-string signatures.
This algorithm is described in the 2017 paper.
formula_4
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\overrightarrow{S_D}=\\bigcup_{t \\in D} \\overrightarrow{S_t}"
},
{
"math_id": 1,
"text": "\\overrightarrow{S_Q}=\\bigcup_{t \\in Q} \\overrightarrow{S_t}"
},
{
"math_id": 2,
"text": "\\overrightarrow{S_Q} \\cap \\overrightarrow{S_D} = \\overrightarrow{S_Q}"
},
{
"math_id": 3,
"text": "M'= \\left\\{ D \\in C \\mid \\overrightarrow{S_Q} \\cap \\overrightarrow{S_D} = \\overrightarrow{S_Q}\\right\\}"
},
{
"math_id": 4,
"text": "\\begin{array}{l}\nM'=\\emptyset \\\\\n\\texttt{for each}\\ D \\in C\\ \\texttt{do} \\\\\n\\qquad \\texttt{if}\\ \\overrightarrow{S_D} \\cap \\overrightarrow{S_Q} = \\overrightarrow{S_Q}\\ \\texttt{then} \\\\\n\\qquad \\qquad M' = M' \\cup \\{ D \\} \\\\\n\\qquad \\texttt{end if} \\\\\n\\texttt{end for}\n\\end{array}"
}
] |
https://en.wikipedia.org/wiki?curid=64256623
|
64257772
|
Laguerre transformations
|
The Laguerre transformations or axial homographies are an analogue of Möbius transformations over the dual numbers. When studying these transformations, the dual numbers are often interpreted as representing oriented lines on the plane. The Laguerre transformations map lines to lines, and include in particular all isometries of the plane.
Strictly speaking, these transformations act on the dual number projective line, which adjoins to the dual numbers a set of points at infinity. Topologically, this projective line is equivalent to a cylinder. Points on this cylinder are in a natural one-to-one correspondence with oriented lines on the plane.
Definition.
A Laguerre transformation is a linear fractional transformation formula_0 where formula_1 are all dual numbers, formula_2 lies on the dual number projective line, and formula_3 is not a zero divisor.
A dual number is a hypercomplex number of the form formula_4 where formula_5 but formula_6. This can be compared to the complex numbers which are of the form formula_7 where formula_8.
The points of the dual number projective line can be defined equivalently in two ways:
Line coordinates.
A line which makes an angle formula_14 with the x-axis, and whose x-intercept is denoted formula_15, is represented by the dual number
formula_16
The above doesn't make sense when the line is parallel to the x-axis. In that case, if formula_17 then set formula_18 where formula_19 is the y-intercept of the line. This may not appear to be valid, as one is dividing by a zero divisor, but this is a valid point on the projective dual line. If formula_20 then set formula_21.
Finally, observe that these coordinates represent "oriented" lines. An oriented line is an ordinary line with one of two possible orientations attached to it. This can be seen from the fact that if formula_14 is increased by formula_22 then the resulting dual number representative is not the same.
Matrix representations.
It's possible to express the above line coordinates as homogeneous coordinates formula_23 where formula_19 is the perpendicular distance of the line from the origin. This representation has numerous advantages: One advantage is that there is no need to break into different cases, such as parallel to the formula_11-axis and non-parallel. The other advantage is that these homogeneous coordinates can be interpreted as vectors, allowing us to multiply them by matrices.
Every Laguerre transformation can be represented as a 2×2 matrix whose entries are dual numbers. The matrix representation of formula_24 is formula_25 (but notice that any non-nilpotent scalar multiple of this matrix represents the same Laguerre transformation). Additionally, as long as the determinant of a 2×2 matrix with dual-number entries is not nilpotent, then it represents a Laguerre transformation.
"(Note that in the above, we represent the homogeneous vector formula_26 as a column vector in the obvious way, instead of as a row vector.)"
Points, oriented lines and oriented circles.
Laguerre transformations do not act on points. This is because if three oriented lines pass through the same point, their images under a Laguerre transformation do not have to meet at one point.
Laguerre transformations can be seen as acting on oriented circles as well as oriented lines. An oriented circle is an ordinary circle with a binary value attached to it, which is either formula_27 or formula_28. The only exception is a circle of radius zero, which has orientation equal to formula_29. A point is defined to be an oriented circle of radius zero. If an oriented circle has orientation equal to formula_27, then the circle is said to be "anti-clockwise" oriented; if it has orientation equal to formula_28 then it is "clockwise" oriented. The radius of an oriented circle is defined to be the radius formula_30 of the underlying unoriented circle multiplied by the orientation.
The image of an oriented circle under a Laguerre transformation is another oriented circle. If two oriented figures – either circles or lines – are tangent to each other then their images under a Laguerre transformation are also tangent. Two oriented circles are defined to be tangent if their underlying circles are tangent and their orientations are equal at the point of contact. Tangency between lines and circles is defined similarly. A Laguerre transformation might map a point to an oriented circle which is no longer a point.
An oriented circle can never be mapped to an oriented line. Likewise, an oriented line can never be mapped to an oriented circle. This is opposite to Möbius geometry, where lines and circles can be mapped to each other, but neither can be mapped to points. Both Möbius geometry and Laguerre geometry are subgeometries of Lie sphere geometry, where points and oriented lines can be mapped to each other, but tangency remains preserved.
The matrix representations of oriented circles (which include points but not lines) are precisely the invertible formula_31 skew-Hermitian dual number matrices. These are all of the form formula_32 (where all the variables are real, and formula_33). The set of oriented lines tangent to an oriented circle is given by formula_34 where formula_35 denotes the projective line over the dual numbers formula_36. Applying a Laguerre transformation represented by formula_37 to the oriented circle represented by formula_38 gives the oriented circle represented by formula_39. The radius of an oriented circle is equal to the half the trace. The orientation is then the sign of the trace.
Profile.
"Note that the animated figures below show some oriented lines, but without any visual indication of a line's orientation (so two lines that differ only in orientation are displayed in the same way); oriented circles are shown as a set of oriented tangent lines, which results in a certain visual effect."
The following can be found in Isaak Yaglom's "Complex numbers in geometry" and a paper by Gutin entitled "Generalizations of singular value decomposition to dual-numbered matrices".
Unitary matrices.
Mappings of the form formula_40 express rigid body motions (sometimes called "direct Euclidean isometries"). The matrix representations of these transformations span a subalgebra isomorphic to the planar quaternions.
The mapping formula_41 represents a reflection about the x-axis.
The transformation formula_42 expresses a reflection about the y-axis.
Observe that if formula_43 is the matrix representation of any combination of the above three transformations, but normalised so as to have determinant formula_27, then formula_43 satisfies formula_44 where formula_45 means formula_46. We will call these "unitary" matrices. Notice though that these are unitary in the sense of the dual numbers and not the complex numbers. The unitary matrices express precisely the Euclidean isometries.
Axial dilation matrices.
An "axial dilation" by formula_47 units is a transformation of the form formula_48. An axial dilation by formula_47 units increases the radius of all oriented circles by formula_47 units while preserving their centres. If a circle has negative orientation, then its radius is considered negative, and therefore for some positive values of formula_47 the circle actually shrinks. An axial dilation is depicted in Figure 1, in which two circles of opposite orientations undergo the same axial dilation.
On lines, an axial dilation by formula_47 units maps any line formula_2 to a line formula_49 such that formula_2 and formula_49 are parallel, and the perpendicular distance between formula_2 and formula_49 is formula_47. Lines that are parallel but have opposite orientations move in opposite directions.
Real diagonal matrices.
The transformation formula_51 for a value of formula_50 that's real preserves the x-intercept of a line, while changing its angle to the x-axis. See Figure 2 to observe the effect on a grid of lines (including the x axis in the middle) and Figure 3 to observe the effect on two circles that differ initially only in orientation (to see that the outcome is sensitive to orientation).
A general decomposition.
Putting it all together, a general Laguerre transformation in matrix form can be expressed as formula_52 where formula_43 and formula_53 are unitary, and formula_54 is a matrix either of the form formula_55 or formula_56 where formula_57 and formula_58 are real numbers. The matrices formula_43 and formula_53 express Euclidean isometries. The matrix formula_54 either represents a transformation of the form formula_51 or an axial dilation. The resemblance to Singular Value Decomposition should be clear.
"Note: In the event that formula_54 is an axial dilation, the factor formula_53 can be set to the identity matrix. This follows from the fact that if formula_53 is unitary and formula_54 is an axial dilation, then it can be seen that formula_59, where formula_60 denotes the transpose of formula_54. So formula_61."
Other number systems and the parallel postulate.
Complex numbers and elliptic geometry.
A question arises: What happens if the role of the dual numbers above is changed to the complex numbers? In that case, the complex numbers represent oriented lines in the elliptic plane (the plane which elliptic geometry takes places over). This is in contrast to the dual numbers, which represent oriented lines in the Euclidean plane. The elliptic plane is essentially a sphere (but where antipodal points are identified), and the lines are thus great circles. We can choose an arbitrary great circle to be the equator. The oriented great circle which intersects the equator at longitude formula_15, and makes an angle formula_14 with the equator at the point of intersection, can be represented by the complex number formula_62. In the case where formula_17 (where the line is literally the same as the equator, but oriented in the opposite direction as when formula_63) the oriented line is represented as formula_64. Similar to the case of the dual numbers, the unitary matrices act as isometries of the elliptic plane. The set of "elliptic Laguerre transformations" (which are the analogues of the Laguerre transformations in this setting) can be decomposed using Singular Value Decomposition of complex matrices, in a similar way to how we decomposed Euclidean Laguerre transformations using an "analogue of Singular Value Decomposition for dual-number matrices".
Split-complex numbers and hyperbolic geometry.
If the role of the dual numbers or complex numbers is changed to the split-complex numbers, then a similar formalism can be developed for representing oriented lines on the hyperbolic plane instead of the Euclidean or elliptic planes: A split-complex number can be written in the form formula_65 because the algebra in question is isomorphic to formula_66. (Notice though that as a *-algebra, as opposed to a mere algebra, the split-complex numbers are not decomposable in this way). The terms formula_57 and formula_58 in formula_65 represent points on the boundary of the hyperbolic plane; they are respectively the starting and ending points of an oriented line. Since the boundary of the hyperbolic plane is homeomorphic to the projective line formula_67, we need formula_57 and formula_58 to belong to the projective line formula_67 instead of the affine line formula_68. Indeed, this hints that formula_69.
The analogue of unitary matrices over the split-complex numbers are the isometries of the hyperbolic plane. This is shown by Yaglom. Furthermore, the set of linear fractional transformations can be decomposed in a way that resembles Singular Value Decomposition, but which also unifies it with the Jordan decomposition.
Summary.
We therefore have a correspondence between the three planar number systems (complex, dual and split-complex numbers) and the three non-Euclidean geometries. The number system that corresponds to Euclidean geometry is the dual numbers.
In higher dimensions.
Euclidean.
n-dimensional Laguerre space is isomorphic to "n" + 1 Minkowski space. To associate a point formula_70 in Minkowski space to an oriented hypersphere, intersect the light cone centred at formula_71 with the formula_72 hyperplane. The group of Laguerre transformations is isomorphic then to the Poincaré group formula_73. These transformations are exactly those which preserve a kind of squared distance between oriented circles called their Darboux product. The "direct Laguerre transformations" are defined as the subgroup formula_74. In 2 dimensions, the direct Laguerre transformations can be represented by 2×2 dual number matrices. If the 2×2 dual number matrices are understood as constituting the Clifford algebra formula_75, then analogous Clifford algebraic representations are possible in higher dimensions.
If we embed Minkowski space formula_76 in the projective space formula_77 while keeping the transformation group the same, then the points at infinity are oriented flats. We call them "flats" because their shape is flat. In 2 dimensions, these are the oriented lines.
As an aside, there are two non-equivalent definitions of a Laguerre transformation: Either as a Lie sphere transformation that preserves oriented flats, or as a Lie sphere transformation that preserves the Darboux product. We use the latter convention in this article. Note that even in 2 dimensions, the former transformation group is more general than the latter: A homothety for example maps oriented lines to oriented lines, but does not in general preserve the Darboux product. This can be demonstrated using the homothety centred at formula_78 by formula_47 units. Now consider the action of this transformation on two circles: One simply being the point formula_78, and the other being a circle of raidus formula_27 centred at formula_78. These two circles have a Darboux product equal to formula_28. Their images under the homothety have a Darboux product equal to formula_79. This therefore only gives a Laguerre transformation when formula_80.
Conformal interpretation.
"In this section, we interpret Laguerre transformations differently from in the rest of the article. When acting on line coordinates, Laguerre transformations are "not" understood to be conformal in the sense described here. This is clearly demonstrated in Figure 2."
The Laguerre transformations preserve angles when the proper angle for the dual number plane is identified. When a ray "y" = "mx", "x" ≥ 0, and the positive x-axis are taken for sides of an angle, the slope "m" is the magnitude of this angle.
This number "m" corresponds to the signed area of the right triangle with base on the interval [(√2,0), (√2, "m" √2)]. The line {1 + "aε": "a" ∈ ℝ}, with the dual number multiplication, forms a subgroup of the unit dual numbers, each element being a shear mapping when acting on the dual number plane. Other angles in the plane are generated by such action, and since shear mapping preserves area, the size of these angles is the same as the original.
Note that the inversion "z" to 1/"z" leaves angle size invariant. As the general Laguerre transformation is generated by translations, dilations, shears, and inversions, and all of these leave angle invariant, the general Laguerre transformation is conformal in the sense of these angles.
|
[
{
"math_id": 0,
"text": "z\\mapsto\\frac{az + b}{cz + d}"
},
{
"math_id": 1,
"text": "a,b,c,d"
},
{
"math_id": 2,
"text": "z"
},
{
"math_id": 3,
"text": "ad-bc"
},
{
"math_id": 4,
"text": "x+y\\varepsilon"
},
{
"math_id": 5,
"text": "\\varepsilon^2=0"
},
{
"math_id": 6,
"text": "\\varepsilon\\neq0"
},
{
"math_id": 7,
"text": "x+yi"
},
{
"math_id": 8,
"text": "i^2=-1"
},
{
"math_id": 9,
"text": "\\{x + y \\varepsilon \\mid x \\in \\mathbb R, y \\in \\mathbb R\\} \\cup \\left\\{\\frac{1}{x \\varepsilon} \\mid x \\in \\mathbb R\\right\\}"
},
{
"math_id": 10,
"text": "\\frac{1}{x \\varepsilon}"
},
{
"math_id": 11,
"text": "x"
},
{
"math_id": 12,
"text": "\\varepsilon"
},
{
"math_id": 13,
"text": "1/\\varepsilon"
},
{
"math_id": 14,
"text": "\\theta"
},
{
"math_id": 15,
"text": "s"
},
{
"math_id": 16,
"text": " z = \\tan(\\theta/2)(1 + \\varepsilon s)."
},
{
"math_id": 17,
"text": "\\theta = \\pi"
},
{
"math_id": 18,
"text": "z = \\frac{-2}{\\varepsilon R}"
},
{
"math_id": 19,
"text": "R"
},
{
"math_id": 20,
"text": "\\theta = 2\\pi"
},
{
"math_id": 21,
"text": "z = \\frac 1 2 \\varepsilon R"
},
{
"math_id": 22,
"text": "\\pi"
},
{
"math_id": 23,
"text": "z = \\left[\\sin\\left(\\frac{\\theta + \\varepsilon R} 2\\right):\\cos\\left(\\frac{\\theta + \\varepsilon R} 2\\right)\\right]"
},
{
"math_id": 24,
"text": "z \\mapsto \\frac{pz + q}{rz + s}"
},
{
"math_id": 25,
"text": "\\begin{pmatrix} p & q \\\\ r & s\\end{pmatrix}"
},
{
"math_id": 26,
"text": "[z:w]"
},
{
"math_id": 27,
"text": "1"
},
{
"math_id": 28,
"text": "-1"
},
{
"math_id": 29,
"text": "0"
},
{
"math_id": 30,
"text": "r"
},
{
"math_id": 31,
"text": "2 \\times 2"
},
{
"math_id": 32,
"text": "H = \\begin{pmatrix} \\varepsilon a & b + c\\varepsilon \\\\ -b + c\\varepsilon & \\varepsilon d \\end{pmatrix}"
},
{
"math_id": 33,
"text": "b \\neq 0"
},
{
"math_id": 34,
"text": "\\{v \\in \\mathbb{DP}^1 \\mid v^* H v = 0\\}"
},
{
"math_id": 35,
"text": "\\mathbb{DP}^1"
},
{
"math_id": 36,
"text": "\\mathbb D"
},
{
"math_id": 37,
"text": "M"
},
{
"math_id": 38,
"text": "H"
},
{
"math_id": 39,
"text": "(M^{-1})^* H M^{-1}"
},
{
"math_id": 40,
"text": "z \\mapsto \\frac{pz - q}{\\bar q z + \\bar p}"
},
{
"math_id": 41,
"text": "z \\mapsto -z"
},
{
"math_id": 42,
"text": "z \\mapsto 1/z"
},
{
"math_id": 43,
"text": "U"
},
{
"math_id": 44,
"text": "UU^* = U^* U = I"
},
{
"math_id": 45,
"text": "U^*"
},
{
"math_id": 46,
"text": "\\overline{U}^\\mathrm{T}"
},
{
"math_id": 47,
"text": "t"
},
{
"math_id": 48,
"text": "\\frac{z + (\\varepsilon t/2)}{(-\\varepsilon t/2) z + 1}"
},
{
"math_id": 49,
"text": "z'"
},
{
"math_id": 50,
"text": "k"
},
{
"math_id": 51,
"text": "z \\mapsto k z"
},
{
"math_id": 52,
"text": "U S V^*"
},
{
"math_id": 53,
"text": "V"
},
{
"math_id": 54,
"text": "S"
},
{
"math_id": 55,
"text": "\\begin{pmatrix} a & 0 \\\\ 0 & b\\end{pmatrix}"
},
{
"math_id": 56,
"text": "\\begin{pmatrix} a & -b\\varepsilon \\\\ b\\varepsilon & a\\end{pmatrix}"
},
{
"math_id": 57,
"text": "a"
},
{
"math_id": 58,
"text": "b"
},
{
"math_id": 59,
"text": "SV = \\begin{cases} V S,& \\det(V) = +1 \\\\ V S^\\mathrm{T},& \\det(V) = -1\\end{cases}"
},
{
"math_id": 60,
"text": "S^\\mathrm{T}"
},
{
"math_id": 61,
"text": "USV^* = \\begin{cases} (U V^*) S,& \\det(V) = +1 \\\\ (U V^*) S^\\mathrm{T},& \\det(V) = -1\\end{cases}"
},
{
"math_id": 62,
"text": "\\tan(\\theta/2)(\\cos(s) + i \\sin(s))"
},
{
"math_id": 63,
"text": "\\theta = 0"
},
{
"math_id": 64,
"text": "\\infty"
},
{
"math_id": 65,
"text": "(a,-b^{-1})"
},
{
"math_id": 66,
"text": "\\mathbb R \\oplus \\mathbb R"
},
{
"math_id": 67,
"text": "\\mathbb{RP}^1"
},
{
"math_id": 68,
"text": "\\mathbb R^1"
},
{
"math_id": 69,
"text": "(\\mathbb R \\oplus \\mathbb R)\\mathbb P^1 \\cong \\mathbb R \\mathbb P^1\\oplus \\mathbb R\\mathbb P^1"
},
{
"math_id": 70,
"text": "P=(x_1,x_2,\\dotsc,x_n,r)"
},
{
"math_id": 71,
"text": "P"
},
{
"math_id": 72,
"text": "t=0"
},
{
"math_id": 73,
"text": "\\mathbb{R}^{n,1} \\rtimes \\operatorname{O}(n, 1)"
},
{
"math_id": 74,
"text": "\\mathbb{R}^{n,1} \\rtimes \\operatorname{O}^+(n, 1)"
},
{
"math_id": 75,
"text": "\\operatorname{Cl}_{2,0,1}(\\mathbb R)"
},
{
"math_id": 76,
"text": "\\mathbb R^{n,1}"
},
{
"math_id": 77,
"text": "\\mathbb{RP}^{n+1}"
},
{
"math_id": 78,
"text": "(0,0)"
},
{
"math_id": 79,
"text": "-t^2"
},
{
"math_id": 80,
"text": "t^2=1"
}
] |
https://en.wikipedia.org/wiki?curid=64257772
|
64258213
|
Polygonalization
|
Polygon through a set of points
In computational geometry, a polygonalization of a finite set of points in the Euclidean plane is a simple polygon with the given points as its vertices. A polygonalization may also be called a polygonization, simple polygonalization, Hamiltonian polygon, non-crossing Hamiltonian cycle, or crossing-free straight-edge spanning cycle.
Every point set that does not lie on a single line has at least one polygonalization, which can be found in polynomial time. For points in convex position, there is only one, but for some other point sets there can be exponentially many. Finding an optimal polygonalization under several natural optimization criteria is a hard problem, including as a special case the travelling salesman problem. The complexity of counting all polygonalizations remains unknown.
Definition.
A polygonalization is a simple polygon having a given set of points in the Euclidean plane as its set of vertices. A polygon may be described by a cyclic order on its vertices, which are connected in consecutive pairs by line segments, the edges of the polygon. A polygon, defined in this way, is "simple" if the only intersection points of these line segments are at shared endpoints.
Some authors only consider polygonalizations for points that are in general position, meaning that no three are on a line. With this assumption, the angle between two consecutive segments of the polygon cannot be 180°. However, when point sets with collinearities are considered, it is generally allowed for their polygonalizations to have 180° angles at some points. When this happens, these points are still considered to be vertices, rather than being interior to edges.
Existence.
observed that every finite point set with no three in a line forms the vertices of a simple polygon. However, requiring no three to be in a line is unnecessarily strong. Instead, all that is required for the existence of a polygonalization (allowing 180° angles) is that the points do not all lie on one line. If they do not, then they have a polygonalization that can be constructed in polynomial time. One way of constructing a polygonalization is to choose any point formula_0 in the convex hull of formula_1 (not necessarily one of the given points). Then radially ordering the points around formula_0 (breaking ties by distance from q) produces the cyclic ordering of a star-shaped polygon through all the given points, with formula_0 in its kernel. The same idea of sorting points radially around a central point is used in some versions of the Graham scan convex hull algorithm, and can be performed in formula_2 time. Polygonalizations that avoid 180° angles do not always exist. For instance, for 3 × 3 and 5 × 5 square grids, all polygonalizations use 180° angles.
As well as star-shaped polygonalizations, every non-collinear set of points has a polygonalization that is a monotone polygon. This means that, with respect to some straight line (which may be taken as the formula_3-axis) every perpendicular line to the reference line intersects the polygon in a single interval, or not at all. A construction of begins by sorting the points by their formula_3-coordinates, and drawing a line through the two extreme points. Because the points are not all in a line, at least one of the two open halfplanes bounded by this line must be non-empty. Grünbaum forms two monotone polygonal chains connecting the extreme points through sorted subsequences of the points: one for the points in this non-empty open halfplane, and the other for the remaining points. Their union is the desired monotone polygon. After the sorting step, the rest of the construction may be performed in linear time.
It is NP-complete to determine whether a set of points has a polygonalization using only axis-parallel edges. However, polygonalizations with the additional constraint that they make a right turn at every vertex, if they exist, are uniquely determined. Each axis-parallel line through a point must pass through an even number of points, and this polygonalization must connect alternating pairs of points on this line. The polygonalization may be found in time formula_2 by grouping the points by equal coordinates and sorting each group by the other coordinate. For any point set, at most one rotation can have a polygonalization of this form, and this rotation can again be found in polynomial time.
Optimization.
<templatestyles src="Unsolved/styles.css" />
Unsolved problem in mathematics:
What is the computational complexity of the longest polygonalization?
Problems of finding an optimal polygonalization (for various criteria of optimality) are often computationally infeasible. For instance, the solution to the travelling salesman problem, for the given points, does not have any crossings. Therefore, it is always a polygonalization, the polygonalization with the minimum perimeter. It is NP-hard to find. Similarly, finding the simple polygonalization with minimum or maximum area is known to be NP-hard, and has been the subject of some computational efforts. The maximum area is always more than half of the area of the convex hull, giving an approximation ratio of 2. The exact complexity of the simple polygonalization with maximum perimeter, and the existence of a constant approximation ratio for this problem, remain unknown. The polygonalization that minimizes the length of its longest edge is also NP-hard to find, and hard to approximate to an approximation ratio better than formula_4; no constant-factor approximation is known.
A non-optimal solution to the travelling salesman problem may have crossings, but it is possible to eliminate all crossings by local optimization steps that reduce the total length. Using steps that also eliminate crossings at each step, this can be done in polynomial time, but without this restriction there exist local optimization sequences that instead use an exponential number of steps.
The shortest bitonic tour (the minimum-perimeter monotone polygon through the given points) is always a polygonalization, and can be found in polynomial time.
Counting.
<templatestyles src="Unsolved/styles.css" />
Unsolved problem in mathematics:
What is the computational complexity of counting polygonalizations?
The problem of counting all polygonalizations of a given point set belongs to #P, the class of counting problems associated with decision problems in NP. However, it is unknown whether it is #P-complete or, if not, what its computational complexity might be. A set of points has exactly one polygonalization if and only if it is in convex position. There exist sets of formula_5 points for which the number of polygonalizations is as large as formula_6, and every set of formula_5 points has at most formula_7 polygonalizations.
Methods applying the planar separator theorem to labeled triangulations of the points can be used to count all polygonalizations of a set of formula_5 points in subexponential time, formula_8. Dynamic programming can be used to count all monotone polygonalizations in polynomial time, and the results of this computation can then be used to generate a random monotone polygonalization.
Generation.
<templatestyles src="Unsolved/styles.css" />
Unsolved problem in mathematics:
Can local moves connect the state space of polygonalizations for every point set?
It is unknown whether it is possible for the system of all polygonalizations to form a connected state space under local moves that change a bounded number of the edges of the polygonalizations. If this were possible, it could be used as part of an algorithm for generating all polygonalizations, by applying a graph traversal to the state space. For this problem, it is insufficient to consider "flips" that remove two edges of a polygonalization and replace them by two other edges, or "VE-flips" that remove three edges, two of which share a vertex, and replace them by three other edges. There exist polygonalizations for which no flip or VE-flip is possible, even though the same point set has other polygonalizations.
The "polygonal wraps", weakly simple polygons that use each given point one or more times as a vertex, include all polygonalizations and are connected by local moves. Another more general class of polygons, the "surrounding polygons", are simple polygons that have some of the given points as vertices and enclose all of the points. They are again locally connected, and can be listed in polynomial time per polygon. The algorithm constructs a tree of polygons, with the convex hull as its root and with the parent of each other surrounding polygon obtained by removing one vertex (proven to be possible by applying the two ears theorem to the exterior of the polygon). It then applies a reverse-search algorithm to this tree to list the polygons. As a consequence of this method, all polygonalizations can be listed in exponential time (formula_9 for formula_5 points) and polynomial space.
Applications.
Classical connect the dots puzzles involve connecting points in sequence to form some unexpected shape, often without crossings. The travelling salesman problem and its variants have many applications. Polygonalization also has applications in the reconstruction of contour lines from scattered data points, and in boundary tracing in image analysis.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "q"
},
{
"math_id": 1,
"text": "P"
},
{
"math_id": 2,
"text": "O(n\\log n)"
},
{
"math_id": 3,
"text": "x"
},
{
"math_id": 4,
"text": "\\sqrt3"
},
{
"math_id": 5,
"text": "n"
},
{
"math_id": 6,
"text": "4.64^n"
},
{
"math_id": 7,
"text": "54.6^n"
},
{
"math_id": 8,
"text": "n^{O(\\sqrt n)}"
},
{
"math_id": 9,
"text": "2^{O(n)}"
}
] |
https://en.wikipedia.org/wiki?curid=64258213
|
6425826
|
Speed Score
|
Speed Score, often simply abbreviated to Spd, is a statistic used in Sabermetric studies to evaluate a baseball player's speed. It was invented by Bill James, and first appeared in the 1987 edition of the "Bill James Baseball Abstract".
Speed score is on a scale of 0 to 10, with zero being the slowest and ten being the fastest. League average is generally around 4.5.
Formula.
Speed Score is calculated using six factors: stolen base percentage, stolen base attempts as a percentage of opportunities, triples, double plays grounded into as a percentage of opportunities, runs scored as a percentage of times on base, and defensive position and range.
Factors.
The individuals factors are calculated as follows:
Factor 1 (Stolen base percentage):
formula_0
Where SB=stolen bases and CS=caught stealings.
Factor 2 (Stolen base attempts):
formula_1
Where 1B=singles, BB=walks and HBP=hit by pitch.
Factor 3 (Triples):
formula_2
Where 3B=triples, AB=at bats, HR=home runs and K=strikeouts.
Factor 4 (Runs scored):
formula_3
Where R=runs and H=hits.
Factor 5 (Grounded into double plays):
formula_4
Where GDP=times ground into double play.
Factor 6 (Defensive position and range):
This is dependent upon the player's primary position as follows:
P: formula_5
C: formula_6
1B: formula_7
2B: formula_8
3B: formula_9
SS: formula_10
OF: formula_11
Where P is pitcher, C is catcher, 1B is first baseman, 2B is second baseman, 3B is third baseman, SS is shortstop, OF is outfielder, PO=putouts, A=assists and G=games played.
Final Calculation.
If any factor is less than zero, it is converted to zero, and if any factor is greater than ten, it is converted to ten.
The average of factors 1 through 6 is taken, and the result is the player's score.
formula_12
Developments.
In 2006, Baseball Prospectus developed their own version of speed score in order to "better [take] advantage of play-by-play data and [ensure] that equal weight is given to the five components." It is on the same zero to ten scale and includes all the original factors (except for the player's defense). 5.0 is set as exactly league average. In general, very fast players tend to score around 7.0 and very slow players around 3.0.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\mathit{F1} = 20 \\cdot \\left({\\mathit{SB} + 3 \\over \\mathit{SB} + \\mathit{CS} +7} - 0.4\\right)"
},
{
"math_id": 1,
"text": "\\mathit{F2} = {1 \\over 0.07} \\cdot \\sqrt{{\\mathit{SB} + \\mathit{CS} \\over \\mathit{1B} + \\mathit{BB} + \\mathit{HBP}}}"
},
{
"math_id": 2,
"text": "\\mathit{F3} = 625 \\cdot {\\mathit{3B} \\over \\mathit{AB} - \\mathit{HR} - \\mathit{K}}"
},
{
"math_id": 3,
"text": "\\mathit{F4} = 25 \\cdot \\left({\\mathit{R} - \\mathit{HR} \\over \\mathit{H} + \\mathit{BB} + \\mathit{HBP} - \\mathit{HR}} - 0.1\\right)"
},
{
"math_id": 4,
"text": "\\mathit{F5} = {1 \\over 0.007} \\cdot \\left(0.063 - {\\mathit{GDP} \\over \\mathit{AB} - \\mathit{HR} - \\mathit{K}}\\right)"
},
{
"math_id": 5,
"text": "\\mathit{F6} = 0"
},
{
"math_id": 6,
"text": "\\mathit{F6} = 1"
},
{
"math_id": 7,
"text": "\\mathit{F6} = 2"
},
{
"math_id": 8,
"text": "\\mathit{F6} = {5 \\over 4} \\cdot {\\mathit{PO} + \\mathit{A} \\over \\mathit{G}}"
},
{
"math_id": 9,
"text": "\\mathit{F6} = {4 \\over 2.65} \\cdot {\\mathit{PO} + \\mathit{A} \\over \\mathit{G}}"
},
{
"math_id": 10,
"text": "\\mathit{F6} = {{7 \\over 4.6}} \\cdot {\\mathit{PO} + \\mathit{A} \\over \\mathit{G}}"
},
{
"math_id": 11,
"text": "\\mathit{F6} = 3 \\cdot {\\mathit{PO} + \\mathit{A} \\over \\mathit{G}}"
},
{
"math_id": 12,
"text": "\\mathit{Spd} = {\\mathit{F1} + \\mathit{F2} + \\mathit{F3} + \\mathit{F4} + \\mathit{F5} + \\mathit{F6} \\over 6}"
}
] |
https://en.wikipedia.org/wiki?curid=6425826
|
64265306
|
József Balogh (mathematician)
|
Hungarian mathematician
József Balogh is a Hungarian-American mathematician, specializing in graph theory and combinatorics.
Education and career.
Balogh grew up in Mórahalom and attended secondary school in Szeged at Ságvári Endre Gyakorló Gimnázium (a special school for mathematics). As a student, he won two silver medals (in 1989 and 1990) at the International Mathematical Olympiad. He studied at the University of Szeged (with one year TEMPUS grant at the University of Ghent), where he received his M.S, in mathematics in 1995 with advisor Péter Hajnal and thesis "On the existence of MDS-cyclic codes". In 2001 Balogh received his doctorate from the University of Memphis with advisor Béla Bollobás and thesis "Graph properties and Bootstrap percolation". As a postdoc Balogh was at AT&T Shannon Laboratories in Florham Park, New Jersey and for several months in 2002 at the Institute for Advanced Study. From 2002 to 2005 he was Zassenhaus Assistant Professor at Ohio State University. At the University of Illinois at Urbana–Champaign he was an assistant professor from 2005 to 2010 and an associate professor from 2010 to 2013 and is since 2013 a full professor. From 2009 to 2011 he was also an associate professor at University of California, San Diego.
Balogh's research deals with extremal and probabilistic combinatorics (especially graph theory) and bootstrap percolation. The latter models the spread of an infection on a d-dimensional grid, whereby nodes are infected in each time step in which at least r neighbors have already been infected. It is based on a randomly chosen starting structure and Bollobás, Balogh, Hugo Duminil-Copin and Robert Morris proved an asymptotic (for large grids) formula for the threshold probability that the whole grid is infected, depending on d and r. He had previously treated the three-dimensional case with r = 3 with Bollobás and Morris.
Recognition.
In 2007 he received an NSF Career Grant. In 2013/14 and 2020 he was a Simons Fellow, in 2013/14 Marie Curie Fellow. In 2016 he received the George Pólya Prize in combinatorics with Robert Morris and Wojciech Samotij. In 2018 Balogh was an invited speaker at the International Congress of Mathematicians in Rio de Janeiro.
He was named to the 2023 class of Fellows of the American Mathematical Society, "for contributions to extremal combinatorics, probability and additive number theory, and for graduate mentoring". In 2024 he was awarded the Leroy P. Steele Prize for Seminal Contribution to Research jointly with Robert Morris and Wojciech Samotij.
|
[
{
"math_id": 0,
"text": "K_{s,t}"
},
{
"math_id": 1,
"text": "K_{r+1}"
}
] |
https://en.wikipedia.org/wiki?curid=64265306
|
6426596
|
Transmission time
|
Time it takes a transmitter to send message
In telecommunication networks, the transmission time is the amount of time from the beginning until the end of a message transmission. In the case of a digital message, it is the time from the first bit until the last bit of a message has left the transmitting node. The packet transmission time in seconds can be obtained from the "packet size" in bit and the bit rate in bit/s as:
Packet transmission time = Packet size / Bit rate
Example: Assuming 100 Mbit/s Ethernet, and the maximum packet size of 1526 bytes, results in
Maximum packet transmission time = 1526×8 bit / (100 × 106 bit/s) ≈ 122 μs
Propagation delay.
The transmission time should not be confused with the propagation delay, which is the time it takes for the first bit to travel from the sender to the receiver (During this time the receiver is unaware that a message is being transmitted). The propagation speed depends on the physical medium of the link (that is, fiber optics, twisted-pair copper wire, etc.) and is in the range of formula_0 meters/sec for copper wires and formula_1 for wireless communication, which is equal to the speed of light. The ratio of actual propagation speed to the speed of light is also called the velocity factor of the medium. The propagation delay of a physical link can be calculated by dividing the distance (the length of the medium) in meter by its propagation speed in m/s.
Propagation time = Distance / propagation speed
Example: Ethernet communication over a UTP copper cable with maximum distance of 100 meter between computer and switching node results in:
Maximum link propagation delay ≈ 100 m / (200 000 000 m/s) = 0.5 μs
Packet delivery time.
The "packet delivery time" or latency is the time from when the first bit leaves the transmitter until the last is received. In the case of a physical link, it can be expressed as:
Packet delivery time = Transmission time + Propagation delay
In case of a network connection mediated by several physical links and forwarding nodes, the network delivery time depends on the sum of the delivery times of each link, and also on the packet queuing time (which is varying and depends on the traffic load from other connections) and the processing delay of the forwarding nodes. In wide-area networks, the delivery time is in the order of milliseconds.
Roundtrip time.
The round-trip time or ping time is the time from the start of the transmission from the sending node until a response (for example an ACK packet or ping ICMP response) is received at the same node. It is affected by packet delivery time as well as the data processing delay, which depends on the load on the responding node. If the sent data packet as well as the response packet have the same length, the roundtrip time can be expressed as:
Roundtrip time = 2 × Packet delivery time + processing delay
In case of only one physical link, the above expression corresponds to:
Link roundtrip time = 2 × packet transmission time + 2 × propagation delay + processing delay
If the response packet is very short, the link roundtrip time can be expressed as close to:
Link roundtrip time ≈ packet transmission time + 2 × propagation delay + processing delay
Throughput.
The network throughput of a connection with flow control, for example a TCP connection, with a certain window size (buffer size), can be expressed as:
Network throughput ≈ Window size / roundtrip time
In case of only one physical link between the sending and transmitting nodes, this corresponds to:
Link throughput ≈ Bitrate × Transmission time / roundtrip time
The "message delivery time" or "latency" over a network depends on the message size in bit, and the network throughput or effective data rate in bit/s, as:
Message delivery time = Message size / Network throughput
|
[
{
"math_id": 0,
"text": "2\\times10^8"
},
{
"math_id": 1,
"text": "3\\times10^8"
}
] |
https://en.wikipedia.org/wiki?curid=6426596
|
64266300
|
Schwartz topological vector space
|
In functional analysis and related areas of mathematics, Schwartz spaces are topological vector spaces (TVS) whose neighborhoods of the origin have a property similar to the definition of totally bounded subsets. These spaces were introduced by Alexander Grothendieck.
Definition.
A Hausdorff locally convex space X with continuous dual formula_0, X is called a Schwartz space if it satisfies any of the following equivalent conditions:
Properties.
Every quasi-complete Schwartz space is a semi-Montel space.
Every Fréchet Schwartz space is a Montel space.
The strong dual space of a complete Schwartz space is an ultrabornological space.
Examples and sufficient conditions.
Counter-examples.
Every infinite-dimensional normed space is "not" a Schwartz space.
There exist Fréchet spaces that are not Schwartz spaces and there exist Schwartz spaces that are not Montel spaces.
References.
<templatestyles src="Reflist/styles.css" />
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "X^{\\prime}"
}
] |
https://en.wikipedia.org/wiki?curid=64266300
|
64266856
|
Distinguished space
|
TVS whose strong dual is barralled
In functional analysis and related areas of mathematics, distinguished spaces are topological vector spaces (TVSs) having the property that weak-* bounded subsets of their biduals (that is, the strong dual space of their strong dual space) are contained in the weak-* closure of some bounded subset of the bidual.
Definition.
Suppose that formula_0 is a locally convex space and let formula_1 and formula_2 denote the strong dual of formula_0 (that is, the continuous dual space of formula_0 endowed with the strong dual topology).
Let formula_3 denote the continuous dual space of formula_2 and let formula_4 denote the strong dual of formula_5
Let formula_6 denote formula_3 endowed with the weak-* topology induced by formula_7 where this topology is denoted by formula_8 (that is, the topology of pointwise convergence on formula_1).
We say that a subset formula_9 of formula_3 is formula_8-bounded if it is a bounded subset of formula_6 and we call the closure of formula_9 in the TVS formula_6 the formula_8-closure of formula_9.
If formula_10 is a subset of formula_0 then the polar of formula_10 is formula_11
A Hausdorff locally convex space formula_0 is called a distinguished space if it satisfies any of the following equivalent conditions:
If in addition formula_0 is a metrizable locally convex topological vector space then this list may be extended to include:
Sufficient conditions.
All normed spaces and semi-reflexive spaces are distinguished spaces.
LF spaces are distinguished spaces.
The strong dual space formula_12 of a Fréchet space formula_0 is distinguished if and only if formula_0 is quasibarrelled.
Properties.
Every locally convex distinguished space is an H-space.
Examples.
There exist distinguished Banach spaces spaces that are not semi-reflexive.
The strong dual of a distinguished Banach space is not necessarily separable; formula_13 is such a space.
The strong dual space of a distinguished Fréchet space is not necessarily metrizable.
There exists a distinguished semi-reflexive non-reflexive non-quasibarrelled Mackey space formula_0 whose strong dual is a non-reflexive Banach space.
There exist H-spaces that are not distinguished spaces.
Fréchet Montel spaces are distinguished spaces.
References.
<templatestyles src="Reflist/styles.css" />
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "X"
},
{
"math_id": 1,
"text": "X^{\\prime}"
},
{
"math_id": 2,
"text": "X^{\\prime}_b"
},
{
"math_id": 3,
"text": "X^{\\prime \\prime}"
},
{
"math_id": 4,
"text": "X^{\\prime \\prime}_b"
},
{
"math_id": 5,
"text": "X^{\\prime}_b."
},
{
"math_id": 6,
"text": "X^{\\prime \\prime}_{\\sigma}"
},
{
"math_id": 7,
"text": "X^{\\prime},"
},
{
"math_id": 8,
"text": "\\sigma\\left(X^{\\prime \\prime}, X^{\\prime}\\right)"
},
{
"math_id": 9,
"text": "W"
},
{
"math_id": 10,
"text": "B"
},
{
"math_id": 11,
"text": "B^{\\circ} := \\left\\{ x^{\\prime} \\in X^{\\prime} : \\sup_{b \\in B} \\left\\langle b, x^{\\prime} \\right\\rangle \\leq 1 \\right\\}."
},
{
"math_id": 12,
"text": "X_b^{\\prime}"
},
{
"math_id": 13,
"text": "l^{1}"
}
] |
https://en.wikipedia.org/wiki?curid=64266856
|
64267369
|
Countably barrelled space
|
In functional analysis, a topological vector space (TVS) is said to be countably barrelled if every weakly bounded countable union of equicontinuous subsets of its continuous dual space is again equicontinuous.
This property is a generalization of barrelled spaces.
Definition.
A TVS "X" with continuous dual space formula_0 is said to be countably barrelled if formula_1 is a weak-* bounded subset of formula_0 that is equal to a countable union of equicontinuous subsets of formula_0, then formula_2 is itself equicontinuous.
A Hausdorff locally convex TVS is countably barrelled if and only if each barrel in "X" that is equal to the countable intersection of closed convex balanced neighborhoods of 0 is itself a neighborhood of 0.
σ-barrelled space.
A TVS with continuous dual space formula_0 is said to be σ-barrelled if every weak-* bounded (countable) sequence in formula_0 is equicontinuous.
Sequentially barrelled space.
A TVS with continuous dual space formula_0 is said to be sequentially barrelled if every weak-* convergent sequence in formula_0 is equicontinuous.
Properties.
Every countably barrelled space is a countably quasibarrelled space, a σ-barrelled space, a σ-quasi-barrelled space, and a sequentially barrelled space.
An H-space is a TVS whose strong dual space is countably barrelled.
Every countably barrelled space is a σ-barrelled space and every σ-barrelled space is sequentially barrelled.
Every σ-barrelled space is a σ-quasi-barrelled space.
A locally convex quasi-barrelled space that is also a 𝜎-barrelled space is a barrelled space.
Examples and sufficient conditions.
Every barrelled space is countably barrelled.
However, there exist semi-reflexive countably barrelled spaces that are not barrelled.
The strong dual of a distinguished space and of a metrizable locally convex space is countably barrelled.
Counter-examples.
There exist σ-barrelled spaces that are not countably barrelled.
There exist normed DF-spaces that are not countably barrelled.
There exists a quasi-barrelled space that is not a 𝜎-barrelled space.
There exist σ-barrelled spaces that are not Mackey spaces.
There exist σ-barrelled spaces that are not countably quasi-barrelled spaces and thus not countably barrelled.
There exist sequentially barrelled spaces that are not σ-quasi-barrelled.
There exist quasi-complete locally convex TVSs that are not sequentially barrelled.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "X^{\\prime}"
},
{
"math_id": 1,
"text": "B^{\\prime} \\subseteq X^{\\prime}"
},
{
"math_id": 2,
"text": "B^{\\prime}"
}
] |
https://en.wikipedia.org/wiki?curid=64267369
|
64267601
|
Buchdahl's theorem
|
Theorem in general relativity
In general relativity, Buchdahl's theorem, named after Hans Adolf Buchdahl, makes more precise the notion that there is a maximal sustainable density for ordinary gravitating matter. It gives an inequality between the mass and radius that must be satisfied for static, spherically symmetric matter configurations under certain conditions. In particular, for areal radius formula_0, the mass formula_1 must satisfy
formula_2
where formula_3 is the gravitational constant and formula_4 is the speed of light. This inequality is often referred to as Buchdahl's bound. The bound has historically also been called Schwarzschild's limit as it was first noted by Karl Schwarzschild to exist in the special case of a constant density fluid. However, this terminology should not be confused with the Schwarzschild radius which is notably smaller than the radius at the Buchdahl bound.
Theorem.
Given a static, spherically symmetric solution to the Einstein equations (without cosmological constant) with matter confined to areal radius formula_5 that behaves as a perfect fluid with a density that does not increase outwards. (An areal radius formula_5 corresponds to a sphere of surface area formula_6. In curved spacetime the proper radius of such a sphere is not necessarily formula_5.) Assumes in addition that the density and pressure cannot be negative. The mass of this solution must satisfy
formula_2
For his proof of the theorem, Buchdahl uses the Tolman-Oppenheimer-Volkoff (TOV) equation.
Significance.
The Buchdahl theorem is useful when looking for alternatives to black holes. Such attempts are often inspired by the information paradox; a way to explain (part of) the dark matter; or to criticize that observations of black holes are based on excluding known astrophysical alternatives (such as neutron stars) rather than direct evidence. However, to provide a viable alternative it is sometimes needed that the object should be extremely compact and in particular violate the Buchdahl inequality. This implies that one of the assumptions of Buchdahl's theorem must be invalid. A classification scheme can be made based on which assumptions are violated.
Special Cases.
Incompressible fluid.
The special case of the incompressible fluid or constant density, formula_7 for formula_8, is a historically important example as, in 1916, Schwarzschild noted for the first time that the mass could not exceed the value formula_9 for a given radius formula_5 or the central pressure would become infinite. It is also a particularly tractable example. Within the star one finds.
formula_10
and using the TOV-equation
formula_11
such that the central pressure, formula_12, diverges as formula_13.
Extensions.
Extensions to Buchdahl's theorem generally either relax assumptions on the matter or on the symmetry of the problem. For instance, by introducing anisotropic matter or rotation. In addition one can also consider analogues of Buchdahl's theorem in other theories of gravity
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "R"
},
{
"math_id": 1,
"text": "M"
},
{
"math_id": 2,
"text": " M < \\frac{4 R c^2}{9G} "
},
{
"math_id": 3,
"text": " G "
},
{
"math_id": 4,
"text": "c"
},
{
"math_id": 5,
"text": " R "
},
{
"math_id": 6,
"text": " 4 \\pi R^2 "
},
{
"math_id": 7,
"text": " \\rho(r) = \\rho_* "
},
{
"math_id": 8,
"text": " r < R "
},
{
"math_id": 9,
"text": " \\frac{4 R c^2}{9G} "
},
{
"math_id": 10,
"text": " m(r) = \\frac{4}{3} \\pi r^3 \\rho_* "
},
{
"math_id": 11,
"text": " p(r) = \\rho_* c^2 \\frac{R \\sqrt{R-2GM/c^2}-\\sqrt{R^3-2GMr^2/c^2}}{\\sqrt{R^3-2GMr^2/c^2}-3R\\sqrt{R-2GM/c^2}} "
},
{
"math_id": 12,
"text": " p(0) "
},
{
"math_id": 13,
"text": " R \\to 9GM/4c^2 "
}
] |
https://en.wikipedia.org/wiki?curid=64267601
|
64267682
|
Bornivorous set
|
A set that can absorb any bounded subset
In functional analysis, a subset of a real or complex vector space formula_0 that has an associated vector bornology formula_1 is called bornivorous and a bornivore if it absorbs every element of formula_2
If formula_0 is a topological vector space (TVS) then a subset formula_3 of formula_0 is bornivorous if it is bornivorous with respect to the von-Neumann bornology of formula_0.
Bornivorous sets play an important role in the definitions of many classes of topological vector spaces, particularly bornological spaces.
Definitions.
If formula_0 is a TVS then a subset formula_3 of formula_0 is called <templatestyles src="Template:Visible anchor/styles.css" />bornivorous and a <templatestyles src="Template:Visible anchor/styles.css" />bornivore if formula_3 absorbs every bounded subset of formula_4
An absorbing disk in a locally convex space is bornivorous if and only if its Minkowski functional is locally bounded (i.e. maps bounded sets to bounded sets).
Infrabornivorous sets and infrabounded maps.
A linear map between two TVSs is called <templatestyles src="Template:Visible anchor/styles.css" />infrabounded if it maps Banach disks to bounded disks.
A disk in formula_0 is called <templatestyles src="Template:Visible anchor/styles.css" />infrabornivorous if it absorbs every Banach disk.
An absorbing disk in a locally convex space is infrabornivorous if and only if its Minkowski functional is infrabounded.
A disk in a Hausdorff locally convex space is infrabornivorous if and only if it absorbs all compact disks (that is, if it is "<templatestyles src="Template:Visible anchor/styles.css" />compactivorous").
Properties.
Every bornivorous and infrabornivorous subset of a TVS is absorbing. In a pseudometrizable TVS, every bornivore is a neighborhood of the origin.
Two TVS topologies on the same vector space have that same bounded subsets if and only if they have the same bornivores.
Suppose formula_5 is a vector subspace of finite codimension in a locally convex space formula_0 and formula_6 If formula_7 is a barrel (resp. bornivorous barrel, bornivorous disk) in formula_5 then there exists a barrel (resp. bornivorous barrel, bornivorous disk) formula_8 in formula_0 such that formula_9
Examples and sufficient conditions.
Every neighborhood of the origin in a TVS is bornivorous.
The convex hull, closed convex hull, and balanced hull of a bornivorous set is again bornivorous.
The preimage of a bornivore under a bounded linear map is a bornivore.
If formula_0 is a TVS in which every bounded subset is contained in a finite dimensional vector subspace, then every absorbing set is a bornivore.
Counter-examples.
Let formula_0 be formula_10 as a vector space over the reals.
If formula_3 is the balanced hull of the closed line segment between formula_11 and formula_12 then formula_3 is not bornivorous but the convex hull of formula_3 is bornivorous.
If formula_13 is the closed and "filled" triangle with vertices formula_14 and formula_12 then formula_13 is a convex set that is not bornivorous but its balanced hull is bornivorous.
References.
<templatestyles src="Reflist/styles.css" />
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "X"
},
{
"math_id": 1,
"text": "\\mathcal{B}"
},
{
"math_id": 2,
"text": "\\mathcal{B}."
},
{
"math_id": 3,
"text": "S"
},
{
"math_id": 4,
"text": "X."
},
{
"math_id": 5,
"text": "M"
},
{
"math_id": 6,
"text": "B \\subseteq M."
},
{
"math_id": 7,
"text": "B"
},
{
"math_id": 8,
"text": "C"
},
{
"math_id": 9,
"text": "B = C \\cap M."
},
{
"math_id": 10,
"text": "\\mathbb{R}^2"
},
{
"math_id": 11,
"text": "(-1, 1)"
},
{
"math_id": 12,
"text": "(1, 1)"
},
{
"math_id": 13,
"text": "T"
},
{
"math_id": 14,
"text": "(-1, -1), (-1, 1),"
}
] |
https://en.wikipedia.org/wiki?curid=64267682
|
64267707
|
Countably quasi-barrelled space
|
In functional analysis, a topological vector space (TVS) is said to be countably quasi-barrelled if every strongly bounded countable union of equicontinuous subsets of its continuous dual space is again equicontinuous.
This property is a generalization of quasibarrelled spaces.
Definition.
A TVS "X" with continuous dual space formula_0 is said to be countably quasi-barrelled if formula_1 is a strongly bounded subset of formula_0 that is equal to a countable union of equicontinuous subsets of formula_0, then formula_2 is itself equicontinuous.
A Hausdorff locally convex TVS is countably quasi-barrelled if and only if each bornivorous barrel in "X" that is equal to the countable intersection of closed convex balanced neighborhoods of 0 is itself a neighborhood of 0.
σ-quasi-barrelled space.
A TVS with continuous dual space formula_0 is said to be σ-quasi-barrelled if every strongly bounded (countable) sequence in formula_0 is equicontinuous.
Sequentially quasi-barrelled space.
A TVS with continuous dual space formula_0 is said to be sequentially quasi-barrelled if every strongly convergent sequence in formula_0 is equicontinuous.
Properties.
Every countably quasi-barrelled space is a σ-quasi-barrelled space.
Examples and sufficient conditions.
Every barrelled space, every countably barrelled space, and every quasi-barrelled space is countably quasi-barrelled and thus also σ-quasi-barrelled space.
The strong dual of a distinguished space and of a metrizable locally convex space is countably quasi-barrelled.
Every σ-barrelled space is a σ-quasi-barrelled space.
Every DF-space is countably quasi-barrelled.
A σ-quasi-barrelled space that is sequentially complete is a σ-barrelled space.
There exist σ-barrelled spaces that are not Mackey spaces.
There exist σ-barrelled spaces (which are consequently σ-quasi-barrelled spaces) that are not countably quasi-barrelled spaces.
There exist sequentially complete Mackey spaces that are not σ-quasi-barrelled.
There exist sequentially barrelled spaces that are not σ-quasi-barrelled.
There exist quasi-complete locally convex TVSs that are not sequentially barrelled.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "X^{\\prime}"
},
{
"math_id": 1,
"text": "B^{\\prime} \\subseteq X^{\\prime}"
},
{
"math_id": 2,
"text": "B^{\\prime}"
}
] |
https://en.wikipedia.org/wiki?curid=64267707
|
64268157
|
Ultrabarrelled space
|
In functional analysis and related areas of mathematics, an ultrabarrelled space is a topological vector spaces (TVS) for which every ultrabarrel is a neighbourhood of the origin.
Definition.
A subset formula_0 of a TVS formula_1 is called an ultrabarrel if it is a closed and balanced subset of formula_1 and if there exists a sequence formula_2 of closed balanced and absorbing subsets of formula_1 such that formula_3 for all formula_4
In this case, formula_2 is called a defining sequence for formula_5
A TVS formula_1 is called ultrabarrelled if every ultrabarrel in formula_1 is a neighbourhood of the origin.
Properties.
A locally convex ultrabarrelled space is a barrelled space.
Every ultrabarrelled space is a quasi-ultrabarrelled space.
Examples and sufficient conditions.
Complete and metrizable TVSs are ultrabarrelled.
If formula_1 is a complete locally bounded non-locally convex TVS and if formula_0 is a closed balanced and bounded neighborhood of the origin, then formula_0 is an ultrabarrel that is not convex and has a defining sequence consisting of non-convex sets.
Counter-examples.
There exist barrelled spaces that are not ultrabarrelled.
There exist TVSs that are complete and metrizable (and thus ultrabarrelled) but not barrelled.
Citations.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "B_0"
},
{
"math_id": 1,
"text": "X"
},
{
"math_id": 2,
"text": "\\left(B_i\\right)_{i=1}^{\\infty}"
},
{
"math_id": 3,
"text": "B_{i+1} + B_{i+1} \\subseteq B_i"
},
{
"math_id": 4,
"text": "i = 0, 1, \\ldots."
},
{
"math_id": 5,
"text": "B_0."
}
] |
https://en.wikipedia.org/wiki?curid=64268157
|
64271048
|
Oracle complexity (optimization)
|
In mathematical optimization, oracle complexity is a standard theoretical framework to study the computational requirements for solving classes of optimization problems. It is suitable for analyzing iterative algorithms which proceed by computing local information about the objective function at various points (such as the function's value, gradient, Hessian etc.). The framework has been used to provide tight worst-case guarantees on the number of required iterations, for several important classes of optimization problems.
Formal description.
Consider the problem of minimizing some objective function formula_0 (over some domain formula_1), where formula_2 is known to belong to some family of functions formula_3. Rather than direct access to formula_4, it is assumed that the algorithm can obtain information about formula_2 via an "oracle" formula_5, which given a point formula_6 in formula_1, returns some local information about formula_2 in the neighborhood of formula_6. The algorithm begins at some initialization point formula_7, uses the information provided by the oracle to choose the next point formula_8, uses the additional information to choose the following point formula_9, and so on.
To give a concrete example, suppose that formula_10 (the formula_11-dimensional Euclidean space), and consider the gradient descent algorithm, which initializes at some point formula_7 and proceeds via the recursive equation
formula_12,
where formula_13 is some step size parameter. This algorithm can be modeled in the framework above, where given any formula_14, the oracle returns the gradient formula_15, which is then used to choose the next point formula_16.
In this framework, for each choice of function family formula_3 and oracle formula_5, one can study how many oracle calls/iterations are required, to guarantee some optimization criterion (for example, ensuring that the algorithm produces a point formula_17 such that formula_18 for some formula_19). This is known as the "oracle complexity" of this class of optimization problems: Namely, the number of iterations such that on one hand, there is an algorithm that provably requires only this many iterations to succeed (for any function in formula_3), and on the other hand, there is a proof that no algorithm can succeed with fewer iterations uniformly for all functions in formula_3.
The oracle complexity approach is inherently different from computational complexity theory, which relies on the Turing machine to model algorithms, and requires the algorithm's input (in this case, the function formula_2) to be represented as a bit of strings in memory. Instead, the algorithm is not computationally constrained, but its access to the function formula_2 is assumed to be constrained. This means that on the one hand, oracle complexity results only apply to specific families of algorithms which access the function in a certain manner, and not any algorithm as in computational complexity theory. On the other hand, the results apply to most if not all iterative algorithms used in practice, do not rely on any unproven assumptions, and lead to a nuanced understanding of how the function's geometry and type of information used by the algorithm affects practical performance.
Common settings.
Oracle complexity has been applied to quite a few different settings, depending on the optimization criterion, function class formula_3, and type of oracle formula_5.
In terms of optimization criterion, by far the most common one is finding a near-optimal point, namely making formula_18 for some small formula_19. Some other criteria include finding an approximately-stationary point (formula_20), or finding an approximate local minima.
There are many function classes formula_3 that have been studied. Some common choices include convex vs. strongly-convex vs. non-convex functions, smooth vs. non-smooth functions (say, in terms of Lipschitz properties of the gradients or higher-order derivatives), domains with bounded dimension formula_11, vs. domains with unbounded dimension, and sums of two or more functions with different properties.
In terms of the oracle formula_5, it is common to assume that given a point formula_6, it returns the value of the function at formula_6, as well as derivatives up to some order (say, value only, value and gradient, value and gradient and Hessian, etc.). Sometimes, one studies more complicated oracles. For example, a stochastic oracle returns the values and derivatives corrupted by some random noise, and is useful for studying stochastic optimization methods. Another example is a proximal oracle, which given a point formula_6 and a parameter formula_21, returns the point formula_22 minimizing formula_23.
Examples of oracle complexity results.
The following are a few known oracle complexity results (up to numerical constants), for obtaining optimization error formula_24 for some small enough formula_24, and over the domain formula_25 where formula_11 is not fixed and can be arbitrarily large (unless stated otherwise). We also assume that the initialization point formula_7 satisfies formula_26 for some parameter formula_27, where formula_28 is some global minimizer of the objective function.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "f:\\mathcal{X}\\rightarrow \\mathbb{R}"
},
{
"math_id": 1,
"text": "\\mathcal{X}"
},
{
"math_id": 2,
"text": "f"
},
{
"math_id": 3,
"text": "\\mathcal{F}"
},
{
"math_id": 4,
"text": "\\mathcal{f}"
},
{
"math_id": 5,
"text": "\\mathcal{O}"
},
{
"math_id": 6,
"text": "\\mathbf{x}"
},
{
"math_id": 7,
"text": "\\mathbf{x}_1"
},
{
"math_id": 8,
"text": "\\mathbf{x}_2"
},
{
"math_id": 9,
"text": "\\mathbf{x}_3"
},
{
"math_id": 10,
"text": "\\mathcal{X}=\\mathbb{R}^d"
},
{
"math_id": 11,
"text": "d"
},
{
"math_id": 12,
"text": " \\mathbf{x}_{t+1} = \\mathbf{x}_t-\\eta\\nabla f(\\mathbf{x}_t)"
},
{
"math_id": 13,
"text": "\\eta"
},
{
"math_id": 14,
"text": "\\mathbf{x_t}"
},
{
"math_id": 15,
"text": "\\nabla f(\\mathbf{x_t})"
},
{
"math_id": 16,
"text": "\\mathbf{x_{t+1}}"
},
{
"math_id": 17,
"text": "\\mathbf{x}_T"
},
{
"math_id": 18,
"text": "f(\\mathbf{x}_T)-\\inf_{\\mathbf{x}\\in\\mathcal{X}}f(\\mathbf{x})\\leq \\epsilon"
},
{
"math_id": 19,
"text": "\\epsilon>0"
},
{
"math_id": 20,
"text": "\\|\\nabla f(\\mathbf{x}_T)\\|\\leq \\epsilon "
},
{
"math_id": 21,
"text": "\\gamma"
},
{
"math_id": 22,
"text": "\\mathbf{y}"
},
{
"math_id": 23,
"text": "f(\\mathbf{y})+\\gamma \\|\\mathbf{y}-\\mathbf{x}\\|^2"
},
{
"math_id": 24,
"text": "\\epsilon"
},
{
"math_id": 25,
"text": "\\mathbb{R}^d"
},
{
"math_id": 26,
"text": "\\|\\mathbf{x}_1-\\mathbf{x}^*\\|\\leq B"
},
{
"math_id": 27,
"text": "B"
},
{
"math_id": 28,
"text": "\\mathbf{x}^*"
}
] |
https://en.wikipedia.org/wiki?curid=64271048
|
64274563
|
Quasi-ultrabarrelled space
|
In functional analysis and related areas of mathematics, a quasi-ultrabarrelled space is a topological vector spaces (TVS) for which every bornivorous ultrabarrel is a neighbourhood of the origin.
Definition.
A subset "B"0 of a TVS "X" is called a bornivorous ultrabarrel if it is a closed, balanced, and bornivorous subset of "X" and if there exists a sequence formula_0 of closed balanced and bornivorous subsets of "X" such that "B""i"+1 + "B""i"+1 ⊆ "B""i" for all "i" = 0, 1, ...
In this case, formula_0 is called a defining sequence for "B"0.
A TVS "X" is called quasi-ultrabarrelled if every bornivorous ultrabarrel in "X" is a neighbourhood of the origin.
Properties.
A locally convex quasi-ultrabarrelled space is quasi-barrelled.
Examples and sufficient conditions.
Ultrabarrelled spaces and ultrabornological spaces are quasi-ultrabarrelled.
Complete and metrizable TVSs are quasi-ultrabarrelled.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\left( B_{i} \\right)_{i=1}^{\\infty}"
}
] |
https://en.wikipedia.org/wiki?curid=64274563
|
64278078
|
Metrizable topological vector space
|
A topological vector space whose topology can be defined by a metric
In functional analysis and related areas of mathematics, a metrizable (resp. pseudometrizable) topological vector space (TVS) is a TVS whose topology is induced by a metric (resp. pseudometric). An LM-space is an inductive limit of a sequence of locally convex metrizable TVS.
Pseudometrics and metrics.
A pseudometric on a set formula_0 is a map formula_1 satisfying the following properties:
A pseudometric is called a metric if it satisfies:
Ultrapseudometric
A pseudometric formula_3 on formula_0 is called a ultrapseudometric or a strong pseudometric if it satisfies:
Pseudometric space
A pseudometric space is a pair formula_4 consisting of a set formula_0 and a pseudometric formula_3 on formula_0 such that formula_0's topology is identical to the topology on formula_0 induced by formula_5 We call a pseudometric space formula_4 a metric space (resp. ultrapseudometric space) when formula_3 is a metric (resp. ultrapseudometric).
Topology induced by a pseudometric.
If formula_3 is a pseudometric on a set formula_0 then collection of open balls:
formula_6 as formula_7 ranges over formula_0 and formula_8 ranges over the positive real numbers,
forms a basis for a topology on formula_0 that is called the formula_3-topology or the pseudometric topology on formula_0 induced by formula_5
Convention: If formula_4 is a pseudometric space and formula_0 is treated as a topological space, then unless indicated otherwise, it should be assumed that formula_0 is endowed with the topology induced by formula_5
Pseudometrizable space
A topological space formula_9 is called pseudometrizable (resp. metrizable, ultrapseudometrizable) if there exists a pseudometric (resp. metric, ultrapseudometric) formula_3 on formula_0 such that formula_10 is equal to the topology induced by formula_5
Pseudometrics and values on topological groups.
An additive topological group is an additive group endowed with a topology, called a group topology, under which addition and negation become continuous operators.
A topology formula_10 on a real or complex vector space formula_0 is called a vector topology or a TVS topology if it makes the operations of vector addition and scalar multiplication continuous (that is, if it makes formula_0 into a topological vector space).
Every topological vector space (TVS) formula_0 is an additive commutative topological group but not all group topologies on formula_0 are vector topologies.
This is because despite it making addition and negation continuous, a group topology on a vector space formula_0 may fail to make scalar multiplication continuous.
For instance, the discrete topology on any non-trivial vector space makes addition and negation continuous but do not make scalar multiplication continuous.
Translation invariant pseudometrics.
If formula_0 is an additive group then we say that a pseudometric formula_3 on formula_0 is translation invariant or just invariant if it satisfies any of the following equivalent conditions:
Value/G-seminorm.
If formula_0 is a topological group the a value or G-seminorm on formula_0 (the "G" stands for Group) is a real-valued map formula_11 with the following properties:
where we call a G-seminorm a G-norm if it satisfies the additional condition:
Properties of values.
If formula_12 is a value on a vector space formula_0 then:
Equivalence on topological groups.
<templatestyles src="Math_theorem/styles.css" />
Theorem — Suppose that formula_0 is an additive commutative group.
If formula_3 is a translation invariant pseudometric on formula_0 then the map formula_14 is a value on formula_0 called the value associated with formula_3, and moreover, formula_3 generates a group topology on formula_0 (i.e. the formula_3-topology on formula_0 makes formula_0 into a topological group).
Conversely, if formula_12 is a value on formula_0 then the map formula_15 is a translation-invariant pseudometric on formula_0 and the value associated with formula_3 is just formula_16
Pseudometrizable topological groups.
<templatestyles src="Math_theorem/styles.css" />
Theorem —
If formula_9 is an additive commutative topological group then the following are equivalent:
If formula_9 is Hausdorff then the word "pseudometric" in the above statement may be replaced by the word "metric."
A commutative topological group is metrizable if and only if it is Hausdorff and pseudometrizable.
An invariant pseudometric that doesn't induce a vector topology.
Let formula_0 be a non-trivial (i.e. formula_17) real or complex vector space and let formula_3 be the translation-invariant trivial metric on formula_0 defined by formula_18 and formula_19 such that formula_20
The topology formula_10 that formula_3 induces on formula_0 is the discrete topology, which makes formula_9 into a commutative topological group under addition but does not form a vector topology on formula_0 because formula_9 is disconnected but every vector topology is connected.
What fails is that scalar multiplication isn't continuous on formula_21
This example shows that a translation-invariant (pseudo)metric is not enough to guarantee a vector topology, which leads us to define paranorms and "F"-seminorms.
Additive sequences.
A collection formula_22 of subsets of a vector space is called additive if for every formula_23 there exists some formula_24 such that formula_25
<templatestyles src="Math_theorem/styles.css" />
Continuity of addition at 0 —
If formula_26 is a group (as all vector spaces are), formula_10 is a topology on formula_27 and formula_28 is endowed with the product topology, then the addition map formula_29 (i.e. the map formula_30) is continuous at the origin of formula_28 if and only if the set of neighborhoods of the origin in formula_9 is additive. This statement remains true if the word "neighborhood" is replaced by "open neighborhood."
All of the above conditions are consequently a necessary for a topology to form a vector topology.
Additive sequences of sets have the particularly nice property that they define non-negative continuous real-valued subadditive functions.
These functions can then be used to prove many of the basic properties of topological vector spaces and also show that a Hausdorff TVS with a countable basis of neighborhoods is metrizable. The following theorem is true more generally for commutative additive topological groups.
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Theorem —
Let formula_31 be a collection of subsets of a vector space such that formula_32 and formula_33 for all formula_34
For all formula_35 let
formula_36
Define formula_37 by formula_38 if formula_39 and otherwise let
formula_40
Then formula_41 is subadditive (meaning formula_42) and formula_43 on formula_44 so in particular formula_45
If all formula_46 are symmetric sets then formula_47 and if all formula_46 are balanced then formula_48 for all scalars formula_49 such that formula_50 and all formula_51
If formula_0 is a topological vector space and if all formula_46 are neighborhoods of the origin then formula_41 is continuous, where if in addition formula_0 is Hausdorff and formula_52 forms a basis of balanced neighborhoods of the origin in formula_0 then formula_53 is a metric defining the vector topology on formula_13
Paranorms.
If formula_0 is a vector space over the real or complex numbers then a paranorm on formula_0 is a G-seminorm (defined above) formula_11 on formula_0 that satisfies any of the following additional conditions, each of which begins with "for all sequences formula_55 in formula_0 and all convergent sequences of scalars formula_56":
A paranorm is called total if in addition it satisfies:
Properties of paranorms.
If formula_12 is a paranorm on a vector space formula_0 then the map formula_1 defined by formula_15 is a translation-invariant pseudometric on formula_0 that defines a vector topology on formula_13
If formula_12 is a paranorm on a vector space formula_0 then:
"F"-seminorms.
If formula_0 is a vector space over the real or complex numbers then an "F"-seminorm on formula_0 (the formula_58 stands for Fréchet) is a real-valued map formula_59 with the following four properties:
An "F"-seminorm is called an "F"-norm if in addition it satisfies:
An "F"-seminorm is called monotone if it satisfies:
"F"-seminormed spaces.
An F"-seminormed space (resp. F"-normed space) is a pair formula_60 consisting of a vector space formula_0 and an "F"-seminorm (resp. "F"-norm) formula_12 on formula_13
If formula_60 and formula_61 are "F"-seminormed spaces then a map formula_62 is called an isometric embedding if formula_63
Every isometric embedding of one "F"-seminormed space into another is a topological embedding, but the converse is not true in general.
Properties of "F"-seminorms.
Every "F"-seminorm is a paranorm and every paranorm is equivalent to some "F"-seminorm.
Every "F"-seminorm on a vector space formula_0 is a value on formula_13 In particular, formula_65 and formula_66 for all formula_51
Topology induced by a single "F"-seminorm.
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Theorem — Let formula_12 be an "F"-seminorm on a vector space formula_13
Then the map formula_67 defined by
formula_15
is a translation invariant pseudometric on formula_0 that defines a vector topology formula_10 on formula_13
If formula_12 is an "F"-norm then formula_3 is a metric.
When formula_0 is endowed with this topology then formula_12 is a continuous map on formula_13
The balanced sets formula_68 as formula_69 ranges over the positive reals, form a neighborhood basis at the origin for this topology consisting of closed set.
Similarly, the balanced sets formula_70 as formula_69 ranges over the positive reals, form a neighborhood basis at the origin for this topology consisting of open sets.
Topology induced by a family of "F"-seminorms.
Suppose that formula_71 is a non-empty collection of "F"-seminorms on a vector space formula_0 and for any finite subset formula_72 and any formula_54 let
formula_73
The set formula_74 forms a filter base on formula_0 that also forms a neighborhood basis at the origin for a vector topology on formula_0 denoted by formula_75 Each formula_76 is a balanced and absorbing subset of formula_13 These sets satisfy
formula_77
Fréchet combination.
Suppose that formula_78 is a family of non-negative subadditive functions on a vector space formula_13
The Fréchet combination of formula_79 is defined to be the real-valued map
formula_80
As an "F"-seminorm.
Assume that formula_78 is an increasing sequence of seminorms on formula_0 and let formula_12 be the Fréchet combination of formula_81
Then formula_12 is an "F"-seminorm on formula_0 that induces the same locally convex topology as the family formula_79 of seminorms.
Since formula_78 is increasing, a basis of open neighborhoods of the origin consists of all sets of the form formula_82 as formula_83 ranges over all positive integers and formula_8 ranges over all positive real numbers.
The translation invariant pseudometric on formula_0 induced by this "F"-seminorm formula_12 is
formula_84
This metric was discovered by Fréchet in his 1906 thesis for the spaces of real and complex sequences with pointwise operations.
As a paranorm.
If each formula_85 is a paranorm then so is formula_12 and moreover, formula_12 induces the same topology on formula_0 as the family formula_79 of paranorms.
This is also true of the following paranorms on formula_0:
Generalization.
The Fréchet combination can be generalized by use of a bounded remetrization function.
A is a continuous non-negative non-decreasing map formula_86 that has a bounded range, is subadditive (meaning that formula_87 for all formula_88), and satisfies formula_89 if and only if formula_90
Examples of bounded remetrization functions include formula_91 formula_92 formula_93 and formula_94
If formula_3 is a pseudometric (respectively, metric) on formula_0 and formula_95 is a bounded remetrization function then formula_96 is a bounded pseudometric (respectively, bounded metric) on formula_0 that is uniformly equivalent to formula_5
Suppose that formula_97 is a family of non-negative "F"-seminorm on a vector space formula_27 formula_95 is a bounded remetrization function, and formula_98 is a sequence of positive real numbers whose sum is finite.
Then
formula_99
defines a bounded "F"-seminorm that is uniformly equivalent to the formula_100
It has the property that for any net formula_101 in formula_27 formula_102 if and only if formula_103 for all formula_104
formula_12 is an "F"-norm if and only if the formula_105 separate points on formula_13
Characterizations.
Of (pseudo)metrics induced by (semi)norms.
A pseudometric (resp. metric) formula_3 is induced by a seminorm (resp. norm) on a vector space formula_0 if and only if formula_3 is translation invariant and absolutely homogeneous, which means that for all scalars formula_49 and all formula_2 in which case the function defined by formula_14 is a seminorm (resp. norm) and the pseudometric (resp. metric) induced by formula_12 is equal to formula_5
Of pseudometrizable TVS.
If formula_9 is a topological vector space (TVS) (where note in particular that formula_10 is assumed to be a vector topology) then the following are equivalent:
Of metrizable TVS.
If formula_9 is a TVS then the following are equivalent:
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Birkhoff–Kakutani theorem —
If formula_9 is a topological vector space then the following three conditions are equivalent:
By the Birkhoff–Kakutani theorem, it follows that there is an equivalent metric that is translation-invariant.
Of locally convex pseudometrizable TVS.
If formula_9 is TVS then the following are equivalent:
Quotients.
Let formula_106 be a vector subspace of a topological vector space formula_21
Examples and sufficient conditions.
If formula_0 is Hausdorff locally convex TVS then formula_0 with the strong topology, formula_109 is metrizable if and only if there exists a countable set formula_110 of bounded subsets of formula_0 such that every bounded subset of formula_0 is contained in some element of formula_111
The strong dual space formula_112 of a metrizable locally convex space (such as a Fréchet space) formula_0 is a DF-space.
The strong dual of a DF-space is a Fréchet space.
The strong dual of a reflexive Fréchet space is a bornological space.
The strong bidual (that is, the strong dual space of the strong dual space) of a metrizable locally convex space is a Fréchet space.
If formula_0 is a metrizable locally convex space then its strong dual formula_112 has one of the following properties, if and only if it has all of these properties: (1) bornological, (2) infrabarreled, (3) barreled.
Normability.
A topological vector space is seminormable if and only if it has a convex bounded neighborhood of the origin.
Moreover, a TVS is normable if and only if it is Hausdorff and seminormable.
Every metrizable TVS on a finite-dimensional vector space is a normable locally convex complete TVS, being TVS-isomorphic to Euclidean space. Consequently, any metrizable TVS that is not normable must be infinite dimensional.
If formula_106 is a metrizable locally convex TVS that possess a countable fundamental system of bounded sets, then formula_106 is normable.
If formula_0 is a Hausdorff locally convex space then the following are equivalent:
and if this locally convex space formula_0 is also metrizable, then the following may be appended to this list:
In particular, if a metrizable locally convex space formula_0 (such as a Fréchet space) is not normable then its strong dual space formula_113 is not a Fréchet–Urysohn space and consequently, this complete Hausdorff locally convex space formula_113 is also neither metrizable nor normable.
Another consequence of this is that if formula_0 is a reflexive locally convex TVS whose strong dual formula_113 is metrizable then formula_113 is necessarily a reflexive Fréchet space, formula_0 is a DF-space, both formula_0 and formula_113 are necessarily complete Hausdorff ultrabornological distinguished webbed spaces, and moreover, formula_113 is normable if and only if formula_0 is normable if and only if formula_0 is Fréchet–Urysohn if and only if formula_0 is metrizable. In particular, such a space formula_0 is either a Banach space or else it is not even a Fréchet–Urysohn space.
Metrically bounded sets and bounded sets.
Suppose that formula_4 is a pseudometric space and formula_114
The set formula_115 is metrically bounded or formula_3-bounded if there exists a real number formula_116 such that formula_117 for all formula_118;
the smallest such formula_95 is then called the diameter or formula_3-diameter of formula_119
If formula_115 is bounded in a pseudometrizable TVS formula_0 then it is metrically bounded;
the converse is in general false but it is true for locally convex metrizable TVSs.
Properties of pseudometrizable TVS.
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Theorem —
All infinite-dimensional separable complete metrizable TVS are homeomorphic.
Completeness.
Every topological vector space (and more generally, a topological group) has a canonical uniform structure, induced by its topology, which allows the notions of completeness and uniform continuity to be applied to it.
If formula_0 is a metrizable TVS and formula_3 is a metric that defines formula_0's topology, then its possible that formula_0 is complete as a TVS (i.e. relative to its uniformity) but the metric formula_3 is not a complete metric (such metrics exist even for formula_120).
Thus, if formula_0 is a TVS whose topology is induced by a pseudometric formula_108 then the notion of completeness of formula_0 (as a TVS) and the notion of completeness of the pseudometric space formula_4 are not always equivalent.
The next theorem gives a condition for when they are equivalent:
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Theorem — If formula_0 is a pseudometrizable TVS whose topology is induced by a translation invariant pseudometric formula_108 then formula_3 is a complete pseudometric on formula_0 if and only if formula_0 is complete as a TVS.
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Theorem (Klee) — Let formula_3 be any metric on a vector space formula_0 such that the topology formula_10 induced by formula_3 on formula_0 makes formula_9 into a topological vector space. If formula_4 is a complete metric space then formula_9 is a complete-TVS.
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Theorem — If formula_0 is a TVS whose topology is induced by a paranorm formula_121 then formula_0 is complete if and only if for every sequence formula_122 in formula_27 if formula_123 then formula_124 converges in formula_13
If formula_106 is a closed vector subspace of a complete pseudometrizable TVS formula_27 then the quotient space formula_107 is complete.
If formula_106 is a complete vector subspace of a metrizable TVS formula_0 and if the quotient space formula_107 is complete then so is formula_13 If formula_0 is not complete then formula_125 but not complete, vector subspace of formula_13
A Baire separable topological group is metrizable if and only if it is cosmic.
Subsets and subsequences.
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Banach-Saks theorem —
If formula_127 is a sequence in a locally convex metrizable TVS formula_9 that converges weakly to some formula_57 then there exists a sequence formula_128 in formula_0 such that formula_129 in formula_9 and each formula_130 is a convex combination of finitely many formula_131
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Mackey's countability condition — Suppose that formula_0 is a locally convex metrizable TVS and that formula_132 is a countable sequence of bounded subsets of formula_13
Then there exists a bounded subset formula_115 of formula_0 and a sequence formula_126 of positive real numbers such that formula_133 for all formula_104
Generalized series
As described in this article's section on generalized series, for any formula_134-indexed family family formula_135 of vectors from a TVS formula_27 it is possible to define their sum formula_136 as the limit of the net of finite partial sums formula_137 where the domain formula_138 is directed by formula_139
If formula_140 and formula_141 for instance, then the generalized series formula_142 converges if and only if formula_143 converges unconditionally in the usual sense (which for real numbers, is equivalent to absolute convergence).
If a generalized series formula_136 converges in a metrizable TVS, then the set formula_144 is necessarily countable (that is, either finite or countably infinite);
in other words, all but at most countably many formula_145 will be zero and so this generalized series formula_146 is actually a sum of at most countably many non-zero terms.
Linear maps.
If formula_0 is a pseudometrizable TVS and formula_147 maps bounded subsets of formula_0 to bounded subsets of formula_64 then formula_147 is continuous.
Discontinuous linear functionals exist on any infinite-dimensional pseudometrizable TVS. Thus, a pseudometrizable TVS is finite-dimensional if and only if its continuous dual space is equal to its algebraic dual space.
If formula_148 is a linear map between TVSs and formula_0 is metrizable then the following are equivalent:
Open and almost open maps
Theorem: If formula_0 is a complete pseudometrizable TVS, formula_149 is a Hausdorff TVS, and formula_150 is a closed and almost open linear surjection, then formula_151 is an open map.
Theorem: If formula_150 is a surjective linear operator from a locally convex space formula_0 onto a barrelled space formula_149 (e.g. every complete pseudometrizable space is barrelled) then formula_151 is almost open.
Theorem: If formula_150 is a surjective linear operator from a TVS formula_0 onto a Baire space formula_149 then formula_151 is almost open.
Theorem: Suppose formula_150 is a continuous linear operator from a complete pseudometrizable TVS formula_0 into a Hausdorff TVS formula_152 If the image of formula_151 is non-meager in formula_149 then formula_150 is a surjective open map and formula_149 is a complete metrizable space.
Hahn-Banach extension property.
A vector subspace formula_106 of a TVS formula_0 has the extension property if any continuous linear functional on formula_106 can be extended to a continuous linear functional on formula_13
Say that a TVS formula_0 has the Hahn-Banach extension property (HBEP) if every vector subspace of formula_0 has the extension property.
The Hahn-Banach theorem guarantees that every Hausdorff locally convex space has the HBEP.
For complete metrizable TVSs there is a converse:
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Theorem (Kalton) — Every complete metrizable TVS with the Hahn-Banach extension property is locally convex.
If a vector space formula_0 has uncountable dimension and if we endow it with the finest vector topology then this is a TVS with the HBEP that is neither locally convex or metrizable.
Notes.
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Proofs
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References.
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"text": "X."
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{
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"text": "p(x) := d(x, 0)"
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{
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"text": "d(x, y) := p(x - y)"
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{
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"text": "X \\neq \\{ 0 \\}"
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"text": "U_{\\bull} = \\left(U_i\\right)_{i=0}^{\\infty}"
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{
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"text": "\\mathbb{S}(u) := \\left\\{ n_{\\bull} = \\left(n_1, \\ldots, n_k\\right) ~:~ k \\geq 1, n_i \\geq 0 \\text{ for all } i, \\text{ and } u \\in U_{n_1} + \\cdots + U_{n_k}\\right\\}."
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{
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{
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{
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"text": "s"
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{
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"text": "Y,"
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"text": "r"
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"text": "\\{x \\in X ~:~ p(x) < r\\},"
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"text": "\\mathcal{L}"
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"text": "\\mathcal{F} \\subseteq \\mathcal{L}"
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"text": "U_{\\mathcal{F}, r/2} + U_{\\mathcal{F}, r/2} \\subseteq U_{\\mathcal{F}, r}."
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"text": "p_{\\bull}"
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"text": "p(x) := \\sum_{i=1}^{\\infty} \\frac{p_i(x)}{2^{i} \\left[ 1 + p_i(x)\\right]}."
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"text": "p_{\\bull}."
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"text": "i"
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"text": "p_i"
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{
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{
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"text": "\\tanh t,"
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{
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{
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"text": "t \\mapsto \\frac{t}{1 + t}."
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{
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"text": "R"
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{
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"text": "R \\circ d"
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{
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"text": "p_\\bull = \\left(p_i\\right)_{i=1}^\\infty"
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{
"math_id": 98,
"text": "r_\\bull = \\left(r_i\\right)_{i=1}^\\infty"
},
{
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"text": "p(x) := \\sum_{i=1}^\\infty r_i R\\left(p_i(x)\\right)"
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{
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"text": "p_\\bull."
},
{
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"text": "x_\\bull = \\left(x_a\\right)_{a \\in A}"
},
{
"math_id": 102,
"text": "p\\left(x_\\bull\\right) \\to 0"
},
{
"math_id": 103,
"text": "p_i\\left(x_\\bull\\right) \\to 0"
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{
"math_id": 104,
"text": "i."
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{
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"text": "M"
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{
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{
"math_id": 110,
"text": "\\mathcal{B}"
},
{
"math_id": 111,
"text": "\\mathcal{B}."
},
{
"math_id": 112,
"text": "X_b^{\\prime}"
},
{
"math_id": 113,
"text": "X^{\\prime}_b"
},
{
"math_id": 114,
"text": "B \\subseteq X."
},
{
"math_id": 115,
"text": "B"
},
{
"math_id": 116,
"text": "R > 0"
},
{
"math_id": 117,
"text": "d(x, y) \\leq R"
},
{
"math_id": 118,
"text": "x, y \\in B"
},
{
"math_id": 119,
"text": "B."
},
{
"math_id": 120,
"text": "X = \\R"
},
{
"math_id": 121,
"text": "p,"
},
{
"math_id": 122,
"text": "\\left(x_i\\right)_{i=1}^{\\infty}"
},
{
"math_id": 123,
"text": "\\sum_{i=1}^{\\infty} p\\left(x_i\\right) < \\infty"
},
{
"math_id": 124,
"text": "\\sum_{i=1}^{\\infty} x_i"
},
{
"math_id": 125,
"text": "M := X,"
},
{
"math_id": 126,
"text": "\\left(r_i\\right)_{i=1}^{\\infty}"
},
{
"math_id": 127,
"text": "\\left(x_n\\right)_{n=1}^{\\infty}"
},
{
"math_id": 128,
"text": "y_{\\bull} = \\left(y_i\\right)_{i=1}^{\\infty}"
},
{
"math_id": 129,
"text": "y_{\\bull} \\to x"
},
{
"math_id": 130,
"text": "y_i"
},
{
"math_id": 131,
"text": "x_n."
},
{
"math_id": 132,
"text": "\\left(B_i\\right)_{i=1}^{\\infty}"
},
{
"math_id": 133,
"text": "B_i \\subseteq r_i B"
},
{
"math_id": 134,
"text": "I"
},
{
"math_id": 135,
"text": "\\left(r_i\\right)_{i \\in I}"
},
{
"math_id": 136,
"text": "\\textstyle\\sum\\limits_{i \\in I} r_i"
},
{
"math_id": 137,
"text": "F \\in \\operatorname{FiniteSubsets}(I) \\mapsto \\textstyle\\sum\\limits_{i \\in F} r_i"
},
{
"math_id": 138,
"text": "\\operatorname{FiniteSubsets}(I)"
},
{
"math_id": 139,
"text": "\\,\\subseteq.\\,"
},
{
"math_id": 140,
"text": "I = \\N"
},
{
"math_id": 141,
"text": "X = \\Reals,"
},
{
"math_id": 142,
"text": "\\textstyle\\sum\\limits_{i \\in \\N} r_i"
},
{
"math_id": 143,
"text": "\\textstyle\\sum\\limits_{i=1}^\\infty r_i"
},
{
"math_id": 144,
"text": "\\left\\{i \\in I : r_i \\neq 0\\right\\}"
},
{
"math_id": 145,
"text": "r_i"
},
{
"math_id": 146,
"text": "\\textstyle\\sum\\limits_{i \\in I} r_i ~=~ \\textstyle\\sum\\limits_{\\stackrel{i \\in I}{r_i \\neq 0}} r_i"
},
{
"math_id": 147,
"text": "A"
},
{
"math_id": 148,
"text": "F : X \\to Y"
},
{
"math_id": 149,
"text": "Y"
},
{
"math_id": 150,
"text": "T : X \\to Y"
},
{
"math_id": 151,
"text": "T"
},
{
"math_id": 152,
"text": "Y."
}
] |
https://en.wikipedia.org/wiki?curid=64278078
|
64280109
|
Transport-of-intensity equation
|
The transport-of-intensity equation (TIE) is a computational approach to reconstruct the phase of a complex wave in optical and electron microscopy. It describes the internal relationship between the intensity and phase distribution of a wave.
The TIE was first proposed in 1983 by Michael Reed Teague. Teague suggested to use the law of conservation of energy to write a differential equation for the transport of energy by an optical field. This equation, he stated, could be used as an approach to phase recovery.
Teague approximated the amplitude of the wave propagating nominally in the z-direction by a parabolic equation and then expressed it in terms of irradiance and phase:
formula_0
where formula_1 is the wavelength, formula_2 is the irradiance at point formula_3, and formula_4 is the phase of the wave. If the intensity distribution of the wave and its spatial derivative can be measured experimentally, the equation becomes a linear equation that can be solved to obtain the phase distribution formula_4.
For a phase sample with a constant intensity, the TIE simplifies to
formula_5
It allows measuring the phase distribution of the sample by acquiring a defocused image, i.e. formula_6.
TIE-based approaches are applied in biomedical and technical applications, such as quantitative monitoring of cell growth in culture, investigation of cellular dynamics and characterization of optical elements. The TIE method is also applied for phase retrieval in transmission electron microscopy.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\frac{2\\pi}{\\lambda} \\frac{\\partial}{\\partial z}I(x,y,z)= -\\nabla_{x,y} \\cdot [I(x,y,z)\\nabla_{x,y}\\Phi],"
},
{
"math_id": 1,
"text": "\\lambda"
},
{
"math_id": 2,
"text": "I(x,y,z)"
},
{
"math_id": 3,
"text": "(x,y,z)"
},
{
"math_id": 4,
"text": "\\Phi"
},
{
"math_id": 5,
"text": "\\frac{d}{dz}I(z) = -\\frac{\\lambda}{2\\pi} I(z) \\nabla_{x,y}^2 \\Phi."
},
{
"math_id": 6,
"text": "I(x,y,z + \\Delta z)"
}
] |
https://en.wikipedia.org/wiki?curid=64280109
|
64281511
|
Quantum jump
|
1913 model of abrupt transitions of quantum systems
A quantum jump is the abrupt transition of a quantum system (atom, molecule, atomic nucleus) from one quantum state to another, from one energy level to another. When the system absorbs energy, there is a transition to a higher energy level (excitation); when the system loses energy, there is a transition to a lower energy level.
The concept was introduced by Niels Bohr, in his 1913 Bohr model.
A quantum jump is a phenomenon that is peculiar to quantum systems and distinguishes them from classical systems, where any transitions are performed gradually. In quantum mechanics, such jumps are associated with the non-unitary evolution of a quantum-mechanical system during measurement.
A quantum jump can be accompanied by the emission or absorption of photons; energy transfer during a quantum jump can also occur by non-radiative resonant energy transfer or in collisions with other particles.
In modern physics, the concept of a quantum jump is rarely used; as a rule scientists speak of transitions between quantum states or energy levels.
Atomic electron transition.
Atomic electron transitions cause the emission or absorption of photons. Their statistics are Poissonian, and the time between jumps is exponentially distributed. The damping time constant (which ranges from nanoseconds to a few seconds) relates to the natural, pressure, and field broadening of spectral lines. The larger the energy separation of the states between which the electron jumps, the shorter the wavelength of the photon emitted.
In an ion trap, quantum jumps can be directly observed by addressing a trapped ion with radiation at two different frequencies to drive electron transitions. This requires one strong and one weak transition to be excited (denoted formula_012 and formula_013 respectively in the figure to the right). The electron energy level, formula_2, has a short lifetime, formula_12 which allows for constant emission of photons at a frequency formula_012 which can be collected by a camera and/or photomultiplier tube. State formula_3 has a relatively long lifetime formula_13 which causes an interruption of the photon emission as the electron gets shelved in state through application of light with frequency formula_013. The ion going dark is a direct observation of quantum jumps.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\omega"
},
{
"math_id": 1,
"text": "\\Gamma"
},
{
"math_id": 2,
"text": "|2\\rangle"
},
{
"math_id": 3,
"text": "|3\\rangle"
}
] |
https://en.wikipedia.org/wiki?curid=64281511
|
64283032
|
List of Egyptian inventions and discoveries
|
Egyptian inventions and discoveries are objects, processes or techniques which owe their existence or first known written account either partially or entirely to an Egyptian person.
Ancient Egypt.
Tools and machines.
Furniture.
Furniture became common first in Ancient Egypt during the Naqada culture. During that period a wide variety of furniture pieces were invented and used.
formula_0.
formula_1
Notes.
<templatestyles src="Reflist/styles.css" />
References.
<templatestyles src="Reflist/styles.css" />
Works cited.
<templatestyles src="Refbegin/styles.css" />
|
[
{
"math_id": 0,
"text": "\\sin^2\\left(\\frac{x}{2}\\right) = \\frac{1 - \\cos(x)}{2}"
},
{
"math_id": 1,
"text": "\\cos a \\cos b = \\frac{\\cos(a+b) + \\cos(a-b)}{2}"
}
] |
https://en.wikipedia.org/wiki?curid=64283032
|
642836
|
Highlife (cellular automaton)
|
2D cellular automaton similar to Conway's Game of Life
Highlife is a cellular automaton similar to Conway's Game of Life. It was devised in 1994 by Nathan Thompson. It is a two-dimensional, two-state cellular automaton in the "Life family" and is described by the rule B36/S23; that is, a cell is born if it has 3 or 6 neighbors and survives if it has 2 or 3 neighbors. Because the rules of HighLife and Conway's Life (rule B3/S23) are similar, many simple patterns in Conway's Life function identically in HighLife. More complicated engineered patterns for one rule, though, typically do not work in the other rule.
Replicator.
The main reason for interest in HighLife comes from the existence of a pattern called the replicator. After running the replicator for twelve generations, the result is two replicators. The replicators will repeatedly reproduce themselves, all on a diagonal line. Whenever two replicators try to expand into each other, the pattern in the middle simply vanishes. The behavior of a row of Replicators interacting with each other in this way simulates the one-dimensional Rule 90 cellular automaton, where a single replicator simulates a nonzero cell of the Rule 90 automaton and a blank space where a replicator could be simulates a zero cell of Rule 90. Replicators can be used to engineer other more complex patterns, such as glider guns and high period oscillators.
A simple c/6 diagonal spaceship, found by Nathan Thompson, is known as the bomber. This pattern consists of a replicator and a blinker; after replicating itself into two replicators, one of the two new replicators reacts with the blinker to "pull" it forward to match the new position of the other new replicator. In this way, the whole pattern repeats with period 48.
It is also possible to make slower spaceships of much larger size that consist of a sequence of replicators between two ends composed of oscillators or still lifes, with the pattern of the replicators carefully chosen so that they interact with the ends of the pattern in such a way as to push the front end and pull the back end at the same speed.
Explicit examples of this design, known as "basilisks", include spaceships of speeds formula_0 (one cell every 24 generations), formula_1, formula_2, and formula_3. A formula_0 basilisk gun has also been constructed.
It had been proven that replicators exist in Conway's Life as well, before an explicit example was found in 2013.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "c/24"
},
{
"math_id": 1,
"text": "c/32"
},
{
"math_id": 2,
"text": "c/63"
},
{
"math_id": 3,
"text": "c/69"
}
] |
https://en.wikipedia.org/wiki?curid=642836
|
6428526
|
Reactive inhibition
|
Reactive inhibition is a phrase coined by Clark L. Hull in his 1943 book titled "Principles of Behavior". He defined it as:
<templatestyles src="Template:Blockquote/styles.css" />Whenever any reaction is evoked in an organism there is left a condition or state which acts as a primary negative motivation in that it has an innate capacity to produce a cessation of the activity which produced the state.
Reactive inhibition is typically studied in the context of drive reduction. Hull likens it to fatigue through which humans become tired over time and thus less accurate and precise within a given task. There is significant debate whether the process of reactive inhibition is due to fatigue or some other process. Nevertheless, it is a factor researchers need to consider in analyses of sustained performance due to its possible role in the results and analysis of research.
Hull goes on to further explain the decay of performance through the use of a decay formula which can estimate the rate of performance deterioration.
Hull explains:
"I" dissipates exponentially with time "t":With the passage of time since its formation "I""R" spontaneously dissipates approximately as a simple decay function of the time "t" elapsed, i.e.,
formula_0 (Hull, 1951, p. 74).Hull's decay formula is somewhat awkward and might give rise to confusion. For example, "I"'"R" does not refer to the derivative of "I""R". A more convenient way of writing the formula would be as follows:
formula_1
with formula_2. formula_3 is the inhibition at the beginning the time interval [0,"t"]. Note that if one takes the natural logarithm of both sides one obtains:
formula_4
where formula_5 and formula_6. The last formula is used in inhibition theory.
Reactive inhibition is distinct from proactive inhibition. Reactive inhibition occurs after an initial response has been activated and set to be carried out. In contrast, proactive inhibition determines whether or not the response process is activated in the future and occurs before initial activation. Reactive inhibition is considered to be a bottom-up processing process and associated with “lower level mechanisms of inhibition”, whereas proactive inhibition is considered more top-down processing and dealing with “higher level mechanisms”.
Applications.
Reactive inhibition may be important in everyday life during a process in which a decline in performance can be detrimental such as driving a car during rush hour. For example, Kathaus, Washcer, & Getzmann (2018) found that older adults who showed a tendency towards reactive inhibition, determined through electroencephalography measures, showed higher “driving lane variability” and more impairment. Although older adults matched younger adults in their lane keeping abilities, they were unable to change lanes as effectively when they relied on reactive inhibition.
Another study also revolving around younger vs older adults in the realm of inhibition found that older adults had decreased reactive inhibition but sustain proactive inhibition overall. By using a smart phone app, participants played a game in two apples were falling from either side of the tree. They were to tap either apple but not press one of the apples if that apple turned brown or “rotten”. This is similar to a Stop Signal Task as described below. Some of these trials were primed for a person to expect a change and others not. What was found was a decreased ability in older adults to inhibit an action when they were not primed thus indicating a deficit in reactive inhibition.
Researchers have also studied reactive inhibition within the context of ADHD. It is commonly accepted that decreased inhibition abilities are a prominent aspect of the symptoms associated with ADHD. Within the context of the Stop Signal Task studies point to an inability to switch attention the signal switches from a go signal, to stop, which can be compared to environmental changes in the world. Further, it is proven that reactive inhibition in particular is affected in individuals with ADHD and related ADHD symptoms, and may not even have an impact on proactive inhibition at all. The ability to inhibit can impact children’s learning abilities and is a lack of reactive inhibition is present in many learning disorders.
Relationship to learning.
Reactive inhibition is also related to repetition performance, including learning. For example, Torok et al. (2017) recorded learning capabilities in 180 adults using the Alternating Serial Reaction Time Test. Results showed reactive inhibition had a profound effect on performance. Specifically, they showed that significantly more learning had occurred than was perceived at the end of the task, for reactive inhibition had effected the individual over time. They concluded that reactive inhibition may affect one’s rate of learning due to how it causes progressive decline within a task. It was stated to be a feature of performance within 90% of the participants, and thus playing a significant role in results.
These findings have caused some researchers to question existing psychological theories. For example, Rickard, Pan, and Albarracín present evidence that even well accepted psychological findings such as memory consolidation during sleep may be incorrect. The increase in “memory” that supposedly occurs after sleep may just be due to reactive inhibition. It may have existed at the end of learning before sleeping occurred, and thus caused seemingly lower memory scores.
Reactive inhibition is often not recognized as a factor of performance in learning based experiments and thus can lead to incorrect results. The presence of reactive inhibition can result in decreased performance over time and thus decrease the level of supposed learning. If tested at a later time, however, when reactive inhibition is not present one may see true measures of learning.
Stop-signal task (SST).
Reactive inhibition within experimental settings is most commonly measured through the stop-signal task (SST). In the SST, a “go-signal” is presented to the participant to indicate that he or she should complete an action. Then, in some instances, a “stop-signal” is also presented to the participant indicate he or she should abandon the previously initiated action. This stop signal is presented within hundreds of milliseconds of receiving the go signal. What is important within this task is the stop signal reaction time, which indicates how long it takes reactive inhibition to be triggered and thus for the action to be ceased. Shorter times indicate a person has better reactive inhibition skills, and thus able to more quickly switch from the activation of some response to the abandoning of that goal through reactive inhibition.
Due to the simplicity of the SST some modern researchers are against its use to make broader assumptions about inhibition. The SST’s demands on attention and inhibition are relatively low and simple in nature, unlike many real life situations, which makes them distrusting of its results. However, the SST is thought to be more indicative by many of reactive inhibition as opposed to proactive. In addition, reactive inhibition is thought to involve mechanisms that are not context dependent but generally the same amongst many conditions in which contexts are changing and the original “go-signal” explicitly or implicitly stated.
Relationship with Parkinson's Disease.
Reactive inhibition is negatively affected by Parkinson's disease. People with Parkinson’s disease have difficulty inhibiting their behaviors. It is proposed that levels of Dopamine are directly associated with one’s ability to inhibit. Proper inhibition is believed to be successful at some desired level of dopamine. Using the simon task, researchers showed that inhibitory processes were significantly depleted in Parkinson’s patients who were withdrawing from their medications, and thus experiencing low levels of dopamine. Performance has been also depleted among high levels of dopamine, indicating that there is an ideal middle ground level of dopamine in which reactive inhibition is most successful. These results suggest an association between dopamine levels and reactive inhibition.
Brain involvement.
Reactive inhibition appears to be related to the subthalamic nucleus (STN), particularly within the active inhibition of “overriding the behavior”. The STN is in charge of sending a signal to “inhibit thalamo-cortical activation”. Thus reaction then causes GABA driven inhibitory signals to be sent to the thalamus which inhibits the behavior. Although reactive inhibition is supported by early STN activity relative to the time of responses, proactive inhibition is defined by more continuous STN activity.
One study has also shown that significant damage to the prefrontal cortex, particularly the right superior medial frontal region, can result in a lack of inhibitory control. When this particular region was damaged patients relied more on last second reactive inhibition to avoid performing inappropriate behaviors.
|
[
{
"math_id": 0,
"text": "I'_R = I_R x 10^{-at}"
},
{
"math_id": 1,
"text": "I(t) = I(0) e^{-bt}"
},
{
"math_id": 2,
"text": "b = a \\ln (10)"
},
{
"math_id": 3,
"text": "I(0)"
},
{
"math_id": 4,
"text": "Y(t) = Y(0) - bt"
},
{
"math_id": 5,
"text": "Y(t) = \\ln I(t)"
},
{
"math_id": 6,
"text": "Y(0) = \\ln I(0)"
}
] |
https://en.wikipedia.org/wiki?curid=6428526
|
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