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Alkaloid
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Biosynthesis
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Biological precursors of most alkaloids are amino acids, such as ornithine, lysine, phenylalanine, tyrosine, tryptophan, histidine, aspartic acid, and anthranilic acid. Nicotinic acid can be synthesized from tryptophan or aspartic acid. Ways of alkaloid biosynthesis are too numerous and cannot be easily classified. However, there are a few typical reactions involved in the biosynthesis of various classes of alkaloids, including synthesis of Schiff bases and Mannich reaction.
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Alkaloid
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Biosynthesis
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Synthesis of Schiff bases Schiff bases can be obtained by reacting amines with ketones or aldehydes. These reactions are a common method of producing C=N bonds.
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Alkaloid
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Biosynthesis
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In the biosynthesis of alkaloids, such reactions may take place within a molecule, such as in the synthesis of piperidine: Mannich reaction An integral component of the Mannich reaction, in addition to an amine and a carbonyl compound, is a carbanion, which plays the role of the nucleophile in the nucleophilic addition to the ion formed by the reaction of the amine and the carbonyl.
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Alkaloid
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Biosynthesis
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The Mannich reaction can proceed both intermolecularly and intramolecularly:
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Alkaloid
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Dimer alkaloids
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In addition to the described above monomeric alkaloids, there are also dimeric, and even trimeric and tetrameric alkaloids formed upon condensation of two, three, and four monomeric alkaloids. Dimeric alkaloids are usually formed from monomers of the same type through the following mechanisms: Mannich reaction, resulting in, e.g., voacamine Michael reaction (villalstonine) Condensation of aldehydes with amines (toxiferine) Oxidative addition of phenols (dauricine, tubocurarine) Lactonization (carpaine).
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Alkaloid
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Dimer alkaloids
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There are also dimeric alkaloids formed from two distinct monomers, such as the vinca alkaloids vinblastine and vincristine, which are formed from the coupling of catharanthine and vindoline. The newer semi-synthetic chemotherapeutic agent vinorelbine is used in the treatment of non-small-cell lung cancer. It is another derivative dimer of vindoline and catharanthine and is synthesised from anhydrovinblastine, starting either from leurosine or the monomers themselves.
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Alkaloid
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Biological role
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Alkaloids are among the most important and best-known secondary metabolites, i.e. biogenic substances not directly involved in the normal growth, development, or reproduction of the organism. Instead, they generally mediate ecological interactions, which may produce a selective advantage for the organism by increasing its survivability or fecundity. In some cases their function, if any, remains unclear. An early hypothesis, that alkaloids are the final products of nitrogen metabolism in plants, as urea and uric acid are in mammals, was refuted by the finding that their concentration fluctuates rather than steadily increasing.Most of the known functions of alkaloids are related to protection. For example, aporphine alkaloid liriodenine produced by the tulip tree protects it from parasitic mushrooms. In addition, the presence of alkaloids in the plant prevents insects and chordate animals from eating it. However, some animals are adapted to alkaloids and even use them in their own metabolism. Such alkaloid-related substances as serotonin, dopamine and histamine are important neurotransmitters in animals. Alkaloids are also known to regulate plant growth. One example of an organism that uses alkaloids for protection is the Utetheisa ornatrix, more commonly known as the ornate moth. Pyrrolizidine alkaloids render these larvae and adult moths unpalatable to many of their natural enemies like coccinelid beetles, green lacewings, insectivorous hemiptera and insectivorous bats. Another example of alkaloids being utilized occurs in the poison hemlock moth (Agonopterix alstroemeriana). This moth feeds on its highly toxic and alkaloid-rich host plant poison hemlock (Conium maculatum) during its larval stage. A. alstroemeriana may benefit twofold from the toxicity of the naturally-occurring alkaloids, both through the unpalatability of the species to predators and through the ability of A. alstroemeriana to recognize Conium maculatum as the correct location for oviposition. A fire ant venom alkaloid known as solenopsin has been demonstrated to protect queens of invasive fire ants during the foundation of new nests, thus playing a central role in the spread of this pest ant species around the world.
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Alkaloid
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Applications
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In medicine Medical use of alkaloid-containing plants has a long history, and, thus, when the first alkaloids were isolated in the 19th century, they immediately found application in clinical practice. Many alkaloids are still used in medicine, usually in the form of salts widely used including the following: Many synthetic and semisynthetic drugs are structural modifications of the alkaloids, which were designed to enhance or change the primary effect of the drug and reduce unwanted side-effects. For example, naloxone, an opioid receptor antagonist, is a derivative of thebaine that is present in opium.
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Alkaloid
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Applications
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In agriculture Prior to the development of a wide range of relatively low-toxic synthetic pesticides, some alkaloids, such as salts of nicotine and anabasine, were used as insecticides. Their use was limited by their high toxicity to humans.
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Alkaloid
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Applications
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Use as psychoactive drugs Preparations of plants containing alkaloids and their extracts, and later pure alkaloids, have long been used as psychoactive substances. Cocaine, caffeine, and cathinone are stimulants of the central nervous system. Mescaline and many indole alkaloids (such as psilocybin, dimethyltryptamine and ibogaine) have hallucinogenic effect. Morphine and codeine are strong narcotic pain killers.There are alkaloids that do not have strong psychoactive effect themselves, but are precursors for semi-synthetic psychoactive drugs. For example, ephedrine and pseudoephedrine are used to produce methcathinone and methamphetamine. Thebaine is used in the synthesis of many painkillers such as oxycodone.
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Alkaloid
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General and cited references
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Aniszewski, Tadeusz (2007). Alkaloids: secrets of life. Amsterdam: Elsevier. ISBN 978-0-444-52736-3.
Begley, Tadhg P. (2009). Encyclopedia of Chemical Biology. pp. 1569–1570. doi:10.1002/cbic.200900262. ISBN 978-0-471-75477-0. {{cite book}}: |journal= ignored (help) Brossi, Arnold (1989). The Alkaloids: Chemistry and Pharmacology. Academic Press.
Dewick, Paul M. (2002). Medicinal Natural Products: A Biosynthetic Approach (Second ed.). Wiley. ISBN 978-0-471-49640-3.
Fattorusso, E.; Taglialatela-Scafati, O. (2008). Modern Alkaloids: Structure, Isolation, Synthesis and Biology. Wiley-VCH. ISBN 978-3-527-31521-5.
Grinkevich NI; Safronich LN, eds. (1983). The chemical analysis of medicinal plants (in Russian). Moscow: Vysshaya Shkola.
Hesse, Manfred (2002). Alkaloids: Nature's Curse or Blessing?. Wiley-VCH. ISBN 978-3-906390-24-6.
Knunyants, IL (1988). Chemical Encyclopedia. Soviet Encyclopedia.
Orekhov, AP (1955). Chemistry alkaloids (Acad. 2nd ed.). Moscow.{{cite book}}: CS1 maint: location missing publisher (link) Plemenkov, VV (2001). Introduction to the Chemistry of Natural Compounds. Kazan.{{cite book}}: CS1 maint: location missing publisher (link) Saxton, J. E. (1971). The Alkaloids: A Specialist Periodical Report. London: The Chemical Society.
Veselovskaya, N. B.; Kovalenko, A. E. (2000). Drugs. Moscow: Triada-X.
Wink, M (2009). "Mode of action and toxicology of plant toxins and poisonous plants". Mitt. Julius Kühn-Inst. 421: 93–112x.
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Criticism of the theory of relativity
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Criticism of the theory of relativity
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Criticism of the theory of relativity of Albert Einstein was mainly expressed in the early years after its publication in the early twentieth century, on scientific, pseudoscientific, philosophical, or ideological bases. Though some of these criticisms had the support of reputable scientists, Einstein's theory of relativity is now accepted by the scientific community.Reasons for criticism of the theory of relativity have included alternative theories, rejection of the abstract-mathematical method, and alleged errors of the theory. According to some authors, antisemitic objections to Einstein's Jewish heritage also occasionally played a role in these objections. There are still some critics of relativity today, but their opinions are not shared by the majority in the scientific community.
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Criticism of the theory of relativity
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Special relativity
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Relativity principle versus electromagnetic worldview Around the end of the 19th century, the view was widespread that all forces in nature are of electromagnetic origin (the "electromagnetic worldview"), especially in the works of Joseph Larmor (1897) and Wilhelm Wien (1900). This was apparently confirmed by the experiments of Walter Kaufmann (1901–1903), who measured an increase of the mass of a body with velocity which was consistent with the hypothesis that the mass was generated by its electromagnetic field. Max Abraham (1902) subsequently sketched a theoretical explanation of Kaufmann's result in which the electron was considered as rigid and spherical. However, it was found that this model was incompatible with the results of many experiments (including the Michelson–Morley experiment, the Experiments of Rayleigh and Brace, and the Trouton–Noble experiment), according to which no motion of an observer with respect to the luminiferous aether ("aether drift") had been observed despite numerous attempts to do so. Henri Poincaré (1902) conjectured that this failure arose from a general law of nature, which he called "the principle of relativity". Hendrik Antoon Lorentz (1904) created a detailed theory of electrodynamics (Lorentz ether theory) that was premised on the existence of an immobile aether and employed a set of space and time coordinate transformations that Poincaré called the Lorentz transformations, including the effects of length contraction and local time. However, Lorentz's theory only partially satisfied the relativity principle, because his transformation formulas for velocity and charge density were incorrect. This was corrected by Poincaré (1905) who obtained full Lorentz covariance of the electrodynamic equations.Criticizing Lorentz's 1904 theory, Abraham (1904) held that the Lorentz contraction of electrons requires a non-electromagnetic force to ensure the electron's stability. This was unacceptable to him as a proponent of the electromagnetic worldview. He continued that as long as a consistent explanation is missing as to how those forces and potentials act together on the electron, Lorentz's system of hypotheses is incomplete and doesn't satisfy the relativity principle. Poincaré (1905) removed this objection by showing that the non-electromagnetic potential ("Poincaré stress") holding the electron together can be formulated in a Lorentz covariant way, and showed that in principle it is possible to create a Lorentz covariant model for gravitation which he considered non-electromagnetic in nature as well. Thus the consistency of Lorentz's theory was proven, but the electromagnetic worldview had to be given up. Eventually, Albert Einstein published in September 1905 what is now called special relativity, which was based on a radical new application of the relativity principle in connection with the constancy of the speed of light. In special relativity, the space and time coordinates depend on the inertial observer's frame of reference, and the luminiferous aether plays no role in the physics. Although this theory was founded on a very different kinematical model, it was experimentally indistinguishable from the aether theory of Lorentz and Poincaré, since both theories satisfy the relativity principle of Poincaré and Einstein, and both employ the Lorentz transformations. After Minkowski's introduction in 1908 of the geometric spacetime model for Einstein's version of relativity, most physicists eventually decided in favor of the Einstein-Minkowski version of relativity with its radical new views of space and time, in which there was no useful role for the aether.
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Criticism of the theory of relativity
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Special relativity
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Claimed experimental refutations Kaufmann–Bucherer–Neumann experiments: To conclusively decide between the theories of Abraham and Lorentz, Kaufmann repeated his experiments in 1905 with improved accuracy. However, in the meantime the theoretical situation had changed. Alfred Bucherer and Paul Langevin (1904) developed another model, in which the electron is contracted in the line of motion, and dilated in the transverse direction, so that the volume remains constant. While Kaufmann was still evaluating his experiments, Einstein published his theory of special relativity. Eventually, Kaufmann published his results in December 1905 and argued that they are in agreement with Abraham's theory and require rejection of the "basic assumption of Lorentz and Einstein" (the relativity principle). Lorentz reacted with the phrase "I am at the end of my Latin", while Einstein did not mention those experiments before 1908. Yet, others started to criticize the experiments. Max Planck (1906) alluded to inconsistencies in the theoretical interpretation of the data, and Adolf Bestelmeyer (1906) introduced new techniques, which (especially in the area of low velocities) gave different results and which cast doubts on Kaufmann's methods. Therefore, Bucherer (1908) conducted new experiments and arrived at the conclusion that they confirm the mass formula of relativity and thus the "relativity principle of Lorentz and Einstein". Yet Bucherer's experiments were criticized by Bestelmeyer leading to a sharp dispute between the two experimentalists. On the other hand, additional experiments of Hupka (1910), Neumann (1914) and others seemed to confirm Bucherer's result. The doubts lasted until 1940, when in similar experiments Abraham's theory was conclusively disproved. (It must be remarked that besides those experiments, the relativistic mass formula had already been confirmed by 1917 in the course of investigations on the theory of spectra. In modern particle accelerators, the relativistic mass formula is routinely confirmed.)In 1902–1906, Dayton Miller repeated the Michelson–Morley experiment together with Edward W. Morley. They confirmed the null result of the initial experiment. However, in 1921–1926, Miller conducted new experiments which apparently gave positive results. Those experiments initially attracted some attention in the media and in the scientific community but have been considered refuted for the following reasons: Einstein, Max Born, and Robert S. Shankland pointed out that Miller had not appropriately considered the influence of temperature. A modern analysis by Roberts shows that Miller's experiment gives a null result, when the technical shortcomings of the apparatus and the error bars are properly considered. Additionally, Miller's result is in disagreement with all other experiments, which were conducted before and after. For example, Georg Joos (1930) used an apparatus of similar dimensions to Miller's, but he obtained null results. In recent experiments of Michelson–Morley type where the coherence length is increased considerably by using lasers and masers the results are still negative.
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Criticism of the theory of relativity
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Special relativity
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In the 2011 Faster-than-light neutrino anomaly, the OPERA collaboration published results which appeared to show that the speed of neutrinos is slightly faster than the speed of light. However, sources of errors were found and confirmed in 2012 by the OPERA collaboration, which fully explained the initial results. In their final publication, a neutrino speed consistent with the speed of light was stated. Also subsequent experiments found agreement with the speed of light, see measurements of neutrino speed.
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Criticism of the theory of relativity
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Special relativity
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Acceleration in special relativity It was also claimed that special relativity cannot handle acceleration, which would lead to contradictions in some situations. However, this assessment is not correct, since acceleration actually can be described in the framework of special relativity (see Acceleration (special relativity), Proper reference frame (flat spacetime), Hyperbolic motion, Rindler coordinates, Born coordinates). Paradoxes relying on insufficient understanding of these facts were discovered in the early years of relativity. For example, Max Born (1909) tried to combine the concept of rigid bodies with special relativity. That this model was insufficient was shown by Paul Ehrenfest (1909), who demonstrated that a rotating rigid body would, according to Born's definition, undergo a contraction of the circumference without contraction of the radius, which is impossible (Ehrenfest paradox). Max von Laue (1911) showed that rigid bodies cannot exist in special relativity, since the propagation of signals cannot exceed the speed of light, so an accelerating and rotating body will undergo deformations.Paul Langevin and von Laue showed that the twin paradox can be completely resolved by consideration of acceleration in special relativity. If two twins move away from each other, and one of them is accelerating and coming back to the other, then the accelerated twin is younger than the other one, since he was located in at least two inertial frames of reference, and therefore his assessment of which events are simultaneous changed during the acceleration. For the other twin nothing changes since he remained in a single frame.Another example is the Sagnac effect. Two signals were sent in opposite directions around a rotating platform. After their arrival a displacement of the interference fringes occurs. Sagnac himself believed that he had proved the existence of the aether. However, special relativity can easily explain this effect. When viewed from an inertial frame of reference, it is a simple consequence of the independence of the speed of light from the speed of the source, since the receiver runs away from one beam, while it approaches the other beam. When viewed from a rotating frame, the assessment of simultaneity changes during the rotation, and consequently the speed of light is not constant in accelerated frames.As was shown by Einstein, the only form of accelerated motion that cannot be non-locally described is the one due to gravitation. Einstein was also unsatisfied with the fact that inertial frames are preferred over accelerated frames. Thus over the course of several years (1908–1915), Einstein developed general relativity. This theory includes the replacement of Euclidean geometry by non-Euclidean geometry, and the resultant curvature of the path of light led Einstein (1912) to the conclusion that (like in extended accelerated frames) the speed of light is not constant in extended gravitational fields. Therefore, Abraham (1912) argued that Einstein had given special relativity a coup de grâce. Einstein responded that within its area of application (in areas where gravitational influences can be neglected) special relativity is still applicable with high precision, so one cannot speak of a coup de grâce at all.
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Criticism of the theory of relativity
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Special relativity
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Superluminal speeds In special relativity, the transfer of signals at superluminal speeds is impossible, since this would violate the Poincaré-Einstein synchronization, and the causality principle. Following an old argument by Pierre-Simon Laplace, Poincaré (1904) alluded to the fact that Newton's law of universal gravitation is founded on an infinitely great speed of gravity. So the clock-synchronization by light signals could in principle be replaced by a clock-synchronization by instantaneous gravitational signals. In 1905, Poincaré himself solved this problem by showing that in a relativistic theory of gravity the speed of gravity is equal to the speed of light. Although much more complicated, this is also the case in Einstein's theory of general relativity.Another apparent contradiction lies in the fact that the group velocity in anomalously dispersive media is higher than the speed of light. This was investigated by Arnold Sommerfeld (1907, 1914) and Léon Brillouin (1914). They came to the conclusion that in such cases the signal velocity is not equal to the group velocity, but to the front velocity which is never faster than the speed of light. Similarly, it is also argued that the apparent superluminal effects discovered by Günter Nimtz can be explained by a thorough consideration of the velocities involved.Also quantum entanglement (denoted by Einstein as "spooky action at a distance"), according to which the quantum state of one entangled particle cannot be fully described without describing the other particle, does not imply superluminal transmission of information (see quantum teleportation), and it is therefore in conformity with special relativity.
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Criticism of the theory of relativity
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Special relativity
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Paradoxes Insufficient knowledge of the basics of special relativity, especially the application of the Lorentz transformation in connection with length contraction and time dilation, led and still leads to the construction of various apparent paradoxes. Both the twin paradox and the Ehrenfest paradox and their explanation were already mentioned above. Besides the twin paradox, also the reciprocity of time dilation (i.e. every inertially moving observer considers the clock of the other one as being dilated) was heavily criticized by Herbert Dingle and others. For example, Dingle wrote a series of letters to Nature at the end of the 1950s. However, the self-consistency of the reciprocity of time dilation had already been demonstrated long before in an illustrative way by Lorentz (in his lectures from 1910, published 1931) and many others—they alluded to the fact that it is only necessary to carefully consider the relevant measurement rules and the relativity of simultaneity. Other known paradoxes are the Ladder paradox and Bell's spaceship paradox, which also can simply be solved by consideration of the relativity of simultaneity.
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Criticism of the theory of relativity
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Special relativity
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Aether and absolute space Many physicists (like Hendrik Lorentz, Oliver Lodge, Albert Abraham Michelson, Edmund Taylor Whittaker, Harry Bateman, Ebenezer Cunningham, Charles Émile Picard, Paul Painlevé) were uncomfortable with the rejection of the aether, and preferred to interpret the Lorentz transformation based on the existence of a preferred frame of reference, as in the aether-based theories of Lorentz, Larmor, and Poincaré. However, the idea of an aether hidden from any observation was not supported by the mainstream scientific community, therefore the aether theory of Lorentz and Poincaré was superseded by Einstein's special relativity which was subsequently formulated in the framework of four-dimensional spacetime by Minkowski.Others such as Herbert E. Ives argued that it might be possible to experimentally determine the motion of such an aether, but it was never found despite numerous experimental tests of Lorentz invariance (see tests of special relativity).
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Criticism of the theory of relativity
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Special relativity
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Also attempts to introduce some sort of relativistic aether (consistent with relativity) into modern physics such as by Einstein on the basis of general relativity (1920), or by Paul Dirac in relation to quantum mechanics (1951), were not supported by the scientific community (see Luminiferous aether#End of aether?).In his Nobel lecture, George F. Smoot (2006) described his own experiments on the Cosmic microwave background radiation anisotropy as "New Aether drift experiments". Smoot explained that "one problem to overcome was the strong prejudice of good scientists who learned the lesson of the Michelson and Morley experiment and Special Relativity that there were no preferred frames of reference." He continued that "there was an education job to convince them that this did not violate Special Relativity but did find a frame in which the expansion of the universe looked particularly simple." Alternative theories The theory of complete aether drag, as proposed by George Gabriel Stokes (1844), was used by some critics as Ludwig Silberstein (1920) or Philipp Lenard (1920) as a counter-model of relativity. In this theory, the aether was completely dragged within and in the vicinity of matter, and it was believed that various phenomena, such as the absence of aether drift, could be explained in an "illustrative" way by this model. However, such theories are subject to great difficulties. Especially the aberration of light contradicted the theory, and all auxiliary hypotheses, which were invented to rescue it, are self-contradictory, extremely implausible, or in contradiction to other experiments like the Michelson–Gale–Pearson experiment. In summary, a sound mathematical and physical model of complete aether drag was never invented, consequently this theory was no serious alternative to relativity.Another alternative was the so-called emission theory of light. As in special relativity the aether concept is discarded, yet the main difference from relativity lies in the fact that the velocity of the light source is added to that of light in accordance with the Galilean transformation. As the hypothesis of complete aether drag, it can explain the negative outcome of all aether drift experiments. Yet, there are various experiments that contradict this theory. For example, the Sagnac effect is based on the independence of light speed from the source velocity, and the image of Double stars should be scrambled according to this model—which was not observed. Also in modern experiments in particle accelerators no such velocity dependence could be observed. These results are further confirmed by the De Sitter double star experiment (1913), conclusively repeated in the X-ray spectrum by K. Brecher in 1977; and the terrestrial experiment by Alväger, et al. (1963);, which all show that the speed of light is independent of the motion of the source within the limits of experimental accuracy.
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Criticism of the theory of relativity
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Special relativity
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Principle of the constancy of the speed of light Some consider the principle of the constancy of the velocity of light insufficiently substantiated. However, as already shown by Robert Daniel Carmichael (1910) and others, the constancy of the speed of light can be interpreted as a natural consequence of two experimentally demonstrated facts: The velocity of light is independent of the velocity of the source, as demonstrated by De Sitter double star experiment, Sagnac effect, and many others (see emission theory).
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Criticism of the theory of relativity
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Special relativity
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The velocity of light is independent of the direction of velocity of the observer, as demonstrated by Michelson–Morley experiment, Kennedy–Thorndike experiment, and many others (see luminiferous aether).Note that measurements regarding the speed of light are actually measurements of the two-way speed of light, since the one-way speed of light depends on which convention is chosen to synchronize the clocks.
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Criticism of the theory of relativity
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General relativity
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General covariance Einstein emphasized the importance of general covariance for the development of general relativity, and took the position that the general covariance of his 1915 theory of gravity ensured implementation of a generalized relativity principle. This view was challenged by Erich Kretschmann (1917), who argued that every theory of space and time (even including Newtonian dynamics) can be formulated in a covariant way, if additional parameters are included, and thus general covariance of a theory would in itself be insufficient to implement a generalized relativity principle. Although Einstein (1918) agreed with that argument, he also countered that Newtonian mechanics in general covariant form would be too complicated for practical uses. Although it is now understood that Einstein's response to Kretschmann was mistaken (subsequent papers showed that such a theory would still be usable), another argument can be made in favor of general covariance: it is a natural way to express the equivalence principle, i.e., the equivalence in the description of a free-falling observer and an observer at rest, and thus it is more convenient to use general covariance together with general relativity, rather than with Newtonian mechanics. Connected with this, also the question of absolute motion was dealt with. Einstein argued that the general covariance of his theory of gravity supports Mach's principle, which would eliminate any "absolute motion" within general relativity. However, as pointed out by Willem de Sitter in 1916, Mach's principle is not completely fulfilled in general relativity because there exist matter-free solutions of the field equations. This means that the "inertio-gravitational field", which describes both gravity and inertia, can exist in the absence of gravitating matter. However, as pointed out by Einstein, there is one fundamental difference between this concept and absolute space of Newton: the inertio-gravitational field of general relativity is determined by matter, thus it is not absolute.
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Criticism of the theory of relativity
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General relativity
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Bad Nauheim Debate In the "Bad Nauheim Debate" (1920) between Einstein and (among others) Philipp Lenard, the latter stated the following objections: He criticized the lack of "illustrativeness" of Einstein's version of relativity, a condition that he suggested could only be met by an aether theory. Einstein responded that for physicists the content of "illustrativeness" or "common sense" had changed in time, so it could no longer be used as a criterion for the validity of a physical theory. Lenard also argued that with his relativistic theory of gravity Einstein had tacitly reintroduced the aether under the name "space". While this charge was rejected (among others) by Hermann Weyl, in an inaugural address given at the University of Leiden in 1920, shortly after the Bad Nauheim debates, Einstein himself acknowledged that according to his general theory of relativity, so-called "empty space" possesses physical properties that influence matter and vice versa. Lenard also argued that Einstein's general theory of relativity admits the existence of superluminal velocities, in contradiction to the principles of special relativity; for example, in a rotating coordinate system in which the Earth is at rest, the distant points of the whole universe are rotating around Earth with superluminal velocities. However, as Weyl pointed out, it is incorrect to handle a rotating extended system as a rigid body (neither in special nor in general relativity)—so the signal velocity of an object never exceeds the speed of light. Another criticism that was raised by both Lenard and Gustav Mie concerned the existence of "fictitious" gravitational fields in accelerating frames, which according to Einstein's Equivalence Principle are no less physically real than those produced by material sources. Lenard and Mie argued that physical forces can only be produced by real material sources, while the gravitational field that Einstein supposed to exist in an accelerating frame of reference has no concrete physical meaning. Einstein responded that, based on Mach's principle, one can think of these gravitational fields as induced by the distant masses. In this respect the criticism of Lenard and Mie has been vindicated, since according to the modern consensus, in agreement with Einstein's own mature views, Mach's principle as originally conceived by Einstein is not actually supported by general relativity, as already mentioned above.
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Criticism of the theory of relativity
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General relativity
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Silberstein–Einstein controversy Ludwik Silberstein, who initially was a supporter of the special theory, objected at different occasions against general relativity. In 1920 he argued that the deflection of light by the sun, as observed by Arthur Eddington et al. (1919), is not necessarily a confirmation of general relativity, but may also be explained by the Stokes-Planck theory of complete aether drag. However, such models are in contradiction with the aberration of light and other experiments (see "Alternative theories"). In 1935, Silberstein claimed to have found a contradiction in the Two-body problem in general relativity. The claim was refuted by Einstein and Rosen (1935).
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Criticism of the theory of relativity
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Philosophical criticism
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The consequences of relativity, such as the change of ordinary concepts of space and time, as well as the introduction of non-Euclidean geometry in general relativity, were criticized by some philosophers of different philosophical schools. Many philosophical critics had insufficient knowledge of the mathematical and formal basis of relativity, which led to the criticisms often missing the heart of the matter. For example, relativity was misinterpreted as some form of relativism. However, this is misleading as it was emphasized by Einstein or Planck. On one hand it's true that space and time became relative, and the inertial frames of reference are handled on equal footing. On the other hand, the theory makes natural laws invariant—examples are the constancy of the speed of light, or the covariance of Maxwell's equations. Consequently, Felix Klein (1910) called it the "invariant theory of the Lorentz group" instead of relativity theory, and Einstein (who reportedly used expressions like "absolute theory") sympathized with this expression as well.Critical responses to relativity were also expressed by proponents of neo-Kantianism (Paul Natorp, Bruno Bauch etc.), and phenomenology (Oskar Becker, Moritz Geiger etc.). While some of them only rejected the philosophical consequences, others rejected also the physical consequences of the theory. Einstein was criticized for violating Immanuel Kant's categoric scheme, i.e., it was claimed that space-time curvature caused by matter and energy is impossible, since matter and energy already require the concepts of space and time. Also the three-dimensionality of space, Euclidean geometry, and the existence of absolute simultaneity were claimed to be necessary for the understanding of the world; none of them can possibly be altered by empirical findings. By moving all those concepts into a metaphysical area, any form of criticism of Kantianism would be prevented. Other pseudo-Kantians like Ernst Cassirer or Hans Reichenbach (1920), tried to modify Kant's philosophy. Subsequently, Reichenbach rejected Kantianism at all and became a proponent of logical positivism.Based on Henri Poincaré's conventionalism, philosophers such as Pierre Duhem (1914) and Hugo Dingler (1920) argued that the classical concepts of space, time, and geometry were, and will always be, the most convenient expressions in natural science, therefore the concepts of relativity cannot be correct. This was criticized by proponents of logical positivism such as Moritz Schlick, Rudolf Carnap, and Reichenbach. They argued that Poincaré's conventionalism could be modified to bring it into accord with relativity. Although it is true that the basic assumptions of Newtonian mechanics are simpler, it can only be brought into accord with modern experiments by inventing auxiliary hypotheses. On the other hand, relativity doesn't need such hypotheses, thus from a conceptual viewpoint, relativity is in fact simpler than Newtonian mechanics.Some proponents of Philosophy of Life, Vitalism, Critical realism (in German speaking countries) argued that there is a fundamental difference between physical, biological and psychological phenomena. For example, Henri Bergson (1921), who otherwise was a proponent of special relativity, argued that time dilation cannot be applied to biological organisms, therefore he denied the relativistic solution of the twin paradox. However, those claims were rejected by Paul Langevin, André Metz and others. Biological organisms consist of physical processes, so there is no reason to assume that they are not subject to relativistic effects like time dilation.Based on the philosophy of Fictionalism, the philosopher Oskar Kraus (1921) and others claimed that the foundations of relativity were only fictitious and even self-contradictory. Examples were the constancy of the speed of light, time dilation, length contraction. These effects appear to be mathematically consistent as a whole, but in reality they allegedly are not true. Yet, this view was immediately rejected. The foundations of relativity (such as the equivalence principle or the relativity principle) are not fictitious, but based on experimental results. Also, effects like constancy of the speed of light and relativity of simultaneity are not contradictory, but complementary to one another.In the Soviet Union (mostly in the 1920s), philosophical criticism was expressed on the basis of dialectic materialism. The theory of relativity was rejected as anti-materialistic and speculative, and a mechanistic worldview based on "common sense" was required as an alternative. Similar criticisms also occurred in the People's Republic of China during the Cultural Revolution. (On the other hand, other philosophers considered relativity as being compatible with Marxism.)
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Criticism of the theory of relativity
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Relativity hype and popular criticism
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Although Planck already in 1909 compared the changes brought about by relativity with the Copernican Revolution, and although special relativity was accepted by most of the theoretical physicists and mathematicians by 1911, it was not before publication of the experimental results of the eclipse expeditions (1919) by a group around Arthur Stanley Eddington that relativity was noticed by the public. Following Eddington's publication of the eclipse results, Einstein was glowingly praised in the mass media, and was compared to Nikolaus Copernicus, Johannes Kepler and Isaac Newton, which caused a popular "relativity hype" ("Relativitätsrummel", as it was called by Sommerfeld, Einstein, and others). This triggered a counter-reaction of some scientists and scientific laymen who could not accept the concepts of modern physics, including relativity theory and quantum mechanics. The ensuing public controversy regarding the scientific status of Einstein's theory of gravity, which was unprecedented, was partly carried out in the press. Some of the criticism was not only directed to relativity, but personally at Einstein as well, who some of his critics accused of being behind the promotional campaign in the German press.
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Criticism of the theory of relativity
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Relativity hype and popular criticism
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Academic and non-academic criticism Some academic scientists, especially experimental physicists such as the Nobel laureates Philipp Lenard and Johannes Stark, as well as Ernst Gehrcke, Stjepan Mohorovičić, Rudolf Tomaschek and others criticized the increasing abstraction and mathematization of modern physics, especially in the form of relativity theory, and later quantum mechanics. It was seen as a tendency to abstract theory building, connected with the loss of intuitive "common sense". In fact, relativity was the first theory, in which the inadequacy of the "illustrative" classical physics was thought to have been demonstrated. Some of Einstein's critics ignored these developments and tried to revitalize older theories, such as aether drag models or emission theories (see "Alternative Theories"). However, those qualitative models were never sufficiently advanced to compete with the success of the precise experimental predictions and explanatory powers of the modern theories. Additionally, there was also a great rivalry between experimental and theoretical physicists, as regards the professorial activities and the occupation of chairs at German universities. The opinions clashed at the "Bad Nauheim debates" in 1920 between Einstein and (among others) Lenard, which attracted much public attention.In addition, there were many critics (with or without physical training) whose ideas were far outside the scientific mainstream. These critics were mostly people who had developed their ideas long before the publication of Einstein's version of relativity, and they tried to resolve in a straightforward manner some or all of the enigmas of the world. Therefore, Wazeck (who studied some German examples) gave to these "free researchers" the name "world riddle solver" ("Welträtsellöser", such as Arvid Reuterdahl, Hermann Fricke or Johann Heinrich Ziegler). Their views had quite different roots in monism, Lebensreform, or occultism. Their views were typically characterized by the fact that they practically rejected the entire terminology and the (primarily mathematical) methods of modern science. Their works were published by private publishers, or in popular and non-specialist journals. It was significant for many "free researchers" (especially the monists) to explain all phenomena by intuitive and illustrative mechanical (or electrical) models, which also found its expression in their defense of the aether. For this reason they objected to the abstractness and inscrutability of the relativity theory, which was considered a pure calculation method that cannot reveal the true reasons underlying the phenomena. The "free researchers" often used Mechanical explanations of gravitation, in which gravity is caused by some sort of "aether pressure" or "mass pressure from a distance". Such models were regarded as an illustrative alternative to the abstract mathematical theories of gravitation of both Newton and Einstein. The enormous self-confidence of the "free researchers" is noteworthy, since they not only believed themselves to have solved the great riddles of the world, but many also seemed to expect that they would rapidly convince the scientific community.Since Einstein rarely defended himself against these attacks, this task was undertaken by other relativity theoreticians, who (according to Hentschel) formed some sort of "defensive belt" around Einstein. Some representatives were Max von Laue, Max Born, etc. and on popular-scientific and philosophical level Hans Reichenbach, André Metz etc., who led many discussions with critics in semi-popular journals and newspapers. However, most of these discussions failed from the start. Physicists like Gehrcke, some philosophers, and the "free researchers" were so obsessed with their own ideas and prejudices that they were unable to grasp the basics of relativity; consequently, the participants of the discussions were talking past each other. In fact, the theory that was criticized by them was not relativity at all, but rather a caricature of it. The "free researchers" were mostly ignored by the scientific community, but also, in time, respected physicists such as Lenard and Gehrcke found themselves in a position outside the scientific community. However, the critics didn't believe that this was due to their incorrect theories, but rather due to a conspiracy of the relativistic physicists (and in the 1920s and 1930s of the Jews as well), which allegedly tried to put down the critics, and to preserve and improve their own positions within the academic world. For example, Gehrcke (1920/24) held that the propagation of relativity is a product of some sort of mass suggestion. Therefore, he instructed a media monitoring service to collect over 5000 newspaper clippings which were related to relativity, and published his findings in a book. However, Gehrcke's claims were rejected, because the simple existence of the "relativity hype" says nothing about the validity of the theory, and thus it cannot be used for or against relativity.Afterward, some critics tried to improve their positions by the formation of alliances. One of them was the "Academy of Nations", which was founded in 1921 in the US by Robert T. Browne and Arvid Reuterdahl. Other members were Thomas Jefferson Jackson See and as well as Gehrcke and Mohorovičić in Germany. It is unknown whether other American critics such as Charles Lane Poor, Charles Francis Brush, Dayton Miller were also members. The alliance disappeared as early as the mid-1920s in Germany and by 1930 in the USA.
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Criticism of the theory of relativity
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Relativity hype and popular criticism
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Chauvinism and antisemitism Shortly before and during World War I, there appeared some nationalistically motivated criticisms of relativity and modern physics. For example, Pierre Duhem regarded relativity as the product of the "too formal and abstract" German spirit, which was in conflict with the "common sense". Similarly, popular criticism in the Soviet Union and China, which partly was politically organized, rejected the theory not because of factual objections, but as ideologically motivated as the product of western decadence.So in those countries, the Germans or the Western civilization were the enemies. However, in Germany the Jewish ancestry of some leading relativity proponents such as Einstein and Minkowski made them targets of racially minded critics, although many of Einstein's German critics did not show evidence of such motives. The engineer Paul Weyland, a known nationalistic agitator, arranged the first public meeting against relativity in Berlin in 1919. While Lenard and Stark were also known for their nationalistic opinions, they declined to participate in Weyland's rallies, and Weyland's campaign eventually fizzled out due to a lack of prominent speakers. Lenard and others instead responded to Einstein's challenge to his professional critics to debate his theories at the scientific conference held annually at Bad Nauheim. While Einstein's critics, assuming without any real justification that Einstein was behind the activities of the German press in promoting the triumph of relativity, generally avoided antisemitic attacks in their earlier publications, it later became clear to many observers that antisemitism did play a significant role in some of the attacks.Reacting to this underlying mood, Einstein himself openly speculated in a newspaper article that in addition to insufficient knowledge of theoretical physics, antisemitism at least partly motivated their criticisms. Some critics, including Weyland, reacted angrily and claimed that such accusations of antisemitism were only made to force the critics into silence. However, subsequently Weyland, Lenard, Stark and others clearly showed their antisemitic biases by beginning to combine their criticisms with racism. For example, Theodor Fritsch emphasized the alleged negative consequences of the "Jewish spirit" within relativity physics, and the far right-press continued this propaganda unhindered. After the murder of Walther Rathenau (1922) and murder threats against Einstein, he left Berlin for some time. Gehrcke's book on "The mass suggestion of relativity theory" (1924) was not antisemitic itself, but it was praised by the far-right press as describing an alleged typical Jewish behavior, which was also imputed to Einstein personally. Philipp Lenard in 1922 spoke about the "foreign spirit" as the foundation of relativity, and afterward he joined the Nazi party in 1924; Johannes Stark did the same in 1930. Both were proponents of the so-called German Physics, which only accepted scientific knowledge based on experiments, and only if accessible to the senses. According to Lenard (1936), this is the "Aryan physics or physics by man of Nordic kind" as opposed to the alleged formal-dogmatic "Jewish physics". Additional antisemitic critics can be found in the writings of Wilhelm Müller, Bruno Thüring and others. For example, Müller erroneously claimed that relativity was a purely "Jewish affair" and it would correspond to the "Jewish essence" etc., while Thüring made comparisons between the Talmud and relativity.
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Criticism of the theory of relativity
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Relativity hype and popular criticism
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Accusations of plagiarism and priority discussions Some of Einstein's critics, like Lenard, Gehrcke and Reuterdahl, accused him of plagiarism, and questioned his priority claims to the authorship of relativity theory. The thrust of such allegations was to promote more traditional alternatives to Einstein's abstract hypothetico-deductive approach to physics, while Einstein himself was to be personally discredited. It was argued by Einstein's supporters that such personal accusations were unwarranted, since the physical content and the applicability of former theories were quite different from Einstein's theory of relativity. However, others argued that between them Poincaré and Lorentz had earlier published several of the core elements of Einstein's 1905 relativity paper, including a generalized relativity principle that was intended by Poincaré to apply to all physics. Some examples: Johann Georg von Soldner (1801) was credited for his calculation of the deflection of light in the vicinity of celestial bodies, long before Einstein's prediction which was based on general relativity. However, Soldner's derivation has nothing to do with Einstein's, since it was fully based on Newton's theory, and only gave half of the value as predicted by general relativity.
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Criticism of the theory of relativity
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Relativity hype and popular criticism
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Paul Gerber (1898) published a formula for the perihelion advance of Mercury, which was formally identical to an approximate solution given by Einstein. However, since Einstein's formula was only an approximation, the solutions are not identical. In addition, Gerber's derivation has no connection with General relativity and was even regarded as meaningless.
Woldemar Voigt (1887) derived a transformation, which is very similar to the Lorentz transformation. As Voigt himself acknowledged, his theory was not based on electromagnetic theory, but on an elastic aether model. His transformation also violates the relativity principle.
Friedrich Hasenöhrl (1904) applied the concept of electromagnetic mass and momentum (which were known long before) to cavity radiation and thermal radiation. Yet, the applicability of Einstein's Mass–energy equivalence goes much further, since it is derived from the relativity principle and applies to all forms of energy.
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Criticism of the theory of relativity
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Relativity hype and popular criticism
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Menyhért Palágyi (1901) developed a philosophical "space-time" model in which time plays the role of an imaginary fourth dimension. Palágyi's model was only a reformulation of Newtonian physics, and had no connection to electromagnetic theory, the relativity principle, or to the constancy of the speed of light.Some contemporary historians of science have revived the question as to whether Einstein was possibly influenced by the ideas of Poincaré, who first stated the relativity principle and applied it to electrodynamics, developing interpretations and modifications of Lorentz's electron theory that appear to have anticipated what is now called special relativity. Another discussion concerns a possible mutual influence between Einstein and David Hilbert as regards completing the field equations of general relativity (see Relativity priority dispute).
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Criticism of the theory of relativity
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Relativity hype and popular criticism
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A Hundred Authors Against Einstein A collection of various criticisms can be found in the book Hundert Autoren gegen Einstein (A Hundred Authors Against Einstein), published in 1931. It contains very short texts from 28 authors, and excerpts from the publications of another 19 authors. The rest consists of a list that also includes people who only for some time were opposed to relativity. From among Einstein's concepts the most targeted one is space-time followed by the speed of light as a constant and the relativity of simultaneity, with other concepts following. Besides philosophic objections (mostly based on Kantianism), also some alleged elementary failures of the theory were included; however, as some commented, those failures were due to the authors' misunderstanding of relativity. For example, Hans Reichenbach wrote a report in the entertainment section of a newspaper, describing the book as “a magnificent collection of naive mistakes” and as “unintended droll literature.” Albert von Brunn interpreted the book as a pamphlet "of such deplorable impotence as occurring elsewhere only in politics" and "a fallback into the 16th and 17th centuries" and concluded “it can only be hoped that German science will not again be embarrassed by such sad scribblings”, and Einstein said, in response to the book, that if he were wrong, then one author would have been enough.According to Goenner, the contributions to the book are a mixture of mathematical–physical incompetence, hubris, and the feelings of the critics of being suppressed by contemporary physicists advocating the new theory. The compilation of the authors show, Goenner continues, that this was not a reaction within the physics community—only one physicist (Karl Strehl) and three mathematicians (Jean-Marie Le Roux, Emanuel Lasker and Hjalmar Mellin) were present—but a reaction of an inadequately educated academic citizenship, which did not know what to do with relativity. As regards the average age of the authors: 57% were substantially older than Einstein, one third was around the same age, and only two persons were substantially younger. Two authors (Reuterdahl, von Mitis) were antisemitic and four others were possibly connected to the Nazi movement. On the other hand, no antisemitic expression can be found in the book, and it also included contributions of some authors of Jewish ancestry (Salomo Friedländer, Ludwig Goldschmidt, Hans Israel, Emanuel Lasker, Oskar Kraus, Menyhért Palágyi).
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Criticism of the theory of relativity
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Status of criticism
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The theory of relativity is considered to be self-consistent, is consistent with many experimental results, and serves as the basis of many successful theories like quantum electrodynamics. Therefore, fundamental criticism (like that of Herbert Dingle, Louis Essen, Petr Beckmann, Maurice Allais and Tom van Flandern) has not been taken seriously by the scientific community, and due to the lack of quality of many critical publications (found in the process of peer review) they were rarely accepted for publication in reputable scientific journals. Just as in the 1920s, most critical works are published in small publication houses, alternative journals (like "Apeiron" or "Galilean Electrodynamics"), or private websites. Consequently, where criticism of relativity has been dealt with by the scientific community, it has mostly been in historical studies.However, this does not mean that there is no further development in modern physics. The progress of technology over time has led to extremely precise ways of testing the predictions of relativity, and so far it has successfully passed all tests (such as in particle accelerators to test special relativity, and by astronomical observations to test general relativity). In addition, in the theoretical field there is continuing research intended to unite general relativity and quantum theory, between which a fundamental incompatibility still remains. The most promising models are string theory and loop quantum gravity. Some variations of those models also predict violations of Lorentz invariance on a very small scale.
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Androgen backdoor pathway
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Androgen backdoor pathway
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The androgen backdoor pathway is a collective name for all metabolic pathways where clinically relevant androgens are synthesized from 21-carbon steroids (pregnanes) by their 5α-reduction with roundabout of testosterone and/or androstenedione. Initially described as pathway where 5α-reduction of 17α-hydroxyprogesterone ultimately leads to 5α-dihydrotestosterone, several other pathways have been since then discovered that lead to 11-oxygenated androgens which are potent agonists of the androgen receptors. A backdoor pathway is an alternative to the conventional, canonical androgenic pathway that involves testosterone and/or androstenedione.
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Androgen backdoor pathway
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Introduction
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The androgen backdoor pathways are critical metabolic processes involved in the synthesis of clinically relevant androgens from 21-carbon steroids (pregnanes) through their 5α-reduction. These pathways occur without the involvement of testosterone and/or androstenedione, which are part of the conventional, canonical androgenic pathway. Initially, the 5α-reduction of 17α-hydroxyprogesterone was described in medical literature as a pathway that ultimately leads to the production of 5α-dihydrotestosterone. However, over the last two decades, several other distinct pathways have been discovered that lead to the synthesis of 11-oxygenated androgens, which are potent agonists of the androgen receptors. The androgen response mechanism occurs through the binding of androgens to cytosolic androgen receptors, which are translocated to the nucleus upon androgen binding and ultimately regulate gene transcription via androgen responsive elements. This response mechanism plays a crucial role in male sexual differentiation and puberty, as well as other tissue types and processes. The discovery of the backdoor pathway to 5α-dihydrotestosterone in the tammar wallaby in 2003 opened new avenues for understanding the biosynthesis of androgens in humans. Subsequently, other backdoor pathways leading to potent 11-oxygenated androgens have also been characterized, providing further insight into the synthesis of androgens in vivo. Understanding these pathways is critical for the development of effective treatments for conditions related to androgen biosynthesis.
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Androgen backdoor pathway
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Biochemistry
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Dihydrotestosterone backdoor biosynthesis The primary feature of the androgen backdoor pathway is that 17α-hydroxyprogesterone (17-OHP) can be 5α-reduced and finally converted to 5α-dihydrotestosterone (DHT) via an alternative route that bypasses the conventional intermediates androstenedione and testosterone.This route is activated during normal prenatal development and leads to early male sexual differentiation. It was first described in the marsupials and later confirmed in humans. Both the canonical and backdoor pathways of DHT biosynthesis are required for normal human male genital development, thus defects in the backdoor pathway from 17-OHP or progesterone (P4) to DHT lead to undervirilization in male fetuses because placental P4 is the precursor of DHT via the backdoor pathway.In 21-hydroxylase deficiency or cytochrome P450 oxidoreductase deficiency, this route may be activated regardless of age and sex by even a mild increase in circulating 17-OHP levels.While 5α-reduction is the last transformation in the classical androgen pathway, it is the first step in the backdoor pathways to 5α-dihydrotestosterone that acts on either 17-OHP or P4 which are ultimately converted to DHT.
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Androgen backdoor pathway
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Biochemistry
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17α-Hydroxyprogesterone pathway The first step of this pathway is the 5α-reduction of 17-OHP to 5α-pregnan-17α-ol-3,20-dione (17OHDHP, since it is also known as 17α-hydroxy-dihydroprogesterone). The reaction is catalyzed by SRD5A1.
17OHDHP is then converted to 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) via 3α-reduction by a 3α-hydroxysteroid dehydrogenase isozyme (AKR1C2 and AKR1C4) or HSD17B6, that also has 3α-reduction activity. 5α-Pdiol is also known as 17α-hydroxyallopregnanolone or 17OH-allopregnanolone.
5α-Pdiol is then converted to 5α-androstan-3α-ol-17-one, also known as androsterone (AST) by 17,20-lyase activity of CYP17A1 which cleaves a side-chain (C17-C20 bond) from the steroid nucleus, converting a C21 steroid (a pregnane) to C19 steroid (an androstane or androgen). AST is 17β-reduced to 5α-androstane-3α,17β-diol (3α-diol) by HSD17B3 or AKR1C3.
The final step is 3α-oxidation of 3α-diol in target tissues to DHT by an enzyme that has 3α-hydroxysteroid oxidase activity, such as AKR1C2, HSD17B6, HSD17B10, RDH16, RDH5, and DHRS9. This oxidation is not required in the classical androgen pathway. The pathway can be summarized as: 17-OHP → 17OHDHP → 5α-Pdiol → AST → 3α-diol → DHT.
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Androgen backdoor pathway
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Biochemistry
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Progesterone pathway The pathway from progesterone (P4) to DHT is similar to that described above from 17-OHP to DHT, but the initial substrate for 5α-reductase here is P4 rather than 17-OHP. Placental P4 in the male fetus is the feedstock for the backdoor pathway found operating in multiple non-gonadal tissues. The first step in this pathway is 5α-reduction of P4 towards 5α-dihydroprogesterone (5α-DHP) by SRD5A1. 5α-DHP is then converted to allopregnanolone (AlloP5) via 3α-reduction by AKR1C2 or AKR1C4. AlloP5 is then converted to 5α-Pdiol by the 17α-hydroxylase activity of CYP17A1. The pathway then proceeds the same way as the pathway that starts from 17-OHP, and can be summarized as: P4 → 5α-DHP → AlloP5 → 5α-Pdiol → AST → 3α-diol → DHT.
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Androgen backdoor pathway
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Biochemistry
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11-Oxygenated androgen backdoor biosynthesis There are two known 11-oxygenated androgens, 11-ketotestosterone (11KT) and 11-ketodihydrotestosterone (11KDHT), which both bind and activate the androgen receptor with affinities, potencies, and efficacies that are similar to that of T and DHT, respectively. Some work suggests that though 11β-hydroxytestosterone (11OHT) and 11β-hydroxydihydrotestosterone (11OHDHT) may not have significant androgenic activity as they were once thought to possess, they may still be important precursors to androgenic molecules. The relative importance of the androgens depends on their activity, circulating levels and stability. The steroids 11OHA4 and 11KA4 have been established as having minimal androgen activity, but remain important molecules in this context since they act as androgen precursors.
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Androgen backdoor pathway
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Biochemistry
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The backdoor pathways to 11-oxygenated androgens can be broadly as two Δ4 steroid entry points (17-OHP and P4) that can undergo a common sequence of three transformations: 11β-hydroxylation by CYP11B1 in the adrenal cortex, 5α-reduction by SRD5A1/SRD5A1, reversible 3α-reduction/oxidation of the ketone/alcohol by AKR1C2 or AKR1C4.
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Androgen backdoor pathway
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Clinical significance
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In congenital adrenal hyperplasia (CAH) due to deficiency of 21-hydroxylase or cytochrome P450 oxidoreductase (POR), the associated elevated 17-OHP levels result in flux through the backdoor pathway to DHT that begins with 5α-reduction of 17-OHP . This pathway may be activated regardless of age and sex. Fetal excess of 17-OHP in CAH may contribute to DHT synthesis that leads to external genital virilization in newborn girls with CAH. P4 levels may also be elevated in CAH, leading to androgen excess via the backdoor pathway from P4 to DHT. 17-OHP and P4 may also serve as substrates to 11-oxygenated androgens in CAH.Serum levels of the C21 11-oxygenated steroids: 21-deoxycorticosterone (11OHP4) and 21-deoxycortisol (21dF), have been known to be elevated in both non-classical and classical forms of CAH since about 1990, and liquid chromatography–mass spectrometry profiles that include these steroids have been proposed for clinical applications. Classical CAH patients receiving glucocorticoid therapy had C19 11-oxygenated steroid serum levels that were elevated 3-4 fold compared to healthy controls. In that same study, the levels of C19 11-oxygenated androgens correlated positively with conventional androgens in women but negatively in men. The levels of 11KT were 4 times higher compared to that of T in women with the condition. In adult women with CAH, the ratio of DHT produced in a backdoor pathway to that produced in a conventional pathway increases as control of androgen excess by glucocorticoid therapy deteriorates. In CAH patients with poor disease control, 11-oxygenated androgens remain elevated for longer than 17-OHP, thus serving as a better biomarker for the effectiveness of the disease control. In males with CAH, 11-oxygenated androgen levels may indicate the presence testicular adrenal rest tumors.
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Androgen backdoor pathway
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Clinical significance
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Both the classical and backdoor androgen pathway to DHT are required for normal human male genital development. Deficiencies in the backdoor pathway to DHT from 17-OHP or from P4 lead to underverilization of the male fetus, as placental P4 is a precursor to DHT in the backdoor pathway.A case study of five 46,XY (male) patients from two families demonstrated that atypical genital appearance were attributed to mutations in AKR1C2 and/or AKR1C4, which operate exclusively in the backdoor pathway to DHT. Mutations in the AKR1C3 and genes involved in the classical androgen pathway were excluded as the causes for the atypical appearance. The 46,XX (female) relatives of affected patients, having the same mutations, were phenotypically normal and fertile. Although both AKR1C2 and AKR1C4 are needed for DHT synthesis in a backdoor pathway, the study found that mutations in AKR1C2 only were sufficient for disruption. However, these AKR1C2/AKR1C4 variants leading to DSD are rare and have been only so far reported in just those two families. This case study emphasizes the role of AKR1C2/4 in the alternative androgen pathways.
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Androgen backdoor pathway
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Clinical significance
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Isolated 17,20-lyase deficiency syndrome due to variants in CYP17A1, cytochrome b5, and POR may also disrupt the backdoor pathway to DHT, as the 17,20-lyase activity of CYP17A1 is required for both classical and backdoor androgen pathways. This rare deficiency can lead to DSD in both sexes, with affected girls being asymptomatic until puberty, when they show amenorrhea.11-oxygenated androgens may play important roles in DSDs. 11-oxygenated androgen fetal biosynthesis may coincide with the key stages of production of cortisol — at weeks 8–9, 13–24, and from 31 and onwards. In these stages, impaired CYP17A1 and CYP21A2 activity lead to increased ACTH due to cortisol deficiency and the accumulation of substrates for CYP11B1 in pathways to 11-oxygenated androgens and could cause abnormal female fetal development.
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Androgen backdoor pathway
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History
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In 1987, Eckstein et al. incubated rat testicular microsomes in the presence of radiolabelled steroids and demonstrated that 5α-androstane-3α,17β-diol is preferentially produced from 17α-hydroxyprogesterone (17-OHP ). While "androstanediol" was used to denote both 5α-androstane-3α,17β-diol and 5α-androstane-3β,17β-diol, "3α-diol" is used here to abbreviate 5α-androstane-3α,17β-diol in this paper as it is a common convention and emphasizes it as the 3α-reduced derivative of DHT. The function of 3α-diol was not known at that time.
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Androgen backdoor pathway
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History
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In 2000, Shaw et al. demonstrated that circulating 3α-diol mediates prostate development in tammar wallaby pouch young via conversion to DHT in target tissues. Tammar wallaby pouch young do not show sexually dimorphic circulating levels of T and DHT during prostate development which suggested that another androgenization mechanism was responsible. While 3α-diol's androgen receptor binding affinity is five orders of magnitude lower than DHT (generally described as AR inactive), it was known 3α-diol can be oxidized back to DHT via the action of a number of dehydrogenases.In 2003, Wilson et al. incubated the testes of tammar wallaby pouch young with radiolabelled progesterone to show that 5α-reductase expression in this tissue enabled a novel pathway from 17-OHP to 3α-diol without T as an intermediate: 17-OHP → 17OHDHP → 5α-Pdiol → AST → 3α-diol.
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Androgen backdoor pathway
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History
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In 2004, Mahendroo et al. demonstrated that an overlapping novel pathway is operating in mouse testes, generalizing what had been demonstrated in tammar wallaby: P4 → 5α-DHP → AlloP5→ 5α-Pdiol → AST → 3α-diol.
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Androgen backdoor pathway
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History
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The term "backdoor pathway" was coined by Auchus in 2004 and defined as a route to DHT that: 1) bypasses conventional intermediates androstenedione (A4) and T; 2) involves 5α-reduction of 21-carbon (C21) pregnanes to 19-carbon (C19) androstanes; and 3) involves the 3α-oxidation of 3α-diol to DHT. The backdoor pathway explained how androgens are produced under certain normal and pathological conditions in humans when the classical androgen pathway cannot fully explain the observed consequences. The pathway Auchus defined adds DHT to the terminus of the pathway described by Wilson et al. in 2003: 17-OHP → 17OHDHP → 5α-Pdiol → AST → 3α-diol → DHT.
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Androgen backdoor pathway
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History
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The clinical relevance of these results was demonstrated in 2012 for the first time when Kamrath et al. attributed the urinary metabolites to the androgen backdoor pathway from 17-OHP to DHT in patients with steroid 21-hydroxylase (encoded by the gene CYP21A2) enzyme deficiency.
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Androgen backdoor pathway
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History
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Barnard et al. in 2017 demonstrated metabolic pathways from C21 steroids to 11KDHT that bypasses A4 and T in vitro in a prostate cancer derived cell line, an aspect that is similar to that of the backdoor pathway to DHT. These newly discovered pathways to 11-oxygenated androgens were also described as "backdoor" pathways due to this similarity, and were further characterized in subsequent studies.
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Androgen backdoor pathway
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Note
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This article was submitted to WikiJournal of Medicine for external academic peer review in 2022 (reviewer reports). The updated content was reintegrated into the Wikipedia page under a CC-BY-SA-3.0 license (2023). The version of record as reviewed is: Maxim Masiutin; Maneesh Yadav; et al. (3 April 2023). "Alternative androgen pathways" (PDF). WikiJournal of Medicine. 10 (1): 3. doi:10.15347/WJM/2023.003. ISSN 2002-4436. Wikidata Q100737840.
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Intercostal nerve block
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Intercostal nerve block
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Intercostal nerve block (abbreviated ICNB) is a nerve block which temporarily or permanently interrupts the flow of signals along an intercostal nerve, usually performed to relieve pain.
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Intercostal nerve block
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Uses
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An ICNB relieves the pain associated with injured intercostal nerves. This pain can arise from chest surgery, physical trauma, aggravation of the shingles virus, or pressure put upon the nerves during pregnancy.
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Intercostal nerve block
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Techniques
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Anesthetic nerve block Injecting local pain relievers and steroids into the injured area alleviates intercostal nerve pain. In this type of nerve block, a needle inserted between two ribs releases a steroid into the area around the nerve. The exact location of injection depends on the underlying cause of the injury. After three to five days, the steroid begins to relieve pain. Depending on the individual, the pain-relieving effects of the steroid last for days to several months.
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Intercostal nerve block
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Techniques
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Risks Injection without a device such as an ultrasound or fluoroscope to guide the needle can cause pneumothorax, a condition where air enters the cavity surrounding the lung or into a blood vessel causing local anesthetic toxicity. Other newer facial plane blocks may be an alternative option due to a preferential safety profile Neurolysis Physicians can also treat intercostal nerve pain by intentionally damaging the intercostal nerves. This process, known as neurolysis, prevents the nerves from sending pain signals. In chemical neurolysis, a needle injects alcohol or phenol into the nerve and prevents the conduction of pain signals. Neurolysis can also be accomplished through a process known as radio-frequency lesioning. In radio-frequency lesioning, a needle transmits radio waves to the nerve and interrupts regular pain signaling.
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SilverFast
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SilverFast
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SilverFast is the name of a family of software for image scanning and processing, including photos, documents and slides, developed by LaserSoft Imaging.
There are also other applications for image processing using digital cameras or printers and for 48-bit raw data image processing.
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SilverFast
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History
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SilverFast was introduced in 1995, it is still under development today. Some scanner manufacturers bundle their hardware with SilverFast software. Some of the features developed for SilverFast, especially in the area of color management, error detection and automatic dust and scratch removal, have been patented. The European Digital Press Association named SilverFast the "Best colour management software of the year 2008" for improving the dynamic range of most scanners and for creating ICC profiles automatically. In 2011 version 8 was introduced, and HDR imaging software followed in 2012. SilverFast 9 has been released in 2020.
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SilverFast
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History
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Patents 2005: Patent granted on barcode technology used for the Auto IT8 Calibration. (Karl-Heinz Zahorsky, EP: 1594301) 2008: Patent granted on technology used for SilverFast Multi-Exposure@. (Karl-Heinz Zahorsky, EP: 1744278, US 8,693,808) 2021: Patent pending for a scanning method that reduces scanning time for flatbed scanners. (Karl-Heinz Zahorsky, German, European and US patents pending)
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SilverFast
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Products
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SilverFast is offered individually adjusted to the respective scanner model. If several scanners are operated, it is therefore necessary to purchase a corresponding number of additional licenses. Upgrade rates are also offered when purchasing a new scanner.
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SilverFast
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Products
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Overview of the available products: 1) including: Multi-Exposure 2) optional: ICC Printer Calibration Optional features As marked in the table above, some products are available with additional features: Multi-Exposure – Multi-Exposure is an exposure blending technique for scanning transparent originals like slides, negatives, and film strips with increased dynamic range. This is accomplished by scanning the original multiple times with different exposure times, increasing the dynamic range and preserving detail in the light and shadow areas of the image. Multi-Exposure does not work with reflective originals and differs from "Multi-Sampling," which also scans multiple times but with unaltered exposure.
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SilverFast
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Products
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ICC Printer Calibration – The SilverFast ICC Printer Calibration calibrates the printer using a previously calibrated flatbed scanner as the measuring device for profiling the printer.
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SilverFast
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Products
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SilverFast Scanner Software It can be used as a stand-alone application, as a Photoshop plug-in, or as a universal TWAIN module. Versions include: SilverFast SE (basic edition) SilverFast SE Plus (plus edition / with Multi-Exposure) SilverFast Ai Studio (premium edition) SilverFast X-Ray (a special version of SilverFast designed to digitize radiographic films for scientific and medical radiography) SilverFast HDR Software SilverFast HDR is a computer program for processing 48-bit raw images. Many newer scanners are able to output the image directly with all existing data instead of breaking it down to 24-bit. This 48-bit raw image can be saved immediately and digital processing performed later.
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SilverFast
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Products
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SilverFast HDR (basic edition) SilverFast HDR Studio (premium edition)SilverFast HDR contains the functionality of SilverFast Ai Studio for 48-bit raw data, such as defining output size and resolution, auto-adjusting of highlights and shadows, three-part histogram, gradation curves, selective color correction, unsharp masking, color cast removal slider, color separation, and CMYK-preview. SilverFast HDR can be used as a native plug-in for Adobe Photoshop, as a universal TWAIN module, or as a stand-alone application.
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SilverFast
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Products
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The HDR Studio has AACO (Auto-Adaptive Contrast Optimization), JPEG 2000, USMPlus (Unsharp Mask Plus), CloneTool and PrinTao.
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SilverFast
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Products
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HDRi (64Bit RAW data with infrared channel) With version 6.6.1 any SilverFast HDR version supports the proprietary RAW data format HDRi. These 64-bit HDRi color files and 32-bit HDRi greyscale files contain additional 16-bit infrared RAW data besides the 48-bit color RAW data and 16-bit greyscale RAW data respectively. Therefore, a scanner with an infrared channel available to the software is necessary. This RAW format can keep any readable image information for later post-processing.
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SilverFast
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Products
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The acronym "HDR" as used by SilverFast is not related to High-dynamic-range imaging, a widely used technique to increase the dynamic range of (digital) images.
The data format is: SilverFast Archiving Software The SilverFast Archive Suite includes SilverFast Ai Studio and SilverFast HDR Studio with an integrated color management system. This package is suitable for archiving slides, negatives, and photos, whereas the post processing can take place anytime after scanning.
SilverFast Archive Suite - consisting of SilverFast Ai Studio and SilverFast HDR Studio (premium edition) PrintTao 8 PrintTao 8 is for Canon and Epson large format printers. It can be used as a stand-alone application or as a plug-in for Adobe Photoshop and Adobe Lightroom.
PrinTao 8 takes over color management and all printer driver settings. It contains various printing templates like pack templates for portrait photographers or gallery wrap templates for printing on canvas.
SRDx Photoshop Plug-in SRDx (Smart Removal of Defects) is a plug-in for Adobe Photoshop to remove defects like dust particles, specks, small scratches and finger prints from digital images.
Most dust and scratch removal tools use blur effects which degrade overall image quality. SRDx works without any unsharping effects. SRDx uses an adjustable automatic detection of dust and scratches that can be fine-tuned manually using a defect marker and a rubber tool.
SilverFast DC SilverFast DC has been discontinued. SilverFast DC was a stand-alone software for digital camera image processing. It contained features for reading the image data from the camera, for processing, optimizing, and archiving the images on the computer, as well as for printing the edited images.
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SilverFast
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IT8 calibration & color management
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Scanner calibration: SilverFast is equipped with a color management system that enables largely automated IT8 color calibration of the scanner using IT8 targets supplied by the manufacturer LaserSoft Imaging.
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SilverFast
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IT8 calibration & color management
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according to ISO standard 12641-1 (from 1993): conventional standard according to ISO standard 12641-2 (from 2020): new standard for a more precise IT8 calibration with about 3 times the number of measuring fields. LaserSoft Imaging cooperated for creating this standard.Printer calibration: SilverFast also enables printer calibration by printing a color table and then scanning it using an already calibrated scanner to create an ICC profile.
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SilverFast
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Supported devices
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SilverFast supports 325 scanners.
Most Microsoft Windows-only 35mm USB scanners sold under various names are not supported. Each scanner model requires a separate license.
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SilverFast
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Heidelberg drum scanner
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SilverFast Ai Studio supports prepress drum scanners made by Heidelberger Druckmaschinen AG (Linotype - Hell) on Microsoft Windows 2000, XP, Vista and 7, as well as Mac OS X 10.3-10.5 operating systems. Supported models include the Chromagraph 3300/3400, Tango/XL, Topaz, Nexscan, and Primescan.
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Grsync
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Grsync
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Grsync is a graphical user interface for rsync. rsync is a differential backup and file synchronization tool widely used in Unix-like operating systems.
Grsync is developed with the GTK widget toolkit. Like rsync, Grsync is free and open-source software licensed under the GNU General Public License.
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Grsync
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About
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Rsync is a tool for creating backups in Linux systems. It supports backing up local folders, SSH tunneling, delta-only synchronization, and so on.
Grsync adds the ability to use such purposes with a graphical user interface, without rsync's need to learn a complex set of command-line arguments. In some cases, it is easier to backup files with grsync than with rsync. Since version 1.3.0, Grsync has GTK-3 compatibility.
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CTAN
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CTAN
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CTAN (an acronym for "Comprehensive TeX Archive Network") is the authoritative place where TeX related material and software can be found for download. Repositories for other projects, such as the MiKTeX distribution of TeX, constantly mirror most of CTAN.
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CTAN
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History
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Before CTAN there were a number of people who made some TeX materials available for public download, but there was no systematic collection. At a podium discussion that Joachim Schrod organized at the 1991 EuroTeX conference, the idea arose to bring together the separate collections. (Joachim was interested in this topic because he is active in the TeX community since 1983 and ran one of the largest ftp servers in Germany at that time.) CTAN was built in 1992, by Rainer Schöpf and Joachim Schrod in Germany, Sebastian Rahtz in the UK, and George Greenwade in the U.S. (George came up with the name). Today, there are still only four people who maintain the archives and the TeX catalogue updates: Erik Braun, Ina Dau, Manfred Lotz, and Petra Ruebe-Pugliese. The site structure was put together at the start of 1992 – Sebastian did the main work – and synchronized at the start of 1993. The TeX Users Group provided a framework, a Technical Working Group, for this task's organization. CTAN was officially announced at the EuroTeX conference at Aston University, 1993. The WEB server itself is maintained by Gerd Neugebauer.
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CTAN
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History
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The English site has been stable since the beginning, but both the American and the German sites have moved thrice. The American site was first at Sam Houston State University under George Greenwade, in 1995 it moved to UMass Boston where it was run by Karl Berry. In 1999 it moved to Saint Michael's College in Colchester, Vermont. There it was announced to go off-line in the end of January 2011. Since January 2013, a mirror has been hosted by the University of Utah (no upload node). The German site was first at the University of Heidelberg, operated by Rainer; in 1999 it moved to the University of Mainz, also operated by Rainer; 2002 to the University of Hamburg, operated by Reinhard Zierke; finally in 2005 it moved to a commercial hosting company since the amount of traffic got too high to get sponsored by a university. The German site is subsidized by DANTE, the Deutschsprachige Anwendervereinigung TeX.
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CTAN
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History
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Today, the main CTAN nodes serve downloads of more than 6 TB per month, not counting its 94 mirror sites worldwide.
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Designing Virtual Worlds
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Designing Virtual Worlds
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Designing Virtual Worlds is a book about the practice of virtual world development by Richard Bartle. It has been noted as an authoritative source regarding the history of world-based online games. College courses have been taught using it.In 2021, the author made the book freely available under a Creative Commons license on his website.
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Designing Virtual Worlds
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Contents
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Designing Virtual Worlds argues that the fundamentals of player relationships to the virtual world and each other are independent of technical issues and are characterized by a blending of online and offline identity. According to the book, it is the designer's role to know what will provide players with a positive game experience, the purpose of virtual worlds is the player's exploration of self, as well as for its expansion of the earlier 4-type Bartle gamer style taxonomy into an 8-type model. The book also focuses on the practicalities of its subject.
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Designing Virtual Worlds
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Reception
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It has been called "the bible of MMORPG design" and spoken of as "excellent", "seminal", "widely read", "the standard text on the subject", "the most comprehensive guide to gaming virtual worlds" and "a foundation text for researchers and developers of virtual worlds" that is "strongly recommended for anyone actually thinking about building one of these places" and "describes the minimum level of competency you should have when discussing design issues for virtual worlds".In less favorable reception, one reviewer, while calling it a "must-read" work, said he found "much that was questionable, incomplete, or just erroneous" in it.
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Tropical Storm Sandy
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Tropical Storm Sandy
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The name Sandy has been used for two tropical cyclones worldwide, one in the Atlantic ocean and one in the Australian Region.
In the Atlantic: Hurricane Sandy (2012) – a Category 3 major hurricane that inflicted nearly $70 billion (2012 USD) in damage and killed 233 people across eight countries from the Caribbean to Canada.The name was retired in the Atlantic after the 2012 hurricane season and was replaced with Sara.
Australian region: Cyclone Sandy (1985) – a severe tropical cyclone that affected Northern Australia.The name was retired in Australian region after the 1984–85 cyclone season.
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Subtitle (titling)
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Subtitle (titling)
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In books and other works, the subtitle is an explanatory title added by the author to the title proper of a work. Another kind of subtitle, often used in the past, is the alternative title, also called alternate title, traditionally denoted and added to the title with the alternative conjunction "or", hence its appellation.As an example, Mary Shelley gave her most famous novel the title Frankenstein; or, The Modern Prometheus, where or, The Modern Prometheus is the alternative title, by which she references the Greek Titan as a hint of the novel's themes.A more modern usage is to simply separate the subtitle by punctuation, making the subtitle more of a continuation or sub-element of the title proper.
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Subtitle (titling)
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Subtitle (titling)
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In library cataloging and in bibliography, the subtitle does not include an alternative title, which is defined as part of the title proper: e.g., One Good Turn: A Natural History of the Screwdriver and the Screw is filed as One Good Turn (title) and A Natural History of the Screwdriver and the Screw (subtitle), while Twelfth Night, or What You Will is filed as Twelfth Night, or What You Will (title).
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Subtitle (titling)
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Literature
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Subtitles and alternative titles for plays were fashionable in the Elizabethan era. William Shakespeare parodied this vogue by giving the comedy Twelfth Night his only subtitle, the deliberately uninformative or What You Will, implying that the subtitle can be whatever the audience wants it to be.In printing, subtitles often appear below the title in a less prominent typeface or following the title after a colon.
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Subtitle (titling)
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Literature
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Some modern publishers choose to forget subtitles when republishing historical works, such as Shelley's famous story, which is often now sold simply as Frankenstein.
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Subtitle (titling)
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Non-fiction
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In political philosophy, for example, the 16th-century theorist Thomas Hobbes named his magnum opus Leviathan or The Matter, Forme and Power of a Common-Wealth Ecclesiasticall and Civil, using the subtitle to explain the subject matter of the book.
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Subtitle (titling)
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Film and other media
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In film, examples of subtitles using "or" include Dr. Strangelove or: How I Learned to Stop Worrying and Love the Bomb and Birdman or (The Unexpected Virtue of Ignorance).
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Subtitle (titling)
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Film and other media
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Subtitles are also used to distinguish different installments in a series, instead of or in addition to a number, such as: Pirates of the Caribbean: Dead Man's Chest, the second in the Pirates of the Caribbean series; Mario Kart: Super Circuit, the third in the Mario Kart series; and Star Trek II: The Wrath of Khan, the second in the Star Trek series.
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Electron beam texturing
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Electron beam texturing
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Electron beam texturing (EBT) is a technology used to apply roughness to the surface of rolling mill cylinders by impinging the surface of these cylinders with a modulated electron beam. The beam locally melts the surface of the cylinder, producing crater-like depressions.
The technology was originally developed by the German company Linotype-Hell for application in the printing industry, but eventually found its successful application in the metal rolling industry, where it is competing with alternative texturing technologies, such as electrodischarge texturing.
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Alzheimer's Research & Therapy
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Alzheimer's Research & Therapy
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Alzheimer's Research & Therapy is a monthly peer-reviewed medical journal covering research on Alzheimer's disease. It was established in 2009 and is published by BioMed Central. The editors-in-chief are Douglas R. Galasko (University of California, San Diego), Todd E. Golde (University of Florida), and Philip Scheltens (VU University Medical Center).
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Alzheimer's Research & Therapy
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Abstracting and indexing
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The journal is abstracted and indexed in Chemical Abstracts Service, Embase, Science Citation Index Expanded, and Scopus. According to the Journal Citation Reports the journal has a 2015 impact factor of 5.197.
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Cumulative hierarchy
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Cumulative hierarchy
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In mathematics, specifically set theory, a cumulative hierarchy is a family of sets Wα indexed by ordinals α such that Wα⊆Wα+1 If λ is a limit ordinal, then {\textstyle W_{\lambda }=\bigcup _{\alpha <\lambda }W_{\alpha }} Some authors additionally require that Wα+1⊆P(Wα) or that W0≠∅ .The union {\textstyle W=\bigcup _{\alpha \in \mathrm {On} }W_{\alpha }} of the sets of a cumulative hierarchy is often used as a model of set theory.The phrase "the cumulative hierarchy" usually refers to the standard cumulative hierarchy Vα of the von Neumann universe with Vα+1=P(Wα) introduced by Zermelo (1930).
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Cumulative hierarchy
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Reflection principle
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A cumulative hierarchy satisfies a form of the reflection principle: any formula in the language of set theory that holds in the union W of the hierarchy also holds in some stages Wα
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Cumulative hierarchy
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Examples
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The von Neumann universe is built from a cumulative hierarchy Vα The sets Lα of the constructible universe form a cumulative hierarchy.
The Boolean-valued models constructed by forcing are built using a cumulative hierarchy.
The well founded sets in a model of set theory (possibly not satisfying the axiom of foundation) form a cumulative hierarchy whose union satisfies the axiom of foundation.
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Redheffer matrix
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Redheffer matrix
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In mathematics, a Redheffer matrix, often denoted An as studied by Redheffer (1977), is a square (0,1) matrix whose entries aij are 1 if i divides j or if j = 1; otherwise, aij = 0. It is useful in some contexts to express Dirichlet convolution, or convolved divisors sums, in terms of matrix products involving the transpose of the nth Redheffer matrix.
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Redheffer matrix
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Variants and definitions of component matrices
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Since the invertibility of the Redheffer matrices are complicated by the initial column of ones in the matrix, it is often convenient to express := Cn+Dn where := [cij] is defined to be the (0,1) matrix whose entries are one if and only if j=1 and i≠1 . The remaining one-valued entries in An then correspond to the divisibility condition reflected by the matrix Dn , which plainly can be seen by an application of Mobius inversion is always invertible with inverse Dn−1=[μ(j/i)Mi(j)] . We then have a characterization of the singularity of An expressed by det det (Dn−1Cn+In).
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Redheffer matrix
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Variants and definitions of component matrices
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If we define the function := if j divides i; otherwise, , then we can define the nth Redheffer (transpose) matrix to be the nxn square matrix Rn=[Mj(i)]1≤i,j≤n in usual matrix notation. We will continue to make use this notation throughout the next sections.
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Redheffer matrix
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Examples
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The matrix below is the 12 × 12 Redheffer matrix. In the split sum-of-matrices notation for 12 := 12 12 , the entries below corresponding to the initial column of ones in Cn are marked in blue. (111111111111110101010101101001001001100100010001100010000100100001000001100000100000100000010000100000001000100000000100100000000010100000000001) A corresponding application of the Mobius inversion formula shows that the nth Redheffer transpose matrix is always invertible, with inverse entries given by Rn−1=[Mj(i)⋅μ(ij)]1≤i,j≤n, where μ(n) denotes the Moebius function. In this case, we have that the 12 12 inverse Redheffer transpose matrix is given by 12 −1=(100000000000−110000000000−1010000000000−10100000000−1000100000001−1−1001000000−100000100000000−10001000000−10000010001−100−10000100−100000000010010−10−1000001)
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Redheffer matrix
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Key properties
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Singularity and relations to the Mertens function and special series Determinants The determinant of the n × n square Redheffer matrix is given by the Mertens function M(n). In particular, the matrix An is not invertible precisely when the Mertens function is zero (or is close to changing signs). As a corollary of the disproof of the Mertens conjecture, it follows that the Mertens function changes sign, and is therefore zero, infinitely many times, so the Redheffer matrix An is singular at infinitely many natural numbers.
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Redheffer matrix
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Key properties
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The determinants of the Redheffer matrices are immediately tied to the Riemann Hypothesis through this relation with the Mertens function, since the Hypothesis is equivalent to showing that M(x)=O(x1/2+ε) for all (sufficiently small) ε>0 Factorizations of sums encoded by these matrices In a somewhat unconventional construction which reinterprets the (0,1) matrix entries to denote inclusion in some increasing sequence of indexing sets, we can see that these matrices are also related to factorizations of Lambert series. This observation is offered in so much as for a fixed arithmetic function f, the coefficients of the next Lambert series expansion over f provide a so-called inclusion mask for the indices over which we sum f to arrive at the series coefficients of these expansions. Notably, observe that ∑d|nf(d)=∑k=1nMk(n)⋅f(k)=[qn](∑n≥1f(n)qn1−qn).
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Redheffer matrix
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Key properties
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Now in the special case of these divisor sums, which we can see from the above expansion, are codified by boolean (zero-one) valued inclusion in the sets of divisors of a natural number n, it is possible to re-interpret the Lambert series generating functions which enumerate these sums via yet another matrix-based construction. Namely, Merca and Schmidt (2017-2018) proved invertible matrix factorizations expanding these generating functions in the form of ∑n≥1f(n)qn1−qn=1(q;q)∞∑n≥1(∑k=1nsn,kf(k))qn, where (q;q)∞ denotes the infinite q-Pochhammer symbol and where the lower triangular matrix sequence is exactly generated as the coefficients of sn,k=[qn]qk1−qk(q;q)∞ , through these terms also have interpretations as differences of special even (odd) indexed partition functions. Merca and Schmidt (2017) also proved a simple inversion formula which allows the implicit function f to be expressed as a sum over the convolved coefficients ℓ(n)=(f∗1)(n) of the original Lambert series generating function in the form of f(n)=∑d|n∑k=1np(d−k)μ(n/d)[∑j≥0k−j≥0ℓ(k−j)[qj](q;q)∞], where p(n) denotes the partition function, μ(n) is the Moebius function, and the coefficients of (q;q)∞ inherit a quadratic dependence on j through the pentagonal number theorem. This inversion formula is compared to the inverses (when they exist) of the Redheffer matrices An for the sake of completion here.
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