signal processing - Calculate autocorrelation using FFT in matlab - Stack Overflow
communicated” (quote from the Webster dictionary), which is thus audible .. quency domain description of noise goes via the Wiener-Khintchine theorem. so, amazingly, the autocorrelation is simply given by the Fourier transform of the absolute square of E_nu. The Wiener-Khinchin theorem is a special case of the. Wiener, and a few years later, Khintchine formulated a fundamental theorem of To quote Levinson “most of Wiener's important work was inspired by physics or.
They describe systems without finite characteristic length scale. Primes indeed seem to be distributed among all natural numbers with all possible gaps between them and there is no natural length scale.
We end this short paper by asking the question: Are the prime numbers in a self-organized critical state? However, surprising connections have appeared betwen number theory and physics The work of Ramanujan in particular has had surprising connections with string theory, conformal field theory and statistical physics.
Gopalkrishna Gadiyar and R. Padma, "Ramanujan—Fourier series, the Wiener—Khintchine formula and the distribution of prime pairs"Physica A — "Consider the integers divisible by both p and q [both primes]. This holds, of course, for any number of primes, and we can say using a picturesque but not very precise language, that the primes play a game of chance! This simple, nearly trivial, observation is the beginning of a new development which links in a significant way number theory on the one hand and probability theory on the other.
It may be that by taking into account divisibility by small primes we can obtain a very accurate picture; or it may be that there are other phenomena disturbing the equi-distribution of primes, that await discovery. Granville"Unexpected irregularities in the distribution of primes numbers" "If you choose a number n and ask how many prime numbers there are less than n it turns out that the answer closely approximates the formula: The formula is not exact, though: Riemann looked at these deviations and saw that they contained periodicities.
Berry likens these to musical harmonics: Amazingly, these harmonics or magic numbers behave exactly like the energy levels in quantum systems that classically would be chaotic. Berry and his collaborator Jon Keating used them to show how techniques in number theory can be applied to problems in quantum chaos and vice versa. In itself such a connection is very tantalising. Although sometimes described as the Queen of mathematics, number theory is often thought of as pretty useless, so this deep connection with physics is quite astonishing.
Berry is also convinced that there must be a particular chaotic system which when quantised would have energy levels that exactly duplicate the Riemann numbers. It could play a fundamental role in describing all kinds of chaos. The search for this model system could be the holy grail of chaos It is a tantalising thought.
Julia"Theories statistique et thermodynamique des nombres", in: Conference de Strasbourg en l'honneur de P. This is especially interesting because number theory also provides such a motivation - indeed much of the theory of the Laplace transform Combining these two connections, one is led to physical interpretations of results in number theory as well as applications of number theoretical results to physics.
For some reasons, reflecting the physical nature of our kind of living matter e. We can equally well spiritually project it upon its non-Archimediean side and calculate most important things arithmetically. The relations between "real" and "arithmetical" pictures of the world is that of complementarity, like the relation between conjugate observables in quantum mechanics.
I believe that techniques and philosophy more organic to Quantum Fields will be most relevant. As this point of view has not so far led to success, I will conclude here with extracts from William Blake's "The Marriage of Heaven and Hell" which, to my mind remarkably illustrate some salient points: But first the notion that man has a body distinct from his soul is to be expunged; this I shall do, by printing in the infernal method, by corrosives, which in Hell are salutary and medicinal, melting apparent surfaces away, and displaying the infinite which was hid.
If the doors of perception were cleansed every thing would appear to man as it is, infinite. There is a photograph of him prominently displayed in one of the hallways, often used in giving directions.
InWiener returned to Europe as a Guggenheim scholar. InWiener's parents arranged his marriage to a German immigrant, Margaret Engemann; they had two daughters. His sister, Constance, married Philip Franklin. It was said that he returned home once to find his house empty. He inquired of a neighborhood girl the reason, and she said that the family had moved elsewhere that day. He thanked her for the information and she replied, "That's why I stayed behind, Daddy! To him is due the now standard practice of modeling an information source as a random process—in other words, as a variety of noise.
His anti-aircraft work eventually led him to formulate cybernetics. These men later made pioneering contributions to computer science and artificial intelligence. Soon after the group was formed, Wiener suddenly ended all contact with its members, mystifying his colleagues.
This emotionally traumatized Pitts, and led to his career decline.
In their biography of Wiener, Conway and Siegelman suggest that Wiener's wife Margaret, who detested McCulloch's bohemian lifestyle, engineered the breach. He discussed the modeling of neurons with John von Neumannand in a letter from November von Neumann presented his thoughts in advance of a meeting with Wiener.Pillai "Power Spectrum of Stationary Stochastic Processes" (1/2)
These included Soviet researchers and their findings. Wiener's acquaintance with them caused him to be regarded with suspicion during the Cold War.
Wiener–Khinchin theorem - Wikipedia
He was a strong advocate of automation to improve the standard of living, and to end economic underdevelopment. His ideas became influential in Indiawhose government he advised during the s.
After the war, Wiener became increasingly concerned with what he believed was political interference with scientific research, and the militarization of science. His article "A Scientist Rebels" from the January issue of The Atlantic Monthly  urged scientists to consider the ethical implications of their work. After the war, he refused to accept any government funding or to work on military projects. The way Wiener's beliefs concerning nuclear weapons and the Cold War contrasted with those of von Neumann is the major theme of the book John Von Neumann and Norbert Wiener.
He died in Marchaged 69, in Stockholmfrom a heart attack.