Mathematical Exploration Problem: The Birthday Paradox

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Mathematical Exploration Problem: The Birthday Paradox



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The Birthday Paradox

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It is convenient to write these effects down in terms of what the user might see or experience in terms of functional failures. Examples of these end effects are: full loss of function x, degraded performance, functions in reversed mode, too late functioning, erratic functioning, etc. These numbers prioritize the failure modes together with probability and detectability. Below a typical classification is given. Other classifications are possible. See also hazard analysis. This is important for maintainability control availability of the system and it is especially important for multiple failure scenarios. This may involve dormant failure modes e. It should be made clear how the failure mode or cause can be discovered by an operator under normal system operation or if it can be discovered by the maintenance crew by some diagnostic action or automatic built in system test.

This type of analysis is useful to determine how effective various test processes are at the detection of latent and dormant faults. The method used to accomplish this involves an examination of the applicable failure modes to determine whether or not their effects are detected, and to determine the percentage of failure rate applicable to the failure modes which are detected. The possibility that the detection means may itself fail latently should be accounted for in the coverage analysis as a limiting factor i. Inclusion of the detection coverage in the FMEA can lead to each individual failure that would have been one effect category now being a separate effect category due to the detection coverage possibilities.

Another way to include detection coverage is for the FTA to conservatively assume that no holes in coverage due to latent failure in the detection method affect detection of all failures assigned to the failure effect category of concern. The FMEA can be revised if necessary for those cases where this conservative assumption does not allow the top event probability requirements to be met. Risk is the combination of End Effect Probability And Severity where probability and severity includes the effect on non-detectability dormancy time. This may influence the end effect probability of failure or the worst case effect Severity.

Preliminary Risk levels can be selected based on a risk matrix like shown below, based on Mil. High risk should be indicated to higher level management, who are responsible for final decision-making. While FMEA identifies important hazards in a system, its results may not be comprehensive and the approach has limitations. Challenges around scoping and organisational boundaries appear to be a major factor in this lack of validity. If used as a top-down tool, FMEA may only identify major failure modes in a system. Fault tree analysis FTA is better suited for "top-down" analysis. When used as a "bottom-up" tool FMEA can augment or complement FTA and identify many more causes and failure modes resulting in top-level symptoms.

It is not able to discover complex failure modes involving multiple failures within a subsystem, or to report expected failure intervals of particular failure modes up to the upper level subsystem or system. Additionally, the multiplication of the severity, occurrence and detection rankings may result in rank reversals, where a less serious failure mode receives a higher RPN than a more serious failure mode. The ordinal rankings only say that one ranking is better or worse than another, but not by how much. For instance, a ranking of "2" may not be twice as severe as a ranking of "1", or an "8" may not be twice as severe as a "4", but multiplication treats them as though they are. See Level of measurement for further discussion.

Various solutions to this problems have been proposed, e. The FMEA worksheet is hard to produce, hard to understand and read, as well as hard to maintain. The use of neural network techniques to cluster and visualise failure modes were suggested starting from The diagrams provide a visualisation of the chains of cause and effect, while the FMEA table provides the detailed information about specific events. From Wikipedia, the free encyclopedia. Systematic technique for identification of potential failure modes in a system and their causes and effects. Design Review Based on Failure Mode Eight disciplines problem solving Failure cause Failure mode, effects, and criticality analysis FMECA — Systematic technique for failure analysis Failure modes, effects, and diagnostic analysis FMEDA Failure rate — Frequency with which an engineered system or component fails Fault tree analysis — Failure analysis system used in safety engineering and reliability engineering Hazard analysis and critical control points — Systematic preventive approach to food safety High availability — systems with high up-time, a.

Fuzzy Optimization and Decision Making. S2CID Koch, John E. Pasadena, California: Jet Propulsion Laboratory. Retrieved Performing a Failure Mode and Effects Analysis pdf. Goddard Space Flight Center. MIL-P — Procedures for performing a failure mode effect and critical analysis. Department of Defense US. Department of Defense USA. Archived from the original on 22 July Westinghouse Electric Corporation Astronuclear Laboratory. General Electric Company. RM 63TMP— National Aeronautics and Space Administration.

Marshall Space Flight Center. Society for Automotive Engineers. Little; Earl G. Hoard; Alfred C. Taylor; Rayford Campbell TM X— United States Environmental Protection Agency. EPA R2—73— December — January FoodSafety Magazine : 42, 44— Matsumoto; Y. Goto SAE Technical Paper Potential Failure Mode and Effect Analysis. Automotive Industry Action Group. ISBN SAE International. Embedded Technology. Archived from the original on Logistics: Principles and Applications.

McGraw Hill. Reliability Analysis Center. PMC PMID Journal of Patient Safety. Journal of Mechanical Design. Bibcode : ISysJ.. Applied Soft Computing. Neural Computing and Applications. Expert Systems with Applications. As Einstein later said, the reason for the development of general relativity was that the preference of inertial motions within special relativity was unsatisfactory, while a theory which from the outset prefers no state of motion even accelerated ones should appear more satisfactory. In that article titled "On the Relativity Principle and the Conclusions Drawn from It", he argued that free fall is really inertial motion, and that for a free-falling observer the rules of special relativity must apply.

This argument is called the equivalence principle. In the same article, Einstein also predicted the phenomena of gravitational time dilation , gravitational redshift and deflection of light. In , Einstein published another article "On the Influence of Gravitation on the Propagation of Light" expanding on the article, in which he estimated the amount of deflection of light by massive bodies. Thus, the theoretical prediction of general relativity could for the first time be tested experimentally. In , Einstein predicted gravitational waves , [] [] ripples in the curvature of spacetime which propagate as waves , traveling outward from the source, transporting energy as gravitational radiation.

The existence of gravitational waves is possible under general relativity due to its Lorentz invariance which brings the concept of a finite speed of propagation of the physical interactions of gravity with it. By contrast, gravitational waves cannot exist in the Newtonian theory of gravitation , which postulates that the physical interactions of gravity propagate at infinite speed. While developing general relativity, Einstein became confused about the gauge invariance in the theory. He formulated an argument that led him to conclude that a general relativistic field theory is impossible. He gave up looking for fully generally covariant tensor equations and searched for equations that would be invariant under general linear transformations only.

In June , the Entwurf 'draft' theory was the result of these investigations. As its name suggests, it was a sketch of a theory, less elegant and more difficult than general relativity, with the equations of motion supplemented by additional gauge fixing conditions. After more than two years of intensive work, Einstein realized that the hole argument was mistaken [] and abandoned the theory in November In , Einstein applied the general theory of relativity to the structure of the universe as a whole. As observational evidence for a dynamic universe was not known at the time, Einstein introduced a new term, the cosmological constant , to the field equations, in order to allow the theory to predict a static universe.

The modified field equations predicted a static universe of closed curvature, in accordance with Einstein's understanding of Mach's principle in these years. This model became known as the Einstein World or Einstein's static universe. Following the discovery of the recession of the nebulae by Edwin Hubble in , Einstein abandoned his static model of the universe, and proposed two dynamic models of the cosmos, The Friedmann-Einstein universe of [] [] and the Einstein—de Sitter universe of In many Einstein biographies, it is claimed that Einstein referred to the cosmological constant in later years as his "biggest blunder".

The astrophysicist Mario Livio has recently cast doubt on this claim, suggesting that it may be exaggerated. In late , a team led by the Irish physicist Cormac O'Raifeartaigh discovered evidence that, shortly after learning of Hubble's observations of the recession of the nebulae, Einstein considered a steady-state model of the universe. For the density to remain constant, new particles of matter must be continually formed in the volume from space.

It thus appears that Einstein considered a steady-state model of the expanding universe many years before Hoyle, Bondi and Gold. General relativity includes a dynamical spacetime, so it is difficult to see how to identify the conserved energy and momentum. Noether's theorem allows these quantities to be determined from a Lagrangian with translation invariance , but general covariance makes translation invariance into something of a gauge symmetry. The energy and momentum derived within general relativity by Noether's prescriptions do not make a real tensor for this reason.

Einstein argued that this is true for a fundamental reason: the gravitational field could be made to vanish by a choice of coordinates. He maintained that the non-covariant energy momentum pseudotensor was, in fact, the best description of the energy momentum distribution in a gravitational field. This approach has been echoed by Lev Landau and Evgeny Lifshitz , and others, and has become standard. In , Einstein collaborated with Nathan Rosen to produce a model of a wormhole , often called Einstein—Rosen bridges.

These solutions cut and pasted Schwarzschild black holes to make a bridge between two patches. If one end of a wormhole was positively charged, the other end would be negatively charged. These properties led Einstein to believe that pairs of particles and antiparticles could be described in this way. In order to incorporate spinning point particles into general relativity, the affine connection needed to be generalized to include an antisymmetric part, called the torsion. This modification was made by Einstein and Cartan in the s. The theory of general relativity has a fundamental law—the Einstein field equations , which describe how space curves.

The geodesic equation , which describes how particles move, may be derived from the Einstein field equations. Since the equations of general relativity are non-linear, a lump of energy made out of pure gravitational fields, like a black hole, would move on a trajectory which is determined by the Einstein field equations themselves, not by a new law. So Einstein proposed that the path of a singular solution, like a black hole, would be determined to be a geodesic from general relativity itself.

This was established by Einstein, Infeld, and Hoffmann for pointlike objects without angular momentum, and by Roy Kerr for spinning objects. In a paper, [] Einstein postulated that light itself consists of localized particles quanta. Einstein's light quanta were nearly universally rejected by all physicists, including Max Planck and Niels Bohr. This idea only became universally accepted in , with Robert Millikan 's detailed experiments on the photoelectric effect, and with the measurement of Compton scattering.

Einstein concluded that each wave of frequency f is associated with a collection of photons with energy hf each, where h is Planck's constant. He does not say much more, because he is not sure how the particles are related to the wave. But he does suggest that this idea would explain certain experimental results, notably the photoelectric effect. In , Einstein proposed a model of matter where each atom in a lattice structure is an independent harmonic oscillator. In the Einstein model, each atom oscillates independently—a series of equally spaced quantized states for each oscillator.

Einstein was aware that getting the frequency of the actual oscillations would be difficult, but he nevertheless proposed this theory because it was a particularly clear demonstration that quantum mechanics could solve the specific heat problem in classical mechanics. Peter Debye refined this model. Throughout the s, quantum mechanics expanded in scope to cover many different systems. After Ernest Rutherford discovered the nucleus and proposed that electrons orbit like planets, Niels Bohr was able to show that the same quantum mechanical postulates introduced by Planck and developed by Einstein would explain the discrete motion of electrons in atoms, and the periodic table of the elements. Einstein contributed to these developments by linking them with the arguments Wilhelm Wien had made.

Wien had shown that the hypothesis of adiabatic invariance of a thermal equilibrium state allows all the blackbody curves at different temperature to be derived from one another by a simple shifting process. Einstein noted in that the same adiabatic principle shows that the quantity which is quantized in any mechanical motion must be an adiabatic invariant. Arnold Sommerfeld identified this adiabatic invariant as the action variable of classical mechanics. In , Einstein received a description of a statistical model from Indian physicist Satyendra Nath Bose , based on a counting method that assumed that light could be understood as a gas of indistinguishable particles.

Einstein also published his own articles describing the model and its implications, among them the Bose—Einstein condensate phenomenon that some particulates should appear at very low temperatures. Einstein's sketches for this project may be seen in the Einstein Archive in the library of the Leiden University. Although the patent office promoted Einstein to Technical Examiner Second Class in , he had not given up on academia. In , he became a Privatdozent at the University of Bern.

This paper introduced the photon concept although the name photon was introduced later by Gilbert N. Lewis in and inspired the notion of wave—particle duality in quantum mechanics. Einstein saw this wave—particle duality in radiation as concrete evidence for his conviction that physics needed a new, unified foundation. In a series of works completed from to , Planck reformulated his quantum theory and introduced the idea of zero-point energy in his "second quantum theory".

Soon, this idea attracted the attention of Einstein and his assistant Otto Stern. Assuming the energy of rotating diatomic molecules contains zero-point energy, they then compared the theoretical specific heat of hydrogen gas with the experimental data. The numbers matched nicely. However, after publishing the findings, they promptly withdrew their support, because they no longer had confidence in the correctness of the idea of zero-point energy. In , at the height of his work on relativity, Einstein published an article in Physikalische Zeitschrift that proposed the possibility of stimulated emission , the physical process that makes possible the maser and the laser.

This paper was enormously influential in the later development of quantum mechanics, because it was the first paper to show that the statistics of atomic transitions had simple laws. Einstein discovered Louis de Broglie 's work and supported his ideas, which were received skeptically at first. In another major paper from this era, Einstein gave a wave equation for de Broglie waves , which Einstein suggested was the Hamilton—Jacobi equation of mechanics. Einstein played a major role in developing quantum theory, beginning with his paper on the photoelectric effect. However, he became displeased with modern quantum mechanics as it had evolved after , despite its acceptance by other physicists. He was skeptical that the randomness of quantum mechanics was fundamental rather than the result of determinism, stating that God "is not playing at dice".

The Bohr—Einstein debates were a series of public disputes about quantum mechanics between Einstein and Niels Bohr , who were two of its founders. Their debates are remembered because of their importance to the philosophy of science. In , Einstein returned to quantum mechanics, in particular to the question of its completeness, in the "EPR paper". No matter how far the two particles were separated, a precise position measurement on one particle would result in equally precise knowledge of the position of the other particle; likewise a precise momentum measurement of one particle would result in equally precise knowledge of the momentum of the other particle, without needing to disturb the other particle in any way.

Given Einstein's concept of local realism , there were two possibilities: 1 either the other particle had these properties already determined, or 2 the process of measuring the first particle instantaneously affected the reality of the position and momentum of the second particle. Einstein rejected this second possibility popularly called "spooky action at a distance". Einstein's belief in local realism led him to assert that, while the correctness of quantum mechanics was not in question, it must be incomplete. But as a physical principle, local realism was shown to be incorrect when the Aspect experiment of confirmed Bell's theorem , which J. Bell had delineated in The results of these and subsequent experiments demonstrate that quantum physics cannot be represented by any version of the picture of physics in which "particles are regarded as unconnected independent classical-like entities, each one being unable to communicate with the other after they have separated.

Although Einstein was wrong about local realism, his clear prediction of the unusual properties of its opposite, entangled quantum states , has resulted in the EPR paper becoming among the top ten papers published in Physical Review. It is considered a centerpiece of the development of quantum information theory. Following his research on general relativity, Einstein attempted to generalize his theory of gravitation to include electromagnetism as aspects of a single entity.

In , he described his " unified field theory " in a Scientific American article titled "On the Generalized Theory of Gravitation". Notably, Einstein's unification project did not accommodate the strong and weak nuclear forces , neither of which were well understood until many years after his death. Although mainstream physics long ignored Einstein's approaches to unification, Einstein's work has motivated modern quests for a theory of everything , in particular string theory , where geometrical fields emerge in a unified quantum-mechanical setting.

Einstein conducted other investigations that were unsuccessful and abandoned. These pertain to force , superconductivity , and other research. In addition to longtime collaborators Leopold Infeld , Nathan Rosen , Peter Bergmann and others, Einstein also had some one-shot collaborations with various scientists. Einstein and De Haas demonstrated that magnetization is due to the motion of electrons, nowadays known to be the spin. In order to show this, they reversed the magnetization in an iron bar suspended on a torsion pendulum.

They confirmed that this leads the bar to rotate, because the electron's angular momentum changes as the magnetization changes. This experiment needed to be sensitive because the angular momentum associated with electrons is small, but it definitively established that electron motion of some kind is responsible for magnetization. Then to each possible quantum motion of a particle in a box associate an independent harmonic oscillator. Quantizing these oscillators, each level will have an integer occupation number, which will be the number of particles in it. This formulation is a form of second quantization , but it predates modern quantum mechanics. This absorption refrigerator was then revolutionary for having no moving parts and using only heat as an input.

Their invention was not immediately put into commercial production, and the most promising of their patents were acquired by the Swedish company Electrolux. While traveling, Einstein wrote daily to his wife Elsa and adopted stepdaughters Margot and Ilse. The letters were included in the papers bequeathed to the Hebrew University of Jerusalem. Margot Einstein permitted the personal letters to be made available to the public, but requested that it not be done until twenty years after her death she died in []. Barbara Wolff, of the Hebrew University's Albert Einstein Archives , told the BBC that there are about 3, pages of private correspondence written between and Einstein's right of publicity was litigated in in a federal district court in California.

Although the court initially held that the right had expired, [] that ruling was immediately appealed, and the decision was later vacated in its entirety. The underlying claims between the parties in that lawsuit were ultimately settled. The right is enforceable, and the Hebrew University of Jerusalem is the exclusive representative of that right. Einstein became one of the most famous scientific celebrities , [] [] beginning with the confirmation of his theory of general relativity in In the period before World War II, The New Yorker published a vignette in their "The Talk of the Town" feature saying that Einstein was so well known in America that he would be stopped on the street by people wanting him to explain "that theory".

He finally figured out a way to handle the incessant inquiries. He told his inquirers "Pardon me, sorry! Always I am mistaken for Professor Einstein. Einstein has been the subject of or inspiration for many novels, films, plays, and works of music. Time magazine's Frederic Golden wrote that Einstein was "a cartoonist's dream come true". Many popular quotations are often misattributed to him. Einstein received numerous awards and honors, and in , he was awarded the Nobel Prize in Physics "for his services to Theoretical Physics, and especially for his discovery of the law of the photoelectric effect". None of the nominations in met the criteria set by Alfred Nobel , so the prize was carried forward and awarded to Einstein in From Wikipedia, the free encyclopedia.

Redirected from Einstein. German-born scientist who developed the theory of relativity — For other uses, see Einstein disambiguation and Albert Einstein disambiguation. Princeton , New Jersey , U. Virtually all modern physicists. See also: Einstein family. Main article: Political views of Albert Einstein. Main article: Religious and philosophical views of Albert Einstein. Main articles: Statistical mechanics , thermal fluctuations , and statistical physics. Main article: Critical opalescence. Main article: History of special relativity. Main article: History of general relativity. See also: Theory of relativity and Einstein field equations.

Main article: Physical cosmology. Main article: Stress—energy—momentum pseudotensor. Main article: Einstein—Cartan theory. Main article: Einstein—Infeld—Hoffmann equations. Main article: Old quantum theory. Main article: Einstein solid. Main article: Adiabatic invariant. Main article: Bose—Einstein statistics. Main article: Bohr—Einstein debates. Main article: Classical unified field theories. Main article: Einstein's unsuccessful investigations. Main article: Einstein—de Haas effect. Main article: Albert Einstein in popular culture. Main article: Einstein's awards and honors. Further information: List of scientific publications by Albert Einstein. Einstein, Albert [Manuscript received: 16 December ].

Written at Zurich, Switzerland. Annalen der Physik in German. Hoboken, New Jersey published 14 March Bibcode : AnP Einstein, Albert a [Manuscript received: 18 March ]. Written at Berne, Switzerland. Hoboken, New Jersey published 10 March Einstein, Albert b [Completed 30 April and submitted 20 July ]. Written at Berne, Switzerland, published by Wyss Buchdruckerei. Einstein, Albert c [Manuscript received: 11 May ]. Einstein, Albert d [Manuscript received: 30 June ]. Annalen der Physik Submitted manuscript in German. Einstein, Albert e [Manuscript received: 27 September ].

Einstein, Albert [Published 25 November ]. Sitzungsberichte in German. Einstein, Albert 22 June Bibcode : SPAW Retrieved 14 November Einstein, Albert a. Einstein, Albert b. Physikalische Zeitschrift in German. Bibcode : PhyZ Einstein, Albert 31 January Einstein, Albert [First published , in English ]. Written at Gothenburg. Nobel Lectures, Physics — in German and English. Stockholm: Nobelprice. Einstein, Albert [Published 10 July ]. Archived from the original Online page images on 14 October First of a series of papers on this topic. Written at Berlin. Die Naturwissenschaften in German. Heidelberg, Germany. Bibcode : NW ISSN S2CID Translated by Cowper, A. US: Dover Publications published ISBN Retrieved 4 January Einstein, Albert Sonderasugabe aus den Sitzungsb.

Einstein, A. Proceedings of the National Academy of Sciences. Bibcode : PNAS PMC PMID Einstein, Albert; Rosen, Nathan Physical Review. Bibcode : PhRv Physical Review Submitted manuscript. Scientific American. Bibcode : SciAm. Ideas and Opinions. New York: Crown Publishers. New York: Three Rivers Press. Munich: Nymphenburger Verlagshandlung. Stachel, John ; Martin J. Klein; A. Kox; Michel Janssen; R. Schulmann; Diana Komos Buchwald; et al. The Collected Papers of Albert Einstein. Princeton University Press. Further information about the volumes published so far can be found on the webpages of the Einstein Papers Project and on the Princeton University Press Einstein Page.

Einstein, Albert; et al. The New York Times. Melville, New York. Archived from the original on 17 December Retrieved 25 May Einstein, Albert May Sweezy, Paul; Huberman, Leo eds. Monthly Review. Reprise ". New York: Monthly Review Foundation. Archived from the original on 11 January Retrieved 16 January — via MonthlyReview. Autobiographical Notes. Paul Arthur Schilpp Centennial ed. Chicago: Open Court. The chasing a light beam thought experiment is described on pages 48— The center was once the Palmer Physical Laboratory. She has chosen the cream of her culture and has suppressed it.

She has even turned upon her most glorious citizen, Albert Einstein, who is the supreme example of the selfless intellectual The man, who, beyond all others, approximates a citizen of the world, is without a home. How proud we must be to offer him temporary shelter. He was quoted as saying that improving the design and changing the types of gases used might allow the design's efficiency to be quadrupled. Oxford University Press. Biographical Memoirs of Fellows of the Royal Society.

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