So physicists are presented with a problem: Singularities must exist as a consequence to the theory of general relativity, but observing these singularities seems to be impossible.
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His interest in singularities was not restricted to black holes.
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You can view them from a theoretical framework as the singularities forming.
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" She added, "Such singularities are useful to the common perception of heroism.
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The structure of these singularities is a subject of contention among physicists.
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Before Süss was an interactive widget, it was a member of the Gallery of Singularities, a collection of algebraic surfaces created and curated in 2006 by Herwig Hauser, a geometer at the University of Vienna, and a master at resolving singularities.
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"I'm inspired by intimacy, sexuality and identity through differences and singularities of human beings," he said.
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In Buenos Aires there are singularities that you don't find in São Paulo, Caracas, Bogota, or Tegucigalpa.
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Many singularities are forecast, but the ones that come true are few, and the timing is always uncertain.
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Millions of singularities have already happened, but we're similar to blind bacteria in our bodies running around cluelessly.
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When singularities break, "they are very difficult to grasp, to study and especially to resolve," Dr. Matt said.
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The process of identifying intriguing algebraic surfaces that possessed singularities sometimes took Dr. Hauser and his students days.
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One of the singularities of Van Buren's sculptural objects is that they look like they were made by nature.
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In this and a number of other examples, time evolution seems to come straight out of symmetries and singularities.
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In one of the singularities of Britain's parliamentary system, bills can be proposed by any member of either house.
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This exceptional case doesn't disprove cosmic censorship as Penrose meant it, because it doesn't suggest naked singularities might actually form.
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Yet general relativity's ability to describe gravity falters on the threshold of singularities, where the curvature of spacetime becomes infinite.
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Despite their differences, the interiors of both AdS and dS universes obey Einstein's classical gravity theory—everywhere outside singularities, that is.
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That theory may reveal what happens, or happened, at other singularities, including the one that begot our universe — the Big Bang.
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Now, new theoretical calculations provide a possible explanation for why naked singularities do not exist—in a particular model universe, at least.
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Given the apparent absence of naked singularities in our universe, physicists will take hints about quantum gravity wherever they can find them.
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These will be transferred to the regions with a two-year lag given the singularities of the regional funding scheme in Spain.
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The places where they cross—called singularities—are especially difficult to analyze (and they correspond to polynomials with a repeated prime factor).
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Always, singularities lay at the centers of black holes—sinkholes in space-time that are so steep that no light can climb out.
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After all, Orlean is drawn to inefficiency, to friction; she gets excited when she encounters people whose indelible singularities don't quite add up.
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The weak cosmic censorship hypothesis, on the other hand, suggests that naked singularities don't exist in the universe, apart from the Big Bang.
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In 20153, the physicists Frans Pretorius and Luis Lehner discovered a mechanism for producing naked singularities in hypothetical universes with five or more dimensions.
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And while this would mean singularities do stay frustratingly hidden, it would also reveal an important feature of the quantum gravity theory that eludes us.
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Hironaka had also been trying for decades to find a proof of a major open problem—what's called the resolution of singularities in characteristic p.
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Black holes, which were predicted by maths before they were discovered in nature, are singularities—points where the familiar laws of physics cease to apply.
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The duo's theoretical investigations of black holes and the mysterious singularities at their centers had turned them on to the question of our cosmic origin.
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His etchings are populated by figures in varying states of metamorphosis; transitioning from human to animal, singularities to pluralities, background to foreground, inanimate to animate.
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Importantly, solving the equation to get these expressions requires considering the equation's singularities: mathematically nonsensical combinations of variables that are equivalent to division by zero.
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The new study offers a solution to this baffling situation, by treating black holes as objects with crystal-like structures, and singularities as tiny geometric defects.
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It also describes the inevitability of singularities, regions of infinite density and zero volume found in the centers of black holes or the start of the universe.
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The other singularities we find in the Universe are black holes, and our hope is that the divergences there will also be cured by the same theories.
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Last year, she curated an exhibition at P21 Gallery called Sudan: Emergence of Singularities, which combined visual arts, theatre, music, design, films, and words from and about Sudan.
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Burt Hasen's (1921-2007) etchings are populated by figures in varying states of metamorphosis; transitioning from human to animal, singularities to pluralities, background to foreground, inanimate to animate.
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The British physicist and mathematician Sir Roger Penrose conjectured in 20063 that visible or "naked" singularities are actually forbidden from forming in nature, in a kind of cosmic censorship.
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The findings indicate that a second, newer conjecture about gravity, if it is true, reinforces Penrose's cosmic censorship conjecture by preventing naked singularities from forming in this model universe.
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If the universe is structured in such a way that naked singularities, wormholes, and/or closed time loops are possible, this could allow for things like traveling backwards in time.
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He and the physicist Roger Penrose described singularities, mind-bending physical concepts where relativity and quantum mechanics collapse inward on each other—as at the heart of a black hole.
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Physicists have wondered for decades whether infinitely dense points known as singularities can ever exist outside black holes, which would expose the mysteries of quantum gravity for all to see.
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Ultimately, though, given its strengths and singularities, whether or not this kind of art is endorsed by a high-art temple like the Met may well be beside the point.
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But Marvin himself was an example of an intelligence so bountiful, unpredictable and sublime that not even a million Singularities could conceivably produce a machine with a mind to match his.
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His best-known work included his collaboration with Roger Penrose on gravitational singularities, the prediction that black holes emit blackbody radiation, and the best-selling book A Brief History of Time.
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As we learned from The Collector in Guardians of the Galaxy, when the universe first formed, the remnants of six "singularities" that predate creation were formed into "concentrated ingots" of immense power.
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He says: Oh, my new friends, before creation itself, there were six singularities, then the universe exploded into existence and the remnants of this system were forged into concentrated ingots ... Infinity Stones.
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But in our current marketplace of ideas, where pivotal singularities sell books — the one event, disease or invention that reputedly changed everything forever — this wide-ranging intensity, free from inflated claims, is refreshing.
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Subsequent calculations by Santos and Crisford supported Vafa's hunch; the simulations they're running now could verify that naked singularities become cloaked in black holes right at the point where gravity becomes the weakest force.
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One of the phenomena predicted by the general theory is the existence of spacetime singularities in black holes, a mass that is so dense that nothing can escape its gravitational effects—not even light.
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This is a good thing because if we could see the singularities at the heart of black hole—what is called a 'naked' singularity—this would destroy the determinism that is fundamental to physics.
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Carolyn See, a novelist and memoirist who built a writing life on the singularities and peculiarities of Los Angeles and its environs, where she lived much of her life, died on Wednesday in Santa Monica, Calif.
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But starting with Manuel in the title role, they don't command the sonic singularities with which pop stars beat off the competition, nor stand out like Gypsy's brassy Ethel Merman or South Pacific tomboy Mary Martin.
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Islands, separated as they are from other people and other species, are both places of great singularities — of flora, fauna, customs and culture — and deeply provincial: places divorced from the concerns and diversity of the rest of the world.
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Preskill and Thorne, both experts in quantum gravity and black holes (Thorne was one of three physicists who founded the black-hole-detecting LIGO experiment), said they felt it might be possible to detect naked singularities and quantum gravity effects.
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And we'll look back, to the origins of automation, to understand that for all the talk of new Skynets and singularities, we've harbored similarly shaped aspirations and fears for thousands of years—back when an automaton was just a man behind the gears.
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The new discovery began to unfold in 2014, when Horowitz, Santos and Benson Way found that naked singularities could exist in a pretend 4-D universe called "anti-de Sitter" (AdS) space whose space-time geometry is shaped like a tin can.
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His other books included "Ingmar Bergman Directs" (1974), "Uneasy Stages: A Chronicle of the New York Theater, 1963-1973" (1975), "Singularities: Essays on the Theater, 1964-1974" (1976) and three collections of his reviews on films, plays and music, all published in 2005.
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Singularities—snags in the otherwise smooth fabric of space and time where Albert Einstein's classical gravity theory breaks down and the unknown quantum theory of gravity is needed—seem to always come cloaked in darkness, hiding from view behind the event horizons of black holes.
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Before Hawking's discovery, black holes were typically thought to be objects into which things go in but never come out (the thinking actually stems in part from Hawking's work describing singularities within black holes.) Essentially, Hawking showed that black holes can, like so many objects in our universe, shrink and die.
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When Hawking wasn't talking about Euclidean quantum gravity, naked singularities, or radiation seeping from black holes, there's a good chance the Cambridge Lucasian Professor of Mathematics was doing his best Chicken Little impersonation, telling a global audience that the sky above would soon give way, should we choose to keep ignoring it.
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Founded in 1940 by Ms. Lambert, a publicist, as a scheme to draw attention to the American fashion industry, the best-dressed list aimed to celebrate both individual style and those who invested in a more classic elegance: the looks of singularities like Diana Vreeland and China Chow, as well as famous designer champions like C.Z. Guest and the Duchess of Windsor.
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In mathematics, canonical singularities appear as singularities of the canonical model of a projective variety, and terminal singularities are special cases that appear as singularities of minimal models. They were introduced by . Terminal singularities are important in the minimal model program because smooth minimal models do not always exist, and thus one must allow certain singularities, namely the terminal singularities.
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In complex analysis, there are several classes of singularities. These include the isolated singularities, the nonisolated singularities and the branch points.
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As alluded to in the last section, one can classify the isolated singularities of harmonic functions as removable singularities, poles, and essential singularities.
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Many important tools of complex analysis such as Laurent series and the residue theorem require that all relevant singularities of the function be isolated. There are three types of isolated singularities: removable singularities, poles and essential singularities.
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Two dimensional terminal singularities are smooth. If a variety has terminal singularities, then its singular points have codimension at least 3, and in particular in dimensions 1 and 2 all terminal singularities are smooth. In 3 dimensions they are isolated and were classified by . Two dimensional canonical singularities are the same as du Val singularities, and are analytically isomorphic to quotients of C2 by finite subgroups of SL2(C).
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Hamilton used the Ricci flow to prove that some compact manifolds were diffeomorphic to spheres and he hoped to apply it to prove the Poincaré Conjecture. He needed to understand the singularities. Hamilton created a list of possible singularities that could form but he was concerned that some singularities might lead to difficulties. He wanted to cut the manifold at the singularities and paste in caps, and then run the Ricci flow again, so he needed to understand the singularities and show that certain kinds of singularities do not occur.
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In short, if one consider a plane projection of a nonsingular curve that has degree d and only ordinary singularities (singularities of multiplicity two with distinct tangents), then the genus is (d − 1)(d − 2)/2 − k, where k is the number of these singularities.
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In summary, singularities are determined by the following characteristics which can vary in strength: # Instability: Singularities are related to effect in which small causes produce great effects. # System relatedness: Singularities represent a peculiarity based on a system and affect its identity. # Uniqueness: Singularities do not stand out from quantitative singularity, but rather by qualitative uniqueness. # Irreversibility: The caused changes of systems are largely irreversible.
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The original singularities are inside the domain of interest. The additional (fictitious) singularities are an artifact needed to satisfy the prescribed but yet unsatisfied boundary conditions.
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The type A2n+1+-singularities are of no interest over the real numbers: they all give an isolated point. Over the complex numbers type A2n+1+-singularities and type A2n+1−-singularities are equivalent: (x,y) → (x, iy) gives the required diffeomorphism of the normal forms.
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Two dimensional log terminal singularities are analytically isomorphic to quotients of C2 by finite subgroups of GL2(C). Two dimensional log canonical singularities have been classified by .
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His research deals with analytic singularities of Feynman integrals, Landau singularities in S-matrix theory, singularities of systems of plane algebraic curves, microlocal analysis, function theory of several complex variables, semiclassical approximations in quantum mechanics, and Sato's hyperfunctions. Pham in the 1960s applied Thom's methods of differential topology to Landau singularities and in the 1970s worked with Bernard Teissier on singularities of systems of plane algebraic curves. In 1970 he was an Invited Speaker at the ICM in Nice with talk (Fractions lipschitziennes et saturation de Zariski des algèbres analytiques complexes).
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The key insight is that, in many cases of interest (such as theta functions), the singularities occur at the roots of unity, and the significance of the singularities is in the order of the Farey sequence. Thus one can investigate the most significant singularities, and, if fortunate, compute the integrals.
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These singularities are located symmetrically with respect to the origin. Their position change when we change equation parameters and the initial condition \theta(0), and therefore, they are called movable singularities due to classification of the singularities of non-linear ordinary differential equations in the complex plane by Paul Painlevé. A similar structure of singularities appears in other non-linear equations that result from the reduction of the Laplace operator in spherical symmetry, e.g., Isothermal Sphere equation .
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The weak and the strong cosmic censorship hypotheses are two mathematical conjectures about the structure of gravitational singularities arising in general relativity. Singularities that arise in the solutions of Einstein's equations are typically hidden within event horizons, and therefore cannot be observed from the rest of spacetime. Singularities that are not so hidden are called naked. The weak cosmic censorship hypothesis was conceived by Roger Penrose in 1969 and posits that no naked singularities exist in the universe.
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The singularities of a curve are not birational invariants. However, locating and classifying the singularities of a curve is one way of computing the genus, which is a birational invariant. For this to work, we should consider the curve projectively and require F to be algebraically closed, so that all the singularities which belong to the curve are considered.
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In order to generalize Sundman's result for the case (or and ) one has to face two obstacles: #As has been shown by Siegel, collisions which involve more than two bodies cannot be regularized analytically, hence Sundman's regularization cannot be generalized. #The structure of singularities is more complicated in this case: other types of singularities may occur (see below). Lastly, Sundman's result was generalized to the case of bodies by Qiudong Wang in the 1990s. Since the structure of singularities is more complicated, Wang had to leave out completely the questions of singularities.
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Shrinking and steady Ricci solitons are fundamental objects in the study of Ricci flow as they appear as blow-up limits of singularities. In particular, it is known that all Type I singularities are modeled on non-collapsed gradient shrinking Ricci solitonsJ. Enders, R. Mueller, P. Topping, "On Type I Singularities in Ricci flow", Communications in Analysis and Geometry, 19 (2011) 905–922 . Type II singularities are expected to be modeled on steady Ricci solitons in general, however to date this has not been proven, even though all known examples are.
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The astroid has four cusp singularities in the real plane, the points on the star. It has two more complex cusp singularities at infinity, and four complex double points, for a total of ten singularities. The dual curve to the astroid is the cruciform curve with equation \textstyle x^2 y^2 = x^2 + y^2. The evolute of an astroid is an astroid twice as large.
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They may alternatively have to do with the presence of singularities. The case of several complex variables is rather different, since singularities then need not be isolated points, and its investigation was a major reason for the development of sheaf cohomology.
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They may alternatively have to do with the presence of mathematical singularities. The case of several complex variables is rather different, since singularities then cannot be isolated points, and its investigation was a major reason for the development of sheaf cohomology.
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Quasi-excellent rings are closely related to the problem of resolution of singularities, and this seems to have been Grothendieck's motivationpg 218 for defining them. Grothendieck (1965) observed that if it is possible to resolve singularities of all complete integral local Noetherian rings, then it is possible to resolve the singularities of all reduced quasi- excellent rings. Hironaka (1964) proved this for all complete integral Noetherian local rings over a field of characteristic 0, which implies his theorem that all singularities of excellent schemes over a field of characteristic 0 can be resolved. Conversely if it is possible to resolve all singularities of the spectra of all integral finite algebras over a Noetherian ring R then the ring R is quasi-excellent.
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The singularities of a projective variety V are canonical if the variety is normal, some power of the canonical line bundle of the non-singular part of V extends to a line bundle on V, and V has the same plurigenera as any resolution of its singularities. V has canonical singularities if and only if it is a relative canonical model. The singularities of a projective variety V are terminal if the variety is normal, some power of the canonical line bundle of the non-singular part of V extends to a line bundle on V, and V the pullback of any section of Vm vanishes along any codimension 1 component of the exceptional locus of a resolution of its singularities.
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Roger Penrose first formulated the cosmic censorship hypothesis in 1969. Since the physical behavior of singularities is unknown, if singularities can be observed from the rest of spacetime, causality may break down, and physics may lose its predictive power. The issue cannot be avoided, since according to the Penrose–Hawking singularity theorems, singularities are inevitable in physically reasonable situations. Still, in the absence of naked singularities, the universe, as described by the general theory of relativity, is deterministic: it is possible to predict the entire evolution of the universe (possibly excluding some finite regions of space hidden inside event horizons of singularities), knowing only its condition at a certain moment of time (more precisely, everywhere on a spacelike three-dimensional hypersurface, called the Cauchy surface).
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Singularities of codimension 2 are of major importance; they are characterized by a single number, the conical angle. The singularities can also studied topologically. Then, for example, there are no topological singularities of codimension 2. In a 3-dimensional polyhedral space without a boundary (faces not glued to other faces) any point has a neighborhood homeomorphic either to an open ball or to a cone over the projective plane.
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Two movable singularities on the imaginary axis are visible. They limit the radius of convergence of the analytical solution around the origin. For different values of initial data and p the location of singularities is different, yet they are located symmetrically on the imaginary axis . The radius of convergence of this series is limited due to existence of two singularities on the imaginary axis in the complex plane.
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Over the complex numbers, some authors consider only the algebraic K3 surfaces. (An algebraic K3 surface is automatically projective.Huybrechts (2016), Remark 1.1.2) Or one may allow K3 surfaces to have du Val singularities (the canonical singularities of dimension 2), rather than being smooth.
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Limitations in the resolution of the optical technique may be to explain for these singularities.
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Toroidal embeddings have recently led to advances in algebraic geometry, in particular resolution of singularities.
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An irreducible plane algebraic curve of degree d has (d − 1)(d − 2)/2 − g singularities, when properly counted. It follows that, if a curve has (d − 1)(d − 2)/2 different singularities, it is a rational curve and, thus, admits a rational parameterization.
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The wave front set is useful, among others, when studying propagation of singularities by pseudodifferential operators.
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In singularity theory the general phenomenon of points and sets of singularities is studied, as part of the concept that manifolds (spaces without singularities) may acquire special, singular points by a number of routes. Projection is one way, very obvious in visual terms when three-dimensional objects are projected into two dimensions (for example in one of our eyes); in looking at classical statuary the folds of drapery are amongst the most obvious features. Singularities of this kind include caustics, very familiar as the light patterns at the bottom of a swimming pool. Other ways in which singularities occur is by degeneration of manifold structure.
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As with the event horizon in the Schwarzschild metric, the apparent singularities at r and r are illusions created by the choice of coordinates (i.e., they are coordinate singularities). In fact, the space-time can be smoothly continued through them by an appropriate choice of coordinates.
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In 1964, Hironaka proved that singularities of algebraic varieties admit resolutions in characteristic zero. This means that any algebraic variety can be replaced by (more precisely is birationally equivalent to) a similar variety which has no singularities. He also introduced Hironaka's example showing that a deformation of Kähler manifolds need not be Kähler. In 2017 he posted to his personal webpage a manuscript that claims to prove the existence of a resolution of singularities in positive characteristic.
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He showed that interplanetary magnetic flux ropes can be understood as extended singularities of the vector potential.
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I: Approximation of metrics with cone singularities. J. Amer. Math. Soc. 28 (2015), no. 1, 183–197.
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For instance, most procedures for resolution of singularities proceed by blowing up singularities until they become smooth. A consequence of this is that blowups can be used to resolve the singularities of birational maps. Classically, blowups were defined extrinsically, by first defining the blowup on spaces such as projective space using an explicit construction in coordinates and then defining blowups on other spaces in terms of an embedding. This is reflected in some of the terminology, such as the classical term monoidal transformation.
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The direction of n-vector corresponds to geodetic latitude A surface position has two degrees of freedom, and thus two parameters are sufficient to represent any position on the surface. On the reference ellipsoid, latitude and longitude are common parameters for this purpose, but like all two-parameter representations, they have singularities. This is similar to orientation, which has three degrees of freedom, but all three- parameter representations have singularities. In both cases the singularities are avoided by adding an extra parameter, i.e.
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For essential singularities, no such simple formula exists, and residues must usually be taken directly from series expansions.
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Journal of Computational Physics 2005; 209(1): 290–321. is introduced and, hence, removes or cancels such singularities.
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The basic questions about singularities (their formation, propagation, and removal, and regularity of solutions) are the same as for linear PDE, but as usual much harder to study. In the linear case one can just use spaces of distributions, but nonlinear PDEs are not usually defined on arbitrary distributions, so one replaces spaces of distributions by refinements such as Sobolev spaces. An example of singularity formation is given by the Ricci flow: Richard S. Hamilton showed that while short time solutions exist, singularities will usually form after a finite time. Grigori Perelman's solution of the Poincaré conjecture depended on a deep study of these singularities, where he showed how to continue the solution past the singularities.
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This solution has a branchpoint at x=c, and so the equation has a movable branchpoint (since it depends on the choice of the solution, i.e. the choice of the constant c). It is a basic feature of linear ordinary differential equations that singularities of solutions occur only at singularities of the equation, and so linear equations do not have movable singularities. When attempting to look for 'good' nonlinear differential equations it is this property of linear equations that one would like to see: asking for no movable singularities is often too stringent, instead one often asks for the so-called Painlevé property: 'any movable singularity should be a pole', first used by Sofia Kovalevskaya.
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Studies of pulsars and some computer simulations (Choptuik, 1997) have been performed. Mathematician Demetrios Christodoulou, a winner of the Shaw Prize, has shown that contrary to what had been expected, singularities which are not hidden in a black hole also occur. However, he then showed that such "naked singularities" are unstable.
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Second microlocalization and propagation of singularities for semi-linear hyperbolic equations. Université de Paris-Sud. Département de Mathématique, 1985.
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In the former case, the point is necessarily a codimension 3 metric singularity. The general problem of topologically classifying singularities in polyhedral spaces is largely unresolved (apart from simple statements that e.g. any singularity is locally a cone over a spherical polyhedral space one dimension less and we can study singularities there).
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The adjunction formula is false when the conormal exact sequence is not a short exact sequence. However, it is possible to use this failure to relate the singularities of X with the singularities of D. Theorems of this type are called inversion of adjunction. They are an important tool in modern birational geometry.
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Another general feature of general relativity is the appearance of spacetime boundaries known as singularities. Spacetime can be explored by following up on timelike and lightlike geodesics—all possible ways that light and particles in free fall can travel. But some solutions of Einstein's equations have "ragged edges"—regions known as spacetime singularities, where the paths of light and falling particles come to an abrupt end, and geometry becomes ill- defined. In the more interesting cases, these are "curvature singularities", where geometrical quantities characterizing spacetime curvature, such as the Ricci scalar, take on infinite values.
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In fact it is more appropriate to speak not about differential equations but about linear systems of differential equations: in order to realise any monodromy by a differential equation one has to admit, in general, the presence of additional apparent singularities, i.e. singularities with trivial local monodromy. In more modern language, the (systems of) differential equations in question are those defined in the complex plane, less a few points, and with a regular singularity at those. A more strict version of the problem requires these singularities to be Fuchsian, i.e.
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Dimca competed in the International Mathematical Olympiad in 1970, 1971, and 1972, earning two bronze medals and one silver medal.. He obtained his PhD in 1981 from the University of Bucharest; his thesis "Stable mappings and singularities", was written under the direction of Gheorghe Galbură. His Google Scholar h-index is 24. Dimca is a distinguished mathematician in algebra, geometry and topology.Bridging Algebra, Geometry, and Topology He has written three important books in this field: Sheaves in Topology, Singularities and Topology of Hypersurfaces and Topics on real and complex singularities.
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Highly anisotropic solids like graphite (quasi-2D) and Bechgaard salts (quasi-1D) show anomalies in spectroscopic measurements that are attributable to the Van Hove singularities. Van Hove singularities play a significant role in understanding optical intensities in single-walled carbon nanotubes (SWNTs) which are also quasi-1D systems. The Dirac point in graphene is a Van-Hove singularity that can be seen directly as a peak in electrical resistance, when the graphene is charge-neutral. Twisted graphene layers also show pronounced Van-Hove singularities in the DOS due to the interlayer coupling.
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The most direct is to split into real and imaginary parts, reducing the problem to evaluating two real-valued line integrals. The Cauchy integral theorem may be used to equate the line integral of an analytic function to the same integral over a more convenient curve. It also implies that over a closed curve enclosing a region where f(z) is analytic without singularities, the value of the integral is simply zero, or in case the region includes singularities, the residue theorem computes the integral in terms of the singularities.
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The watcher measure imagines the world line of an eternal "watcher" that passes through an infinite number of Big Crunch singularities.
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Mallios and Raptis use ADG to avoid the singularities in general relativity and propose this as a route to quantum gravity.
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In algebraic geometry, a singularity of an algebraic variety is a point of the variety where the tangent space may not be regularly defined. The simplest example of singularities are curves that cross themselves. But there are other types of singularities, like cusps. For example, the equation defines a curve that has a cusp at the origin .
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However, near singularities small actuator torques result in a large end-effector wrench. Thus near singularity configurations robots have large mechanical advantage.
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Marie-Hélène Schwartz (1913 – 5 January 2013) was a French mathematician, known for her work on characteristic numbers of spaces with singularities...
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Richard S. Hamilton. The formation of singularities in the Ricci flow. Surveys in differential geometry, Vol. II (Cambridge, MA, 1993), 7–136. Int.
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Additionally, in the Star Trek: Deep Space Nine episode "Visionary", the side effects from quantum singularities cause Miles O'Brien to shift through time.
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Kähler- Einstein metrics on Fano manifolds. I: Approximation of metrics with cone singularities. J. Amer. Math. Soc. 28 (2015), no. 1, 183–197.
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In 1979-1980 she visited Harvard University where she learned gauge field theory from Clifford Taubes. This lead results about point singularities in the Yang- Mills equation and the Yang–Mills–Higgs equations. Her interest in singularities soon brought her deeper into geometry, leading to a classification of singular connections and to a condition for removing two- dimensional singularities in work with Robert Sibner. Realizing that instantons could under certain circumstances be viewed as monopoles, the Sibners and Uhlenbeck constructed non-minimal unstable critical points of the Yang-Mills functional over the four-sphere in 1989.
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Astronomical observations of black holes indicate that their rate of rotation falls below the threshold to produce a naked singularity (spin parameter 1). GRS 1915+105 comes closest to the limit, with a spin parameter of 0.82-1.00. According to the cosmic censorship hypothesis, gravitational singularities may not be observable. If loop quantum gravity is correct, naked singularities may be possible in nature.
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A similar analysis using homogeneous polynomial can be carried out to determine the stability of plane cubics. The Hilbert–Mumford criterion shows that a plane cubic is stable if and only if it is smooth; it is semi-stable if and only if it admits at worst ordinary double points as singularities; a cubic with worse singularities (e.g. a cusp) is unstable.
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He received in 1983 his Ph.D from Brown University under Robert MacPherson with thesis The Intersection Homology D-module on Hypersurfaces with Isolated Singularities.
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One can speak of the singularities of an improper integral, meaning those points of the extended real number line at which limits are used.
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Such singularities in algebraic geometry are the easiest in principle to study, since they are defined by polynomial equations and therefore in terms of a coordinate system. One can say that the extrinsic meaning of a singular point isn't in question; it is just that in intrinsic terms the coordinates in the ambient space don't straightforwardly translate the geometry of the algebraic variety at the point. Intensive studies of such singularities led in the end to Heisuke Hironaka's fundamental theorem on resolution of singularities (in birational geometry in characteristic 0). This means that the simple process of "lifting" a piece of string off itself, by the "obvious" use of the cross-over at a double point, is not essentially misleading: all the singularities of algebraic geometry can be recovered as some sort of very general collapse (through multiple processes).
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In algebraic geometry, a logarithmic pair consists of a variety, together with a divisor along which one allows mild logarithmic singularities. They were studied by .
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The occurrence of such singularities was first analyzed by the Belgian physicist Léon Van Hove in 1953 for the case of phonon densities of states.
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In mathematics, a pseudomanifold is a special type of topological space. It looks like a manifold at most of its points, but it may contain singularities. For example, the cone of solutions of z^2=x^2+y^2 forms a pseudomanifold. Figure 1: A pinched torus A pseudomanifold can be regarded as a combinatorial realisation of the general idea of a manifold with singularities.
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Chaos in the colloquial sense of complete disorder or confusion, however, is to be found. It is often the basis for singularities, where cause-and-effect relationships are not clear. There are already numerous examples of singularities in social systems with Maxwell and Poincaré. Maxwell states that a word can start a war and all the great discoveries of man based on singular states.
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Since , is symmetrical around the line , and has singular points at the repeated roots of the classical modular polynomial, where it crosses itself in the complex plane. These are not the only singularities, and in particular when , there are two singularities at infinity, where and , which have only one branch and hence have a knot invariant which is a true knot, and not just a link.
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There are different types of singularities, each with different physical features which have characteristics relevant to the theories from which they originally emerged, such as the different shape of the singularities, conical and curved. They have also been hypothesized to occur without Event Horizons, structures which delineate one spacetime section from another in which events cannot affect past the horizon; these are called naked.
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7508; doi:10.1073/pnas.83.21.8390; doi:10.1073/pnas.83.22.8779 . The resulting columnar arrangement contains fractures and "pinwheel" singularities of the same types as those found experimentally.
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In geometry, the tangent cone is a generalization of the notion of the tangent space to a manifold to the case of certain spaces with singularities.
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Hamilton, Richard S. The formation of singularities in the Ricci flow. Surveys in differential geometry, Vol. II (Cambridge, MA, 1993), 7–136, Int. Press, Cambridge, MA, 1995.
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In more complicated cases it could have singularities. The limits M+ and M− could be classical and continuous or they could be taken in the L2 sense.
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I was a failsafe in case you managed to kill her. I was her final weapon. (Branch A, Chapter 5, Verse 4) Around this central timeline, various "branches" appear, caused by "singularities", namely Zero, her sisters and the Disciples.Accord: In the flow of post-Cataclysm history, if a unique set of conditions known as "singularities" come together, splits occur in time, resulting in the multiple world divergence phenomenon.
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However, deterministic chaos is just a special case of a singularity, in which a small cause produces a large observable effect due to a nonlinear dynamic behavior. In contrast the singularities raised by Maxwell, such as a loose rock at a singular point on a slope, show a linear dynamic behavior as it was demonstrated by Poincaré. Singularities are the common staple of the chaos theory, catastrophe theory and bifurcation theory.
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He also showed that, contrary to what had been expected, singularities which are not hidden in a black hole also occur. However, he then showed that such "naked singularities" are unstable. In 2000, Christodoulou published a book on general systems of partial differential equations deriving from a variational principle (or "action principle"). In 2007, he published a book on the formation of shock waves in 3-dimensional fluids.
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The Glasser effect describes the creation of singularities in the flow field of a magnetically confined plasma when small resonant perturbations modify the gradient of the pressure field.
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It was shown that in non-local gravity, Schwarzschild singularities are stable to small perturbations. Further stability analysis of black holes was carried out by Myung and Park.
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In 2013, she completed a Master of Fine Arts in creative writing at the University of Maryland, College Park. Her MFA thesis, Singularities, was directed by Maud Casey.
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In algebraic geometry, a Du Val singularity, also called simple surface singularity, Kleinian singularity, or rational double point, is an isolated singularity of a complex surface which is modeled on a double branched cover of the plane, with minimal resolution obtained by replacing the singular point with a tree of smooth rational curves, with intersection pattern dual to a Dynkin diagram of A-D-E singularity type. They are the canonical singularities (or, equivalently, rational Gorenstein singularities) in dimension 2. They were studied by and Felix Klein. The Du Val singularities also appear as quotients of C2 by a finite subgroup of SL2(C); equivalently, a finite subgroup of SU(2), which are known as binary polyhedral groups.
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However, if some elements of or are not zero, then function has multitudes of additional singularities and cutlines in the complex plane, due to the exponential growth of sin and cos along the imaginary axis; the smaller the coefficients and are, the further away these singularities are from the real axis. The extension of tetration into the complex plane is thus essential for the uniqueness; the real-analytic tetration is not unique.
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Until the early 1990s, it was widely believed that general relativity hides every singularity behind an event horizon, making naked singularities impossible. This is referred to as the cosmic censorship hypothesis. However, in 1991, physicists Stuart Shapiro and Saul Teukolsky performed computer simulations of a rotating plane of dust that indicated that general relativity might allow for "naked" singularities. What these objects would actually look like in such a model is unknown.
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Nonetheless, these problems are important for the study of the reflection of singularities of solutions to various other PDEs. Moreover, they arise in the pricing problem for certain financial instruments.
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To avoid complications arising from the presence of singularities, usually one requires the vector field to be nonvanishing. If we add more mathematical structure, our congruence may acquire new significance.
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Zhang, Effectivity of Iitaka fibrations and pluricanonical systems of polarized pairs. To appear in Pub. Math IHES. In more recent work, Birkar studied Fano varieties and singularities of linear systems.
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Terence Gaffney (born 9 March 1948) is an American mathematician who has made fundamental contributions to singularity theory – in particular, to the fields of singularities of maps and equisingularity theory..
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Piranian's dissertation was entitled A Study of the Position and Nature of the Singularities of Functions Given by Their Taylor Series. Piranian joined the faculty at University of Michigan in 1945.
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This does not solve the problem of resolution of singularities because in dimensions greater than 1 the Zariski–Riemann space is not locally affine and in particular is not a scheme.
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The rings of invariant polynomials of these finite group actions were computed by Klein, and are essentially the coordinate rings of the singularities; this is a classic result in invariant theory.
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Pankaj S Joshi. Global Aspects in Gravitation and Cosmology. Oxford: Oxford University Press, 1996. Section 3.5.Pankaj S Joshi. Gravitational Collapse and Spacetime Singularities. Cambridge: Cambridge University Press, 2007. Section 2.7.6.
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Among the essential contributions one should mention the application of the operator product expansion, as was suggested by Parisi. Recently a proof was suggested for absence of renormalon singularities in \phi^4 theory and a general criterion for their existence was formulated in terms of the asymptotic behavior of the Gell-Mann - Low function \beta(g). Analytical results for asymptotics of \beta(g) in \phi^4 theory and QED indicate the absence of renormalon singularities in these theories.
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The author show that image points alone do not make good features due to the occurrence of singularities. The paper continues, by discussing the possible additional checks to prevent singularities namely, condition numbers of J_s and Jˆ+_s, to check the null space of ˆ J_s and J^T_s . One main point that the author highlights is the relation between local minima and unrealizable image feature motions. Over the years many hybrid techniques have been developed.
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In the 1970s he introduced a coordinate transformation (now known as the McGehee transformation) which he used to regularize singularities arising in the Newtonian three-body problem. In 1975 he, with John N. Mather, proved that for the Newtonian collinear four-body problem there exist solutions which become unbounded in a finite time interval. In 1978 he was an Invited Speaker on the subject of Singularities in classical celestial mechanics at the International Congress of Mathematicians in Helsinki.
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Saito received in 1971 his promotion Ph.D. from the University of Göttingen under Egbert Brieskorn, with thesis Quasihomogene isolierte Singularitäten von Hyperflächen (Quasihomogeneous isolated singularities of hypersurfaces. Saito is a professor at the Research Institute for Mathematical Sciences (RIMS) of Kyoto University. Saito's research deals with the interplay among Lie algebras, reflection groups (Coxeter groups), braid groups, and singularities of hypersurfaces. From the 1980s, he did research on underlying symmetries of period integrals in complex hypersurfaces.
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Many theories in physics have mathematical singularities of one kind or another. Equations for these physical theories predict that the ball of mass of some quantity becomes infinite or increases without limit. This is generally a sign for a missing piece in the theory, as in the ultraviolet catastrophe, re-normalization, and instability of a hydrogen atom predicted by the Larmor formula. Some theories, such as the theory of loop quantum gravity, suggest that singularities may not exist.
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Infinite derivative gravity is a theory of gravity which attempts to remove cosmological and black hole singularities by adding extra terms to the Einstein–Hilbert action, which weaken gravity at short distances.
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3, 81–82.Victor Goryunov, Gábor Lippner, "Simple framed curve singularities" in Majid Gazor, Pei Yu, "Spectral sequences and parametric normal forms", Journal of Differential Equations 252 (2012) no. 2, 1003–1031.
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SWNTs have unique optical and spectroscopic properties largely due to one-dimensional confinement of electronic and phonon states, resulting in so-called van Hove singularities in the nanotube density of states (DOS).
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According to Schmutzer's theory the singularities of General relativity theory are smoothed, so that instead of a "Big bang" singularity, matter begins with a more gentle progression which Schmutzer terms an "Urstart".
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Compactification of moduli spaces generally require allowing certain degeneracies – for example, allowing certain singularities or reducible varieties. This is notably used in the Deligne–Mumford compactification of the moduli space of algebraic curves.
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It has been argued that elementary particles are fundamentally not material, either, but are localized properties of spacetime.. In quantum gravity, singularities may in fact mark transitions to a new phase of matter.
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A "pole" (or isolated singularity) of a function is a point where the function's value becomes unbounded, or "blows up". If a function has such a pole, then one can compute the function's residue there, which can be used to compute path integrals involving the function; this is the content of the powerful residue theorem. The remarkable behavior of holomorphic functions near essential singularities is described by Picard's Theorem. Functions that have only poles but no essential singularities are called meromorphic.
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It may be especially convenient to remove all singularities to obtain a space with a flat Riemannian metric and to study the holonomies there. One concepts thus arising are polyhedral Kähler manifolds, when the holonomies are contained in a group, conjugate to the unitary matrices. In this case, the holonomies also preserve a symplectic form, together with a complex structure on this polyhedral space (manifold) with the singularities removed. All the concepts such as differential form, L2 differential form, etc.
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Jeff Xia's 5-body configuration consists of five point masses, with two pairs in eccentric elliptic orbits around each other and one mass moving along the line of symmetry. Xia proved that for certain initial conditions the final mass will be accelerated to infinite velocity in finite time. This proves the Painlevé conjecture for five bodies and upwards. In physics, the Painlevé conjecture is a theorem about singularities among the solutions to the n-body problem: there are noncollision singularities for n ≥ 4.
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The theoretical existence of naked singularities is important because their existence would mean that it would be possible to observe the collapse of an object to infinite density. It would also cause foundational problems for general relativity, because general relativity cannot make predictions about the future evolution of space-time near a singularity. In generic black holes, this is not a problem, as an outside viewer cannot observe the space-time within the event horizon. Naked singularities have not been observed in nature.
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In characteristic 2, :z2 = f(x, y) , for a sufficiently general polynomial f(x, y) of degree 6, defines a surface with 21 isolated singularities. The smooth projective minimal model of such a surface is a unirational K3 surface, and hence a K3 surface with Picard number 22. The largest Artin invariant here is 10. Similarly, in characteristic 3, :z3 = g(x, y) , for a sufficiently general polynomial g(x, y) of degree 4, defines a surface with 9 isolated singularities.
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David Bradley Massey (born August 24, 1959) is an American mathematician. He completed both his undergraduate studies and his doctoral research work at Duke University, receiving his Ph.D. in 1986 for his results in the area of complex analytic singularities under the direction of William L. Pardon. In 1988, he was awarded a National Science Foundation Postdoctoral Research Fellowship, and went to conduct research on singularities at Northeastern University. In 1991, he assumed a regular faculty position in the Mathematics Department at Northeastern.
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At least three single-axis CMGs are necessary for control of spacecraft attitude. However, no matter how many CMGs a spacecraft uses, gimbal motion can lead to relative orientations that produce no usable output torque along certain directions. These orientations are known as singularities and are related to the kinematics of robotic systems that encounter limits on the end-effector velocities due to certain joint alignments. Avoiding these singularities is naturally of great interest, and several techniques have been proposed.
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This view was held in particular by Vladimir Belinsky, Isaak Khalatnikov, and Evgeny Lifshitz, who tried to prove that no singularities appear in generic solutions. However, in the late 1960s Roger Penrose and Stephen Hawking used global techniques to prove that singularities appear generically. For this work, Penrose received half of the 2020 Nobel Prize in Physics, Hawking having died in 2018. Work by James Bardeen, Jacob Bekenstein, Carter, and Hawking in the early 1970s led to the formulation of black hole thermodynamics.
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Based on his monotonicity formula, Huisken showed that many of these regions, specifically those known as type I singularities, are modeled in a precise way by self-shrinking solutions of the mean curvature flow. There is now a reasonably complete understanding of the rescaling process in the setting of mean curvature flows which only involve hypersurfaces whose mean curvature is strictly positive. Following provisional work by Huisken, Tobias Colding and William Minicozzi have shown that (with some technical conditions) the only self-shrinking solutions of mean curvature flow which have nonnegative mean curvature are the round cylinders, hence giving a complete local picture of the type I singularities in the "mean-convex" setting.Tobias H. Colding and William P. Minicozzi, II. Generic mean curvature flow I: generic singularities. Ann.
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Geroch obtained his Ph.D. degree from Princeton University in 1967 under the supervision of John Archibald Wheeler, with a thesis on Singularities in the spacetime of general relativity: their definition, existence, and local characterization.
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An important object associated to an unfolding is its bifurcation set. This set lives in the parameter space of the unfolding, and gives all parameter values for which the resulting function has degenerate singularities.
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An example of a rational singularity is the singular point of the quadric cone :x^2 + y^2 + z^2 = 0. \, showed that the rational double points of algebraic surfaces are the Du Val singularities.
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In 1983, Terao asked whether the freeness of an arrangement is determined from its intersection lattice.Hiroaki. Terao, The exponents of a free hypersurface, Singularities, Part 2 (Arcata, Calif., 1981), 561–566, Proc. Sympos. Pure Math.
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The Worldwide Center of Mathematics (or Center of Math) is an American education technology company that publishes mathematics textbooks and produces educational videos and mathematical research. Since 2010, it has published the Journal of Singularities.
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In string theory, a Whitney brane is a D7-brane wrapping a variety whose singularities are locally modeled by the Whitney umbrella. Whitney branes appear naturally when taking Sen's weak coupling limit of F-theory.
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The singularities nearest 0, which is the center of the power series expansion, are at ±2i. The distance from the center to either of those points is 2, so the radius of convergence is 2\.
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Another direction of research are developments of dynamical billiards in polyhedral spaces, e.g. of nonpositive curvature (hyperbolic billiards). Positively curved polyhedral spaces arise also as links of points (typically metric singularities) in Euclidean polyhedral spaces.
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In algebraic geometry, the Bott–Samelson resolution of a Schubert variety is a resolution of singularities. It was introduced by in the context of compact Lie groups. The algebraic formulation is independently due to and .
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The Tanh-Sinh method is quite insensitive to endpoint behavior. Should singularities or infinite derivatives exist at one or both endpoints of the (−1, +1) interval, these are mapped to the (−∞,+∞) endpoints of the transformed interval, forcing the endpoint singularities and infinite derivatives to vanish. This results in a great enhancement of the accuracy of the numerical integration procedure, which is typically performed by the Trapezoidal Rule. In most cases, the transformed integrand displays a rapid roll-off (decay), enabling the numerical integrator to quickly achieve convergence.
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Perelman rejected that prize as well. Perelman proved the conjecture by deforming the manifold using the Ricci flow (which behaves similarly to the heat equation that describes the diffusion of heat through an object). The Ricci flow usually deforms the manifold towards a rounder shape, except for some cases where it stretches the manifold apart from itself towards what are known as singularities. Perelman and Hamilton then chop the manifold at the singularities (a process called "surgery") causing the separate pieces to form into ball-like shapes.
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A direct result of this folding is that the nanotube density of states has a number of sharp peaks due to 1-D van Hove singularities, which are absent in graphene and graphite. Despite the presence of these singularities, the overall density of states is similar at high energies, so that the high temperature specific heat should be roughly equal as well. This is to be expected: the high-energy phonons are more reflective of carbon–carbon bonding than the geometry of the graphene sheet.
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Levy's theories rely heavily on the technological developments of the 1990s, particularly the rise of biotechnology, nanotechnology, the Internet, new media and information technologies. In chapter 3, he describes how technologies have made a shift from the molar to the molecular (a move which makes literal a distinction by Delueze and Guattari) in that technologies now handle units as individuals (his term is "singularities") rather than in mass. He suggests that this mirrors our rising recognition of the individuals as singularities rather than massive conglomerated groups.
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The map from the surface to the flat torus obtained by identifying all squares is a branched covering with branch points the singularities (the cone angle at a singularity is proportional to the degree of branching).
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However, similarly to the use of Euler angles as a formalism for representing rotations, using only the minimum number of parameters gives singularities, and thus three parameters are required for the horizontal position to avoid this.
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Polini is currently an Associate Editor for the Journal of Commutative Algebra. She co-authored the research monograph A Study of Singularities on Rational Curves Via Syzygies with David Cox, Andrew R. Kustin, and Bernd Ulrich.
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The quantum theory of magnetic charge started with a paper by the physicist Paul Dirac in 1931.Paul Dirac, "Quantised Singularities in the Electromagnetic Field". Proc. Roy. Soc. (London) A 133, 60 (1931). Journal Site, Free Access .
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Lautrup has noted that there exist individual diagrams giving approximately the same contribution. In principle, it is possible that such diagrams are automatically taken into account in Lipatov’s calculation, because its interpretation in terms of diagrammatic technique is problematic. However, 't Hooft put forward a conjecture that Lipatov's and Lautrup's contributions are related with different types of singularities in the Borel plane, the former with instanton ones and the latter with renormalon ones. Existence of instanton singularities is beyond any doubt, while existence of renormalon ones was never proved rigorously in spite of numerous efforts.
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Although folk tales such as St Swithun's Day generally have little credibility, some of these events have a more solid basis. Early scientific investigation involved the creation of calendars of singularities based on temperature and rainfall anomalies. Later and more successful work by Hubert Lamb of the Climatic Research Unit was based on air circulation patterns. Lamb's work analysed daily frequency of airflow categories between 1898 and 1947.Lamb H.H (1950) Types and spells of weather around the year in the British Isles: Annual trends, seasonal structure of the year, singularities. Quart.
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A singularity is a weather phenomenon likely to occur with reasonable regularity around a specific approximate calendar date,Barry R.G. & Chorley R.J. (1987), Atmosphere, Weather & Climate, 5th ed, Routledge, outside of more general seasonal weather patterns (e.g., that May Day is usually warmer than New Year's Day in northern locales). The existence of singularities is disputed, some considering them due to seeing patterns in noise and statistical artifacts from small samples. In North America, the most significant purported singularities are January thaw (warmer weather around January 25) and Indian summer (warmer weather in mid-autumn).
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Furthermore, there are so-called singularity theorems which predict that such singularities must exist within the universe if the laws of general relativity were to hold without any quantum modifications. The best-known examples are the singularities associated with the model universes that describe black holes and the beginning of the universe.With a focus on string theory, the search for quantum gravity is described in ; for an account from the point of view of loop quantum gravity, see . Other attempts to modify general relativity have been made in the context of cosmology.
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This is to be contrasted with the highly sensitive and fading footprint of hyperbolic LCSs away from strongly hyperbolic regions in diffusive tracer patterns. Under variable endpoint boundary conditions, initial positions of parabolic LCSs turn out to be alternating chains of shrink lines and stretch lines that connect singularities of these line fields. These singularities occur at points where \lambda_1(x_0)=\lambda_2(x_0), and hence no infinitesimal deformation takes place between the two time instances t_0 and t_1. Fig. 14b shows an example of parabolic LCSs in Jupiter's atmosphere, located using this variational theory.
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The quad sphere projection does not produce singularities at the poles or elsewhere, as do some other equal-area mapping schemes. Distortion is moderate over the entire sphere, so that at no point are shapes altered beyond recognition.
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In mathematics, Reeb sphere theorem, named after Georges Reeb, states that : A closed oriented connected manifold M n that admits a singular foliation having only centers is homeomorphic to the sphere Sn and the foliation has exactly two singularities.
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Department of Physics and Astronomy, University of New Mexico. April 1, 2018. Accessed January 18, 2019. It treats spinors, the variational- principle formulation, the initial-value formulation, (exact) gravitational waves, singularities, Penrose diagrams, Hawking radiation, and black-hole thermodynamics.
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World Scientific. With a suitably enhanced notion of satellite operation called splicing, the JSJ decomposition gives a proper uniqueness theorem for satellite knots.Eisenbud, D. Neumann, W. Three- dimensional link theory and invariants of plane curve singularities. Ann. of Math. Stud.
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In algebraic geometry a normal crossing singularity is a singularity similar to a union of coordinate hyperplanes. The term can be confusing because normal crossing singularities are not usually normal schemes (in the sense of the local rings being integrally closed).
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Newcome records that "among other of his singularities he made the sophisters say their positions without book". cites Newcome Autobiography, p. 9. He was regarded as "a man of very great eminency in learning, strictness in religion, unblamableness in conversation".
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2, 267–275. F. Diacu, On the singularities of the curved n-body problem, Transactions of the American Mathematical Society 363 (2011), no. 4, 2249–2264. These equations provide a new criterion for determining the geometrical nature of the physical space.
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A naked singularity could allow scientists to observe an infinitely dense material, which would under normal circumstances be impossible by the cosmic censorship hypothesis. Without an event horizon of any kind, some speculate that naked singularities could actually emit light.
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XLIV-19, ISSN 1029-8479 [hep-th/0106099] [abs] 13\. W Pardon and M Stern, Pure hodge structure on the L2-cohomology of varieties with isolated singularities, Journal fur die Reine und Angewandte Mathematik, vol. 533 (2001), pp. 55–80 14\.
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The foreword for the Russian translation of his book Curves and Singularities was written by V. I. Arnold. Giblin has also authored or co–authored over a hundred peer reviewed published articles. The first of these was published in 1968.
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David Bailey and others have argued (in patents and in academic publications) that merely avoiding the "divide by zero" error that is associated with these singularities is sufficient. Two more recent patents summarize competing approaches.US Patent 7246776 See also Gimbal lock.
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In complex analysis, a branch of mathematics, the Casorati–Weierstrass theorem describes the behaviour of holomorphic functions near their essential singularities. It is named for Karl Theodor Wilhelm Weierstrass and Felice Casorati. In Russian literature it is called Sokhotski's theorem.
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Therefore, it is necessary to study the possible singularities of the 3-body problems. As it will be briefly discussed below, the only singularities in the 3-body problem are binary collisions (collisions between two particles at an instant) and triple collisions (collisions between three particles at an instant). Collisions, whether binary or triple (in fact, any number), are somewhat improbable, since it has been shown that they correspond to a set of initial conditions of measure zero. However, there is no criterion known to be put on the initial state in order to avoid collisions for the corresponding solution.
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The quest for a quantum version of general relativity addresses one of the most fundamental open questions in physics. While there are promising candidates for such a theory of quantum gravity, notably string theory and loop quantum gravity, there is at present no consistent and complete theory. It has long been hoped that a theory of quantum gravity would also eliminate another problematic feature of general relativity: the presence of spacetime singularities. These singularities are boundaries ("sharp edges") of spacetime at which geometry becomes ill-defined, with the consequence that general relativity itself loses its predictive power.
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Immanuel Kant unwittingly proposed one answer to this question in his Critique of Pure Reason by stating that there must "exist" a "thing in itself" – a thing which is the cause of phenomena, but not a phenomenon itself. But, so to speak, "approximations" of "things in themselves" crop up in many models of empirical phenomena. Singularities in physics, such as gravitational singularities, certain aspects of which (e.g., their unquantifiability) can seem almost to mirror various "aspects" of the proposed "thing in itself", are generally eliminated (or attempts are made at eliminating them) in newer, more precise models of the universe.
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For smooth, but not necessarily affine varieties, there is an isomorphism relating the hypercohomology of algebraic the de Rham complex to the singular cohomology. A proof of this comparison result using the concept of a Weil cohomology was given by . Counter-examples in the singular case can be found with non-Du Bois singularities such as the graded ring k[x,y]/(y^2-x^3) with y where \deg(y)=3 and \deg(x)=2. Other counterexamples can be found in algebraic plane curves with isolated singularities whose Milnor and Tjurina numbers are non-equal.
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Perelman's major achievement was to show that, if one takes a certain perspective, if they appear in finite time, these singularities can only look like shrinking spheres or cylinders. With a quantitative understanding of this phenomena, he cuts the manifold along the singularities, splitting the manifold into several pieces, and then continues with the Ricci flow on each of these pieces. This procedure is known as Ricci flow with surgery. Perelman provided a separate argument based on curve shortening flow to show that, on a simply-connected compact 3-manifold, any solution of the Ricci flow with surgery becomes extinct in finite time.
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The term quantum singularity is used to refer to many different phenomena in fiction. They often only approximate a gravitational singularity in the scientific sense in that they are massive, localized distortions of space and time. The name invokes one of the most fundamental problems remaining in modern physics: the difficulty in uniting Einstein's theory of relativity, which includes singularities within its models of black holes, and quantum mechanics. In fact, since according to relativity, singularities, by definition, are infinitely small, and expected to be quantum mechanical by nature, a theory of quantum gravity would be required to describe their behaviour.
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The knowledge space is an emerging anthropological space which, while it has always existed (139), is only now coming into fruition as a guiding space of humanity. In this space, singularities (individuals) are recognized as singularities and knowledge becomes the guiding value for humanity. Since all human experience represents unique knowledge, within the knowledge space all individuals are valued for their unique knowledge regardless of race (earth space), nationality (territorial space), or economic status (commodity space). Within this space static identity gives way to the "quantum identities" as individuals become participates and the distinction between of "us" and "them" disappears (159).
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The unifying feature of Vasy's work is the application of tools from microlocal analysis to problems in hyperbolic partial differential or pseudo-differential equations. He analyzed the propagation of singularities for solutions of wave equations on manifolds with cornersAndrás Vasy, "Propagation of singularities for the wave equation on manifolds with corners", Annals of Mathematics 168, 749-812 (2008) or more complicated boundary structures, partially in joint work with Richard Melrose and Jared Wunsch.Jared Wunsch, András Vasy, and Richard B. Melrose, "Propagation of singularities for the wave equation on edge manifolds", Duke Math. J. 144(1), pp. 109-193 (2008) For his paper on a unified approach to scattering theory on asymptotically hyperbolic spaces and spacetimes arising in Einstein's theory of general relativity such as de Sitter space and Kerr-de Sitter spacetimes,András Vasy, "Microlocal analysis of asymptotically hyperbolic and Kerr-de Sitter spaces (with an appendix by Semyon Dyatlov)", Inventiones Mathematicae 194(2), pp.
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In mathematics, a point source is a singularity from which flux or flow is emanating. Although singularities such as this do not exist in the observable universe, mathematical point sources are often used as approximations to reality in physics and other fields.
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Thus a pole is a certain type of singularity of a function, nearby which the function behaves relatively regularly, in contrast to essential singularities, such as 0 for the logarithm function, and branch points, such as 0 for the complex square root function.
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Pinwheel formations in the primary visual cortex with singularities in the center. Each color represents an orientation column of a specific line phase. Adapted image from fMRI studies. Using 2D optical techniques, pinwheel formations (also known as whorls) of orientation columns were discovered.
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Hélène Esnault Hélène Esnault (born 1953 in Paris) is a French and German mathematician, specializing in algebraic geometry. She received her PhD in 1976 under Professor Lê Dũng Tráng, writing her dissertation on Singularites rationnelles et groupes algebriques (Rational singularities and algebraic groups).
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Aimed at beginning graduate students, it covers spinors, the variational-principle formulation, the initial-value formulation, (exact) gravitational waves, singularities, Penrose diagrams, Hawking radiation, and black-hole thermodynamics.A Guide to Relativity Books. John C. Baez et al. University of California, Riverside. September 1998.
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The earlier work of Uhlmann was in microlocal analysis and propagation of singularities for equations with multiple characteristics, in particular in understanding the phenomenon of conical refraction."Light intensity distribution in conical refraction", Comm. on Pure and Appl. Math., 35 (1982), 69-80.
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Subsequent research has illustrated the more general use of the method for problems involving singularities, material interfaces, regular meshing of microstructural features such as voids, and other problems where a localized feature can be described by an appropriate set of basis functions.
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Buz M. Walker achieved worldwide distinction in 1906 with a University of Chicago dissertation “On the Resolution of the Higher Singularities of Algebraic Curves Into Ordinary Nodes”, supervised by Oskar Bolza. Walker was an Invited Speaker of the ICM in 1936 in Oslo.
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An analytic space is a generalization of an analytic manifold that allows singularities. An analytic space is a space that is locally the same as an analytic variety. They are prominent in the study of several complex variables, but they also appear in other contexts.
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Use of this (or equivalent) equipment and following an appropriate protocol can yield better than 10% repeatability in absolute measurements for the rate of Raman scattering. This can be useful with resonance Raman for accurately determining optical transitions in structures with strong Van Hove singularities.
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The Journal of Singularities is a peer-reviewed open-access scientific journal which publishes research in the area of singularity theory. It was established in 2010 by David B. Massey, who remains editor-in-chief , and is published by the Worldwide Center of Mathematics.
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Bill Pardon, Mark A Stern, Pure Hodge structures on the L2-cohomology of varieties with isolated singularities., J. Reine Angew. Math. 533 (2001) 55-80. 28\. Sonia Paban, Savdeep Sethi, and Mark A. Stern, Summing Instantons in 3 dimensional Yang- Mills theories, Adv. Theor. Math.
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West of Alzey, in Rhenish Hesse, at an elevation of 250 m lies Erbes-Büdesheim, a place marked by distinctive geological features, botanical singularities and a great number of surprising historical facts. It belongs to the Verbandsgemeinde of Alzey-Land, whose seat is in Alzey.
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While it is perfectly determined, possessing certain dimensions, it has no other determinations than its formal singularities, equidistant vertices and center, no other contents or occupants than the four similar protagonists who traverse it ceaselessly. It is a closed, globally defined, any-space-whatever.
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At this time, he began his work with Abe Gelbart on what would eventually develop into the theory of pseudoanalytic functions. Through the 1940s and 1950s he continued to develop this theory, and to use it to study the planar elliptic partial differential equations associated with subsonic flows. Another of his major results in this time concerned the singularities of the partial differential equations defining minimal surfaces. Bers proved an extension of Riemann's theorem on removable singularities, showing that any isolated singularity of a pencil of minimal surfaces can be removed; he spoke on this result at the 1950 International Congress of Mathematicians and published it in Annals of Mathematics.
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In mathematics, precisely in the theory of functions of several complex variables, Hartogs's extension theorem is a statement about the singularities of holomorphic functions of several variables. Informally, it states that the support of the singularities of such functions cannot be compact, therefore the singular set of a function of several complex variables must (loosely speaking) 'go off to infinity' in some direction. More precisely, it shows that an isolated singularity is always a removable singularity for any analytic function of complex variables. A first version of this theorem was proved by Friedrich Hartogs,See the original paper of and its description in various historical surveys by , and .
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Petters is known for his work in the mathematical theory of gravitational lensing. Over the ten-year period from 1991 to 2001, Petters systematically developed a mathematical theory of weak-deflection gravitational lensing, beginning with his 1991 MIT Ph.D. thesis on "Singularities in Gravitational Microlensing". A. O. Petters, Ph.D. Thesis, MIT, Department of Mathematics (1991): "Singularities in Gravitational Microlensing." In a series of papers, he and his collaborators resolved an array of theoretical problems in weak-deflection gravitational lensing covering image counting, fixed-point images, image magnification, image time delays, local geometry of caustics, global geometry of caustics, wavefronts, caustic surfaces, and caustic surfing.
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The extension of the problem to higher dimensions (that is, for k-dimensional surfaces in n-dimensional space) turns out to be much more difficult to study. Moreover, while the solutions to the original problem are always regular, it turns out that the solutions to the extended problem may have singularities if k \leq n - 2. In the hypersurface case where k = n - 1, singularities occur only for n \geq 8. To solve the extended problem in certain special cases, the theory of perimeters (De Giorgi) for codimension 1 and the theory of rectifiable currents (Federer and Fleming) for higher codimension have been developed.
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This leads to Walternate's creation of the shapeshifters that can cross between the two universes to act as his agents in the prime universe. His actions are further spurred by the damage of Walter's crossing, creating numerous singularities that tear at the parallel universe's fabric of reality. According to Nina, William had warned her that the "great storm" would cause one of the two universes to be completely destroyed by the other. Such events are shown to come to pass in the future Peter witnesses while using the Machine in "The Day We Died", in which the parallel universe had been fully obliterated by the singularities.
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Loop quantum cosmology (LQC) is a finite, symmetry-reduced model of loop quantum gravity (LQG) that predicts a "quantum bridge" between contracting and expanding cosmological branches. The distinguishing feature of LQC is the prominent role played by the quantum geometry effects of loop quantum gravity (LQG). In particular, quantum geometry creates a brand new repulsive force which is totally negligible at low space-time curvature but rises very rapidly in the Planck regime, overwhelming the classical gravitational attraction and thereby resolving singularities of general relativity. Once singularities are resolved, the conceptual paradigm of cosmology changes and one has to revisit many of the standard issues—e.g.
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The Fourier transformation being (well-)defined for compactly supported generalized functions (component-wise), one can apply the same construction as for distributions, and define Lars Hörmander's wave front set also for generalized functions. This has an especially important application in the analysis of propagation of singularities.
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In algebraic geometry, the abundance conjecture is a conjecture in birational geometry, more precisely in the minimal model program, stating that for every projective variety X with Kawamata log terminal singularities over a field k if the canonical bundle K_X is nef, then K_X is semi-ample.
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The improper integral can also be defined for functions of several variables. The definition is slightly different, depending on whether one requires integrating over an unbounded domain, such as \R^2, or is integrating a function with singularities, like f(x,y)=\log(x^2+y^2).
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A graduate of Bronx High School of Science, he received a B.S. in mathematics from Massachusetts Institute of Technology in 1960, and M.A. and Ph.D. degrees in mathematics from Brandeis University, respectively in 1963 and 1972. His dissertation was about singularities in analytic partial differential equations.
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In mathematical analysis, more precisely in microlocal analysis, the wave front (set) WF(f) characterizes the singularities of a generalized function f, not only in space, but also with respect to its Fourier transform at each point. The term "wave front" was coined by Lars Hörmander around 1970.
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The game also includes 25 different weapons, including Flame-throwers, Vortex Singularities, Plasma Rifles, Mini-guns, Ion Cannons and Nukes. Sabotage and espionage missions allow units to deposit mines inside enemy buildings, steal research, and pass back vital information. Machines is multiplayer over a LAN or the internet.
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As a mathematical tool, Colombeau algebras can be said to combine a treatment of singularities, differentiation and nonlinear operations in one framework, lifting the limitations of distribution theory. These algebras have found numerous applications in the fields of partial differential equations, geophysics, microlocal analysis and general relativity so far.
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Bliss's Lectures more or less constitutes the culmination of the classic calculus of variations of Weierstrass, Hilbert, and Bolza. Subsequent work on variational problems would strike out in new directions, such as Morse theory, optimal control, and dynamic programming. Bliss also studied singularities of real transformations in the plane.
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Technically this involves group actions of Lie groups on spaces of jets; in less abstract terms Taylor series are examined up to change of variable, pinning down singularities with enough derivatives. Applications, according to Arnold, are to be seen in symplectic geometry, as the geometric form of classical mechanics.
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The bicuspid is a quartic plane curve with the equation :(x^2-a^2)(x-a)^2+(y^2-a^2)^2=0 \, where a determines the size of the curve. The bicuspid has only the two nodes as singularities, and hence is a curve of genus one.
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In mathematics, in the theory of algebraic curves, a delta invariant measures the number of double points concentrated at a point.John Milnor, Singular Points of Hypersurfaces, p. 85 It is a non-negative integer. Delta invariants are discussed in the "Classification of singularities" section of the algebraic curve article.
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In algebraic geometry, general elephant is an idiosyncratic name for a general element of the anticanonical system of a variety, introduced by . For 3-folds the general elephant problem (or conjecture) asks whether general elephants have at most du Val singularities; this has been proved in several cases.
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This application requires the same evaluation of DFT term X[k], as discussed in the previous section, using a real-valued or complex-valued input stream. Then the signal phase can be evaluated as taking appropriate precautions for singularities, quadrant, and so forth when computing the inverse tangent function.
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Einstein and coworker Leopold Infeld managed to demonstrate that, in Einstein's final theory of the unified field, true singularities of the field did have trajectories resembling point particles. However, singularities are places where the equations break down, and Einstein believed that in an ultimate theory the laws should apply everywhere, with particles being soliton-like solutions to the (highly nonlinear) field equations. Further, the large-scale topology of the universe should impose restrictions on the solutions, such as quantization or discrete symmetries. The degree of abstraction, combined with a relative lack of good mathematical tools for analyzing nonlinear equation systems, make it hard to connect such theories with the physical phenomena that they might describe.
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A projective variety X is called minimal if the canonical bundle KX is nef. For X of dimension 2, it is enough to consider smooth varieties in this definition. In dimensions at least 3, minimal varieties must be allowed to have certain mild singularities, for which KX is still well-behaved; these are called terminal singularities. That being said, the minimal model conjecture would imply that every variety X is either covered by rational curves or birational to a minimal variety Y. When it exists, Y is called a minimal model of X. Minimal models are not unique in dimensions at least 3, but any two minimal varieties which are birational are very close.
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This also cast doubt on the physicality of the third (transverse–transverse) type that Eddington showed always propagate at the speed of light regardless of coordinate system. In 1936, Einstein and Nathan Rosen submitted a paper to Physical Review in which they claimed gravitational waves could not exist in the full general theory of relativity because any such solution of the field equations would have a singularity. The journal sent their manuscript to be reviewed by Howard P. Robertson, who anonymously reported that the singularities in question were simply the harmless coordinate singularities of the employed cylindrical coordinates. Einstein, who was unfamiliar with the concept of peer review, angrily withdrew the manuscript, never to publish in Physical Review again.
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The goal is to prove asymptotic behavior of a series: to show that a_n\sim F(n) for some function. This is done by taking the generating function of the series, then computing the residues about zero (essentially the Fourier coefficients). Technically, the generating function is scaled to have radius of convergence 1, so it has singularities on the unit circle – thus one cannot take the contour integral over the unit circle. The circle method is specifically how to compute these residues, by partitioning the circle into minor arcs (the bulk of the circle) and major arcs (small arcs containing the most significant singularities), and then bounding the behavior on the minor arcs.
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In the Voyager episode "Hunters", the crew discover a Hirogen relay station almost 100,000 years old, powered by a quantum singularity, also referred to by Tom Paris as a black hole. The word "tiny" being used to describe a quantum singularity, about "a centimeter" in diameter, making it relatively large, although it is more likely that the stated diameter instead refers to the singularity's event horizon. In the episode "Scorpion", Species 8472 and the Borg, make use of quantum singularities to travel to and from fluidic space. Artificial quantum singularities are also used to power Romulan Warbirds as first described in the Star Trek: The Next Generation episode "Face of the Enemy".
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There are efficient algorithms for both conversions, that is for computing the recurrence relation from the differential equation, and vice versa. It follows that, if one represents (in a computer) holonomic functions by their defining differential equations and initial conditions, most calculus operations can be done automatically on these functions, such as derivative, indefinite and definite integral, fast computation of Taylor series (thanks of the recurrence relation on its coefficients), evaluation to a high precision with certified bound of the approximation error, limits, localization of singularities, asymptotic behavior at infinity and near singularities, proof of identities, etc.Benoit, A., Chyzak, F., Darrasse, A., Gerhold, S., Mezzarobba, M., & Salvy, B. (2010, September). The dynamic dictionary of mathematical functions (DDMF).
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A toric variety is nonsingular if its cones of maximal dimension are generated by a basis of the lattice. This implies that every toric variety has a resolution of singularities given by another toric variety, which can be constructed by subdividing the maximal cones into cones of nonsingular toric varieties.
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In mathematics, a Severi variety is an algebraic variety in a Hilbert scheme that parametrizes curves in projective space with given degree and geometric genus and at most node singularities. Its dimension is 3d + g − 1\. It is a theorem that Severi varieties are algebraic varieties, i.e. it is irreducible.
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Beverly K. Berger is an American physicist known for her work on gravitational physics, especially gravitational waves, gravitons, and gravitational singularities. Alongside Berger's more serious physics research, she is also known for noticing that vibrational patterns caused by local ravens were interfering with observations at the Laser Interferometer Gravitational-Wave Observatory.
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A theorem of Hassler Whitney Th. Bröcker, Differentiable Germs and Catastrophes, London Mathematical Society. Lecture Notes 17. Cambridge, (1975)Bruce and Giblin, Curves and singularities, (1984, 1992) , (paperback) states :Theorem. Any closed set in Rn occurs as the solution set of f −1(0) for some smooth function f:Rn→R.
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Borel summation finds application in perturbation expansions in quantum field theory. In particular in 2-dimensional Euclidean field theory the Schwinger functions can often be recovered from their perturbation series using Borel summation . Some of the singularities of the Borel transform are related to instantons and renormalons in quantum field theory .
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Andrei Gabrielov is a mathematician who is a professor at Purdue University. He is a fellow of the American Mathematical Society since 2016, for "contributions to real algebraic and analytic geometry, and the theory of singularities, and for contributions to geophysics." He obtained his Ph.D. from Moscow State University in 1973.
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He also worked with hypergeometric differential equations in 1857 using complex analytical methods and presented the solutions through the behavior of closed paths about singularities (described by the monodromy matrix). The proof of the existence of such differential equations by previously known monodromy matrices is one of the Hilbert problems.
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Pinwheels are the location where multiple orientation columns converge. Orientation columns are organized radially around a point known as a singularity. The arrangement, around the singularity, can be observed to be in both a counter-clockwise or clockwise fashion. It is suggested that an artifact of the optical recordings may cause these singularities.
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Gravitationally-interacting massive particles (GIMPs) are a set of particles theorised to explain the dark matter in our universe, as opposed to an alternative theory based on weakly-interacting massive particles (WIMPs). The proposal makes dark matter a form of singularities in dark energy, described by Einstein's gravitational field equations for General Relativity.
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It is interesting to study the curvature of polyhedral spaces (the curvature in the sense of Alexandrov spaces), specifically polyhedral spaces of nonnegative and nonpositive curvature. Nonnegative curvature on singularities of codimension 2 implies nonnegative curvature overall. However, this is false for nonpositive curvature. For example, consider R^3 with one octant removed.
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Double entendres also accentuate the gospel's theological singularities. For example, the narrator uses the verb “to be lifted up” (Greek: ὑψωθῆναι, hypsōthēnai) to describe Jesus’ crucifixion in John 3:14, 8:28, and 12:32. In each instance, it has a second, theological meaning: He is exalted or glorified in this act.
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His research deals with microlocal analysis, partial differential equations and potential theory. In 1981 he published important results on paradifferential operators, extending the theory of pseudifferential operators published by Ronald Coifman and Yves Meyer in 1979. Bony applied his theory to the propagation of singularities in solutions of semilinear wave equations.Bony, J-M.
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The cone is oriented if and only if S is oriented and not zero-dimensional. If we consider the (closed) cone with bottom, we get a stratifold with boundary S. Other examples for stratifolds are one-point compactifications and suspensions of manifolds, (real) algebraic varieties with only isolated singularities and (finite) simplicial complexes.
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Unable to deal with the guilt of allowing millions to die by not acting sooner, she goes into hiding. Bette is discovered by the Plutonian in Irredeemable #29, living a hedonistic lifestyle while awaiting death. Bette is later possessed by Modeus. In Issue #35, Modeus unlocks Bette's true potential, physics altering gravity singularities.
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Since such quantities become infinite at the singularity, the laws of normal spacetime break down. Gravitational singularities are mainly considered in the context of general relativity, where density apparently becomes infinite at the center of a black hole, and within astrophysics and cosmology as the earliest state of the universe during the Big Bang. Physicists are undecided whether the prediction of singularities means that they actually exist (or existed at the start of the Big Bang), or that current knowledge is insufficient to describe what happens at such extreme densities. General relativity predicts that any object collapsing beyond a certain point (for stars this is the Schwarzschild radius) would form a black hole, inside which a singularity (covered by an event horizon) would be formed.
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Properties of the original curve correspond to dual properties on the dual curve. In the image at right, the red curve has three singularities – a node in the center, and two cusps at the lower right and lower left. The black curve has no singularities, but has four distinguished points: the two top-most points have the same tangent line (a horizontal line), while there are two inflection points on the upper curve. The two top-most points correspond to the node (double point), as they both have the same tangent line, hence map to the same point in the dual curve, while the inflection points correspond to the cusps, corresponding to the tangent lines first going one way, then the other (slope increasing, then decreasing).
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Libgober, Homotopy groups of the complements to singular hypersurfaces II, Annals of Mathematics (2) 139 (1994), no. 1, 117-144 and explicit relations between these fundamental groups, the position of singularities, and local invariants of singularities (the constants of quasi-adjunction). Later he introduced the characteristic varieties of the fundamental groups, providing a multivariable extension of Alexander polynomials, and applied these methods to the study of homotopy groups of the complements to hypersurfaces in projective spaces and the topology of arrangements of hyperplanes. In the early 90s he started work on interactions between algebraic geometry and physics, providing mirror symmetry predictions for the count of rational curves on complete intersections A.Libgober, J.Teitelbaum, Lines on Calabi-Yau complete intersections, mirror symmetry, and Picard-Fuchs equations. Internat. Math. Res.
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Associated with such singularities, renormalon contributions are discussed in the context of quantum chromodynamics (QCD) and usually have the power-like form \left(\Lambda/Q\right)^p as functions of the momentum Q (here \Lambda is the momentum cut-off). They are cited against the usual logarithmic effects like \ln\left(\Lambda/Q\right).
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Another parametrization was discovered by Rob Kusner and Robert Bryant.. Boy's surface is one of the two possible immersions of the real projective plane which have only a single triple point. Unlike the Roman surface and the cross-cap, it has no other singularities than self-intersections (that is, it has no pinch-points).
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The first two terms of this expansion give the Euler equations and the Navier-Stokes equations. The higher terms have singularities. The problem of developing mathematically the limiting processes, which lead from the atomistic view (represented by Boltzmann's equation) to the laws of motion of continua, is an important part of Hilbert's sixth problem.
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By considering data associated with families of Riemann surfaces branched over the singularities, we can consider the isomonodromy equations as nonhomogeneous Gauss–Manin connections. This leads to alternative descriptions of the isomonodromy equations in terms of abelian functions - an approach known to Fuchs and Painlevé, but lost until rediscovery by Yuri Manin in 1996.
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If this quintom energy exists, it may indicate how the universe avoids Big Bangs, Big Rips, and other time-like singularities. The Quintom Scenario was applied in 2008 to produce a model of inflationary cosmology with a Big Bounce instead of a Big Bang singularity. It was also applied in 2007 to a Big Bounce model of the universe.
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The Leroy P Steele Prize of the AMS, MacTutor history of mathematics archive, retrieved 2015-04-24. He proved theorems in Schubert calculus about singularities of Schubert varieties. The Carrell–Liebermann theorem on the zero set of a holomorphic vector field is used in complex algebraic geometry. He is a fellow of the American Mathematical Society.
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Educated at Swarthmore College and Harvard University, MacPherson received his Ph.D. from Harvard in 1970. His thesis, written under the direction of Raoul Bott, was entitled Singularities of Maps and Characteristic Classes. Among his many Ph.D students are Kari Vilonen and Mark Goresky. In 1992 MacPherson was awarded the NAS Award in Mathematics from the National Academy of Sciences.
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Since 2009, he is an academician of the Serbian Academy of Sciences and Arts. His research interests include functional analysis, generalized functions and hyperfunctions, pseudo-differential operators, time–frequency analysis, linear and nonlinear equations with singularities. Probability theory and stochastic processes. Moreover, he is also interested in applications of mathematics in mechanics with applications in medicine.
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Like the heat flow, Ricci flow tends towards uniform behavior. Unlike the heat flow, the Ricci flow could run into singularities and stop functioning. A singularity in a manifold is a place where it is not differentiable: like a corner or a cusp or a pinching. The Ricci flow was only defined for smooth differentiable manifolds.
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Complexity that is kind of a breeding ground for singularities, shows the downfall of ancient cultures. Causes such as intruders, internal conflicts or natural disasters are not sufficient alone to justify the destruction of a culture. Rather requirement is an increasing complexity and associated declining marginal returns.Tainter, J.A.: The Collapse of Complex Societies, Cambridge, New York u.a.
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Thus, Penrose diagrams are also useful in the study of asymptotic properties of spacetimes and singularities. An infinite static Minkowski universe, coordinates (x, t) is related to Penrose coordinates (u, v) by: :\tan(u \pm v) = x \pm t The corners of the Penrose diamond, which represent the spacelike and timelike conformal infinities, are \pi /2 from the origin.
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In differential topology, a branch of mathematics, a stratifold is a generalization of a differentiable manifold where certain kinds of singularities are allowed. More specifically a stratifold is stratified into differentiable manifolds of (possibly) different dimensions. Stratifolds can be used to construct new homology theories. For example, they provide a new geometric model for ordinary homology.
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Varadarajan's early work, including his doctoral thesis, was in the area of probability theory. He then moved into representation theory where he has done some of his best known work. In the 1980s, he wrote a series of papers with Donald Babbitt on the theory of differential equations with irregular singularities. His latest work has been in supersymmetry.
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He was born and studied in Turin. Corrado Segre, his uncle, also served as his doctoral advisor. Among his main contributions to algebraic geometry are studies of birational invariants of algebraic varieties, singularities and algebraic surfaces. His work was in the style of the old Italian School, although he also appreciated the greater rigour of modern algebraic geometry.
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This was opened in 1974. Bowen also instigated rain-making experiments in Australia in 1947, and continued after he retired in 1971. He was also interested in the phenomenon of Climatic Singularities, suggesting that they might be related to the Earth's passage through belts of meteor dust – whose particles then acted as ice-nuclei for seeding clouds.
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Bernstein introduced a similar approach in the polynomial coefficients case. Kashiwara's master thesis states the foundations of D-module theory. His PhD thesis proves the rationality of the roots of b-functions (Bernstein–Sato polynomials), using D-module theory and resolution of singularities. He is a member of the French Academy of Sciences and of the Japan Academy.
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He has remained at Northeastern University ever since, where he is now a Full Professor. He has published over 35 research- level papers and two research-level books and written four Calculus textbooks. He is the editor-in-chief of an open-access journal, the Journal of Singularities.Journal of Singularities home page and Scimago report, accessed 2016-07-07.
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If one is going to solve the Einstein field equations using approximate methods such as the post-Newtonian expansion, then one should try to choose a coordinate condition which will make the expansion converge as quickly as possible (or at least prevent it from diverging). Similarly, for numerical methods one needs to avoid caustics (coordinate singularities).
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For irreducible curves and polynomials, the number of singular points is at most (d−1)(d−2)/2, because of the formula expressing the genus in term of the singularities (see below). The maximum is reached by the curves of genus zero whose all singularities have multiplicity two and distinct tangents (see below). The equation of the tangents at a singular point is given by the nonzero homogeneous part of the lowest degree in the Taylor series of the polynomial at the singular point. When one changes the coordinates to put the singular point at the origin, the equation of the tangents at the singular point is thus the nonzero homogeneous part of the lowest degree of the polynomial, and the multiplicity of the singular point is the degree of this homogeneous part.
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American Physical Society which can potentially resolve the Schwarzschild singularity for mini black holes, yielding a non-singular compact object without event horizon, and cosmological singularities. He has also conjectured with Koshelev that astrophysical black hole has no curvature singularity and devoid of an event horizon,Alexey Koshelev and Anupam Mazumdar (31 October 2017)massive compact objects without event horizon exist in infinite derivative gravity? American Physical Society in infinite derivative theories of gravity, because the scale of non-locality in gravitational interaction can engulf the gravitational radius of the compact object. At time scales and at distances below the effective scale of non-locality the gravitational interaction weakens sufficiently enough that a finite pressure from normal matter satisfying null, strong and weak energy conditions can avoid forming blackhole with event horizon and cosmological singularities.
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The CMG control program was responsible for making sure that the gimbals never hit the stops, by redistributing angular momentum between the three rotors to bring large gimbal angles closer to zero. Since the total angular momentum to be stored had only three degrees of freedom, while the control program could change six independent variables (the three pairs of gimbal angles), the program had sufficient freedom of action to do this while still obeying other constraints such as avoiding anti-parallel alignments. One advantage of limited gimbal movement such as Skylab's is that singularities are less of a problem. If Skylab's inner gimbals had been able to reach 90 degrees or more away from zero, then the 'North and South poles' could have become singularities; the gimbal stops prevented this.
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Nemo is available in various configurations. Global configurations use ORCA tripolar grids, which allow the entire oceanic domain to be covered without singularity points. In fact, the grid formed by the meridians and the parallels has two singularities: the North Pole and the South Pole. Near these two points the mesh size tends to zero, making the use of modeling equations problematic.
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Complex analysis shows some features not apparent in real analysis. For example, any two holomorphic functions and that agree on an arbitrarily small open subset of C necessarily agree everywhere. Meromorphic functions, functions that can locally be written as with a holomorphic function , still share some of the features of holomorphic functions. Other functions have essential singularities, such as at .
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Here if the curve does not contain the center of inversion and if the center of inversion is a nonsingular point on it; similarly the circular points, , are singularities of order on . The value can be eliminated from these relations to show that the set of -circular curves of degree , where may vary but is a fixed positive integer, is invariant under inversion.
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Then on the edges of this octant (singularities of codimension 2) the curvature is nonpositive (because of branching geodesics), yet it is not the case at the origin (singularity of codimension 3), where a triangle such as (0,0,e), (0,e,0), (e,0,0) has a median longer than would be in the Euclidean plane, which is characteristic of nonnegative curvature.
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A surface is a two- dimensional, topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3—for example, the surface of a ball. On the other hand, there are surfaces, such as the Klein bottle, that cannot be embedded in three-dimensional Euclidean space without introducing singularities or self- intersections.
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For the academic year 1992–1993 he was a Guggenheim Fellow. He was in 1978 an invited speaker (Singularities of solutions of boundary value problems) at the ICM in Helsinki and in 1990 a plenary speaker (Pseudodifferential operators, corners and singular limits) at the ICM in Kyoto. His doctoral students include Mark S. Joshi, John M. Lee, András Vasy, and Maciej Zworski.
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In a series of papers he explored propagation of singularities of symmetric hyperbolic systems inside of the domain and near the boundary. He was invited to give a talk at ICM—1978, Helsinki but was not granted an exit visa by the Soviet authorities;International Congress of Mathematicians#Soviet participation however his talk was published in the Proceedings of the Congress.
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Rahman, 2009) using the zig-zag construction of MacPherson-Vilonen (R. MacPherson & K. Vilonen, 1986). This perverse sheaf proved the Hübsch conjecture for isolated conic singularities, satisfied Poincarè duality, and aligned with some of the properties of the Kähler package. Satisfaction of all of the Kähler package by this Perverse sheaf for higher codimension strata is still an open problem.
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Felice Casorati (17 December 1835 – 11 September 1890) was an Italian mathematician who studied at the University of Pavia. He was born in Pavia and died in Casteggio. He is best known for the Casorati–Weierstrass theorem in complex analysis. The theorem, named for Casorati and Karl Theodor Wilhelm Weierstrass, describes the remarkable behavior of holomorphic functions near essential singularities.
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The cosmic censorship hypothesis says that a gravitational singularity would remain hidden by the event horizon. LIGO events, including GW150914, are consistent with these predictions. Although data anomalies would have resulted in the case of a singularity, the nature of those anomalies remains unknown. Some research has suggested that if loop quantum gravity is correct, then naked singularities could exist in nature,M.
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A further development of Maxwell's thoughts in relation to dynamic systems was carried out first by the French mathematician Henri Poincaré. Poincaré distinguished four different simple singularities (points singuliers) of differential equations. These are the node (les noeuds), the saddle (les cols), the focus (les foyers) and the center (les centers). In recent times, the chaos theory found special attention.
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In the theory of several complex variables and complex manifolds in mathematics, a Stein manifold is a complex submanifold of the vector space of n complex dimensions. They were introduced by and named after . A Stein space is similar to a Stein manifold but is allowed to have singularities. Stein spaces are the analogues of affine varieties or affine schemes in algebraic geometry.
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In transverse aspect, one hemisphere becomes the Adams hemisphere-in-a-square projection (the pole is placed at the corner of the square). Its four singularities are at the North Pole, the South Pole, on the equator at 25°W, and on the equator at 155°E, in the Arctic, Atlantic, and Pacific oceans, and in Antarctica. Carlos A. Furuti. Map Projections:Conformal Projections.
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In mathematics and string theory, a conifold is a generalization of a manifold. Unlike manifolds, conifolds can contain conical singularities, i.e. points whose neighbourhoods look like cones over a certain base. In physics, in particular in flux compactifications of string theory, the base is usually a five-dimensional real manifold, since the typically considered conifolds are complex 3-dimensional (real 6-dimensional) spaces.
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Under the viscosity solution concept, u does not need to be everywhere differentiable. There may be points where either Du or D^2 u does not exist and yet u satisfies the equation in an appropriate generalized sense. The definition allows only for certain kind of singularities, so that existence, uniqueness, and stability under uniform limits, hold for a large class of equations.
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From 1946 to 1949, Villars worked as a research assistant at the Swiss Federal Institute. While there, he collaborated with Wolfgang Pauli on work in quantum electrodynamics. They developed a method of dealing with mathematical singularities in quantum field theory, in order to extract finite physical results. This method, Pauli–Villars regularization, is used by physicists when working with field theory.
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In 1957, he suggested a straightforward criterion for ferromagnetism from observations of magnetic isotherms. This method was called Arrott plots. In collaboration with Murray J. Press, he gave a description of surface singularities in liquid-crystal droplets. A lot of works are devoted to the properties of ferromagnetic samples (for example the so-called Arrott's cylinder) with micrometer and sub-micrometer sizes.
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In algebra, the Hochster–Roberts theorem, introduced by , states that rings of invariants of linearly reductive groups acting on regular rings are Cohen–Macaulay. In other words, :If V is a rational representation of a linearly reductive group G over a field k, then there exist algebraically independent invariant homogeneous polynomials f_1, \cdots, f_d such that k[V]^G is a free finite graded module over k[f_1, \cdots, f_d]. proved that if a variety over a field of characteristic 0 has rational singularities then so does its quotient by the action of a reductive group; this implies the Hochster–Roberts theorem in characteristic 0 as rational singularities are Cohen–Macaulay. In characteristic p>0, there are examples of groups that are reductive (or even finite) acting on polynomial rings whose rings of invariants are not Cohen–Macaulay.
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The development of systems provides the science currently so before that by a singular Big Bang uniformly dispersed plasma spread after the creation of the universe in space, which is cooled with increasing expansion, so that formed atoms and finally for very small (singular) fluctuations in the uniform density inhomogeneities created self-reinforcing. They subsequently led to the formation of galaxies, stars and other systems in the universe, from which humans emerged at the end. Even if the singularity of the Big Bang can be avoided in the mathematical models, singularities remain an essential element of history. The evolutionary history shows that not only successful mutations can be perceived as positive singularities, but the humanization and the human becoming, the singular most important event in the evolution and represents a jump from the continuum of past evolutionary development of the planet Earth.
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The thermodynamic metric can then be used with different thermodynamic potentials without changing the geometric properties of the equilibrium manifold. One expects the geometric properties of the equilibrium manifold to be related to the macroscopic physical properties. The details of this relation can be summarized in three main points: #Curvature is a measure of the thermodynamical interaction. #Curvature singularities correspond to curvature phase transitions.
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Porteous began teaching at the University of Liverpool as a lecturer in 1959, becoming senior lecturer in 1972. During a year (1961–62) at Columbia University in New York, Porteous was influenced by Serge Lang. He continued to do research on manifolds in differential geometry. In 1971 his article "The normal singularities of a submanifold" was published in Journal of Differential Geometry 5:543–64.
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In this sense, they speak of a BwO of "the earth". "The Earth," they write, "is a body without organs. This body without organs is permeated by unformed, unstable matters, by flows in all directions, by free intensities or nomadic singularities, by mad or transitory particles" (A Thousand Plateaus, p. 40). That is, we usually think of the world as composed of relatively stable entities ("bodies," beings).
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The behavior of solutions to the Novikov–Veselov equation depends essentially on the regularity of the scattering data for this solution. If the scattering data are regular, then the solution vanishes uniformly with time. If the scattering data have singularities, then the solution may develop solitons. For example, the scattering data of the Grinevich–Zakharov soliton solutions of the Novikov–Veselov equation have singular points.
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In mathematics, and in particular differential geometry and complex geometry, a complex analytic variety or complex analytic space is a generalization of a complex manifold which allows the presence of singularities. Complex analytic varieties are locally ringed spaces which are locally isomorphic to local model spaces, where a local model space is an open subset of the vanishing locus of a finite set of holomorphic functions.
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And of course, 0+1D models cannot capture any nontrivial aspect of relativity because there is no space at all. This class of models retains just enough complexity to include among its solutions black holes, their formation, FRW cosmological models, gravitational singularities, etc. In the quantized version of such models with matter fields, Hawking radiation also shows up, just as in higher-dimensional models.
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Applying a linear change of basis shows that the integral of the exponential of a homogeneous polynomial in n variables may depend only on SL(n)-invariants of the polynomial. One such invariant is the discriminant, zeros of which mark the singularities of the integral. However, the integral may also depend on other invariants. Exponentials of other even polynomials can numerically be solved using series.
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An important topic in potential theory is the study of the local behavior of harmonic functions. Perhaps the most fundamental theorem about local behavior is the regularity theorem for Laplace's equation, which states that harmonic functions are analytic. There are results which describe the local structure of level sets of harmonic functions. There is Bôcher's theorem, which characterizes the behavior of isolated singularities of positive harmonic functions.
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Perelman discovered the singularities were all very simple: essentially three-dimensional cylinders made out of spheres stretched out along a line. An ordinary cylinder is made by taking circles stretched along a line. Perelman proved this using something called the "Reduced Volume" which is closely related to an eigenvalue of a certain elliptic equation. Sometimes an otherwise complicated operation reduces to multiplication by a scalar (a number).
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Hubsch, 1997) conjectured what this cohomology theory should be for singular target spaces. Tristan Hubsch and Abdul Rahman (T. Hubsch and A. Rahman, 2005) worked to solve the Hubsch conjecture by analyzing the non-transversal case of Witten's gauged linear sigma model (E. Witten,1993) which induces a stratification of these algebraic varieties (termed the ground state variety) in the case of isolated conical singularities.
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In 2003, Peter Lynds has put forward a new cosmology model in which time is cyclic. In his theory our Universe will eventually stop expanding and then contract. Before becoming a singularity, as one would expect from Hawking's black hole theory, the universe would bounce. Lynds claims that a singularity would violate the second law of thermodynamics and this stops the universe from being bounded by singularities.
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Philosophy has an affinity with despotism, due to its > predilection for Platonic-fascist top-down solutions that always screw up > viciously. Schizoanalysis works differently. It avoids Ideas, and sticks to > diagrams: networking software for accessing bodies without organs. BwOs, > machinic singularities, or tractor fields emerge through the combination of > parts with (rather than into) their whole; arranging composite > individuations in a virtual/actual circuit.
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An important reason why singularities cause problems in mathematics is that, with a failure of manifold structure, the invocation of Poincaré duality is also disallowed. A major advance was the introduction of intersection cohomology, which arose initially from attempts to restore duality by use of strata. Numerous connections and applications stemmed from the original idea, for example the concept of perverse sheaf in homological algebra.
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The term singularity for an explanation of unstable systems was first, and in a most general meaning used in 1873 by James Clerk Maxwell. Maxwell does not differentiate between dynamical systems and social systems. Therefore, a singularity refers to a context in which a small change can cause a large effect. The existence of singularities is primarily an argument against determinism and absolute causality for Maxwell.
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Abhyankar was appointed the Marshall Distinguished Professor of Mathematics at Purdue in 1967. His research topics include algebraic geometry (particularly resolution of singularities, a field in which he made significant progress over fields of finite characteristic), commutative algebra, local algebra, valuation theory, theory of functions of several complex variables, quantum electrodynamics, circuit theory, invariant theory, combinatorics, computer-aided design, and robotics. He popularized the Jacobian conjecture.
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Linear elliptic partial differential equations can be characterized as those whose principal symbol is nowhere zero. In the study of hyperbolic and parabolic partial differential equations, zeros of the principal symbol correspond to the characteristics of the partial differential equation. Consequently, the symbol is often fundamental for the solution of such equations, and is one of the main computational devices used to study their singularities.
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These prescriptions are known as Feynman rules. Internal lines correspond to virtual particles. Since the propagator does not vanish for combinations of energy and momentum disallowed by the classical equations of motion, we say that the virtual particles are allowed to be off shell. In fact, since the propagator is obtained by inverting the wave equation, in general, it will have singularities on the shell.
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This is perhaps the most important property of a solution to the isomonodromic deformation equations. This means that all essential singularities of the solutions are fixed, although the positions of poles may move. It was proved by Bernard Malgrange for the case of Fuchsian systems, and by Tetsuji Miwa in the general setting. Indeed, suppose we are given a partial differential equation (or a system of them).
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For , , so, as in the 2-D case, this corresponds to 3-D dual tree CWT. But the case of gives rise to a new DHWT transform. The combination of 3-D HWT wavelets is done in a manner to ensure that the resultant wavelet is lowpass along 1-D and bandpass along 2-D. In, this was used to detect line singularities in 3-D space.
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A typical use of cutoffs is to prevent singularities from appearing during calculation. If some quantities are computed as integrals over energy or another physical quantity, these cutoffs determine the limits of integration. The exact physics is reproduced when the appropriate cutoffs are sent to zero or infinity. However, these integrals are often divergent – see IR divergence and UV divergence – and a cutoff is needed.
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Libgober's early work studies the diffeomorphism type of complete intersections in complex projective space. This later led to the discovery of relations between Hodge and Chern numbers.A.Libgober, J.Wood, Differentiable structures on complete intersections I, Topology, 21 (1982),469-482 He introduced the technique of Alexander polynomialA.Libgober,Development of the theory of Alexander invariants in algebraic geometry, Topology of algebraic varieties and singularities, 3–17, Contemp. Math.
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These are fixed points of the flow. (A flow is a one- dimensional group of diffeomorphisms; a flow defines an action by the one- dimensional Lie group R, having locally nice geometric properties.) These two singularities correspond to two points, rather than two curves. In this example, the other integral curves are all simple closed curves. Many flows are considerably more complicated than this.
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Sabir Gusein-Zade (2010), El Escorial Sabir Medgidovich Gusein-Zade (; born 29 July 1950 in MoscowHome page of Sabir Gusein-Zade) is a Russian mathematician and a specialist in singularity theory and its applications.. He studied at Moscow State University, where he earned his Ph.D. in 1975 under the joint supervision of Sergei Novikov and Vladimir Arnold. Before entering the university, he had earned a gold medal at the International Mathematical Olympiad. Gusein-Zade co-authored with V. I. Arnold and A. N. Varchenko the textbook Singularities of Differentiable Maps (published in English by Birkhäuser). A professor in both the Moscow State University and the Independent University of Moscow, Gusein-Zade also serves as co-editor-in- chief for the Moscow Mathematical Journal.. He shares credit with Norbert A'Campo for results on the singularities of plane curves... Translated from the German original by John Stillwell, 2012 reprint of the 1986 edition.
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Any reduced scheme X has a unique normalization: a normal scheme Y with an integral birational morphism Y → X. (For X a variety over a field, the morphism Y → X is finite, which is stronger than "integral".Eisenbud, D. Commutative Algebra (1995). Springer, Berlin. Corollary 13.13) The normalization of a scheme of dimension 1 is regular, and the normalization of a scheme of dimension 2 has only isolated singularities.
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This is the case c>s=0, the case without saddles. Theorem:. Let M^n be a closed oriented connected manifold of dimension n\ge 2. Assume that M^n admits a C^1-transversely oriented codimension one foliation F with a non empty set of singularities all of them centers. Then the singular set of F consists of two points and M^n is homeomorphic to the sphere S^n.
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With Caffarelli, they studied the Yamabe equation on Euclidean space, proving a positive mass-style theorem on the asymptotic behavior of isolated singularities. In 1974, Spruck and David Hoffman extended a mean curvature-based Sobolev inequality of James H. Michael and Leon Simon to the setting of submanifolds of Riemannian manifolds.Michael, J.H.; Simon, L.M. Sobolev and mean-value inequalities on generalized submanifolds of . Comm. Pure Appl. Math. 26 (1973), 361–379.
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Quasi-excellent rings are conjectured to be the base rings for which the problem of resolution of singularities can be solved; showed this in characteristic 0, but the positive characteristic case is (as of 2016) still a major open problem. Essentially all Noetherian rings that occur naturally in algebraic geometry or number theory are excellent; in fact it is quite hard to construct examples of Noetherian rings that are not excellent.
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Several spherically symmetric solutions to ae-theory have been found. Most recently Christopher Eling and Ted Jacobson have found solutions resembling stars and solutions resembling black holes. In particular, they demonstrated that there are no spherically symmetric solutions in which stars are constructed entirely from the aether. Solutions without additional matter always have either naked singularities or else two asymptotic regions of spacetime, resembling a wormhole but with no horizon.
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31—59 led to the work of Victor Pereyra error- correcting algorithms for boundary-value problems and Stetter's results on defect correction and the resulting order of convergence. Fox was also interested in the treatment of singularities in partial differential equations, the Stefan problem and other cases of free and moving boundaries. Many of these problems arose from his collaboration with mathematicians in industry through the Oxford Study Groups.
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Both books have been described as "pop economics". The Tipping Point focuses on how ideas and behaviors reach critical mass, such as how Hush Puppies rapidly grew popular in the 1990s. Blink explains "what happens during the first two seconds we encounter something, before we actually start to think". All Gladwell's books focus on singularities: singular events in The Tipping Point, singular moments in Blink, and singular people in Outliers.
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The personality psychologist John D. Mayer (University of New Hampshire) published an essay in 1993 in which he suggested an independent psychiatric category for destructive personalities like Hitler: A dangerous leader disorder (DLD). Mayer identified three groups of symptomatic behavioral singularities: 1. indifference (becoming manifest for example in murder of opponents, family members or citizens, or in genocide); 2. intolerance (practicing press censorship, running a secret police or condoning torture); 3.
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In the mathematical subfield of numerical analysis, numerical stability is a generally desirable property of numerical algorithms. The precise definition of stability depends on the context. One is numerical linear algebra and the other is algorithms for solving ordinary and partial differential equations by discrete approximation. In numerical linear algebra the principal concern is instabilities caused by proximity to singularities of various kinds, such as very small or nearly colliding eigenvalues.
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They are additive rather than > substitutive, and immanent rather than transcendent: executed by functional > complexes of currents, switches, and loops, caught in scaling > reverberations, and fleeing through intercommunications, from the level of > the integrated planetary system to that of atomic assemblages. > Multiplicities captured by singularities interconnect as desiring-machines; > dissipating entropy by dissociating flows, and recycling their machinism as > self-assembling chronogenic circuitry.Nick Land, Fanged Noumena (2011, 442).
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This result is often implicitly used to extend affine geometry to projective geometry: it is entirely typical for an affine variety to acquire singular points on the hyperplane at infinity, when its closure in projective space is taken. Resolution says that such singularities can be handled rather as a (complicated) sort of compactification, ending up with a compact manifold (for the strong topology, rather than the Zariski topology, that is).
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Valentini has been described as an "ardent admirer of de Broglie". He noted that "de Broglie (rather like Maxwell) emphasized an underlying 'mechanical' picture: particles were assumed to be singularities of physical waves in space".Antony Valentini: Pilot-wave theory of fields, gravitation and cosmology, in: James T. Cushing, Arthur Fine, Sheldon Goldstein (eds.): Bohmian mechanics and quantum theory: an appraisal, Kluwer Academic Publishers, 1996, p. 45–66, p. 47.
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Zamora increases its population 5 times, up to 300.000 people during the festival. The singularities of this celebration include the medieval set up of some of the parades where the brotherhoods use monk´s robes instead of the most usual nazareno´s conical hat, torch fire instead of candles or male choirs instead of marching bands. Semana Santa in Zamora was declared Fiesta of International Tourist Interest of Spain in 1986.
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A gravastar is an object hypothesized in astrophysics by Pawel O. Mazur and Emil Mottola as an alternative to the black hole theory. It has usual black hole metric outside of the horizon, but de Sitter metric inside. On the horizon there is a thin shell of matter. The term "gravastar" is a portmanteau of the words "gravitational vacuum star".. This solution of Einstein equations is stable and has no singularities.
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Although the above correspondence with holomorphic functions only holds for functions of two real variables, harmonic functions in n variables still enjoy a number of properties typical of holomorphic functions. They are (real) analytic; they have a maximum principle and a mean- value principle; a theorem of removal of singularities as well as a Liouville theorem holds for them in analogy to the corresponding theorems in complex functions theory.
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Due to Walter's crossing, the parallel universe (and the prime universe, to a limited extent), suffers from weaknesses in the fabric of space-time, potentially creating gravitational singularities. "Amber 31422" is a gaseous mixture deployed over an area which solidifies to seal off these weaknesses. Living creatures are entrapped inside the solid, like "an insect trapped in amber". The amber is believed to originally have been developed by Walternate.
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More modern CMGs such as the four units installed on the ISS in 2000 have unlimited gimbal travel and therefore no 'blind areas'. Thus they do not have to be mounted facing along mutually perpendicular directions; the four units on the ISS all face the same way. The control program need not concern itself with gimbal stops, but on the other hand it must pay more attention to avoiding singularities.
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There are a number of examples of climate change impacts that may be irreversible, at least over the timescale of many human generations. These include the large-scale singularities such as the melting of the Greenland and West Antarctic ice sheets, and changes to the AMOC. In biological systems, the extinction of species would be an irreversible impact. In social systems, unique cultures may be lost due to climate change.
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This breakdown, however, is expected; it occurs in a situation where quantum effects should describe these actions, due to the extremely high density and therefore particle interactions. To date, it has not been possible to combine quantum and gravitational effects into a single theory, although there exist attempts to formulate such a theory of quantum gravity. It is generally expected that such a theory will not feature any singularities.
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From the fact that the group of conformal transforms is infinite-dimensional in two dimensions and finite-dimensional for more than two dimensions, one can surmise that potential theory in two dimensions is different from potential theory in other dimensions. This is correct and, in fact, when one realizes that any two-dimensional harmonic function is the real part of a complex analytic function, one sees that the subject of two-dimensional potential theory is substantially the same as that of complex analysis. For this reason, when speaking of potential theory, one focuses attention on theorems which hold in three or more dimensions. In this connection, a surprising fact is that many results and concepts originally discovered in complex analysis (such as Schwarz's theorem, Morera's theorem, the Weierstrass-Casorati theorem, Laurent series, and the classification of singularities as removable, poles and essential singularities) generalize to results on harmonic functions in any dimension.
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In particular, for sufficiently well-behaved generating functions, Cauchy's integral formula can be used to recover the power series coefficients (the real object of study) from the generating function, and knowledge of the singularities of the function can be used to derive accurate estimates of the resulting integrals. After an introductory chapter and a chapter giving examples of the possible behaviors of rational functions and meromorphic functions, the remaining chapters of this part discuss the way the singularities of a function can be used to analyze the asymptotic behavior of its power series, apply this method to a large number of combinatorial examples, and study the saddle-point method of contour integration for handling some trickier examples. The final part investigates the behavior of random combinatorial structures, rather than the total number of structures, using the same toolbox. Beyond expected values for combinatorial quantities of interest, it also studies limit theorems and large deviations theory for these quantities.
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Walter also struggles with adjusting to normal life in Peter's care after living seventeen years in a mental institution while hiding the fact that Peter is from the parallel universe, "his" Peter having died as a child. In Season 2, the occurrences are found to be in conjunction with activities of a parallel universe, which is plagued by singularities occurring at weakened points of the fabric between worlds; over there, scientists have developed an amber-like substance that isolates these singularities as well as any innocent people caught in the area on its release. The Fringe team deals with more cases that are leading to a "great storm" as the parallel universe appears to be at war with the prime one, engineered by human-machine hybrid shapeshifters from the parallel universe. Walter is forced to tell Peter that he is from the parallel universe, a replacement for his own Peter, who died from a genetic disease.
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In the mathematical discipline of complex analysis, the analytic capacity of a compact subset K of the complex plane is a number that denotes "how big" a bounded analytic function on C \ K can become. Roughly speaking, γ(K) measures the size of the unit ball of the space of bounded analytic functions outside K. It was first introduced by Ahlfors in the 1940s while studying the removability of singularities of bounded analytic functions.
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Mathematician Nicholas Michael John Woodhouse at Oxford University considered this book to be an authoritative treatise that could become a classic. He observed that the authors begin with axioms of geometry and physics then derive the consequences in a rigorous fashion. Various well- known exact solutions to Einstein's field equations and their physical meaning are explored. In particular, Hawking and Ellis show that singularities and black holes arise in a large class of plausible solutions.
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A life is subjectless, neutral, and preceding all individuation and stratification, is present in all things, and thus always immanent to itself. "A life is everywhere ...: an immanent life carrying with it the events and singularities that are merely actualized in subjects and objects."Deleuze, Pure Immanence, p.29 An ethics of immanence will disavow its reference to judgments of good and evil, right and wrong, as according to a transcendent model, rule or law.
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The arithmetic Riemann–Roch theorem then describes how the Chern class behaves under pushforward of vector bundles under a proper map of arithmetic varieties. A complete proof of this theorem was only published recently by Gillet, Rössler and Soulé. Arakelov's intersection theory for arithmetic surfaces was developed further by . The theory of Bost is based on the use of Green functions which, up to logarithmic singularities, belong to the Sobolev space L^2_1.
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For instance, Legendre's differential equation can be shown to be a special case of the hypergeometric differential equation. Hence, by solving the hypergeometric differential equation, one may directly compare its solutions to get the solutions of Legendre's differential equation, after making the necessary substitutions. For more details, please check the hypergeometric differential equation. We shall prove that this equation has three singularities, namely at x = 0, x = 1 and around x = infinity.
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The proposed identification of dark matter with GIMPs makes dark matter a form of dark energy filled with singularities, i.e., “entangled” dark energy. This would roughly affirm Einstein's hope in 1919 that all particles in the universe would follow the traceless version of his equation. If we identify all matter as the sum of dark energy plus dark matter in the form of GIMPs, his expectation would turn out to have been almost right.
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After being declared insane, Walker spent some years in the care of Dr. Belcombe, and because of her mental state, was unable to make a valid will. She died in 1854 at her childhood home, Cliff Hill in Lightcliffe, West Yorkshire. More than 40 years after her death, while reporting on a dispute over the ownership of Shibden Hall, the Leeds Times in 1882 stated, "Miss Lister's masculine singularities of character are still remembered".
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In Palatini () gravity, one treats the metric and connection independently and varies the action with respect to each of them separately. The matter Lagrangian is assumed to be independent of the connection. These theories have been shown to be equivalent to Brans–Dicke theory with . Due to the structure of the theory, however, Palatini () theories appear to be in conflict with the Standard Model, may violate Solar system experiments, and seem to create unwanted singularities.
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DeLanda describes how social and economic formations influence war machines, i.e. the form of armies, in each historical period. He draws on chaos theory to show how the biosphere reaches singularities (or bifurcations) which mark self-organization thresholds where emergent properties are displayed and claims that the "mecanosphere", constituted by the machinic phylum, possesses similar qualities. He argues for example how a certain level of population growth may induce invasions and others may provoke wars.
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This serves as basis for his works on subjectivation, creativity and singularization of the social life. In 2017 he published his work on the structure of the current late modern society,Die Gesellschaft der Singularitäten. Zum Strukturwandel der Moderne, which was published in English in 2020 as The Society of Singularities. In this book he analyses how economy, work, information technlogy, lifestyle, classes and politics follow a system which values singularity and devalues non-sigularity.
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In recent years, his research has focused on nonlinear hyperbolic systems of conservation laws whose solutions spontaneously develop singularities propagating as shock waves. In particular, he is studying the interplay between thermodynamics and analysis in the theory of these systems and he is analyzing the fundamental role of entropy as a stabilizing agent. In 2012 he became a fellow of the American Mathematical Society.List of Fellows of the American Mathematical Society, retrieved 2012-11-10.
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Susan Marjorie Scott is an Australian physicist whose work concerns general relativity, gravitational singularities, and black holes. She is a professor of quantum science at the Australian National University (ANU). At ANU, she is the leader of the General Relativity Theory and Data Analysis Group, part of the LIGO Scientific Collaboration that has discovered gravity waves from collisions involving black holes and neutron stars, and is a member of the LIGO Scientific Collaboration Council.
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Jens G. Eggers from the University of Bristol, was awarded the status of Fellow in the American Physical Society, after they were nominated by their Division of Fluid Dynamics in 2009, for applications of the ideas of singularities to free-boundary problems such as jet breakup, drop formation, air entrainment, thin-film dynamics including wetting, dewetting and contact line motions, and with further applications to polymeric flows and models for granular dynamics.
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In general relativity, the de Sitter–Schwarzschild solution describes a black hole in a causal patch of de Sitter space. Unlike a flat-space black hole, there is a largest possible de Sitter black hole, which is the Nariai spacetime. The Nariai limit has no singularities, the cosmological and black hole horizons have the same area, and they can be mapped to each other by a discrete reflection symmetry in any causal patch.
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The weak and the strong cosmic censorship hypotheses are two conjectures concerned with the global geometry of spacetimes. The weak cosmic censorship hypothesis asserts there can be no singularity visible from future null infinity. In other words, singularities need to be hidden from an observer at infinity by the event horizon of a black hole. Mathematically, the conjecture states that, for generic initial data, the maximal Cauchy development possesses a complete future null infinity.
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In those models there is no dynamic interaction between the universe at t < 0 and t > 0. , allowing an exchange of matter between the two conjugated sheets, based on an idea after Igor Dmitriyevich Novikov. Novikov called such singularities a collapse and an anticollapse, which are an alternative to the couple black hole and white hole in the wormhole model. Sakharov also proposed the idea of induced gravity as an alternative theory of quantum gravity.
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Among Yau's research interests are bioinformatics, complex algebraic geometry, singularities theory, and nonlinear filtering. He published nearly 300 papers and established the "Yau algebra" and the "Yau number". He served as Chairman of the IEEE International Conference on Control and Information and co-founded the Journal of Algebraic Geometry in 1991. He founded the journal Communications in Information and Systems in 2000, and has served as Editor-in-Chief since its inception.
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Numerical relativity is the sub-field of general relativity which seeks to solve Einstein's equations through the use of numerical methods. Finite difference, finite element and pseudo-spectral methods are used to approximate the solution to the partial differential equations which arise. Novel techniques developed by numerical relativity include the excision method and the puncture method for dealing with the singularities arising in black hole spacetimes. Common research topics include black holes and neutron stars.
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The mathematical counterpart of physical diffraction is the Fourier transform and the qualitative description of a diffraction picture as 'clear cut' or 'sharp' means that singularities are present in the Fourier spectrum. There are different methods to construct model quasicrystals. These are the same methods that produce aperiodic tilings with the additional constraint for the diffractive property. Thus, for a substitution tiling the eigenvalues of the substitution matrix should be Pisot numbers.
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In mathematics, an ancient solution to a differential equation is a solution that can be extrapolated backwards to all past times, without singularities. That is, it is a solution "that is defined on a time interval of the form ." The term was introduced in Grigori Perelman's research on the Ricci flow,. and has since been applied to other geometric flows.... as well as to other systems such as the Navier–Stokes equations.. and heat equation..
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A Van Hove singularity is a singularity (non-smooth point) in the density of states (DOS) of a crystalline solid. The wavevectors at which Van Hove singularities occur are often referred to as critical points of the Brillouin zone. For three-dimensional crystals, they take the form of kinks (where the density of states is not differentiable). The most common application of the Van Hove singularity concept comes in the analysis of optical absorption spectra.
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One of the original motivations for studying such maps was in the study of knots constructed by taking an \varepsilon-ball around a singular point of a plane curve, which is isomorphic to a real 4-dimensional ball, and looking at the knot inside the boundary, which is a 1-manifold inside of a 3-sphere. Since this concept could be generalized to hypersurfaces with isolated singularities, Milnor introduced the subject and proved his theorem.
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In mathematics, and especially affine differential geometry, the affine focal set of a smooth submanifold M embedded in a smooth manifold N is the caustic generated by the affine normal lines. It can be realised as the bifurcation set of a certain family of functions. The bifurcation set is the set of parameter values of the family which yield functions with degenerate singularities. This is not the same as the bifurcation diagram in dynamical systems.
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In fluid dynamics, aerodynamic potential flow codes or panel codes are used to determine the fluid velocity, and subsequently the pressure distribution, on an object. This may be a simple two-dimensional object, such as a circle or wing, or it may be a three-dimensional vehicle. A series of singularities as sources, sinks, vortex points and doublets are used to model the panels and wakes. These codes may be valid at subsonic and supersonic speeds.
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GRADELA is a simple gradient elasticity model involving one internal length in addition to the two Lamé parameters. It allows to eliminate elastic singularities and discontinuities and to interpret elastic size effects. This model has been suggested by Elias C. Aifantis. The main advantage of GRADELA over Mindlin's elasticity models (which contains five extra constants) is the fact that solutions of boundary value problems can be found in terms of corresponding solutions of classical elasticity by operator splitting method.
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Deligne has proved that the nth cohomology group of an arbitrary algebraic variety has a canonical mixed Hodge structure. This structure is functorial, and compatible with the products of varieties (Künneth isomorphism) and the product in cohomology. For a complete nonsingular variety X this structure is pure of weight n, and the Hodge filtration can be defined through the hypercohomology of the truncated de Rham complex. The proof roughly consists of two parts, taking care of noncompactness and singularities.
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He is part of a group of British science fiction writers who specialise in hard science fiction and space opera. His contemporaries include Stephen Baxter, Iain M. Banks, Paul J. McAuley, Alastair Reynolds, Adam Roberts, Charles Stross, Richard Morgan, and Liz Williams. His science fiction novels often explore socialist, communist, and anarchist political ideas, especially Trotskyism and anarcho-capitalism (or extreme economic libertarianism). Technical themes encompass singularities, divergent human cultural evolution, and post-human cyborg-resurrection.
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However, he is finally defeated when Luthor (having escaped surgery through undisclosed means) hacks and shuts off Brainiac from the inside. His body is promptly destroyed by Superman. In one last act of spite, Brainiac's death automatically triggers his ship's self-destruct, which, according to Luthor, would eradicate the entire Earth and everything within a 15,000,000 mile radius. As the gravitational singularities powering Brainiac's ship threaten to explode, Superman rockets it into outer space, where it blows up.
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The only conditions being that the feature points being tracked never leave the field of view and that a depth estimate be predetermined by some off-line technique. 2-1/2-D servoing has been shown to be more stable than the techniques that preceded it. Another interesting observation with this formulation is that the authors claim that the visual Jacobian will have no singularities during the motions. The hybrid technique developed by Corke and Hutchinson,P.
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In 2001 Smith won the Ruth Lyttle Satter Prize in Mathematics for her development of tight closure methods, introduced by Hochster and Huneke, in commutative algebra and her application of these methods in algebraic geometry. The prize committee specifically cited her papers "Tight closure of parameter ideals" (Inventiones Mathematicae 1994), "F-rational rings have rational singularities" (American J. Math. 1997, and, with Gennady Lyubeznik, "Weak and strong F-regularity are equivalent in graded rings" (American J. Math., 1999).
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He generalized the classical approach to the classification of algebraic surfaces to the classification of algebraic three- folds. The classical approach used the concept of minimal models of algebraic surfaces. He found that the concept of minimal models can be applied to three- folds as well if we allow some singularities on them. The extension of Mori's results to dimensions higher than three is called the minimal model program and is an active area of research in algebraic geometry.
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When he traveled, he carried no luggage other than a box of Bibles to be given away. Throughout most of his life, what little money he ever collected was either given away to the poor or used to purchase Bibles. In his later years, he did accumulate a bit of money from the sales of his autobiography and religious writings. His singularities of manner and of dress excited prejudices against him, and counteracted the effect of his eloquence.
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Esole works on F-theory, a branch of string theory at the interface with mathematics. He joined the Department of Physics at Harvard University as a postdoctoral research fellow in 2008. He moved to the Department of Mathematics in 2013, and was appointed Benjamin Peirce Fellow working with the Fields Medal winner Shing-Tung Yau. He worked on SU(5) models and opened the door to the systematic use of crepant resolutions of singularities in F-theory.
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Here also parametrised KAM theory comes into play. Apart from developing theory, together with colleagues, he also worked on applications in the fields of classical and quantum mechanics, population dynamics and climate modelling. Here often computational tools were employed, which sometimes led to experimental mathematics. Broer was granted a doctorate in the faculty of mathematics and natural sciences in 1979 under the supervision of Floris Takens for a thesis entitled Bifurcations of singularities in volume preserving vector fields.
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This work also gave rise to the ideas of an algebraic space and algebraic stack, and has proved very influential in moduli theory. Additionally, he has made important contributions to the deformation theory of algebraic varieties. With Peter Swinnerton-Dyer, he provided a resolution of the Shafarevich-Tate conjecture for elliptic K3 surfaces and the pencil of elliptic curves over finite fields. Artin contributed to the theory of surface singularities which are both fundamental and seminal.
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Intense gravitational distortions, similar to quantum singularities, make travel through the Expanse extremely difficult, as it seems that space does not obey the known laws of physics in this region. Travelers risk injury, disfigurement, and death if their vessels are not lined with the protective metal Trellium-D. Species native to the Expanse include the Loque'eque, the Skagarans, the Triannon, and the Xindi. Featured locations of the Expanse include Azati Prime, the Calindra system, Oran'taku, Triannon, and Xindus.
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The stochastic interpretation rests upon the assumption that the quantum vacuum consists of real fields. Their fluctuations might explain the "dark energy" recently discovered in cosmology; see Astrophysics and Space Science, 332, 423-435 (2011). Another subject of interest has been the study of relativistic stars in order to see whether some modifications of general relativity (possibly of quantum origin) might prevent the collapse to singularities ("black holes"); see Astrophysics and Space Science 341, 411-416 (2012).
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Then a homotopy between the two systems is considered. It consists, for example, of the straight line between the two systems, but other paths may be considered, in particular to avoid some singularities, in the system :(1-t)g_1+t f_1=0,\, \ldots,\, (1-t)g_n+t f_n=0. The homotopy continuation consists in deforming the parameter t from 0 to 1 and following the N solutions during this deformation. This gives the desired solutions for t = 1.
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Closely related is the notion of singularity with the concept of complexity. J.C. Maxwell has already pointed out that a system has all the more singular points, the more complex it is. Complexity is also the basis of perceived chaos and singularities. Suppose a seemingly insignificant event that produces a great effect, even in a simple context, how difficult would it be to detect the reason in a complex situation with tremendously many elements and relationships.
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This was, however, an approximation, using singularities to represent the vortices that generated lift, and Weber was tasked with improving it. She realised that some of her work overlapped with Küchemann's research on jet engine intakes. They teamed up, with Weber doing the theoretical development and wind tunnel testing, and Küchemann setting the direction of their research based on his consultation with manufacturers. Over the period of the Second World War, they created a substantial body of work.
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In differential geometry, the Angenent torus is a smooth embedding of the torus into three-dimensional Euclidean space, with the property that it remains self-similar as it evolves under the mean curvature flow. Its existence shows that, unlike the one-dimensional curve-shortening flow (for which every embedded closed curve converges to a circle as it shrinks to a point), the two-dimensional mean-curvature flow has embedded surfaces that form more complex singularities as they collapse.
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Irregular points (also called singularities) in the flow field occur when streamlines have kinks in them (the derivative doesn't exist at a point). This can happen where the bend is outward (e.g., the bottom of the cutoff wall in the figure above), and there is infinite flux at a point, or where the bend is inward (e.g., the corner just above and to the left of the cutoff wall in the figure above) where the flux is zero.
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Amal Kumar Raychaudhuri (; 14 September 1923 – 18 June 2005) was an Indian physicist, known for his research in general relativity and cosmology. His most significant contribution is the eponymous Raychaudhuri equation, which demonstrates that singularities arise inevitably in general relativity and is a key ingredient in the proofs of the Penrose–Hawking singularity theorems. Raychaudhuri was also revered as a teacher during his tenure at Presidency College, Kolkata. Many of his students have gone on to become established scientists.
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In mathematics, Milnor maps are named in honor of John Milnor, who introduced them to topology and algebraic geometry in his book Singular Points of Complex Hypersurfaces (Princeton University Press, 1968) and earlier lectures. The most studied Milnor maps are actually fibrations, and the phrase Milnor fibration is more commonly encountered in the mathematical literature. These were introduced to study isolated singularities by constructing numerical invariants related to the topology of a smooth deformation of the singular space.
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Steven "Steve" Morris Zelditch (born 13 September 1953) is an American mathematician, specializing in global analysis, complex geometry, and mathematical physics (e.g. quantum chaos). Steve Zelditch, Berkeley 1986 Zelditch received in 1975 from Harvard University his bachelor's degree in mathematics and in 1981 from the University of California, Berkeley his Ph.D. under Alan Weinstein with thesis Reconstruction of singularities of solutions for Schrödinger's equations. From 1981 to 1985 Zelditch was Ritt Assistant Professor at Columbia University.
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'I met George in 1970 when he burst on the algebraic geometry scene with a spectacular PhD thesis. His thesis gave a wonderful analysis of the singularities of the subvarieties $W_r$ of the Jacobian of a curve obtained by adding the curve to itself $r$ times inside its Jacobian. This was one of the major themes that he pursued throughout his career: understanding the interaction of a curve with its Jacobian and especially to the map from the $r$-fold symmetric product of the curve to the Jacobian. In his thesis he gave a determinantal representation both of $W_r$ and of its tangent cone at all its singular points, which gives you a complete understanding of the nature of these singularities' - D. Mumford 'One of the things that distinguished his work was the total mastery with which he used higher cohomology. A paper which, I believe, every new student of algebraic geometry should read, is his elementary proof of the Riemann-Roch theorem on curves: “Algebraic Curves” in Crelle, 1977.
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After graduation in 2001, he started a reflection on the human body as a unit of architecture, referring to Vitruvius, Leonardo da Vinci, Le Corbusier, to determine the archetype to place at the heart of contemporary architecture. But rather than marring an ideal, a perfect modular, his quest is more about the definition of a variable geometry model, incorporating therein the particulars and deformities of all. He questions the differences that make humanity what it is, i.e. a set of non- interchangeable singularities.
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His results on 0-cycles on algebraic varieties with isolated singularities effectively reduces their study to the corresponding study on the desingularization, together with information about multiples of the exceptional divisors. This allows the complete calculation of the Chow group of 0-cycles on an algebraic variety in many cases, like the case of rational varieties or cones. Working initially with Levine, and later with Park, Krishna built up the original constructions of Bloch-Esnault on additive Chow groups into a full theory.
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Kosiński was an editor of many volumes of collective works and conference materials and an author of two monographs: # W. Kosiński: Field Singularities and Wave Analysis in Continuum Mechanics. Ellis Horwood Series: Mathematics and Its Applications, Ellis Horwood Ltd., Chichester, Halsted Press: a Division of John Wiley & Sons, New York Chichester Brisbane Toronto, PWN – Polish Scientific Publishers, Warsaw (1986) # W. Kosiński: Wstęp do teorii osobliwości pola i analizy fal. PWN, Warsaw – Poznań (1981) as well as over 230 of other scientific publications.
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Normalization is not usually used for resolution of singularities for schemes of higher dimension. To define the normalization, first suppose that X is an irreducible reduced scheme X. Every affine open subset of X has the form Spec R with R an integral domain. Write X as a union of affine open subsets Spec Ai. Let Bi be the integral closure of Ai in its fraction field. Then the normalization of X is defined by gluing together the affine schemes Spec Bi.
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Observation of gravitational waves from binary black hole merger GW150914 General relativity has emerged as a highly successful model of gravitation and cosmology, which has so far passed many unambiguous observational and experimental tests. However, there are strong indications the theory is incomplete.; The problem of quantum gravity and the question of the reality of spacetime singularities remain open.section Quantum gravity, above Observational data that is taken as evidence for dark energy and dark matter could indicate the need for new physics.
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In Schwarzschild's original paper, he put what we now call the event horizon at the origin of his coordinate system. In this paper he also introduced what is now known as the Schwarzschild radial coordinate ( in the equations above), as an auxiliary variable. In his equations, Schwarzschild was using a different radial coordinate that was zero at the Schwarzschild radius. A more complete analysis of the singularity structure was given by David Hilbert in the following year, identifying the singularities both at and .
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P. Van Alstine and H.W. Crater, Journal of Mathematical Physics 23, 1697 (1982). Their structures, unlike the more familiar two-body Dirac equation of Breit, which is a single equation, are that of two simultaneous quantum relativistic wave equations. A single two- body Dirac equation similar to the Breit equation can be derived from the TBDE. Unlike the Breit equation, it is manifestly covariant and free from the types of singularities that prevent a strictly nonperturbative treatment of the Breit equation.
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If the Roman surface were to be inscribed inside the tetrahedron with least possible volume, one would find that each edge of the tetrahedron is tangent to the Roman surface at a point, and that each of these six points happens to be a Whitney singularity. These singularities, or pinching points, all lie at the edges of the three lines of double points, and they are defined by this property: that there is no plane tangent to any surface at the singularity.
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A "Round Earth" data type (GEOGRAPHY) uses an ellipsoidal model in which the Earth is defined as a single continuous entity which does not suffer from the singularities such as the international dateline, poles, or map projection zone "edges". Approximately 70 methods are available to represent spatial operations for the Open Geospatial Consortium Simple Features for SQL, Version 1.1. SQL Server includes better compression features, which also helps in improving scalability. It enhanced the indexing algorithms and introduced the notion of filtered indexes.
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Whitney umbrella as a ruled surface, generated by a moving straight line Whitney umbrella made with a single string inside a plastic cube Whitney's umbrella is a ruled surface and a right conoid. It is important in the field of singularity theory, as a simple local model of a pinch point singularity. The pinch point and the fold singularity are the only stable local singularities of maps from R2 to R3. It is named after the American mathematician Hassler Whitney.
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Progress in solving the field equations and understanding the solutions has been ongoing. The solution for a spherically symmetric charged object was discovered by Reissner and later rediscovered by Nordström, and is called the Reissner–Nordström solution. The black hole aspect of the Schwarzschild solution was very controversial, and Einstein did not believe that singularities could be real. However, in 1957 (two years after Einstein's death in 1955), Martin Kruskal published a proof that black holes are called for by the Schwarzschild solution.
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For avoiding ambiguity, the term irreducible hypersurface is often used. As for algebraic varieties, the coefficients of the defining polynomial may belong to any fixed field , and the points of the hypersurface are the zeros of in the affine space K^n, where is an algebraically closed extension of . A hypersurface may have singularities, which are the common zeros, if any, of the defining polynomial and its partial derivatives. In particular, a real algebraic hypersurface is not necessarily a manifold.
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Since around 2003, various condensed-matter physics groups have used the term “magnetic monopole” to describe a different and largely unrelated phenomenon. A true magnetic monopole would be a new elementary particle, and would violate Gauss's law for magnetism . A monopole of this kind, which would help to explain the law of charge quantization as formulated by Paul Dirac in 1931,"Quantised Singularities in the Electromagnetic Field" Paul Dirac, Proceedings of the Royal Society, May 29, 1931. Retrieved February 1, 2014.
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So Sundman's strategy consisted of the following steps: # Using an appropriate change of variables to continue analyzing the solution beyond the binary collision, in a process known as regularization. # Proving that triple collisions only occur when the angular momentum vanishes. By restricting the initial data to , he removed all real singularities from the transformed equations for the 3-body problem. # Showing that if , then not only can there be no triple collision, but the system is strictly bounded away from a triple collision.
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One such model consists of a combination of harmonic attraction and a repulsive inverse-cube force. This model is considered nontrivial since it is associated with a set of nonlinear differential equations containing singularities (compared with, e.g., harmonic interactions alone, which lead to an easily solved system of linear differential equations). In these two respects it is analogous to (insoluble) models having Coulomb interactions, and as a result has been suggested as a tool for intuitively understanding physical systems like the helium atom.
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The Fuchsian theory of linear differential equations, which is named after Lazarus Immanuel Fuchs, provides a characterization of various types of singularities and the relations among them. At any ordinary point of a homogeneous linear differential equation of order n there exists a fundamental system of n linearly independent power series solutions. A non-ordinary points is called a singularity. At a singularity the maximal number of linearly independent power series solutions may be less than the order of the differential equation.
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Notable accomplishments here include a proof of Langlands' conjecture on the discrete series, along with a later proof (joint with Michael Atiyah) constructing all such discrete series representations on spaces of harmonic spinors. Schmid along with his student Henryk Hecht proved Blattner's conjecture in 1975. In the 1970s he described the singularities of the Griffith's period map by applying Lie-theoretic methods to problems in algebraic geometry. Schmid has been very involved in K–12 mathematics education both nationally and internationally.
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Under certain conditions it was determined that this ground state variety was a conifold (P. Green & T.Hubsch, 1988; T. Hubsch, 1992) with isolated conic singularities over a certain base with a 1-dimensional exocurve (termed exo-strata) attached at each singular point. T. Hubsch and A. Rahman determined the (co)-homology of this ground state variety in all dimensions, found it compatible with Mirror symmetry and String Theory but found an obstruction in the middle dimension (T. Hubsch and A. Rahman, 2005).
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While nothing in this guarantees that the analytical method will work, it does explain the rationale of using a Farey series-type criterion on roots of unity. In the case of Waring's problem, one takes a sufficiently high power of the generating function to force the situation in which the singularities, organised into the so-called singular series, predominate. The less wasteful the estimates used on the rest, the finer the results. As Bryan Birch has put it, the method is inherently wasteful.
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Vladimir Arnold defines the main goal of singularity theory as describing how objects depend on parameters, particularly in cases where the properties undergo sudden change under a small variation of the parameters. These situations are called perestroika (), bifurcations or catastrophes. Classifying the types of changes and characterizing sets of parameters which give rise to these changes are some of the main mathematical goals. Singularities can occur in a wide range of mathematical objects, from matrices depending on parameters to wavefronts.
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The theory mentioned above does not directly relate to the concept of mathematical singularity as a value at which a function is not defined. For that, see for example isolated singularity, essential singularity, removable singularity. The monodromy theory of differential equations, in the complex domain, around singularities, does however come into relation with the geometric theory. Roughly speaking, monodromy studies the way a covering map can degenerate, while singularity theory studies the way a manifold can degenerate; and these fields are linked.
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In contrast to Bode plots, it can handle transfer functions with right half-plane singularities. In addition, there is a natural generalization to more complex systems with multiple inputs and multiple outputs, such as control systems for airplanes. The Nyquist criterion is widely used in electronics and control system engineering, as well as other fields, for designing and analyzing systems with feedback. While Nyquist is one of the most general stability tests, it is still restricted to linear, time-invariant (LTI) systems.
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They are also called conformal diagrams, or simply spacetime diagrams (although the latter may refer to Minkowski diagrams). Two lines drawn at 45° angles should intersect in the diagram only if the corresponding two light rays intersect in the actual spacetime. So, a Penrose diagram can be used as a concise illustration of spacetime regions that are accessible to observation. The diagonal boundary lines of a Penrose diagram correspond to the "infinity" or to singularities where light rays must end.
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That great circle divides the traditional Western and Eastern hemispheres. In oblique aspect (45 degrees) of one hemisphere becomes the Guyou hemisphere-in-a-square projection (the pole is placed in the middle of the edge of the square). Its four singularities are at 45 degrees north and south latitude on the great circle composed of the 20°W meridian and the 160°E meridians, in the Atlantic and Pacific oceans. That great circle divides the traditional western and eastern hemispheres.
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An algebraic curve is called p-circular if it contains the points (1, i, 0) and (1, −i, 0) when considered as a curve in the complex projective plane, and these points are singularities of order at least p. The terms bicircular, tricircular, etc. apply when p = 2, 3, etc. In terms of the polynomial F given above, the curve F(x, y) = 0 is p-circular if Fn−i is divisible by (x2 + y2)p−i when i < p.
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The Weyl curvature represents such gravitational effects as tidal fields and gravitational radiation. Mathematical treatments of Penrose's ideas on the Weyl curvature hypothesis have been given in the context of isotropic initial cosmological singularities e.g. in the articles. Penrose views the Weyl curvature hypothesis as a physically more credible alternative to cosmic inflation (a hypothetical phase of accelerated expansion in the early life of the universe) in order to account for the presently observed almost spatial homogeneity and isotropy of our universe.
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In representation theory, Zhu’s work is focused on classical groups and their smooth representations. Jointly with Sun Binyong, he proved multiplicity at most one for the branching (also called strong Gelfand pair property) of irreducible Casselman-Wallach representations of classical groups in the Archimedean case, and the conservation relation conjecture of Stephen S. Kudla and Stephen Rallis. He has also applied Howe correspondence to the structural study of degenerate representations and to the understanding of singularities for infinite-dimensional representations.
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DM stacks are similar to schemes, but they permit singularities that cannot be described solely in terms of polynomials. They play the same role for schemes that orbifolds do for manifolds. For example, the quotient of the affine plane by a finite group of rotations around the origin yields a Deligne–Mumford stack that is not a scheme or an algebraic space. Away from the origin, the quotient by the group action identifies finite sets of equally spaced points on a circle.
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These branes can be viewed as objects with a morphism between them. In this case, the morphism will be the state of a string that stretches between brane A and brane B. Singularities are avoided because the observed consequences of "Big Crunches" never reach zero size. In fact, should the universe begin a "big crunch" sort of process, string theory dictates that the universe could never be smaller than the size of one string, at which point it would actually begin expanding.
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Standard ideas can be used in singularity theory to classify, up to local diffeomorphism, the affine focal set. If the family of affine distance functions can be shown to be a certain kind of family then the local structure is known. The family of affine distance functions should be a versal unfolding of the singularities which arise. The affine focal set of a plane curve will generically consist of smooth pieces of curve and ordinary cusp points (semi-cubical parabolae).
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One might be tempted to use the data for a precision measurement of . However, while the asymptotic solution, defined properly at higher order, appears superficially very sensitive to , spurious singularities at small require either technical ad-hoc regularizations or the switching to the evolution from pre-fixed initial conditions at small . Both techniques reduce the sensitivity to . Nevertheless, values of : \alpha_s(M_Z) = 0.1198 \pm 0.0028(ex) \pm 0.0040(th) in analyses of the QCD coupling along these lines agree well with other experimental methods.
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Sternglass also wrote the book Before the Big Bang: the Origins of the Universe, in which he offers an argument for the Lemaître theory of the primeval atom. He offers technical data showing the plausibility of an original super massive relativistic electron-positron pair. This particle contained the entire mass of the universe and through a series of 270 divisions created everything that now exists. If true, this would help ameliorate some of the problems with the current models, namely inflation and black hole singularities.
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The simplest example of a translation surface is obtained by gluing the opposite sides of a parallelogram. It is a flat torus with no singularities. If P is a regular 4g-gon then the translation surface obtained by gluing opposite sides is of genus g with a single singular point, with angle (2g-1)2\pi. If P is obtained by putting side to side a collection of copies of the unit square then any translation surface obtained from P is called a square-tiled surface.
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When two D-branes approach each other the interaction is captured by the one loop annulus amplitude of strings between the two branes. The scenario of two parallel branes approaching each other at a constant velocity can be mapped to the problem of two stationary branes that are rotated relative to each other by some angle. The annulus amplitude yields singularities that correspond to the on-shell production of open strings stretched between the two branes. This is true irrespective of the charge of the D-branes.
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Using faster-than-light technology, the human race has colonized dozens of star systems. Ubu Roy and his sister Beautiful Maria are down-on-their-luck traders on the edge of human space. In a last-ditch effort to make money to pay their debts and repair their aging spacecraft, they are reduced to searching uncharted space for singularities to capture and sell. In a rare stroke of luck, Ubu and Maria happen upon an alien spacecraft and make humanity's first contact with an alien race.
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Most special functions are considered as a function of a complex variable. They are analytic; the singularities and cuts are described; the differential and integral representations are known and the expansion to the Taylor series or asymptotic series are available. In addition, sometimes there exist relations with other special functions; a complicated special function can be expressed in terms of simpler functions. Various representations can be used for the evaluation; the simplest way to evaluate a function is to expand it into a Taylor series.
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In the study of polyhedral spaces (particularly of those that are also topological manifolds) metric singularities play a central role. Let a polyhedral space be an n-dimensional manifold. If a point in a polyhedral space that is an n-dimensional topological manifold has no neighborhood isometric to a Euclidean neighborhood in R^n, this point is said to be a metric singularity. It is a singularity of codimension k, if it has a neighborhood isometric to R^{n-k} with a metric cone.
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Thus, one can start from immersions and try to eliminate multiple points, seeing if one can do this without introducing other singularities – studying "multiple disjunctions". This was first done by André Haefliger, and this approach is fruitful in codimension 3 or more – from the point of view of surgery theory, this is "high (co)dimension", unlike codimension 2 which is the knotting dimension, as in knot theory. It is studied categorically via the "calculus of functors" by Thomas Goodwillie, John Klein, and Michael S. Weiss.
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After discovering his concession might have been premature, a new and more refined wager was made. This one specified that such singularities would occur without extra conditions. The same year, Thorne, Hawking and Preskill made another bet, this time concerning the black hole information paradox. Thorne and Hawking argued that since general relativity made it impossible for black holes to radiate and lose information, the mass-energy and information carried by Hawking radiation must be "new", and not from inside the black hole event horizon.
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Infinity is reached, that is, there are non-collision singularities. The Painleve conjecture is an important conjecture in the field of power systems proposed in 1895. A very important recent development for the 4-body problem is that Xue Jinxin and Dolgopyat proved a non-collision singularity in a simplified version of the 4-body system around 2013. In addition, in 2007, Shen Weixiao and Kozlovski, Van-Strien proved the Real Fatou conjecture: Real hyperbolic polynomials are dense in the space of real polynomials with fixed degree.
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A singular target space means that only the CY manifold is singular as Minkowski space is smooth. Such a singular CY manifold is called a conifold as it is a CY manifold that admits conical singularities. Andrew Strominger observed (A. Strominger, 1995) that conifolds correspond to massless blackholes. Conifolds are important objects in string theory: Brian Greene explains the physics of conifolds in Chapter 13 of his book The Elegant Universe —including the fact that the space can tear near the cone, and its topology can change.
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The singularities correspond to points where some gluons are massless, and so could not be integrated out. In the full quantum moduli space is nonsingular, and its structure depends on the relative values of M and N. For example, when M is less than or equal to N+1, the theory exhibits confinement. When M is less than N, the effective action differs from the classical action. More precisely, while the perturbative nonrenormalization theory forbids any perturbative correction to the superpotential, the superpotential receives nonperturbative corrections.
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In the mathematical description, curvature is a property of a particular point and can be divorced from some invisible surface to which curved points in spacetime meld themselves. So for example, concepts like singularities (the most widely known of which in general relativity is the black hole) which cannot be expressed completely in a real world geometry, can correspond to particular states of a mathematical equation. The full mathematical description also captures some subtle distinctions made in general relativity between space-like dimensions and time-like dimensions.
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Consani and Scholten define their hypersurface from the (projectivized) set of solutions to the equation :P(x,y)=P(z,w) in four complex variables, where :P(x,y)=x^5+y^5-(5xy-5)(x^2+y^2-x-y). In this form the resulting hypersurface is singular: it has 120 double points. Its Hodge diamond is The Consani–Scholton quintic itself is the non-singular hypersurface obtained by blowing up these singularities. As a non-singular quintic threefold, it is a Calabi–Yau manifold.
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His primary research was on nanoscale phenomena at metal-ceramic interfaces using a combination of microscopy techniques. At the MPI Carroll first began working with carbon nanotubes and their variants. Specifically, Carroll was the first to identify the signature for one-dimensional behavior in multiwalled nanotubes (the so-called van Hove Singularities) as well as the signatures for defect states for those systems. This work helped to open the door to the use of scanning probe spectroscopies in understanding the electronics of low-dimensional systems.
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There are two classes of the concept of discriminant. The first class is the discriminant of an algebraic number field, which, in some cases including quadratic fields, is the discriminant of a polynomial defining the field. Discriminants of the second class arise for problems depending on coefficients, when degenerate instances or singularities of the problem are characterized by the vanishing of a single polynomial in the coefficients. This is the case for the discriminant of a polynomial, which is zero when two roots collapse.
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The possibility of traveling to another universe is, however, only theoretical since any perturbation would destroy this possibility. It also appears to be possible to follow closed timelike curves (returning to one's own past) around the Kerr singularity, which leads to problems with causality like the grandfather paradox. It is expected that none of these peculiar effects would survive in a proper quantum treatment of rotating and charged black holes. The appearance of singularities in general relativity is commonly perceived as signaling the breakdown of the theory.
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Suppose the base field is perfect. Any affine curve X is isomorphic to an open subset of an integral projective (hence complete) curve. Taking the normalization (or blowing up the singularities) of the projective curve then gives a smooth completion of X. Their points correspond to the discrete valuations of the function field that are trivial on the base field. By construction, the smooth completion is a projective curve which contains the given curve as an everywhere dense open subset, and the added new points are smooth.
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"Resurgent functions" are divergent power series whose Borel transforms converge in a neighborhood of the origin and give rise, by means of analytic continuation, to (usually) multi-valued functions, but these multi-valued functions have merely isolated singularities without singularities that form cuts with dimension one or greater.Sauzin Resurgent functions and splitting theorem , 2007Boris Sternin, Victor Shatalov Borel-Laplace Transform and Asymptotic Theory: Introduction to Resurgent Analysis , CRC Press 1996Bernard Malgrange Introduction aux travaux de J. Écalle , L'Enseignement Mathématique, 31, 1985, 261-282 Écalle's theory has important applications to solutions of generalizations of Abel's integral equation; the method of resurgent functions provides for such solutions a (Borel) resummation method for dealing with divergent series arising from semiclassical asymptotic developments in quantum theory.Frédéric Pham Introduction à la résurgence quantique, d'après Écalle et Voros, Séminaire Bourbaki 656, 1985/86 He applied his theory to dynamic systems Bernard Malgrange, Travaux d'Écalle et Martinet-Ramis sur les systèmes dynamiques, Séminaire Bourbaki 582, 1981/82 and to the interplay between diophantine small denominators and resonance involved in problems of germs of vector fields.Écalle Singularités non abordables par la géométrie, Ann. Inst.
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A.T. Schafer (1944) "Two singularities of space curves", Duke Mathematical Journal 11; 655–70 Alice continued her investigations into curves near an undulation point, publishing in American Journal of Mathematics in 1948.A. T. Schafer (1948) "The neighborhood of an undulation point on a space curve", American Journal of Mathematics 70: 351–63 When she was completing her studies at Chicago, she met Richard Schafer, who was also completing his Ph.D. in mathematics at Chicago. In 1942 Turner married Richard Schafer, after both had completed their doctorates. They had two sons.
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This problem, which was considered before to be mathematically non-tractable was settled in the paper [10] by Bunimovich with Ya.G. Sinai for diffusion of mass in periodic Lorentz gas. In their previous paper was constructed the first infinite Markov partition for chaotic systems with singularities which allowed to transform this deterministic problem into probabilistic one. Then in Bunimovich's paper with H.Spohn [11] diffusion of shear and bulk viscosities in deterministic periodic fluid was rigorously derived. The paper by Bunimovich with Ya.G. Sinai [12] pioneered rigorous studies of the space-time chaos.
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In order for a mesh to appear attractive when rendered, it is desirable that it be non-self- intersecting, meaning that no edge passes through a polygon. Another way of looking at this is that the mesh cannot pierce itself. It is also desirable that the mesh not contain any errors such as doubled vertices, edges, or faces. For some purposes it is important that the mesh be a manifold – that is, that it does not contain holes or singularities (locations where two distinct sections of the mesh are connected by a single vertex).
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One might wonder which ring-theoretic property of A=k[x_1,\ldots,x_n] causes the Hilbert syzygy theorem to hold. It turns out that this is regularity, which is an algebraic formulation of the fact that affine -space is a variety without singularities. In fact the following generalization holds: Let A be a Noetherian ring. Then A has finite global dimension if and only if A is regular and the Krull dimension of A is finite; in that case the global dimension of A is equal to the Krull dimension.
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In the mid-1970s, advances in the technologies of astronomical observations – radio, infrared, and X-ray astronomy – opened up the Universe of exploration. New tools became necessary. In this book, Hawking and Ellis attempt to establish the axiomatic foundation for the geometry of four-dimensional spacetime as described by Albert Einstein's general theory of relativity and to derive its physical consequences for singularities, horizons, and causality. Whereas the tools for studying Euclidean geometry were a straightedge and a compass, those needed to investigate curved spacetime are test particles and light rays.
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In particular, the ' is directly related to the energy and momentum of whatever matter and radiation are present. The relation is specified by the Einstein field equations, a system of partial differential equations. Some predictions of general relativity differ significantly from those of classical physics, especially concerning the passage of time, the geometry of space, the motion of bodies in free fall, and the propagation of light. Examples of such differences include gravitational time dilation, gravitational lensing, the gravitational redshift of light, the gravitational time delay and singularities/black holes.
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Under broad assumptions, an initial/boundary-value problem for a linear parabolic PDE has a solution for all time. The solution u(x,t), as a function of x for a fixed time t > 0, is generally smoother than the initial data u(x,0) = u_0(x). For a nonlinear parabolic PDE, a solution of an initial/boundary-value problem might explode in a singularity within a finite amount of time. It can be difficult to determine whether a solution exists for all time, or to understand the singularities that do arise.
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These constructions work over the moduli stack of smooth elliptic curves, and they also work for the Deligne- Mumford compactification of this moduli stack, in which elliptic curves with nodal singularities are included. TMF is the spectrum that results from the global sections over the moduli stack of smooth curves, and tmf is the spectrum arising as the global sections of the Deligne–Mumford compactification. TMF is a periodic version of the connective tmf. While the ring spectra used to construct TMF are periodic with period 2, TMF itself has period 576.
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Ordinary cusps appear when the tangent to the curve is parallel to the direction of projection (that is when the tangent projects on a single point). More complicated singularities occur when several phenomena occurs simultaneously. For example, rhamphoid cusps occur for inflection points (and for undulation points) for which the tangent is parallel to the direction of projection. In many cases, and typically in computer vision and computer graphics, the curve that is projected is the curve of the critical points of the restriction to a (smooth) spatial object of the projection.
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In mathematics, global analysis, also called analysis on manifolds, is the study of the global and topological properties of differential equations on manifolds and vector bundles. Global analysis uses techniques in infinite- dimensional manifold theory and topological spaces of mappings to classify behaviors of differential equations, particularly nonlinear differential equations. These spaces can include singularities and hence catastrophe theory is a part of global analysis. Optimization problems, such as finding geodesics on Riemannian manifolds, can be solved using differential equations so that the calculus of variations overlaps with global analysis.
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An n-ellipse is the set of points all having the same sum of distances to n foci. (The n=2 case is the conventional ellipse.) The concept of a focus can be generalized to arbitrary algebraic curves. Let C be a curve of class m and let I and J denote the circular points at infinity. Draw the m tangents to C through each of I and J. There are two sets of m lines which will have m2 points of intersection, with exceptions in some cases due to singularities, etc.
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Polini's mother was a school teacher, and before Polini reached school age herself she was already solving the mathematics problems in her mother's lessons. She graduated from the University of Padua in 1990, and completed her Ph.D. at Rutgers University in 1995, with Wolmer Vasconcelos as her doctoral advisor. Her dissertation was Studies on Singularities. After postdoctoral research at Michigan State University, she became an assistant professor at Hope College in Michigan in 1998, then moved to the University of Oregon in 2000 and to Notre Dame in 2001.
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A curvelet transform differs from other directional wavelet transforms in that the degree of localisation in orientation varies with scale. In particular, fine-scale basis functions are long ridges; the shape of the basis functions at scale j is 2^{-j} by 2^{-j/2} so the fine-scale bases are skinny ridges with a precisely determined orientation. Curvelets are an appropriate basis for representing images (or other functions) which are smooth apart from singularities along smooth curves, where the curves have bounded curvature, i.e. where objects in the image have a minimum length scale.
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In 1983, the National Science Foundation awarded Berger a Visiting Professorship for Women in Science and Engineering. In 1998, she was named a Fellow of the American Physical Society, after a nomination by the APS Division of Gravitational Physics, "for her pioneering contributions to global issues in classical general relativity, particularly the analysis of the nature of cosmological singularities, and for founding the Topical Group on Gravitation of the APS". On the occasion of her retirement in 2012, she was honored by a special session of the April 2012 APS meeting.
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In his work, and in collaboration with Penrose, Hawking extended the singularity theorem concepts first explored in his doctoral thesis. This included not only the existence of singularities but also the theory that the universe might have started as a singularity. Their joint essay was the runner-up in the 1968 Gravity Research Foundation competition. In 1970, they published a proof that if the universe obeys the general theory of relativity and fits any of the models of physical cosmology developed by Alexander Friedmann, then it must have begun as a singularity.
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Laffoley first began to organize his ideas in a format related to eastern mandalas, partially inspired by the late night patterns he watched for Warhol on late night television. This quickly developed into four general subcategories of paintings: operating systems, psychotronic devices, meta- energy, time travel, and lucid dreaming. Conceived of as "structured singularities", Laffoley never works in series, but rather approaches each project as a unique schematic. Working in a solitary lifestyle, each 73 ½ x 73 ½ inch canvas would take up to three years to paint and code.
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Saito introduced higher-dimensional generalizations of elliptic integrals. These generalizations are integrals of "primitive forms", first considered in the study of the unfolding of isolated singularities of complex hypersurfaces, associated with infinite-dimensional Lie algebras. He also studied the corresponding new automorphic forms.Kyoji Saito at the Kavli Institute for the Physics and Mathematics of the Universe The theory has a geometric connection to "flat structures" (now called "Saito Frobenius manifolds"), mirror symmetry, Frobenius manifolds, and Gromov–Witten theory in algebraic geometry and various topics in mathematical physics related to string theory.
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Major steps in the proof involve showing how manifolds behave when they are deformed by the Ricci flow, examining what sort of singularities develop, determining whether this surgery process can be completed and establishing that the surgery need not be repeated infinitely many times. The first step is to deform the manifold using the Ricci flow. The Ricci flow was defined by Richard S. Hamilton as a way to deform manifolds. The formula for the Ricci flow is an imitation of the heat equation which describes the way heat flows in a solid.
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This helped him eliminate some of the more troublesome singularities that had concerned Hamilton, particularly the cigar soliton solution, which looked like a strand sticking out of a manifold with nothing on the other side. In essence Perelman showed that all the strands that form can be cut and capped and none stick out on one side only. Completing the proof, Perelman takes any compact, simply connected, three-dimensional manifold without boundary and starts to run the Ricci flow. This deforms the manifold into round pieces with strands running between them.
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Section A, "Research Period 1991–2001," Curriculum Vitae of Arlie O. Petters, Department of Mathematics, Duke University His work culminated in book, entitled Singularity Theory and Gravitational Lensing (Springer 2012), which he co-authored with Harold Levine and Joachim Wambganns. This book, which addressed the question, "What is the universe made of?", systematically created a framework of stability and genericity for k-plane gravitational lensing. The book drew upon powerful tools from the theory of singularities and put the subject of weak-deflection k-plane gravitational lensing on a rigorous and unified mathematical foundation.
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When discussing mathematical analysis in general, or more specifically real analysis or complex analysis or differential equations, it is common for a function which contains a mathematical singularity to be referred to as a 'singular function'. This is especially true when referring to functions which diverge to infinity at a point or on a boundary. For example, one might say, "1/x becomes singular at the origin, so 1/x is a singular function." Advanced techniques for working with functions that contain singularities have been developed in the subject called distributional or generalized function analysis.
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Atiyah used Hironaka's resolution of singularities to answer this affirmatively. An ingenious and elementary solution was found at about the same time by J. Bernstein, and discussed by Atiyah. As an application of the equivariant index theorem, Atiyah and Hirzebruch showed that manifolds with effective circle actions have vanishing Â-genus. (Lichnerowicz showed that if a manifold has a metric of positive scalar curvature then the Â-genus vanishes.) With Elmer Rees, Atiyah studied the problem of the relation between topological and holomorphic vector bundles on projective space.
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In general relativity, conformal maps are the simplest and thus most common type of causal transformations. Physically, these describe different universes in which all the same events and interactions are still (causally) possible, but a new additional force is necessary to effect this (that is, replication of all the same trajectories would necessitate departures from geodesic motion because the metric tensor is different). It is often used to try to make models amenable to extension beyond curvature singularities, for example to permit description of the universe even before the Big Bang.
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Vladimir Mikhailovich Zakalyukin (in Russian: Владимир Михайлович Закалюкин; 9 July 1951 – 30 December 2011) was a Russian mathematician known for his research on singularity theory, differential equations, and optimal control theory. He obtained his Ph.D. at Moscow State University in 1977 (the thesis: "Lagrangian and Legendrian singularities"). His thesis advisor was Vladimir Arnold.A. A. Agrachev, D. V. Anosov, I. A. Bogaevskii, A. S. Bortakovskii, A. B. Budak, V. A. Vasil’ev, V. V. Goryunov, S. M. Gusein-Zade, A. A. Davydov, V. K. Zarodov, V. D. Sedykh, D. V. Treshchev, and V. N. Chubarikov.
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Caustics produced by a glass of water In optics, a caustic or caustic network is the envelope of light rays reflected or refracted by a curved surface or object, or the projection of that envelope of rays on another surface. The caustic is a curve or surface to which each of the light rays is tangent, defining a boundary of an envelope of rays as a curve of concentrated light. Therefore, in the photo on the side, caustics can be seen as patches of light or their bright edges. These shapes often have cusp singularities.
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Multigrade CLR images produce certain images on the screen of the controlling device, which differ in the first and minus first orders of laser light diffraction. As a variant, a hidden image which is both negative and positive, in plus one and minus one order respectively, may be created. More recently, novel computer-generated holograms have been proposed working with structured light carrying phase singularities . Such optical elements further improve the security level, since the encoded information only appears when the input illumination is endowed with the correct intensity and phase distribution.
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Alice Merchant (Phyllis Somerville). When Olivia and Peter enter the apartment, Olivia (but not Peter) can see a shimmering figure that Alice claims is the ghost of her late husband Derek (Ken Pogue). Walter surmises that the figure is a parallel Derek seen across a crack between the universes. If the crack widens, Walter predicts they would see occurrences of the same singularities that have plagued the parallel universe, and suggests the use of the same amber-like compound they had previously recovered ("The Ghost Network") to limit the damage.
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Brill and Max Noether developed alternative proofs using algebraic methods for much of Riemann's work on Riemann surfaces. Brill–Noether theory went further by estimating the dimension of the space of maps of given degree d from an algebraic curve to projective space Pn. In birational geometry, Noether introduced the fundamental technique of blowing up in order to prove resolution of singularities for plane curves. Noether made major contributions to the theory of algebraic surfaces. Noether's formula is the first case of the Riemann-Roch theorem for surfaces.
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The method of images (or method of mirror images) is a mathematical tool for solving differential equations, in which the domain of the sought function is extended by the addition of its mirror image with respect to a symmetry hyperplane. As a result, certain boundary conditions are satisfied automatically by the presence of a mirror image, greatly facilitating the solution of the original problem. The domain of the function is not extended. The function is made to satisfy given boundary conditions by placing singularities outside the domain of the function.
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Maria Hoffmann-Ostenhof (née Bauer, born 1947) is an Austrian mathematician known for her work on the behavior of the Schrödinger equation, and particularly on its asymptotic analysis, nodal lines, and behavior near its singularities. Hoffmann-Ostenhof was born on 12 January 1947 in Vienna. She studied mathematics at the University of Vienna, with a year visiting the University of Zurich, and completed her Ph.D. in 1973 at the University of Vienna. Her dissertation, Über Kongruenzverbände universaler Algebren und binärer Systeme [On congruence relations of universal algebras and binary systems] was supervised by .
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In mathematics, the signature defect of a singularity measures the correction that a singularity contributes to the signature theorem. introduced the signature defect for the cusp singularities of Hilbert modular surfaces. defined the signature defect of the boundary of a manifold as the eta invariant, the value as s = 0 of their eta function, and used this to show that Hirzebruch's signature defect of a cusp of a Hilbert modular surface can be expressed in terms of the value at s = 0 or 1 of a Shimizu L-function.
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A pleated surface A "crumpled" but not pleated surface with more complicated singularities: there are points where different folds meet. In geometry, a pleated surface is roughly a surface that may have simple folds but is not crumpled in more complicated ways. More precisely, a pleated surface is an isometry from a complete hyperbolic surface S to a hyperbolic 3-fold such that every point of S is in the interior of a geodesic that is mapped to a geodesic. They were introduced by , where they were called uncrumpled surfaces.
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Although the M-D CWT provides one with oriented wavelets, these orientations are only appropriate to represent the orientation along the (m-1)th dimension of a signal with dimensions. When singularities in manifold of lower dimensions are considered, such as a bee moving in a straight line in the 4-D space-time, oriented wavelets that are smooth in the direction of the manifold and change rapidly in the direction normal to it are needed. A new transform, Hypercomplex Wavelet transform was developed in order to address this issue.
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After being a lecturer and then senior lecturer at the University of Warwick from 2002 to 2006, she returned to Germany as a professor at the University of Augsburg, where she held the Chair for Analysis and Geometry. She moved to Freiburg in 2011. In 2009, Wendland was given the Medal for special merits for Bavaria in a united Europe by the Bavarian government. In 2010 she was an invited speaker at the International Congress of Mathematicians, with a talk entitled "On the geometry of singularities in quantum field theories".
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A useful tool for the analysis of derivatised nanotubes is Raman spectroscopy which shows a G-band (G for graphite) for the native nanotubes at 1580 cm−1 and a D-band (D for defect) at 1350 cm−1 when the graphite lattice is disrupted with conversion of sp² to sp³ hybridized carbon. The ratio of both peaks ID/IG is taken as a measure of functionalization. Other tools are UV spectroscopy where pristine nanotubes show distinct Van Hove singularities where functionalized tubes do not, and simple TGA analysis.
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In topology, a branch of mathematics, a topologically stratified space is a space X that has been decomposed into pieces called strata; these strata are manifolds and are required to fit together in a certain way. Topologically stratified spaces provide a purely topological setting for the study of singularities analogous to the more differential-geometric theory of Whitney. They were introduced by René Thom, who showed that every Whitney stratified space was also a topologically stratified space, with the same strata. Another proof was given by John Mather in 1970, inspired by Thom's proof.
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A paper published in 1938 by Albert Einstein, Leopold Infeld and Banesh Hoffmann showed that if elementary particles are treated as singularities in spacetime, it is unnecessary to postulate geodesic motion as part of general relativity. The electron may be treated as such a singularity. If one ignores the electron's angular momentum and charge, as well as the effects of quantum mechanics, one can treat the electron as a black hole and attempt to compute its radius. The Schwarzschild radius of a mass is the radius of the event horizon for a non-rotating, uncharged black hole of that mass.
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In physics, a renormalon (a term suggested by 't Hooft't Hooft G, in: The whys of subnuclear physics (Erice, 1977), ed. A Zichichi, Plenum Press, New York, 1979.) is a particular source of divergence seen in perturbative approximations to quantum field theories (QFT). When a formally divergent series in a QFT is summed using Borel summation, the associated Borel transform of the series can have singularities as a function of the complex transform parameter. The renormalon is a possible type of singularity arising in this complex Borel plane, and is a counterpart of an instanton singularity.
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A function f is said to be of type Ak± if it lies in the orbit of x2 ± yk+1, i.e. there exists a diffeomorphic change of coordinate in source and target which takes f into one of these forms. These simple forms x2 ± yk+1 are said to give normal forms for the type Ak±-singularities. A curve with equation f = 0 will have a tacnode, say at the origin, if and only if f has a type A3−-singularity at the origin. Notice that a node (x2 − y2 = 0) corresponds to a type A1−-singularity.
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This works well up to some accuracy and some range for the input values, but some interesting phenomena such as solitons, chaos,Nonlinear Dynamics I: Chaos at MIT's OpenCourseWare and singularities are hidden by linearization. It follows that some aspects of the dynamic behavior of a nonlinear system can appear to be counterintuitive, unpredictable or even chaotic. Although such chaotic behavior may resemble random behavior, it is in fact not random. For example, some aspects of the weather are seen to be chaotic, where simple changes in one part of the system produce complex effects throughout.
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General relativity, upon which the FRW metric is based, is an extremely successful theory which has met every observational test so far. However, at a fundamental level it is incompatible with quantum mechanics, and by predicting singularities, it also predicts its own breakdown. Any alternative theory of gravity would imply immediately an alternative cosmological theory since current modeling is dependent on general relativity as a framework assumption. There are many different motivations to modify general relativity, such as to eliminate the need for dark matter or dark energy, or to avoid such paradoxes as the firewall.
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Another way to characterize an essential singularity is that the Laurent series of f at the point a has infinitely many negative degree terms (i.e., the principal part of the Laurent series is an infinite sum). A related definition is that if there is a point a for which no derivative of f(z)(z-a)^n converges to a limit as z tends to a, then a is an essential singularity of f(z). The behavior of holomorphic functions near their essential singularities is described by the Casorati–Weierstrass theorem and by the considerably stronger Picard's great theorem.
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Laurent series are the complex-valued equivalent to Taylor series, but can be used to study the behavior of functions near singularities through infinite sums of more well understood functions, such as polynomials. A bounded function that is holomorphic in the entire complex plane must be constant; this is Liouville's theorem. It can be used to provide a natural and short proof for the fundamental theorem of algebra which states that the field of complex numbers is algebraically closed. If a function is holomorphic throughout a connected domain then its values are fully determined by its values on any smaller subdomain.
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De Lellis has given a number of remarkable contributions in different fields related to partial differential equations. In geometric measure theory he has been interested in the study of regularity and singularities of minimising hypersufaces, pursuing a program aimed at disclosing new aspects of the theory started by Almgren in his "Big regularity paper". There Almgren proved his famous regularity theorem asserting that the singular set of an m-dimensional mass-minimizing surface has dimension at most m − 2\. De Lellis has also worked on various aspects of the theory of hyperbolic systems of conservation laws and of incompressible fluid dynamics.
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Review of The recognition theorem for graded Lie algebras in prime characteristic by Murray R. Bremner (2009), In the early 90s Benkart and Efim Zelmanov started to work on classification of root-graded Lie algebras and intersection matrix algebras. The latter were introduced by P. Slodowy in his work on singularities. Berman and Moody recognized that these algebras (generalizations of affine Kac–Moody algebras ) are universal root graded Lie algebras and classified them for simply laced root systems. Benkart and Zelmanov tackled the remaining cases involving the so-called Magic Freudenthal–Tits “Square” and extended this square to exceptional Lie superalgebras.
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1\. Among the many singularities related of Radcliffe, it has been noticed that, when he was in a convivial party, he was unwilling to leave it, even though sent for by persons of the highest distinction. Whilst he was thus deeply engaged at a tavern, he was called on by a grenadier, who desired his immediate attendance on his colonel; but no entreaties could prevail on the physician to postpone his revelry. :"Sir," quoth the soldier, "my orders are to bring you." And being a very powerful man, he took him up in his arms, and carried him off per force.
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The main reason is the singularities at the Poles, which makes longitude undefined at these points. Also near the poles the latitude/longitude grid is highly non-linear, and several errors may occur in calculations that are sufficiently accurate on other locations. Another problematic area is the meridian at ±180° longitude, where the longitude has a discontinuity, and hence specific program code must often be written to handle this. An example of the consequences of omitting such code is the crash of the navigation systems of twelve F-22 Raptor fighter aircraft while crossing this meridian.
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In their work, Seiden and Woeginger refer to Sylvester's sequence as "Salzer's sequence" after the work of on closest approximation. Znám's problem concerns sets of numbers such that each number in the set divides but is not equal to the product of all the other numbers, plus one. Without the inequality requirement, the values in Sylvester's sequence would solve the problem; with that requirement, it has other solutions derived from recurrences similar to the one defining Sylvester's sequence. Solutions to Znám's problem have applications to the classification of surface singularities (Brenton and Hill 1988) and to the theory of nondeterministic finite automata.
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Such an interaction is conjectured to replace the singular Big Bang with a cusp-like Big Bounce at a minimum but finite scale factor, before which the observable universe was contracting. This scenario also explains why the present Universe at largest scales appears spatially flat, homogeneous and isotropic, providing a physical alternative to cosmic inflation. Torsion allows fermions to be spatially extended instead of "pointlike", which helps to avoid the formation of singularities such as black holes and removes the ultraviolet divergence in quantum field theory. According to general relativity, the gravitational collapse of a sufficiently compact mass forms a singular black hole.
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In this page we provide information regarding filter banks, multidimensional filter banks and different methods to design multidimensional filters. Also we talked about NDFB, which is built upon an efficient tree-structured construction, which leads to a low redundancy ratio and refinable angular resolution. By combining the NDFB with a new multiscale pyramid, we can construct the surfacelet transform, which has potential in efficiently capturing and representing surface-like singularities in multidimensional signals. AS mentioned above NDFB and surfacelet transform have applications in various areas that involve the processing of multidimensional volumetric data, including video processing, seismic image processing, and medical image analysis.
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She joined the Edinburgh Mathematical Society where she presented several of her papers including 'The equation of telegraphy' and 'The equation of conduction of heat'. She was elected to the Committee of the Society in November 1923 and continued as a member throughout her career. In 1924 she travelled to the United States under the assistance of both a British graduates scholarship and a Carnegie scholarship to attend Bryn Mawr College, Pennsylvania from where she gained a PhD under the supervision of Anna Johnson Pell Wheeler. Her research topic was 'A boundary value problem of ordinary self-adjoint differential equations with singularities'.
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In the field of differential geometry in mathematics, mean curvature flow is an example of a geometric flow of hypersurfaces in a Riemannian manifold (for example, smooth surfaces in 3-dimensional Euclidean space). Intuitively, a family of surfaces evolves under mean curvature flow if the normal component of the velocity of which a point on the surface moves is given by the mean curvature of the surface. For example, a round sphere evolves under mean curvature flow by shrinking inward uniformly (since the mean curvature vector of a sphere points inward). Except in special cases, the mean curvature flow develops singularities.
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For example, a result about the singularities of harmonic functions would be said to belong to potential theory whilst a result on how the solution depends on the boundary data would be said to belong to the theory of the Laplace equation. This is not a hard and fast distinction, and in practice there is considerable overlap between the two fields, with methods and results from one being used in the other. Modern potential theory is also intimately connected with probability and the theory of Markov chains. In the continuous case, this is closely related to analytic theory.
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His books appeal to expert players rather than beginners: they contain many long analyses of variations in critical positions, and "singularities and exceptions were his forte, not rules and simplifications". Although Alekhine was declared an enemy of the Soviet Union after his anti-Bolshevik statement in 1928, he was gradually rehabilitated by the Soviet chess elite following his death in 1946. Alexander Kotov's research on Alekhine's games and career, culminating in a biography,Kotov 1975 led to a Soviet series of Alekhine Memorial tournaments. The first of these, at Moscow 1956, was won jointly by Botvinnik and Vasily Smyslov.
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Inspired by Roger Penrose's theorem of a spacetime singularity in the centre of black holes, Hawking applied the same thinking to the entire universe; and, during 1965, he wrote his thesis on this topic. Hawking's thesis was approved in 1966. There were other positive developments: Hawking received a research fellowship at Gonville and Caius College at Cambridge; he obtained his PhD degree in applied mathematics and theoretical physics, specialising in general relativity and cosmology, in March 1966; and his essay "Singularities and the Geometry of Space-Time" shared top honours with one by Penrose to win that year's prestigious Adams Prize.
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Solem discovered a strange polarization of the hydrogen atom that, contrary to intuition, drove electron orbits perpendicular to an applied electric field (1987). He elucidated the interpretation of geometric phase in quantum mechanics by showing the invalidity of superposition of quantal states, the distinction between rays and vectors in projective Hilbert space, and the meaning of resultant singularities (1993b). Using the symmetry of the Kepler Orbital Problem in operator formalism for both classical and quantum mechanics, Solem predicted a previously unknown elastic scattering process that will rotate the linear polarization of the scattered photon by ½\pi (1997a).
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Arlie Oswald Petters, MBE (born February 8, 1964) is a Belizean-American mathematical physicist, who is the Benjamin Powell Professor of Mathematics and a Professor of Physics and Economics at Duke University. Petters will become the Provost at New York University Abu Dhabi effective September 1, 2020. Petters is a founder of mathematical astronomy, focusing on problems connected to the interplay of gravity and light and employing tools from astrophysics, cosmology, general relativity, high energy physics, differential geometry, singularities, and probability theory. His monograph "Singularity Theory and Gravitational Lensing" developed a mathematical theory of gravitational lensing.
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In this page we provide information regarding filter banks, multidimensional filter banks and different methods to design multidimensional filters. Also we talked about MDFB, which is built upon an efficient tree- structured construction, which leads to a low redundancy ratio and refinable angular resolution. By combining the MDFB with a new multiscale pyramid, we can constructed the surfacelet transform, which has potentials in efficiently capturing and representing surface-like singularities in multidimensional signals. MDFB and surfacelet transform have applications in various areas that involve the processing of multidimensional volumetric data, including video processing, seismic image processing, and medical image analysis.
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Later, from about 1960, and largely led by Grothendieck, the idea of schemes was worked out, in conjunction with a very refined apparatus of homological techniques. After a decade of rapid development the field stabilized in the 1970s, and new applications were made, both to number theory and to more classical geometric questions on algebraic varieties, singularities, moduli, and formal moduli. An important class of varieties, not easily understood directly from their defining equations, are the abelian varieties, which are the projective varieties whose points form an abelian group. The prototypical examples are the elliptic curves, which have a rich theory.
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Wavelet-based contourlet packet using 3 dyadic wavelet levels and 8 directions at the finest level. Although the wavelet transform is not optimal in capturing the 2-D singularities of images, it can take the place of LP decomposition in the double filter bank structure to make the contourlet transform a non-redundant image transform. The wavelet-based contourlet transform is similar to the original contourlet transform, and it also consists of two filter bank stages. In the first stage, the wavelet transform is used to do the sub-band decomposition instead of the Laplacian pyramid (LP) in the contourlet transform.
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Much of his subsequent research was driven by a desire to understand this relationship and to develop new direct and simple methods for studying the Painlevé equations. Kruskal was rarely satisfied with the standard approaches to differential equations. The six Painlevé equations have a characteristic property called the Painlevé property: their solutions are single-valued around all singularities whose locations depend on the initial conditions. In Kruskal's opinion, since this property defines the Painlevé equations, one should be able to start with this, without any additional unnecessary structures, to work out all the required information about their solutions.
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With three symphonies transcribed, Liszt set aside the work for another 23 years. It was not until 1863 that Breitkopf & Härtel suggested to Liszt that he transcribe the complete set for a future publication. For this work, Liszt recycled his previous transcriptions by simplifying passages, stating that "the more intimately acquainted one becomes with Beethoven, the more one clings to certain singularities and finds that even insignificant details are not without their value". He would note down the names of the orchestral instruments for the pianist to imitate, and also add pedal marks and fingerings for amateurs and sight readers.
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One could define the -axis as a tangent at this point, but this definition can not be the same as the definition at other points. In fact, in this case, the -axis is a "double tangent." For affine and projective varieties, the singularities are the points where the Jacobian matrix has a rank which is lower than at other points of the variety. An equivalent definition in terms of commutative algebra may be given, which extends to abstract varieties and schemes: A point is singular if the local ring at this point is not a regular local ring.
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Penrose diagrams are frequently used to illustrate the causal structure of spacetimes containing black holes. Singularities are denoted by a spacelike boundary, unlike the timelike boundary found on conventional space-time diagrams. This is due to the interchanging of timelike and spacelike coordinates within the horizon of a black hole (since space is uni-directional within the horizon, just as time is uni-directional outside the horizon). The singularity is represented by a spacelike boundary to make it clear that once an object has passed the horizon it will inevitably hit the singularity even if it attempts to take evasive action.
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A variety of plots gravitating around the original conspiracy theory were imagined by activists, one stating for instance that as many as 4,000 Jews were warned of the September 11 attacks before they happened. Believers also claim that ZOG-like forces control the American foreign policy. Despite their own singularities, most ZOG theories involve the idea of a Jewish power over the finance or banking, including one imagining a Jewish control on the Federal Reserve. Neo-Nazi David Lane developed his version of the white genocide conspiracy theory in his White Genocide Manifesto, the origin of the later use of the term.
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Together with Felipe Voloch, in 1992 he succeeded in making progress toward proving the Mordell-Lang Conjecture in characteristic p (the full proof came later from Ehud Hrushovski). He has been a guest scholar at, among other institutions, the Hebrew University of Jerusalem, the Max Planck Institute for Mathematics in Bonn, the Mathematical Sciences Research Institute (MSRI), the Pierre and Marie Curie University in Paris and at IHES. He was an invited speaker at ICM 2018 (Resolution of singularities of complex algebraic varieties and their families).Arxiv From 1996 to 1998 he was a Sloan Fellow.
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A geodesic grid is a global Earth reference that uses triangular tiles based on the subdivision of a polyhedron (usually the icosahedron, and usually a Class I subdivision) to subdivide the surface of the Earth. Such a grid does not have a straightforward relationship to latitude and longitude, but conforms to many of the main criteria for a statistically valid discrete global grid. Primarily, the cells' area and shape are generally similar, especially near the poles where many other spatial grids have singularities or heavy distortion. The popular Quaternary Triangular Mesh (QTM) falls into this category.
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In physics, especially quantum field theory, regularization is a method of modifying observables which have singularities in order to make them finite by the introduction of a suitable parameter called regulator. The regulator, also known as a "cutoff", models our lack of knowledge about physics at unobserved scales (e.g. scales of small size or large energy levels). It compensates for (and requires) the possibility that "new physics" may be discovered at those scales which the present theory is unable to model, while enabling the current theory to give accurate predictions as an "effective theory" within its intended scale of use.
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These new transcendental functions, solving the remaining six equations, are called the Painlevé transcendents, and interest in them has revived recently due to their appearance in modern geometry, integrable systems Ablowitz, M. J. and Clarkson, P.A. (1991) Solitons, nonlinear evolution equations and inverse scattering. Cambridge University Press and statistical mechanics. In 1895 he gave a series of lectures at Stockholm University on differential equations, at the end stating the Painlevé conjecture about singularities of the n-body problem. In the 1920s, Painlevé briefly turned his attention to the new theory of gravitation, general relativity, which had recently been introduced by Albert Einstein.
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The Angenent torus can be used to prove the existence of certain other kinds of singularities of the mean curvature flow. For instance, if a dumbbell shaped surface, consisting of a thin cylindrical "neck" connecting two large volumes, can have its neck surrounded by a disjoint Angenent torus, then the two surfaces of revolution will remain disjoint under the mean curvature flow until one of them reaches a singularity; if the ends of the dumbbell are large enough, this implies that the neck must pinch off, separating the two spheres from each other, before the torus surrounding the neck collapses..
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In 1993, he published a book coauthored with Klainerman in which their proof of the stability result is laid out in detail. In that year, he was named a MacArthur Fellow. In 1991, he published a paper which shows that the test masses of a gravitational wave detector suffer permanent relative displacements after the passage of a gravitational wave train, an effect which has been named "nonlinear memory effect". In the period 1987–1999 he published a series of papers on the gravitational collapse of a spherically symmetric self-gravitating scalar field and the formation of black holes and associated spacetime singularities.
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Using the orientation of X one may assign to each of these points a sign; in other words intersection yields a 0-dimensional cycle. One may prove that the homology class of this cycle depends only on the homology classes of the original i- and (n-i)-dimensional cycles; one may furthermore prove that this pairing is perfect. When X has singularities--that is, when the space has places that do not look like \R^n--these ideas break down. For example, it is no longer possible to make sense of the notion of "general position" for cycles.
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R. Bryant's parameterization) Werner Boy (; 4 May 1879 − 6 September 1914) was a German mathematician. He was the discoverer and eponym of Boy's surface—a three-dimensional projection of the real projective plane without singularities, the first of its kind. He discovered it in 1901 after his thesis adviser, David Hilbert, asked him to prove that it was not possible to immerse the real projective plane in three-dimensional space. Boy sketched several models of the surface, and discovered that it could have 3-fold rotational symmetry, but was unable to find a parametric model for the surface.
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Jean Écalle (born 1950) is a French mathematician, specializing in dynamic systems, perturbation theory, and analysis. Écalle received, in 1974 from the University of Paris-Saclay in Orsay, a doctorate under the supervision of Hubert Delange with Thèse d'État entitled La théorie des invariants holomorphes. He is a directeur de recherché (senior researcher) of the Centre national de la recherche scientifique (CNRS) and is a professor at the University of Paris-Saclay. He developed a theory of so-called "resurgent functions", analytic functions with isolated singularities, which have a special algebra of derivatives (Alien calculus, Calcul différentiel étranger).
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Kill-T-Gang have access to special abilities known as "singularities" that differ on the individual but have the common trait of both sharing memories and communicating telepathically through kissing. Because the Kill-T-Gang's true forms absorb libido through proximity, humanity would be wiped out should even one make it to the Earth, necessitating the use of Impacters. ; : :The humanoid form of the Type-1 Kill- T-Gang robot . He is the leader of the invasion force he calls the "Planetary Gears," and sees the eternal lives of the Kill-T-Gang as their strength.
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Together with Paolo Cascini, Christopher Hacon and James McKernan, Birkar settled several conjectures including existence of log flips, finite generation of log canonical rings, and existence of minimal models for varieties of log general type, building upon earlier work of Vyacheslav Shokurov and of Hacon and McKernan.C. Birkar, P. Cascini, C. Hacon, J. McKernan Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23 (2010), 405–468. In the setting of log canonical singularities, he proved existence of log flips along with key cases of the minimal model and abundance conjectures.
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He proved several fundamental problems such as Shokurov's conjecture on boundedness of complements and Borisov–Alexeev–Borisov conjecture on boundedness of Fano varieties.C. Birkar, Anti-pluricanonical systems on Fano varieties. arXiv:1603.05765C. Birkar, Singularities of linear systems and boundedness of Fano varieties. arXiv:1609.05543. In 2018, Birkar was given the Fields Medal for his Fano varieties and his other contributions the minimal model problem. In a video made available by the Simons Foundation, Birkar expressed hope that his Fields Medal will put “just a little smile on the lips” of the world's estimated 40 million Kurds.
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Taylor has made contributions to quantum field theory and the physics of elementary particles. His contributions include: the discovery (also made independently by Lev Landau) of singularities in the analytical structure of the Feynman integrals for processes in quantum field theory, the PCAC nature of radioactive decay of the pion and the discovery in 1971 of the so-called Slavnov–Taylor identities, which control symmetry and renormalisation of gauge theories. With various collaborators, in 1980 he discovered that real and virtual infrared divergences do not cancel in QCD as they do in QED. They also showed how these infrared divergences exponentiate.
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The problem concerns the existence of singularities of non-collision character in the N-body problem in three-dimensional space; Xia proved existence for N \geq 5 . For the existence proof he constructed an example of five masses, of which four are separated into two pairs which revolve around each other in eccentric elliptical orbits about the z-axis of symmetry, and a fifth mass moves along the z-axis. For selected initial conditions, the fifth mass can be accelerated to an infinite velocity in a finite time interval (without any collision between the bodies involved in the example).
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In 1908 Edvard Hugo von Zeipel proved the surprising fact that the existence of a non-collision singularity in the N-body problem necessarily causes the velocity of at least one particle to become unbounded. The case N = 4 is open.Joseph L. Gerver gave arguments (a heuristic model) for the existence of a non-collision singularity for the planar Newtonian 4-body problem — however, there is still no rigorous proof. See For N = 3 Painlevé had proved that the singularities (points of the orbit in which accelerations become infinite in a finite time interval) must be of the collision type.
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Donal O'Shea Donal O'Shea is a Canadian mathematician, who is also noted for his bestselling books. He is currently the fifth president of New College of Florida in Sarasota, a position to which he was named on July 1, 2012. Before coming to New College, he served in various roles at Mount Holyoke College, including professor of mathematics, dean of faculty, and vice president for academic affairs. O'Shea graduated with a B.Sc. from Harvard College, and received a Ph.D. in mathematics from Queen's University in Kingston, Ontario in 1981; his thesis, titled On μ-Equivalent Families of Singularities, was written under the direction of Albert John Coleman.
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But it is well understood today (and was understood well by some even then) that collapse cannot happen through stationary states the way Einstein imagined. Nevertheless, the extent to which the models of black holes in classical general relativity correspond to physical reality remains unclear, and in particular the implications of the central singularity implicit in these models are still not understood. Closely related to his rejection of black holes, Einstein believed that the exclusion of singularities might restrict the class of solutions of the field equations so as to force solutions compatible with quantum mechanics, but no such theory has ever been found.
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The study of the analytic structure of an algebraic curve in the neighborhood of a singular point provides accurate information of the topology of singularities. In fact, near a singular point, a real algebraic curve is the union of a finite number of branches that intersect only at the singular point and look either as a cusp or as a smooth curve. Near a regular point, one of the coordinates of the curve may be expressed as an analytic function of the other coordinate. This is a corollary of the analytic implicit function theorem, and implies that the curve is smooth near the point.
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The compact set K is called removable if, whenever Ω is an open set containing K, every function which is bounded and holomorphic on the set Ω \ K has an analytic extension to all of Ω. By Riemann's theorem for removable singularities, every singleton is removable. This motivated Painlevé to pose a more general question in 1880: "Which subsets of C are removable?" It is easy to see that K is removable if and only if γ(K) = 0. However, analytic capacity is a purely complex-analytic concept, and much more work needs to be done in order to obtain a more geometric characterization.
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In principle, such methods may be applied to any system, given sufficient computer resources, and may address fundamental questions such as naked singularities. Approximate solutions may also be found by perturbation theories such as linearized gravityFor instance and its generalization, the post-Newtonian expansion, both of which were developed by Einstein. The latter provides a systematic approach to solving for the geometry of a spacetime that contains a distribution of matter that moves slowly compared with the speed of light. The expansion involves a series of terms; the first terms represent Newtonian gravity, whereas the later terms represent ever smaller corrections to Newton's theory due to general relativity.
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X. Chen, S. K. Donaldson, and S. Sun. Kähler-Einstein metrics and stability. International Mathematics Research Notices, 1(8):2119–2125.X. Chen, S. K. Donaldson, and S. Sun. Kähler-Einstein metrics on Fano manifolds. I: Approximation of metrics with cone singularities. Journal of the American Mathematical Society, 28(1):183–197.X. Chen, S. K. Donaldson, and S. Sun. Kähler-Einstein metrics on Fano manifolds. II: Limits with cone angle less than 2π. Journal of the American Mathematical Society, 28(1):199–234.X. Chen, S. K. Donaldson, and S. Sun. Kähler-Einstein metrics on Fano manifolds. III: Limits as cone angle approaches 2π and completion of the main proof.
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He is also good friends with the second-in command of the Fringe Division, Colonel Phillip Broyles. He identified means of stopping the singularities caused by Walter's crossing in 1985 by using an amber-like substance to seal the area around these, regardless of the innocent lives trapped. He has also engineered the shapeshifters to cross to the prime universe and perform actions that ultimately allow him to cross over and convince Peter to return to the parallel universe. He shows Peter the doomsday device that they have found and reconstructed, and urges Peter to figure out how it works as the device only reacts to his biological signature.
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The name of umbral moonshine derives from the use of shadows in the theory of mock modular forms. Other moonlight-related words like 'lambency' were given technical meanings (in this case, the genus zero group attached to a shadow SX, whose level is the dual Coxeter number of the root system X) by Cheng, Duncan, and Harvey to continue the theme. Although the umbral moonshine conjecture has been settled, there are still many questions that remain. For example, connections to geometry and physics are still not very solid, although there is work relating umbral functions to duVal singularities on K3 surfaces by Cheng and Harrison.
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Applications for behavioral economics include the modeling of the consumer decision-making process for applications in artificial intelligence and machine learning. The Silicon Valley-based start- up Singularities is using the AGM postulates proposed by Alchourrón, Gärdenfors, and Makinson—the formalization of the concepts of beliefs and change for rational entities—in a symbolic logic to create a "machine learning and deduction engine that uses the latest data science and big data algorithms in order to generate the content and conditional rules (counterfactuals) that capture customer's behaviors and beliefs." Applications of behavioral economics also exist in other disciplines, for example in the area of supply chain management.
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In 1996, de Jong developed his theory of alterations which was used by Fedor Bogomolov and Tony Pantev (1996) and Dan Abramovich and de Jong (1997) to prove resolution of singularities in characteristic 0 and to prove a weaker result for varieties of all dimensions in characteristic p which is strong enough to act as a substitute for resolution for many purposes. In 2005, de Jong started the Stacks Project, "an open source textbook and reference work on algebraic stacks and the algebraic geometry needed to define them." The book that the project has generated currently runs to more than 6,000 pages as of March 2018.
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The strategy leads to extra computational costs and makes the method is not as efficient as expected compared to the MFS. The second approachChen W, Gu Y, "Recent advances on singular boundary method", Joint International Workshop on Trefftz Method VI and Method of Fundamental Solution II, Taiwan 2011.Gu Y, Chen, W, "Improved singular boundary method for three dimensional potential problems", Chinese Journal of Theoretical and Applied Mechanics, 2012, 44(2): 351-360 (in Chinese) is to employ a regularization technique to cancel the singularities of the fundamental solution and its derivatives. Consequently, the origin intensity factors can be determined directly without using any sample nodes.
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In physical cosmology, imaginary time may be incorporated into certain models of the universe which are solutions to the equations of general relativity. In particular, imaginary time can help to smooth out gravitational singularities, where known physical laws break down, to remove the singularity and avoid such breakdowns (see Hartle–Hawking state). The Big Bang, for example, appears as a singularity in ordinary time but, when modelled with imaginary time, the singularity can be removed and the Big Bang functions like any other point in four-dimensional spacetime. Any boundary to spacetime is a form of singularity, where the smooth nature of spacetime breaks down.
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The peridynamic theory is based on integral equations, in contrast with the classical theory of continuum mechanics, which is based on partial differential equations. Since partial derivatives do not exist on crack surfaces and other singularities, the classical equations of continuum mechanics cannot be applied directly when such features are present in a deformation. The integral equations of the peridynamic theory can be applied directly, because they do not require partial derivatives. The ability to apply the same equations directly at all points in a mathematical model of a deforming structure helps the peridynamic approach avoid the need for the special techniques of fracture mechanics.
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Using Sard's theorem, whose hypothesis is a special case of the transversality of maps, it can be shown that transverse intersections between submanifolds of a space of complementary dimensions or between submanifolds and maps to a space are themselves smooth submanifolds. For instance, if a smooth section of an oriented manifold's tangent bundle—i.e. a vector field—is viewed as a map from the base to the total space, and intersects the zero-section (viewed either as a map or as a submanifold) transversely, then the zero set of the section—i.e. the singularities of the vector field—forms a smooth 0-dimensional submanifold of the base, i.e.
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Gregory Lawrence Eyink is an American mathematical physicist at Johns Hopkins University. He received his bachelor’s degree in mathematics and philosophy (1981) and Doctor of Philosophy (1987) from Ohio State University. He now holds joint appointments in the departments of Physics and Astronomy, Mathematics, and Mechanical Engineering at Johns Hopkins. He was awarded the status of Fellow of the American Physical Society , after being nominated by their Topical Group on Statistical and Nonlinear Physics in 2003, for his work in nonequilibrium statistical mechanics, in particular on the foundation of transport laws in chaotic dynamical systems, on field-theoretic methods in statistical hydrodynamics and on singularities and dissipative anomalies in fluid turbulence.
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In 1935, following a controversial debateP. Havas, The General- Relativistic Two-Body Problem and the Einstein–Silberstein Controversy, in with Albert Einstein, Silberstein published a solution of Einstein's field equations that appeared to describe a static, axisymmetric metric with only two point singularities representing two point masses. Such a solution clearly violates our understanding of gravity: with nothing to support them and no kinetic energy to hold them apart, the two masses should fall towards each other due to their mutual gravity, in contrast with the static nature of Silberstein's solution. This led Silberstein to claim that A. Einstein's theory was flawed, in need of a revision.
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The place of anointment also differs, with Mark and Matthew stating that it was over the head, with John and Luke recording an anointing of feet and wiping with hair. :The central message of the stories in Matthew, Mark, and John is very similar with some minor differences such as "The poor you will always have with you" and "She poured perfume on my body beforehand to prepare for my burial". These are not in Luke, who instead records comments on hospitality and forgiveness of sins that are not in the other accounts.See for all points Hornsby, 339 Several singularities set the Lukan narrative apart from other gospel accounts.
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Brian Thomas Swimme (born 1950) is a professor at the California Institute of Integral Studies, in San Francisco, where he teaches evolutionary cosmology to graduate students in the Philosophy, Cosmology, and Consciousness program. He received his Ph.D. (1978) from the department of mathematics at the University of Oregon for work with Richard Barrar on singularity theory, with a dissertation entitled Singularities in the N-Body Problem. Swimme was a faculty member in the department of mathematics at the University of Puget Sound in Tacoma, Washington, 1978–81. He was a member of the faculty at the Institute in Culture and Creation Spirituality at Holy Names University in Oakland, California, 1983–89.
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The external body forces will appear as the independent ("right-hand side") term in the differential equations, while the concentrated forces appear as boundary conditions. An external (applied) surface force, such as ambient pressure or friction, can be incorporated as an imposed value of the stress tensor across that surface. External forces that are specified as line loads (such as traction) or point loads (such as the weight of a person standing on a roof) introduce singularities in the stress field, and may be introduced by assuming that they are spread over small volume or surface area. The basic stress analysis problem is therefore a boundary-value problem.
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Clarence Lemuel Elisha Moore (12 May 1876, Bainbridge, Ohio – 5 December 1931) was an American mathematics professor, specializing in algebraic geometry and Riemannian geometry. He is chiefly remembered for the memorial eponymous C. L. E. Moore instructorship at the Massachusetts Institute of Technology; this prestigious instructorship has produced many famous mathematicians, including three Fields medal winners: Paul Cohen, Daniel Quillen, and Curtis T. McMullen. C. L. E. Moore received his B.Sc. from Ohio State University (1901) and then his A.M. (1902) and Ph.D. (1904) from Cornell University. His doctoral dissertation was entitled Classification of the surfaces of singularities of the quadratic spherical complex with Virgil Snyder as thesis advisor.
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In analyzing the Planck Satellite data from 2013, Anna Ijjas and Paul Steinhardt showed that the simplest textbook inflationary models were eliminated and that the remaining models require exponentially more tuned starting conditions, more parameters to be adjusted, and less inflation. Later Planck observations reported in 2015 confirmed these conclusions. A 2014 paper by Kohli and Haslam called into question the viability of the eternal inflation theory, by analyzing Linde's chaotic inflation theory in which the quantum fluctuations are modeled as Gaussian white noise. They showed that in this popular scenario, eternal inflation in fact cannot be eternal, and the random noise leads to spacetime being filled with singularities.
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A curve with double point A curve with a cusp Historically, singularities were first noticed in the study of algebraic curves. The double point at (0, 0) of the curve :y^2 = x^2 + x^3 and the cusp there of :y^2 = x^3\ are qualitatively different, as is seen just by sketching. Isaac Newton carried out a detailed study of all cubic curves, the general family to which these examples belong. It was noticed in the formulation of Bézout's theorem that such singular points must be counted with multiplicity (2 for a double point, 3 for a cusp), in accounting for intersections of curves.
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While Thom was an eminent mathematician, the subsequent fashionable nature of elementary catastrophe theory as propagated by Christopher Zeeman caused a reaction, in particular on the part of Vladimir Arnold. He may have been largely responsible for applying the term singularity theory to the area including the input from algebraic geometry, as well as that flowing from the work of Whitney, Thom and other authors. He wrote in terms making clear his distaste for the too-publicised emphasis on a small part of the territory. The foundational work on smooth singularities is formulated as the construction of equivalence relations on singular points, and germs.
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The Riemann singularity theorem was extended by George Kempf in 1973, building on work of David Mumford and Andreotti - Mayer, to a description of the singularities of points p = class(D) on Wk for 1 ≤ k ≤ g − 1\. In particular he computed their multiplicities also in terms of the number of independent meromorphic functions associated to D (Riemann-Kempf singularity theorem).Griffiths and Harris, p.348 More precisely, Kempf mapped J locally near p to a family of matrices coming from an exact sequence which computes h0(O(D)), in such a way that Wk corresponds to the locus of matrices of less than maximal rank.
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A lifting modifies biorthogonal filters in order to increase the number of vanishing moments of the resulting biorthogonal wavelets, and hopefully their stability and regularity. Increasing the number of vanishing moments decreases the amplitude of wavelet coefficients in regions where the signal is regular, which produces a more sparse representation. However, increasing the number of vanishing moments with a lifting also increases the wavelet support, which is an adverse effect that increases the number of large coefficients produced by isolated singularities. Each lifting step maintains the filter biorthogonality but provides no control on the Riesz bounds and thus on the stability of the resulting wavelet biorthogonal basis.
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Walter had crossed over on the frozen ice of Reiden Lake in 1985 to administer the cure for the alternate version of Peter, but, after accidentally destroying a dose of the cure upon transport, he instead brought the boy across. On return, they fell through the ice but were saved by the Observer September (Michael Cerveris), who told Walter of the importance of "the boy", which Walter took to mean Peter. Walter's crossing is what caused the singularities in the parallel universe, with Reiden Lake at their center. Walter has been looking for a sign of forgiveness in the form of a white tulip.
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Although the Kerr solution appears to be singular at the roots of Δ = 0, these are actually coordinate singularities, and, with an appropriate choice of new coordinates, the Kerr solution can be smoothly extended through the values of r corresponding to these roots. The larger of these roots determines the location of the event horizon, and the smaller determines the location of a Cauchy horizon. A (future-directed, time-like) curve can start in the exterior and pass through the event horizon. Once having passed through the event horizon, the r coordinate now behaves like a time coordinate, so it must decrease until the curve passes through the Cauchy horizon.
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Abramovich received the bachelor's degree at Tel Aviv University in 1987 and completed the doctorate at Harvard University in 1991 under Joe Harris (Subvarieties of abelian varieties and of Jacobians of curves). From 1991 to 1994 he was Moore Instructor at the Massachusetts Institute of Technology. Thereafter he held faculty positions at Boston University from 1994 to 1999 and since 2003 has been Professor at Brown University. Among other topics, he has dealt with birational geometry, the resolution of singularities, subvarieties of Abelian varieties, limits for the torsion of elliptic curves, rational and integer points on algebraic varieties and moduli spaces of vector bundles on curves.
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Using his only Noble Phantasm to erase himself from existence and weaken Goetia, he then said his goodbyes to Ritsuka and in order for him to defeat Goetia once and for all. Upon the final Singularity being restored to order, Mash is revived via the power of Beast IV, who is revealed to be Fou, a dog-like creature that has journeyed alongside Fujimaru and their party through the entirety of the game. Upon completion of the singularities, Ritsuka Fujimaru is awarded the rank of Cause, by the Mages Association. In the aftermath of the singularity crisis Chaldea would be tasked with handling Singularity Subspecies.
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In the multivariate case, the consideration of complex zeros is also needed, but not sufficient: one must also consider zeros at infinity. For example, Bézout's theorem asserts that the intersection of two plane algebraic curves of respective degrees and consists of exactly points if one consider complex points in the projective plane, and if one counts the points with their multiplicity. Another example is the genus–degree formula that allows computing the genus of a plane algebraic curve form its singularities in the complex projective plane. So a projective variety is the set of points in a projective space, whose homogeneous coordinates are common zeros of a set of homogeneous polynomials.
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A resolution of singularities :f:X\to Y of a complex variety Y is called a small resolution if for every r > 0, the space of points of Y where the fiber has dimension r is of codimension greater than 2r. Roughly speaking, this means that most fibers are small. In this case the morphism induces an isomorphism from the (intersection) homology of X to the intersection homology of Y (with the middle perversity). There is a variety with two different small resolutions that have different ring structures on their cohomology, showing that there is in general no natural ring structure on intersection (co)homology.
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In 1963, Roy Kerr found the exact solution for a rotating black hole. Two years later, Ezra Newman found the axisymmetric solution for a black hole that is both rotating and electrically charged. Through the work of Werner Israel, Brandon Carter, and David Robinson the no- hair theorem emerged, stating that a stationary black hole solution is completely described by the three parameters of the Kerr–Newman metric: mass, angular momentum, and electric charge. At first, it was suspected that the strange features of the black hole solutions were pathological artifacts from the symmetry conditions imposed, and that the singularities would not appear in generic situations.
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Simulation of two black holes colliding Penrose demonstrated that once an event horizon forms, general relativity without quantum mechanics requires that a singularity will form within. Shortly afterwards, Hawking showed that many cosmological solutions that describe the Big Bang have singularities without scalar fields or other exotic matter (see "Penrose–Hawking singularity theorems"). The Kerr solution, the no-hair theorem, and the laws of black hole thermodynamics showed that the physical properties of black holes were simple and comprehensible, making them respectable subjects for research. Conventional black holes are formed by gravitational collapse of heavy objects such as stars, but they can also in theory be formed by other processes.
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The East Hesse Highlands is bounded immediately to the east by the West Hesse Highlands and Lowlands. Almost all of the region is formed by Bunter sandstone and this defines both its relief and the surface of the land apart from occasional layers of overlying volcanic basalt. All the prominent ridges are, at least partly, characterised by volcanic features. Between the Hoher Meissner (754 m) and Kaufungen Forest (up to 643 m high) in the north, the Knüll (636 m) in the centre, the Vogelsberg (773 m) in the southwest and the Rhön (950 m) in the southeast, there are numerous individual singularities which catalogue the volcanic activity between the two Central Uplands regions.
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A transformed bicorn with a = 1 The bicorn is a plane algebraic curve of degree four and genus zero. It has two cusp singularities in the real plane, and a double point in the complex projective plane at x=0, z=0 . If we move x=0 and z=0 to the origin substituting and perform an imaginary rotation on x bu substituting ix/z for x and 1/z for y in the bicorn curve, we obtain :(x^2-2az+a^2z^2)^2 = x^2+a^2z^2.\, This curve, a limaçon, has an ordinary double point at the origin, and two nodes in the complex plane, at x = ± i and z=1.
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Although probably best known for his Star Trek covers for Bantam Books and almost fifty covers for Sphere Books, much of his output was for German publishers, including more than 100 covers for Bastei Lübbe's science fiction imprint and over 500 for Terra Astra magazine. The Science Fiction Writers of America described Jones as "the precursor to a generation of artists that helped define the look of early '70s SF illustration". He was nominated for a Hugo Award for Best Professional Artist in 1970 and 1971. In Larry Niven's short story "Singularities Make Me Nervous", from Convergent Series, the protagonist, speaking in the future, describes his apartment as containing "Eddie Jones originals".
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He gave elegant proofs for the spin-statistics theorem and the CPT theorem, and noted that the field algebra led to anomalous Schwinger terms in various classical identities, because of short distance singularities. These were foundational results in field theory, instrumental for the proper understanding of anomalies. In other notable early work, Rarita and Schwinger formulated the abstract Pauli and Fierz theory of the spin-3/2 field in a concrete form, as a vector of Dirac spinors, Rarita–Schwinger equation. In order for the spin-3/2 field to interact consistently, some form of supersymmetry is required, and Schwinger later regretted that he had not followed up on this work far enough to discover supersymmetry.
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In 1924–25, while researching at the University of Dhaka, Prof Satyendra Nath Bose well known for his works in quantum mechanics, provided the foundation for Bose–Einstein statistics and the theory of the Bose–Einstein condensate. Meghnad Saha was the first scientist to relate a star's spectrum to its temperature, developing thermal ionization equations (notably the Saha ionization equation) that have been foundational in the fields of astrophysics and astrochemistry. Amal Kumar Raychaudhuri was a physicist, known for his research in general relativity and cosmology. His most significant contribution is the eponymous Raychaudhuri equation, which demonstrates that singularities arise inevitably in general relativity and is a key ingredient in the proofs of the Penrose–Hawking singularity theorems.
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The evolute of a curve will generically have a cusp when the curve has a vertex; other, more degenerate and non-stable singularities may occur at higher-order vertices, at which the osculating circle has contact of higher order than four. Although a single generic curve will not have any higher-order vertices, they will generically occur within a one-parameter family of curves, at the curve in the family for which two ordinary vertices coalesce to form a higher vertex and then annihilate. The symmetry set of a curve has endpoints at the cusps corresponding to the vertices, and the medial axis, a subset of the symmetry set, also has its endpoints in the cusps.
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After the war Eshelby returned to Bristol University to study for a PhD and taught himself the theory of elasticity for his thesis on "Stationary and moving dislocations". He obtained his PhD in 1950 under Neville Mott. In 1951 he moved to the University of Illinois at Urbana as a Research Associate, where he stayed until 1953 when he was appointed a lecturer at the University of Birmingham , where he taught from 1953 to 1964 at the Department of Metallurgy. During this time, he worked on point defects and dislocations, developing the method of 'transformation strains' and studying the Eshelby inclusion problems for the first time, as well as the study of forces on elastic singularities.
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In his keynote speech to the 2005 Art Association of Australia & New Zealand Conference, Nicolas Bourriaud explained: > Artists are looking for a new modernity that would be based on translation: > What matters today is to translate the cultural values of cultural groups > and to connect them to the world network. This “reloading process” of > modernism according to the twenty-first-century issues could be called > altermodernism, a movement connected to the creolisation of cultures and the > fight for autonomy, but also the possibility of producing singularities in a > more and more standardized world. Altermodern can essentially be read as an artist working in a hypermodern world or with supermodern ideas or themes.
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In the modern cosmological models, most energy in the universe is in forms that have never been detected directly, namely dark energy and dark matter. There have been several controversial proposals to remove the need for these enigmatic forms of matter and energy, by modifying the laws governing gravity and the dynamics of cosmic expansion, for example modified Newtonian dynamics.For dark matter, see ; for dark energy, Beyond the challenges of quantum effects and cosmology, research on general relativity is rich with possibilities for further exploration: mathematical relativists explore the nature of singularities and the fundamental properties of Einstein's equations,See . and ever more comprehensive computer simulations of specific spacetimes (such as those describing merging black holes) are run.
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The Fallen is a fictional robot supervillain in the Transformers robot superhero franchise. According to Hasbro, he was formerly known as Megatronus Prime, and was a multiversal singularity, meaning that while he exists across the multiverse, this is no longer the case because of an event called The Shrouding, where all multiversal singularities within the Hasbro-Takara multiversal now exist as separate beings within each universe. As explained in more detail in the appropriate sections below, The Fallen has been given different origin stories in several of the different continuities in which he has appeared. Although the Fallen's origins are only suggested in his comic book appearance, they would be fully explained in Dorling Kindersley's Transformers: The Ultimate Guide.
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Shiffman received in 1964 from Massachusetts Institute of Technology (MIT) a bachelor's degree and in 1968 from the University of California, Berkeley a PhD under Shiing-Shen Chern with thesis On the removal of singularities in several complex variables. Shiffman was at MIT a C.L.E. Moore Instructor from 1968 to 1970 and at Yale University an assistant professor from 1970 to 1973. At Johns Hopkins University he was from 1973 to 1977 an associate professor and is from 1977 a full professor; he was the chair of the department of mathematics from 1990 to 1993 and again from 2012 to 2014. He has held visiting positions in the US, France, Germany, and Sweden.
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The formula as given can be applied more widely; for example if f(z) is a meromorphic function, it makes sense at all complex values of z at which f has neither a zero nor a pole. Further, at a zero or a pole the logarithmic derivative behaves in a way that is easily analysed in terms of the particular case :zn with n an integer, n ≠ 0\. The logarithmic derivative is then :n/z; and one can draw the general conclusion that for f meromorphic, the singularities of the logarithmic derivative of f are all simple poles, with residue n from a zero of order n, residue −n from a pole of order n. See argument principle.
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By eliminating variables (by any tool of elimination theory), an algebraic curve may be projected onto a plane algebraic curve, which however may introduce new singularities such as cusps or double points. A plane curve may also be completed in a curve in the projective plane: if a curve is defined by a polynomial of total degree , then simplifies to a homogeneous polynomial of degree . The values of such that are the homogeneous coordinates of the points of the completion of the curve in the projective plane and the points of the initial curve are those such that is not zero. An example is the Fermat curve , which has an affine form .
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More precisely, the extrinsic geometry is controlled by the extrinsic geometry of the isometric embedding uniquely determined by the intrinsic geometry. Shi and Tam's proof adopts a method, due to Robert Bartnik, of using parabolic partial differential equations to construct noncompact Riemannian manifolds-with-boundary of nonnegative scalar curvature and prescribed boundary behavior. By combining Bartnik's construction with the given compact manifold-with-boundary, one obtains a complete Riemannian manifold which is non-differentiable along a closed and smooth hypersurface. By using Bartnik's method to relate the geometry near infinity to the geometry of the hypersurface, and by proving a positive energy theorem in which certain singularities are allowed, Shi and Tam's result follows.
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She was made chair of Geometry at the Politecnico in 1948 and retained that post until her retirement in 1969. In addition to lecturing on descriptive geometry in the mathematics department, she taught projective geometry to the students in the architectural and engineering departments. Biggiogero's research produced a large body of work on algebraic geometry, including research on the shapes and bundles of algebraic curves, tensorial calculations, Hessian singularities of curves and the construction of the triple and quadruple planes. She and Chisini co-published several works together, including two textbooks Lezioni di geometria descrittiva (Lessons of Descriptive Geometry) published in 1941 and Esercizi di geometria descrittiva (Exercises of Descriptive Geometry), produced in 1946.
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The wavelet transform modulus maxima (WTMM) is a method for detecting the fractal dimension of a signal. More than this, the WTMM is capable of partitioning the time and scale domain of a signal into fractal dimension regions, and the method is sometimes referred to as a "mathematical microscope" due to its ability to inspect the multi-scale dimensional characteristics of a signal and possibly inform about the sources of these characteristics. The WTMM method uses continuous wavelet transform rather than Fourier transforms to detect singularities singularity – that is discontinuities, areas in the signal that are not continuous at a particular derivative. In particular, this method is useful when analyzing multifractal signals, that is, signals having multiple fractal dimensions.
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Zeeman's main contributions to mathematics were in topology, particularly in knot theory, the piecewise linear category, and dynamical systems. His 1955 thesis at the University of Cambridge described a new theory termed "dihomology", an algebraic structure associated to a topological space, containing both homology and cohomology, introducing what is now known as the Zeeman spectral sequence. This was studied by Clint McCrory in his 1972 Brandeis thesis following a suggestion of Dennis Sullivan that one make "a general study of the Zeeman spectral sequence to see how singularities in a space perturb Poincaré duality". This in turn led to the discovery of intersection homology by Robert MacPherson and Mark Goresky at Brown University where McCrory was appointed in 1974.
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The community resembles a "plural multitude" (of people) instead of the working class in traditional Marxist theory. This plural multitude is the pure constituted power and reclaims this power by searching and creating mutual understandings within the community. This strand of radical democracy challenges the traditional thinking about equality and freedom in liberal democracies by stating that individual equality can be found in the singularities within the multitude, equality overall is created by an all- inclusive multitude and freedom is created by restoring the multitude in its pure constituted power. This strand of radical democracy is often a term used to refer to the post-Marxist perspectives of Italian radicalism - for example Paolo Virno.
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A boundary Q-divisor on a variety is a Q-divisor D of the form ΣdiDi where the Di are the distinct irreducible components of D and all coefficients are rational numbers with 0≤di≤1. A logarithmic pair, or log pair for short, is a pair (X,D) consisting of a normal variety X and a boundary Q-divisor D. The log canonical divisor of a log pair (X,D) is K+D where K is the canonical divisor of X. A logarithmic 1-form on a log pair (X,D) is allowed to have logarithmic singularities of the form d log(z) = dz/z along components of the divisor given locally by z=0.
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British Rail's efforts realized profits, particularly due to the Impressionist portfolio, but the collection was liquidated because it came to be seen as an illegitimate investment area, particularly as alternative investments became available. Also, because original artworks are not fungible like stocks, they have valuation challenges not similarly affecting securities,Baumol, William J., Unnatural Value: Or Art Investment as Floating Crap Game , The American Economic Review, 76:2 (May 1986), pp. 10-14, Papers and Proceedings of the Ninety-Eighth Annual Meeting of the American Economic Association,; Journal of Arts Management and Law (now The Journal of Arts Management, Law, and Society), 15:3, 1985, pp. 47-60. . with dynamics of what Karpik calls singularities.
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In 1991 he became a full professor at the University of Cologne and since 1999 he has been a professor at the University of Regensburg. Jannsen's research deals with, among other topics, the Galois theory of algebraic number fields, the theory of motives in algebraic geometry, the Hasse principle (local-global principle), and resolution of singularities. In particular, he has done research on a cohomology theory for algebraic varieties, involving their extension in mixed motives as a development of research by Pierre Deligne, and a motivic cohomology as a development of research by Vladimir Voevodsky. In the 1980s with Kay Wingberg he completely described the absolute Galois group of p-adic number fields, i.e.
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The essential idea is that a Calabi-Yau manifold with complex dimension three should be foliated by "special Lagrangian" tori, which are certain types of three-dimensional minimal submanifolds of the six-dimensional Riemannian manifold underlying the Calabi-Yau structure. Given one three- dimensional Calabi-Yau manifold, one constructs its "mirror" by looking its torus foliation, dualizing each torus, and reconstructing the three- dimensional Calabi-Yau manifold, which will now have a new structure. The Strominger-Yau-Zaslow (SYZ) proposal, although not stated very precisely, is now understood to be overly optimistic. One must allow for various degenerations and singularities; even so, there is still no single precise form of the SYZ conjecture.
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I can prove that to build a finite time machine, you need negative energy." This result comes from Hawking's 1992 paper on the chronology protection conjecture, where he examines "the case that the causality violations appear in a finite region of spacetime without curvature singularities" and proves that "there will be a Cauchy horizon that is compactly generated and that in general contains one or more closed null geodesics which will be incomplete. One can define geometrical quantities that measure the Lorentz boost and area increase on going round these closed null geodesics. If the causality violation developed from a noncompact initial surface, the averaged weak energy condition must be violated on the Cauchy horizon.
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Gravity's Rainbow describes many varieties of sexual fetishism (including sado-masochism, coprophilia and a borderline case of tentacle erotica), and features numerous episodes of drug use, most notably cannabis but also cocaine, naturally occurring hallucinogens, and the mushroom Amanita muscaria. Gravity's Rainbow also derives much from Pynchon's background in mathematics: at one point, the geometry of garter belts is compared with that of cathedral spires, both described as mathematical singularities. Mason & Dixon explores the scientific, theological, and socio-cultural foundations of the Age of Reason while also depicting the relationships between actual historical figures and fictional characters in intricate detail and, like Gravity's Rainbow, is an archetypal example of the genre of historiographic metafiction.
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Let G=G(z,\zeta) be the associated fundamental solution of the PDE satisfied by u. In the case of straight edges, Green's representation theorem leads to Due to the orthogonality of the Legendre polynomials, for a given z=x+iy, the integrals in the above representation are Legendre expansion coefficients of certain analytic functions (written in terms of G). Hence the integrals can be computed rapidly (all at once) by expanding the functions in a Chebyshev basis (using the FFT) and then converting to a Legendre basis. This can also be used to approximate the `smooth' part of the solution after adding global singular functions to take care of corner singularities.
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In mathematics, the spectral theory of ordinary differential equations is the part of spectral theory concerned with the determination of the spectrum and eigenfunction expansion associated with a linear ordinary differential equation. In his dissertation Hermann Weyl generalized the classical Sturm–Liouville theory on a finite closed interval to second order differential operators with singularities at the endpoints of the interval, possibly semi-infinite or infinite. Unlike the classical case, the spectrum may no longer consist of just a countable set of eigenvalues, but may also contain a continuous part. In this case the eigenfunction expansion involves an integral over the continuous part with respect to a spectral measure, given by the Titchmarsh–Kodaira formula.
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Spectral theory for second order ordinary differential equations on a compact interval was developed by Jacques Charles François Sturm and Joseph Liouville in the nineteenth century and is now known as Sturm–Liouville theory. In modern language it is an application of the spectral theorem for compact operators due to David Hilbert. In his dissertation, published in 1910, Hermann Weyl extended this theory to second order ordinary differential equations with singularities at the endpoints of the interval, now allowed to be infinite or semi-infinite. He simultaneously developed a spectral theory adapted to these special operators and introduced boundary conditions in terms of his celebrated dichotomy between limit points and limit circles.
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The Raychaudhuri equation > hit the zenith of fame as it was a key tool in the hands of young > relativists like Stephen Hawking and Roger Penrose in the middle of the late > 1960s in their attempt to answer the question on the existence of space-time > singularities and to explain the theory of universe. In fact, this equation > is important as a fundamental lemma for the Penrose-Hawking singularity > theorems. There is such wide acceptability of this equation like other > notable equations in physics like ‘the Dirac equation and Schroedinger > equation’ that nobody cares about its origin or date of publication. The > Raychaudhuri equation paved the way for later research into the singularity > problem.
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Mingione received his Ph.D. in mathematics from the University of Naples Federico II in 1999 having Nicola Fusco as advisor; he is professor of mathematics at the University of Parma. He has mainly worked on regularity aspects of the Calculus of Variations, solving a few longstanding questions about the Hausdorff dimension of the singular sets of minimisers of vectorial integral functionals and the boundary singularities of solutions to nonlinear elliptic systems. This connects to the work of authors as Almgren, De Giorgi, Morrey, Giusti, who proved theorems asserting regularity of solutions outside a singular set (i.e. a closed subset of null measure) both in geometric measure theory and for variational systems of partial differential equations.
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Mircea Immanuel Mustață (born 1971 in Romania) is a Romanian-American mathematician, specializing in algebraic geometry. Mustață received from the University of Bucharest a bachelor's degree in 1995 and a master's degree in 1996 and from the University of California, Berkeley a Ph.D. in 2001 with thesis advisor David Eisenbud and thesis Singularities and Jet Schemes. As a postdoc he was at the University of Nice Sophia Antipolis (Fall 2001), at the Isaac Newton Institute (Spring 2002), and at Harvard University (2002–2004); he was from 2001 to 2004 a Clay Research Fellow. At the University of Michigan in Ann Arbor he became in 2004 an associate professor and in 2008 a full professor.
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A much anticipated feature of a theory of quantum gravity is that it will not feature singularities or event horizons and thus black holes would not be real artifacts. For example, in the fuzzball model based on string theory, the individual states of a black hole solution do not generally have an event horizon or singularity, but for a classical/semi-classical observer the statistical average of such states appears just as an ordinary black hole as deduced from general relativity. A few theoretical objects have been conjectured to match observations of astronomical black hole candidates identically or near-identically, but which function via a different mechanism. These include the gravastar, the black star, and the dark-energy star.
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The Haldane prior probability distribution Beta(0,0) is an "improper prior" because its integration (from 0 to 1) fails to strictly converge to 1 due to the singularities at each end. However, this is not an issue for computing posterior probabilities unless the sample size is very small. Furthermore, Zellner points out that on the log-odds scale, (the logit transformation ln(p/1−p)), the Haldane prior is the uniformly flat prior. The fact that a uniform prior probability on the logit transformed variable ln(p/1−p) (with domain (-∞, ∞)) is equivalent to the Haldane prior on the domain [0, 1] was pointed out by Harold Jeffreys in the first edition (1939) of his book Theory of Probability ( p. 123).
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In mathematics, a branched manifold is a generalization of a differentiable manifold which may have singularities of very restricted type and admits a well-defined tangent space at each point. A branched n-manifold is covered by n-dimensional "coordinate charts", each of which involves one or several "branches" homeomorphically projecting into the same differentiable n-disk in Rn. Branched manifolds first appeared in the dynamical systems theory, in connection with one-dimensional hyperbolic attractors constructed by Smale and were formalized by R. F. Williams in a series of papers on expanding attractors. Special cases of low dimensions are known as train tracks (n = 1) and branched surfaces (n = 2) and play prominent role in the geometry of three-manifolds after Thurston.
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The Camassa–Holm equation models breaking waves: a smooth initial profile with sufficient decay at infinity develops into either a wave that exists for all times or into a breaking wave (wave breaking being characterized by the fact that the solution remains bounded but its slope becomes unbounded in finite time). The fact that the equations admits solutions of this type was discovered by Camassa and Holm and these considerations were subsequently put on a firm mathematical basis. It is known that the only way singularities can occur in solutions is in the form of breaking waves., Moreover, from the knowledge of a smooth initial profile it is possible to predict (via a necessary and sufficient condition) whether wave breaking occurs or not.
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In mathematics, differential forms on a Riemann surface are an important special case of the general theory of differential forms on smooth manifolds, distinguished by the fact that the conformal structure on the Riemann surface intrinsically defines a Hodge star operator on 1-forms (or differentials) without specifying a Riemannian metric. This allows the use of Hilbert space techniques for studying function theory on the Riemann surface and in particular for the construction of harmonic and holomorphic differentials with prescribed singularities. These methods were first used by in his variational approach to the Dirichlet principle, making rigorous the arguments proposed by Riemann. Later found a direct approach using his method of orthogonal projection, a precursor of the modern theory of elliptic differential operators and Sobolev spaces.
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In algebraic geometry, a Zariski–Riemann space or Zariski space of a subring k of a field K is a locally ringed space whose points are valuation rings containing k and contained in K. They generalize the Riemann surface of a complex curve. Zariski–Riemann spaces were introduced by who (rather confusingly) called them Riemann manifolds or Riemann surfaces. They were named Zariski–Riemann spaces after Oscar Zariski and Bernhard Riemann by who used them to show that algebraic varieties can be embedded in complete ones. Local uniformization (proved in characteristic 0 by Zariski) can be interpreted as saying that the Zariski–Riemann space of a variety is nonsingular in some sense, so is a sort of rather weak resolution of singularities.
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Like Gaussian quadrature, Tanh-Sinh quadrature is well suited for arbitrary-precision integration, where an accuracy of hundreds or even thousands of digits is desired. The convergence is exponential (in the discretization sense) for sufficiently well-behaved integrands: doubling the number of evaluation points roughly doubles the number of correct digits. Tanh-Sinh quadrature is not as efficient as Gaussian quadrature for smooth integrands, but unlike Gaussian quadrature, tends to work equally well with integrands having singularities or infinite derivatives at one or both endpoints of the integration interval as already noted. Furthermore, Tanh-Sinh quadrature can be implemented in a progressive manner, with the step size halved each time the rule level is raised, and reusing the function values calculated on previous levels.
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Peter, from this action, deduces that he is from the parallel universe, and furious at Walter for hiding this information, leaves on his own. While hiding in the Pacific Northwest, Peter meets Mr. Secretary - Walternate, his true father, who offers to take him back to the parallel universe, which Peter accepts. Olivia and Walter are alerted by September that Walternate plans to use Peter to initiate the operation of a strange device that threatens to destroy the prime universe, and the two launch a rescue attempt. In the parallel universe, they find that it suffers from singularities caused by Walter's crossing in 1985, forcing Walternate's Fringe team to use an amber-like substance to surround and quarantine such areas, regardless of innocent lives trapped within.
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Chandler is the recipient of the 2016 Richard Wilbur Award for her book The Frangible Hour, University of Evansville Press. She also won the 2010 Howard Nemerov Sonnet Award for her poem "Coming to Terms", the final judge being A.E. Stallings. She was also a finalist for the Nemerov award in 2008 ("Missing"), 2009 ("Singularities"), 2012 ("Composure"), 2013 ("The Watchers at Punta Ballena, Uruguay"), 2014 ("Afterwords"), 2015 ("Oleka"), 2016 ("Family at Sunset Beach, California"), and 2017 ("Celebration"), and won The Lyric Quarterly Prize in 2004 ("Franconia") and the Leslie Mellichamp Award in 2015 ("Chiaroscuro"). Eight of her poems, including "66", "Body of Evidence" and "Writ" received nine Pushcart Prize nominations, and her poem, "66" was a finalist for the Best of the Net award in 2006.
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One can simply pick arbitrary values for one of the two sets of unknowns, use them to estimate the second set, then use these new values to find a better estimate of the first set, and then keep alternating between the two until the resulting values both converge to fixed points. It's not obvious that this will work, but it can be proven that in this context it does, and that the derivative of the likelihood is (arbitrarily close to) zero at that point, which in turn means that the point is either a maximum or a saddle point. In general, multiple maxima may occur, with no guarantee that the global maximum will be found. Some likelihoods also have singularities in them, i.e.
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Other writers who have contributed to the discipline's emergence include Eric Kluitenberg, Anne Friedberg, Friedrich Kittler, and Jonathan Crary. New media theorist Jussi Parikka defines media archaeology as follows: :Media archaeology exists somewhere between materialist media theories and the insistence on the value of the obsolete and forgotten through new cultural histories that have emerged since the 1980s. I see media archaeology as a theoretically refined analysis of the historical layers of media in their singularity—a conceptual and practical exercise in carving out the aesthetic, cultural, and political singularities of media. And it's much more than paying theoretical attention to the intensive relations between new and old media mediated through concrete and conceptual archives; increasingly, media archaeology is a method for doing media design and art.
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However the lines do usually have well determined directions at essential singularities of the function, and there is sometimes a natural choice of these lines as follows. The asymptotic expansion of a function is often given by a linear combination of functions of the form f(z)e±g(z) for functions f and g. The Stokes lines can then be taken as the zeros of the imaginary part of g, and the anti-Stokes lines as the zeros of the real part of g. (This is not quite canonical, because one can add a constant to g, changing the lines.) If the lines are defined like this then they are orthogonal where they meet, unless g has a multiple zero.
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In real analysis, this example shows that there are infinitely differentiable functions whose Taylor series are not equal to even if they converge. By contrast, the holomorphic functions studied in complex analysis always possess a convergent Taylor series, and even the Taylor series of meromorphic functions, which might have singularities, never converge to a value different from the function itself. The complex function , however, does not approach 0 when approaches 0 along the imaginary axis, so it is not continuous in the complex plane and its Taylor series is undefined at 0. More generally, every sequence of real or complex numbers can appear as coefficients in the Taylor series of an infinitely differentiable function defined on the real line, a consequence of Borel's lemma.
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The HAM is an analytic approximation method designed for the computer era with the goal of "computing with functions instead of numbers." In conjunction with a computer algebra system such as Mathematica or Maple, one can gain analytic approximations of a highly nonlinear problem to arbitrarily high order by means of the HAM in only a few seconds. Inspired by the recent successful applications of the HAM in different fields, a Mathematica package based on the HAM, called BVPh, has been made available online for solving nonlinear boundary-value problems . BVPh is a solver package for highly nonlinear ODEs with singularities, multiple solutions, and multipoint boundary conditions in either a finite or an infinite interval, and includes support for certain types of nonlinear PDEs.
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In the years that followed, not only did computers become more powerful, but also various research groups developed alternate techniques to improve the efficiency of the calculations. With respect to black hole simulations specifically, two techniques were devised to avoid problems associated with the existence of physical singularities in the solutions to the equations: (1) Excision, and (2) the "puncture" method. In addition the Lazarus group developed techniques for using early results from a short-lived simulation solving the nonlinear ADM equations, in order to provide initial data for a more stable code based on linearized equations derived from perturbation theory. More generally, adaptive mesh refinement techniques, already used in computational fluid dynamics were introduced to the field of numerical relativity.
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The rational maps play a role analogous to the meromorphic functions of complex analysis, and indeed in the case of compact Riemann surfaces the two concepts coincide. If V is now taken to be of dimension 1, we get a picture of a typical algebraic curve C presented 'over' P1(K). Assuming C is non-singular (which is no loss of generality starting with K(C)), it can be shown that such a rational map from C to P1(K) will in fact be everywhere defined. (That is not the case if there are singularities, since for example a double point where a curve crosses itself may give an indeterminate result after a rational map.) This gives a picture in which the main geometric feature is ramification.
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This allows defining surfaces in spaces of dimension higher than three, and even abstract surfaces, which are not contained in any other space. On the other hand, this excludes surfaces that have singularities, such as the vertex of a conical surface or points where a surface crosses itself. In classical geometry, a surface is generally defined as a locus of a point or a line. For example, a sphere is the locus of a point which is at a given distance of a fixed point, called the center; a conical surface is the locus of a line passing through a fixed point and crossing a curve; a surface of revolution is the locus of a curve rotating around a line.
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Using the formula, Varchenko constructed a counterexample to V. I. Arnold's semicontinuity conjecture that the brightness of light at a point on a caustic is not less than the brightness at the neighboring points. Varchenko formulated a conjecture on the semicontinuity of the spectrum of a critical point under deformations of the critical point and proved it for deformations of low weight of quasi- homogeneous singularities. Using the semicontinuity, Varchenko gave an estimate from above for the number of singular points of a projective hypersurface of given degree and dimension. Varchenko introduced the asymptotic mixed Hodge structure on the cohomology, vanishing at a critical point of a function, by studying asymptotics of integrals of holomorphic differential forms over families of vanishing cycles.
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However, this is considered a last resort, as destroying the bridge will affect the healing of the singularities in the parallel universe. When a second simultaneous set of earthquakes occur, the parallel universe's version of Nick Lane (David Call) approaches Agent Lincoln Lee (Seth Gabel) of the prime universe, believing him to be the parallel universe's version of Lee. Lee feigns familiarity, learning that Nick had visions of being at the epicenter of the quake before it began. When Lee reports this to Olivia Dunham (Anna Torv) in the prime universe, she suddenly recalls her Cortexiphan trials including fellow subject Nick Lane, and with the team's help, identifies that other Cortexiphan subjects are the epicenter of these quakes, linking to their parallel universe versions to achieve synchronization.
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He became a well-known scientific figure in the 1970s, and was the subject of a short documentary film completed shortly before his death."Documentary on Amal Kumar Raychaudhuri, the renowned theoretical physicist from Kolkata" Dipayan Pal wrote of Raychaudhuri for Science Reporter (CSIR, NISCAIR) in 2018: > In general relativity, the Raychaudhuri equation plays a significant role to > explain the space-time singularities and gravitational focusing properties > in cosmology. He aimed to address the fundamental question of singularity in > the most simple and general form with no reference to any symmetry and any > specific property of space-time and energy distribution. The first mention > of the term ‘Raychaudhuri Equation’ appeared in a research paper published > in 1965 by George F.R. Ellis and Stephen Hawking.
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Xia's construction proving the Painlevé conjecture Xia received in 1982 from Nanjing University a bachelor's degree in astronomy and in 1988 a PhD in mathematics from Northwestern University with thesis advisor Donald G. Saari and thesis The Existence of the Non-Collision Singularities. From 1988 to 1990 Xia was an assistant professor at Harvard University and from 1990 to 1994 an associate professor at Georgia Institute of Technology (and Institute Fellow). In 1994 he became a full professor at Northwestern University and since 2000 he has been there the Arthur and Gladys Pancoe Professor of Mathematics. His research deals with celestial mechanics, dynamical systems, Hamiltonian dynamics, and ergodic theory. In his dissertation he solved the Painlevé conjecture, a long- standing problem posed in 1895 by Paul Painlevé.
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The reason why until now the geometry of curves of constant width could not be put to practical use in the gear design is that no conventional gear structure with the regular rolling on of the gears would permit the exact rolling-on of the singularities. The RKMs solve this problem by introducing the inversely conjugated gear system, which makes it possible to have singular trajectories of the axes of rolling-on gears and, thus, allows the transfer of the angular momentum during the passage of the piston through its stop positions.RKM's - Rotating Piston Machines - Scientific and Technical Comments In simple words, the gear mechanism introduces corrections to the piston's motion, correcting the axis of rotation as it leaves the stop positions, so as to create a smooth motion.
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In 2016 he was announced as a winner of the ICTP Ramanujan Prize for that year, "in recognition of Xu's outstanding works in algebraic geometry, notably in the area of birational geometry, including works both on log canonical pairs and on Q-Fano varieties, and on the topology of singularities and their dual complexes." He is one of five winners of the 2019 New Horizons Prize for Early-Career Achievement in Mathematics, associated with the Breakthrough Prize in Mathematics for his research in the minimal model program and applications to the moduli of algebraic varieties. He was elected as a Fellow of the American Mathematical Society in the 2020 Class, for "contributions to algebraic geometry, in particular the minimal model program and the K-stability of Fano varieties".
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Traces of representations of the corresponding curves on the surfaces generate a Poisson algebra, whose Lie bracket has a topological description in terms of the intersections of curves. Furthermore, the Hamiltonian vector fields of these trace functions define flows generalizing the Fenchel–Nielsen flows on Teichmüller space. This symplectic structure is invariant under the natural action of the mapping class group, and using the relationship between Dehn twists and the generalized Fenchel–Nielsen flows, he proved the ergodicity of the action of the mapping class group on the SU(2)-character variety with respect to symplectic Lebesgue measure. Following suggestions of Pierre Deligne, he and John Millson proved that the variety of representations of the fundamental group of a compact Kähler manifold has singularities defined by systems of homogeneous quadratic equations.
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The Schwarzschild solution is named in honour of Karl Schwarzschild, who found the exact solution in 1915 and published it in January 1916, For a translation, see a little more than a month after the publication of Einstein's theory of general relativity. It was the first exact solution of the Einstein field equations other than the trivial flat space solution. Schwarzschild died shortly after his paper was published, as a result of a disease he contracted while serving in the German army during World War I. Johannes Droste in 1916 independently produced the same solution as Schwarzschild, using a simpler, more direct derivation. In the early years of general relativity there was a lot of confusion about the nature of the singularities found in the Schwarzschild and other solutions of the Einstein field equations.
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Much of his work on stratified sets was developed so as to understand the notion of topologically stable maps, and to eventually prove the result that the set of topologically stable mappings between two smooth manifolds is a dense set. Thom's lectures on the stability of differentiable mappings, given at the University of Bonn in 1960, were written up by Harold Levine and published in the proceedings of a year long symposium on singularities at Liverpool University during 1969-70, edited by C. T. C. Wall. The proof of the density of topologically stable mappings was completed by John Mather in 1970, based on the ideas developed by Thom in the previous ten years. A coherent detailed account was published in 1976 by Christopher Gibson, Klaus Wirthmüller, Andrew du Plessis, and Eduard Looijenga.
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Lev Davidovich Landau (; 22 January 1908 – 1 April 1968) was a Soviet physicist who made fundamental contributions to many areas of theoretical physics. His accomplishments include the independent co-discovery of the density matrix method English translation reprinted in: in quantum mechanics (alongside John von Neumann), the quantum mechanical theory of diamagnetism, the theory of superfluidity, the theory of second-order phase transitions, the Ginzburg–Landau theory of superconductivity, the theory of Fermi liquid, the explanation of Landau damping in plasma physics, the Landau pole in quantum electrodynamics, the two-component theory of neutrinos, and Landau's equations for S matrix singularities. He received the 1962 Nobel Prize in Physics for his development of a mathematical theory of superfluidity that accounts for the properties of liquid helium II at a temperature below ().
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The second paper of Kazhdan and Lusztig established a geometric setting for definition of Kazhdan–Lusztig polynomials, namely, the geometry of singularities of Schubert varieties in the flag variety. Much of the later work of Lusztig explored analogues of Kazhdan–Lusztig polynomials in the context of other natural singular algebraic varieties arising in representation theory, in particular, closures of nilpotent orbits and quiver varieties. It turned out that the representation theory of quantum groups, modular Lie algebras and affine Hecke algebras are all tightly controlled by appropriate analogues of Kazhdan–Lusztig polynomials. They admit an elementary description, but the deeper properties of these polynomials necessary for representation theory follow from sophisticated techniques of modern algebraic geometry and homological algebra, such as the use of intersection cohomology, perverse sheaves and Beilinson–Bernstein–Deligne decomposition.
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Clemens received in 1961 his bachelor's degree from College of the Holy Cross and in 1966 his Ph.D. from the University of California, Berkeley under Phillip Griffiths with thesis Picard–Lefschetz Theorem for Families of Algebraic Varieties Acquiring Certain Singularities. In 1970 he became an assistant professor at Columbia University and went on to become an associate professor before leaving in 1975 to become an associate professor at the University of Utah where he became a full professor in 1976 and a Distinguished Professor in 2001. In 2002 he left Utah to become a professor of mathematics and mathematics education at the Ohio State University. Clemens was a visiting scholar at the Institute for Advanced Study from September 1968 to March 1970 and from September 2001 to June 2003.
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Sachrajda has made contributions to the development of quantum chromodynamics (QCD), the theory of the strong interactions. His work on the factorisation of mass singularities led to the perturbative prediction of many physical quantities in strong interaction physics. With others, he pioneered the calculation of higher-order corrections to deep inelastic structure functions, which has led to detailed tests of QCD and the determination of the momentum distribution of quarks and gluons inside the proton. He has played a leading role in the development of the lattice formulation of QCD into a quantitative non- perturbative technique and in the use of this formulation to compute, from first principles, a number of physical quantities, including deep inelastic structure functions, electromagnetic form factors of hadrons and semi-leptonic decays of charmed mesons.
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This allows high precision and high speed of movements, and motivates the use of parallel manipulators in flight simulators (high speed with rather large masses) and electrostatic or magnetic lenses in particle accelerators (very high precision in positioning large masses). five-bar parallel robot Sketchy, a portrait-drawing delta robot A drawback of parallel manipulators, in comparison to serial manipulators, is their limited workspace. As for serial manipulators, the workspace is limited by the geometrical and mechanical limits of the design (collisions between legs maximal and minimal lengths of the legs). The workspace is also limited by the existence of singularities, which are positions where, for some trajectories of the movement, the variation of the lengths of the legs is infinitely smaller than the variation of the position.
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Examples of large human "buy-ins" into technology include the computer revolution, as well as massive government projects like the Manhattan Project and the Human Genome Project. The foundation organizing the Methuselah Mouse Prize believes aging research could be the subject of such a massive project if substantial progress is made in slowing or reversing cellular aging in mice. Both Theodore Modis and Jonathan Huebner have argued—each from different perspectives—that the rate of technological innovation has not only ceased to rise, but is actually now declining. In fact, "technological singularity" is just one of a few singularities detected through the analysis of a number of characteristics of the World System development, for example, with respect to the world population, world GDP, and some other economic indices.e.g.
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A proponent of this theory can thus ask the direct realist why he or she thinks it is necessary to move to taking the imagining of singularity for real when there is no practical difference in the outcome in action. Therefore, although there are selections from our sensory fields which for the time being we treat as if they were objects, they are only provisional, open to corrections at any time, and, hence, far from being direct representations of pre-existing singularities, they retain an experimental character. Virtual constructs or no, they remain, however, selections that are causally linked to the real and can surprise us at any time—which removes any danger of solipsism in this theory. This approach dovetails with the philosophy known as social constructivism.
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Titans act as the "masters" of the techno-organic beings known as Phalanx. They are singularities of consciousness so vast and dense, that they have caved in on their own combined intelligence to form black holes. They are also singular intellects and not a collective or a group. A single black hole is a Titan intelligence, up to five black holes confined to a galactic cluster or a dense collection of stacked galaxies becomes a Stronghold, warring factions seeking to actively destroy or absorb other Strongholds in order to achieve Dominion status which is when 10 or more of these incomprehensible cosmologically-scaled beings act in unison to control a particular sector or sectors of space in both the area and epochs of time, becoming galaxy-spanning, interconnected tears in the fabric of existence.
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Mas dissolved the Catalan Parliament, called for new elections, and promised to conduct a referendum on independence within the next four years. Rajoy's government declared that they would use all "legal instruments"—current legislation requires the central executive government or the Congress of Deputies to call for or sanction a binding referendum— to block any such attempt. The Spanish Socialist Workers' Party and its counterpart in Catalonia proposed to reopen the debate on the territorial organization of Spain, changing the constitution to create a true federal system to "better reflect the singularities" of Catalonia, as well as to modify the current taxation system. On Friday 27 of October 2017 the Catalan Parliament voted the independence of Catalonia; the result was 70 in favor, 10 against, 2 neither, with 53 representatives not present in protest.
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The aim of the exhibition is to trace out a map of the main results of the artistic research conducted in Italy during the two decades prior to 2008 as well as of the experiences considered representative of that period, devoting particular attention to the artists in the middle of their careers and the younger ones, indicative of the possible future evolution of the Italian artistic panorama. The exhibition registers the different tendencies of contemporary art in which conceptual art, minimalism, and the various tendencies of the pictorial and photographic image are confronted in a further attempt to identify the possible singularities of the Italian situation in the international system of art. 15th Rome Quadriennale, press release URL retrieved 24 May 2009. Accessed 2009-05-24. Archived 2009-05-27.
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Eva Silverstein (born October 24, 1970) is an American theoretical physicist, cosmologist, and string theorist. She is best known for her work on early universe cosmology, developing the structure of inflation and its range of signatures, as well as extensive contributions to string theory and gravitational physics. Her early work included control of tachyon condensation in string theory and resulting resolution of some spacetime singularities (with Joseph Polchinski and others). Other significant research contributions include the construction of the first models of dark energy in string theory, some basic extensions of the AdS/CFT correspondence to more realistic field theories (with Shamit Kachru), as well as the discovery of a predictive new mechanism for cosmic inflation involving D-brane dynamics (with David Tong) which helped motivate more systematic analyses of primordial non-Gaussianity.
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The deputy prime minister of Spain, Soraya Sáenz de Santamaría, declared that the central government would exercise all "legal instruments" (current legislation requires the executive government or the Congress of Deputies to call for or sanction a binding referendum) to block any such attempt. The leaders of the opposition, in the Catalan Parliament, in the Cortes Generales, and from the Socialist Party, do not support Catalan secession, but instead favor changing the constitution to modify the current taxation system and to create a true federal system in Spain, to "better reflect the singularities" of Catalonia. In December 2012, an opposing rally was organised by the Partido Popular and Ciutadans, which drew 30,000-160,000 people in one of Barcelona's main squares under a large flag of Spain and Catalonia.
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In March 2019, Uhlenbeck became the first woman to receive the Abel Prize, with the award committee citing the decision for "her pioneering achievements in geometric partial differential equations, gauge theory and integrable systems, and for the fundamental impact of her work on analysis, geometry and mathematical physics." Hans Munthe-Kaas, who chairs the award committee, stated that "Her theories have revolutionised our understanding of minimal surfaces, such as more general minimisation problems in higher dimensions". Uhlenbeck also won the National Medal of Science in 2000,. and the Leroy P. Steele Prize for Seminal Contribution to Research of the American Mathematical Society in 2007, "for her foundational contributions in analytic aspects of mathematical gauge theory", based on her 1982 papers "Removable singularities in Yang–Mills fields" and "Connections with bounds on curvature".
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In terms of the completion of some of the underlying Grothendieck program of research, he defined absolute Hodge cycles, as a surrogate for the missing and still largely conjectural theory of motives. This idea allows one to get around the lack of knowledge of the Hodge conjecture, for some applications. The theory of mixed Hodge structures, a powerful tool in algebraic geometry that generalizes classical Hodge theory, was created by applying weight filtration, Hironaka's resolution of singularities and other methods, which he then used it to prove the Weil conjectures. He reworked the Tannakian category theory in his 1990 paper for the "Grothendieck Festschrift", employing Beck's theorem – the Tannakian category concept being the categorical expression of the linearity of the theory of motives as the ultimate Weil cohomology.
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On further talks with Captain Lee, Agent Lee is surprised to find their backgrounds were exactly the same, deviating only when Captain Lee decided to become more assertive in more recent years. As the investigation continues, Agent Lee learns of how amber was used to quarantine areas of the parallel universe to protect it from singularities, but with the introduction of the bridge through The Machine, these areas are healing themselves, allowing the amber to be removed. In one such region, two workers discover nearly two dozen bodies in various states of decay in a church, and these are quickly identified as the missing criminals. The Fringe division discover telltale signs of shapeshifter extraction marks, though more advanced than the means used by the initial models created by Walternate (Walter's doppelganger).
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In 1977, Schwartz pointed out that the hypercolumn model of Hubel and Weisel implied the existence of a periodic vortex like pattern of orientation singularities across the surface of visual cortex. Specifically, the angular part of the complex logarithm function, viewed as a spatial map provided a possible explanation of the hypercolumn structure, which in current language is termed the "pinwheel" structure of visual cortex . In 1990, together with Alan Rojer, Schwartz showed that such "vortex" or "pinwheel" structures, together with the associated ocular dominance column pattern in cortex, could be caused by spatial filtering of random vector or scalar spatial noise, respectively. Prior to this work, most modeling of cortical columns was in terms of somewhat opaque and clumsy "neural network" models—bandpass-filtered noise quickly became a standard modeling technique for cortical columnar structure.
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Johnson lived in Montserrat for 10 years. Johnson's research focus is in superstring theory and particle physics, specifically related to strongly coupled phenomena.Strings Link the Ultracold With the Superhot Science News, 25 April 2009,String Theory Officially Useful, May Not Represent Reality Ars Technica, 17 February 2009 He has previously worked at the Kavli Institute for Theoretical Physics at the University of California, Santa Barbara, the Institute for Advanced Study and Princeton University. He received the 2005 Maxwell Medal and Prize from the Institute of Physics, "For his outstanding contribution to string theory, quantum gravity and its interface with strongly coupled field theory, in particular for his work on understanding the censorship of singularities and the thermodynamic properties of quantum spacetime."Recipients of the Maxwell Medal and Prize Institute of PhysicsFaces and Places Cern Courier 4 October 2004U.
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It encompasses the Xeelee Sequence, which of nine novels (including the Destiny's Children trilogy and Reboot Duology), plus three volumes collecting the 52 short pieces (short stories and novellas) in the series, all of which fit into a single timeline stretching from the Big Bang singularity of the past to his Timelike Infinity singularity of the future. These stories begin in the present day and end when the Milky Way galaxy collides with Andromeda five billion years in the future. The central narrative is that of Humanity rising and evolving to become the second most powerful race in the universe, next to the god-like Xeelee. Character development tends to take second place to the depiction of advanced theories and ideas, such as the true nature of the Great Attractor, naked singularities and the great battle between Baryonic and Dark Matter lifeforms.
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A lemma of Brian White shows that the minimum area double bubble must be a surface of revolution. For, if not, it would be possible to find two orthogonal planes that bisect both volumes, replace surfaces in two of the four quadrants by the reflections of the surfaces in the other quadrants, and then smooth the singularities at the reflection planes, reducing the total area. Based on this lemma, Michael Hutchings was able to restrict the possible shapes of non-standard optimal double bubbles, to consist of layers of toroidal tubes.. Additionally, Hutchings showed that the number of toroids in a non-standard but minimizing double bubble could be bounded by a function of the two volumes. In particular, for two equal volumes, the only possible nonstandard double bubble consists of a single central bubble with a single toroid around its equator.
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In the Star Trek universe, wormhole theory states that if a section in the fabric of spacetime joins together with another section of spacetime, a direct connection can be made between the two, allowing speedy travel between the two (normally unrelated) spacetime coordinates. Black holes are one such way of stretching the fabric of spacetime; so it's theoretically possible to create wormholes using a pair of singularities, at least in the fictional universe of Star Trek. The NASA Web site has a somewhat dated article called "The Science of Star Trek", by physicist David Allen Batchelor (5 May 2009), which considers some of the implementations in Star Trek. He says it's "the only science fiction series crafted with such respect for real science and intelligent writing", with some "imaginary science" mixed in; and considers it to be the "only science fiction series that many scientists watch regularly", like himself.
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The theoretical activity is being developed in parallel to the experimental one, a fundamental theme being the quantum treatment of phenomena such asthose observed in the study of elementary chemical processes, where the motion of nuclei enters into play: their behaviour is at the borderline of classical mechanics (semi-classical regime). On this theme (quantum mechanics in the short-wave limit) he has studied non- adiabatic processes, the role of singularities (catastrophes), the chaotic regime, and has also contributed to the historical – epistemological debate. His major theoretical effort has concerned the formulation and implementation of the treatment of the dynamics of processes involving few bodies, that encounter as a quantum mechanical principal obstacle the necessity to explicitly treat the coupling of angular momenta and spin, with electronic, rotational and orbital momenta. He has contributed in this area to the introduction of hyperspherical coordinates and harmonics, developing analytical tools and original algorithms.
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Among the highlights of this theory is the formula for the Kolmogorov-Sinai entropy of the system (also known as Pesin entropy formula). His main article on this topic "Characteristic Lyapunov exponents and smooth ergodic theory" (Russian Mathematical Surveys, 1977) has a very high number of citations in mathematical literature and beyond (in physics, biology, etc.). 3) Pesin's later work on non-uniform hyperbolicity includes establishing presence of systems with non-zero Lyapunov exponents on any manifold; a proof of the Eckmann—Ruelle conjecture; the study of the essential coexistence phenomenon of regular and chaotic dynamics; constructions of SRB measures for hyperbolic attractors with singularities, partially hyperbolic and non-uniformly hyperbolic attractors; and effecting thermodynamic formalism for some classes of non-uniformly hyperbolic dynamical systems. 4) Pesin designed a construction (known also as the Caratheodory-Pesin construction) that allows one to introduce and study various dimension-type characteristics of dynamical systems.
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In this way, Beltrami attempted to demonstrate that two-dimensional non-Euclidean geometry is as valid as the Euclidean geometry of the space, and in particular, that Euclid's parallel postulate could not be derived from the other axioms of Euclidean geometry. It is often stated that this proof was incomplete due to the singularities of the pseudosphere, which means that geodesics could not be extended indefinitely. However, John Stillwell remarks that Beltrami must have been well aware of this difficulty, which is also manifested by the fact that the pseudosphere is topologically a cylinder, and not a plane, and he spent a part of his memoir designing a way around it. By a suitable choice of coordinates, Beltrami showed how the metric on the pseudosphere can be transferred to the unit disk and that the singularity of the pseudosphere corresponds to a horocycle on the non-Euclidean plane.
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Therefore, fusion reactions are possible to be obtained in ultra-miniaturized devices (driven by generators of 0.1J for example), as well as they are obtained in bigger devices (driven by generators of 1MJ). However, the stability of the plasma pinch highly depends on the size and energy of the device. A rich plasma phenomenology it has been observed in the table-top plasma focus devices developed by Soto´s group: filamentary structures, toroidal singularities, plasma bursts and plasma jets generations. In addition, possible applications are explored using these kind of small plasma devices: development of portable generator as non-radioactive sources of neutrons and x-rays for field applications, pulsed radiation applied to biological studies, plasma focus as neutron source for nuclear fusion-fission hybrid reactors, and the use of plasma focus devices as plasma accelerators for studies of materials under intense fusion-relevant pulses.
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Penrose diagrams are often used to illustrate the hypothetical Einstein–Rosen bridge connecting two separate universes in the maximally extended Schwarzschild black hole solution. The precursors to the Penrose diagrams were Kruskal-Szekeres diagrams. (The Penrose diagram adds to Kruskal and Szekeres' diagram the conformal crunching of the regions of flat space-time far from the hole.) These introduced the method of aligning the event horizon into past and future horizons oriented at 45° angles (since one would need to travel faster than light to cross from the Schwarzschild radius back into flat spacetime); and splitting the singularity into past and future horizontally-oriented lines (since the singularity "cuts off" all paths into the future once one enters the hole). The Einstein–Rosen bridge closes off (forming "future" singularities) so rapidly that passage between the two asymptotically flat exterior regions would require faster-than-light velocity, and is therefore impossible.
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The United States in the parallel universe only consists of 48 states; while there exists a North and South Texas, other single states appear to take the place of two separate states in the prime universe, such as both North and South Carolina being the single "Carolina" in the parallel universe. Furthermore, much of western California has been lost, suggesting that a large earthquake along the San Andreas Fault has caused much of the coastal region to sink below sea level. Other effects on the global scale have caused sheep to become extinct, and made coffee and avocados valuable rarities. The singularities that plague the parallel universe can lead to destructive voids; to prevent these, the Fringe division there uses a fast-setting amber-like substance to prevent weakened areas from becoming destructive, but with no regard for innocents that may be trapped within it.
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Accordingly, there has as yet been no experimental verification of the prediction. A recent analysis argued that the Scharnhorst effect cannot be used to send information backwards in time with a single set of plates since the plates' rest frame would define a "preferred frame" for FTL signalling. However, with multiple pairs of plates in motion relative to one another the authors noted that they had no arguments that could "guarantee the total absence of causality violations", and invoked Hawking's speculative chronology protection conjecture which suggests that feedback loops of virtual particles would create "uncontrollable singularities in the renormalized quantum stress-energy" on the boundary of any potential time machine, and thus would require a theory of quantum gravity to fully analyze. Other authors argue that Scharnhorst's original analysis, which seemed to show the possibility of faster-than-c signals, involved approximations which may be incorrect, so that it is not clear whether this effect could actually increase signal speed at all.
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In the years that followed he studied the theory of conversion groups and analytical mechanics in Leipzig, and differential equations in Göttingen. In 1891 he was awarded a PhD (under M. Sophius Lie) in Leipzig for his thesis on the applications of group conversion theory to differential geometry. In 1892 he became a lecturer at the Polytechnic Higher School of Lwów where he taught mathematics and, in 1893, assumed the Chair of Mechanical Science. In 1893, Żorawski received a doctorate in mathematics from Jagiellonian University in Kraków, and in 1895 he traveled to Berlin to study higher level geodesy. He later returned to Kraków where, he was named assistant professor and, in 1898, full professor of mathematics at Jagiellonian where he taught higher analysis, geometry (analytic, differential and projective), theory of algebraic curves and theory of singularities. In 1900 he was elected a member of the Academy of Learning (from 1919 Polish Academy of Learning) in Kraków.
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In some cases Hamilton was able to show that this works; for example, his original breakthrough was to show that if the Riemannian manifold has positive Ricci curvature everywhere, then the above procedure can only be followed for a bounded interval of parameter values, t\in [0,T) with T<\infty, and more significantly, that there are numbers c_t such that as t earrow T, the Riemannian metrics c_tg(t) smoothly converge to one of constant positive curvature. According to classical Riemannian geometry, the only simply-connected compact manifold which can support a Riemannian metric of constant positive curvature is the sphere. So, in effect, Hamilton showed a special case of the Poincaré conjecture: if a compact simply-connected 3-manifold supports a Riemannian metric of positive Ricci curvature, then it must be diffeomorphic to the 3-sphere. If, instead, one only has an arbitrary Riemannian metric, the Ricci flow equations must lead to more complicated singularities.
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Ideally one would like to describe the (moduli) space of all solutions explicitly, and for some very special PDEs this is possible. (In general this is a hopeless problem: it is unlikely that there is any useful description of all solutions of the Navier–Stokes equation for example, as this would involve describing all possible fluid motions.) If the equation has a very large symmetry group, then one is usually only interested in the moduli space of solutions modulo the symmetry group, and this is sometimes a finite- dimensional compact manifold, possibly with singularities; for example, this happens in the case of the Seiberg–Witten equations. A slightly more complicated case is the self dual Yang–Mills equations, when the moduli space is finite-dimensional but not necessarily compact, though it can often be compactified explicitly. Another case when one can sometimes hope to describe all solutions is the case of completely integrable models, when solutions are sometimes a sort of superposition of solitons; this happens e.g.
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I can prove that to build a finite time machine, you need negative energy." Hawking's argument appears in his 1992 paper on the chronology protection conjecture (though the argument is distinct from the conjecture itself, since the argument asserts that classical general relativity predicts a finite region containing closed timelike curves can only be created if there is a violation of the weak energy condition in that region, whereas the conjecture predicts that closed timelike curves will prove to be impossible in a future theory of quantum gravity which replaces general relativity). In the paper, he examines "the case that the causality violations appear in a finite region of spacetime without curvature singularities" and proves that "[t]here will be a Cauchy horizon that is compactly generated and that in general contains one or more closed null geodesics which will be incomplete. One can define geometrical quantities that measure the Lorentz boost and area increase on going round these closed null geodesics.
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The relation of concepts and cases in Rabinow’s work distinguishes itself from the more common mode of social science work predicated on using examples to test general theories or philosophical practice that seeks analytic clarity about universals or general (often highly abstract) cases. In contrast, Rabinow argues that work on concepts opens up and orients inquiry into the concrete features of distinctive cases, whereas the use of ostensibly timeless theory or universal concepts are unlikely to be very helpful in drawing attention to particularities and singularities. Given this goal, such traditional approaches can function as a real impediment to inquiry. Rabinow defines concept work as “constructing, elaborating and testing a conceptual inventory as well as specifying and experimenting with multi-dimensional diagnostic and analytic frames.”Paul Rabinow and Gaymon Bennett, “Toward Synthetic Anthropos: Remediating Concepts” online at www.anthropos-lab.net/documents In that sense, Rabinow’s work continues with appropriate modifications a social scientific tradition stretching from Max Weber through Clifford Geertz.
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In addition, highly blue-shifted light rays (called a "blue sheet") would make it impossible for anyone to pass through. Penrose Diagrams of various black hole solutions The maximally extended solution does not describe a typical black hole created from the collapse of a star, as the surface of the collapsed star replaces the sector of the solution containing the past-oriented "white hole" geometry and other universe. While the basic space-like passage of a static black hole cannot be traversed, the Penrose diagrams for solutions representing rotating and/or electrically charged black holes illustrate these solutions' inner event horizons (lying in the future) and vertically oriented singularities, which open up what is known as a time-like "wormhole" allowing passage into future universes. In the case of the rotating hole, there is also a "negative" universe entered through a ring-shaped singularity (still portrayed as a line in the diagram) that can be passed through if entering the hole close to its axis of rotation.
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In 1982, William Thurston published his renowned geometrization conjecture, asserting that in an arbitrary closed 3-manifold, one could find embedded two-dimensional spheres and tori which disconnect the 3-manifold into pieces which admit uniform "geometric" structures. In the same year, Richard Hamilton published his epochal work on the Ricci flow, using a convergence theorem for a parabolic partial differential equation to prove that certain non-uniform geometric structures on 3-manifolds could be deformed into uniform geometric structures. Although it is often attributed to Hamilton, he has observed that Yau is responsible for the insight that a precise understanding of the failure of convergence for Hamilton's differential equation could suffice to prove the existence of the relevant spheres and tori in Thurston's conjecture. This insight stimulated Hamilton's further research in the 1990s on singularities of the Ricci flow, and culminated with Grigori Perelman's preprints on the problem in 2002 and 2003.
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Uhlenbeck is one of the founders of the field of geometric analysis, a discipline that uses differential geometry to study the solutions to differential equations and vice versa. She has also contributed to topological quantum field theory and integrable systems.. Together with Jonathan Sacks in the early 1980s, Uhlenbeck established regularity estimates that have found applications to studies of the singularities of harmonic maps and the existence of smooth local solutions to the Yang–Mills–Higgs equations in gauge theory. In particular, Donaldson describes their joint 1981 paper The existence of minimal immersions of 2-spheres as a "landmark paper... which showed that, with a deeper analysis, variational arguments can still be used to give general existence results" for harmonic map equations. Building on these ideas, Uhlenbeck initiated a systematic study of the moduli theory of minimal surfaces in hyperbolic 3-manifolds (also called minimal submanifold theory) in her 1983 paper, Closed minimal surfaces in hyperbolic 3-manifolds.
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Nataša Pavlović is a Serbian mathematician who works as a professor of mathematics at the University of Texas at Austin. Her research concerns fluid dynamics and nonlinear dispersive partial differential equations.. She is known for her work with Nets Katz pioneering an approach to constructing singularities in equations resembling the Navier–Stokes equations, by transferring a finite amount of energy through an infinitely decreasing sequence of time and length scales.. Pavlović earned a bachelor's degree in mathematics from the University of Belgrade in 1996, and completed her doctorate from the University of Illinois at Chicago in 2002 under the joint supervision of Susan Friedlander and Nets Katz. After temporary positions at the Clay Mathematics Institute, Princeton University, Institute for Advanced Study, and Mathematical Sciences Research Institute, she joined the Princeton faculty in 2005, and moved to the University of Texas in 2007. She was a Sloan Research Fellow from 2008 to 2012.
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The Xeelee Sequence deals with many concepts stemming from the fringe of theoretical physics and futurology, such as exotic-matter physics, naked singularities, closed timelike curves, multiple universes, hyperadvanced computing and artificial intelligence, faster-than-light travel, and the upper echelons of the Kardashev scale. Thematically, the series deals heavily with certain existential and social philosophical issues, such as striving for survival and relevance in a harsh and unknowable universe and the effects of war and militarism on society. As of August 2018, the series is composed of 9 novels and 53 short pieces (short stories and novellas, with most collected in 3 anthologies), all of which fit into a fictional timeline stretching from the Big Bang's singularity of the past to the eventual heat death of the universe and Timelike Infinitys singularity of the future. An omnibus edition of the first four Xeelee novels (Raft, Timelike Infinity, Flux, and Ring), entitled Xeelee: An Omnibus, was released in January 2010.
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Its partisan structure exhibits some singularities, such as the creation of "brigades" that group their militants according to environment of activity; brigades that live together organically, and brigades of militant youths such as the Confederacy of the Socialist Youth, and the Confederacy of Socialist Women. In the later 1930s they included the "Left Communist" faction, formed by a split of the Communist Party of Chile, headed by Manuel Noble Plaza and comprising the journalist Oscar Waiss, the lawyer Tomás Chadwick and the first secretary of the PS, Ramón Sepúlveda Loyal, among others. In 1934 the Socialists, along with the Radical-Socialist Party and the Democratic Party constituted the "Leftist Bloc". In the first parliamentary election (March 1937) they obtained 22 representatives (19 representatives and 3 senators), among them its Secretary general Oscar Schnake Vergara, elected senator of Tarapacá-Antofagasta, placed by the PS in a noticeable place inside the political giants of the epoch.
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The demand for consistency between a quantum description of matter and a geometric description of spacetime,Put simply, matter is the source of spacetime curvature, and once matter has quantum properties, we can expect spacetime to have them as well. Cf. as well as the appearance of singularities (where curvature length scales become microscopic), indicate the need for a full theory of quantum gravity: for an adequate description of the interior of black holes, and of the very early universe, a theory is required in which gravity and the associated geometry of spacetime are described in the language of quantum physics. Despite major efforts, no complete and consistent theory of quantum gravity is currently known, even though a number of promising candidates exist.A timeline and overview can be found in Projection of a Calabi–Yau manifold, one of the ways of compactifying the extra dimensions posited by string theory Attempts to generalize ordinary quantum field theories, used in elementary particle physics to describe fundamental interactions, so as to include gravity have led to serious problems.
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Classical black holes create a problem for physics known as the black hole information paradox, an issue first raised in 1972 by Jacob Bekenstein and later popularized by Stephen Hawking. The information paradox is born out of the realization that all the quantum nature (information) of the matter and energy that falls into a classic black hole is thought to entirely vanish from existence into the zero-volume singularity at its heart. For instance, a black hole that is feeding on the stellar atmosphere (protons, neutrons, and electrons) from a nearby companion star should, if it obeyed the known laws of quantum mechanics, technically grow to be increasingly different in composition from one that is feeding on light (photons) from neighboring stars. Yet, the implications of classic black hole theory are inescapable: other than the fact that the two classic black holes would become increasingly massive due to the infalling matter and energy, they would undergo zero change in their relative composition because their singularities have no composition.
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An exception-handling style enabled by the use of status flags involves: first computing an expression using a fast, direct implementation; checking whether it failed by testing status flags; and then, if necessary, calling a slower, more numerically robust, implementation. The IEEE 754 standard uses the term "trapping" to refer to the calling of a user-supplied exception-handling routine on exceptional conditions, and is an optional feature of the standard. The standard recommends several usage scenarios for this, including the implementation of non-default pre-substitution of a value followed by resumption, to concisely handle removable singularities. The default IEEE 754 exception handling behaviour of resumption following pre- substitution of a default value avoids the risks inherent in changing flow of program control on numerical exceptions. For example, in 1996 the maiden flight of the Ariane 5 (Flight 501) ended in a catastrophic explosion due in part to the Ada programming language exception handling policy of aborting computation on arithmetic error, which in this case was a 64-bit floating point to 16-bit integer conversion overflow.
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In May 2026, the prime universe is suffering from the same singularities that have already destroyed the parallel universe, as a result of the two universes being inextricably linked together. Though the Fringe Division that developed in this universe has been able to use amber to contain these vortices, a group called the "End of Dayers", led by a man named Moreau (Brad Dourif), attempts to breach the fabric of reality at soft spots and create more vortices. After one such incident at a theater, Peter and Olivia (Anna Torv), now married, along with Astrid (Jasika Nicole) and Ella (Emily Meade), Olivia's niece and now a rookie Fringe agent, find an unactivated container that they believe the End of Dayers used to trigger the breach. Fringe is unable to determine how the container works, and Peter convinces Broyles (Lance Reddick), now a senator, to allow him to release his father Walter (Noble), currently in maximum security prison as punishment for activating the doomsday device, to help identify its workings.
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A rich plasma phenomenology it has been observed in the table-top plasma focus devices developed at the Chilean Nuclear Energy Commission: filamentary structures, toroidal singularities, plasma bursts and plasma jets generations. In addition, possible applications are explored using these kind of small plasma devices: development of portable generator as non- radioactive sources of neutrons and x-rays for field applications, pulsed radiation applied to biological studies, plasma focus as neutron source for nuclear fusion-fission hybrid reactors, and the use of plasma focus devices as plasma accelerators for studies of materials under intense fusion-relevant pulses. In addition, Chilean Nuclear Energy Commission currently operates the facility SPEED-2, the largest Plasma Focus facility of the southern hemisphere. Since the beginning of 2009, a number of new plasma focus machines have been/are being commissioned including the INTI Plasma Focus in Malaysia, the NX3 in Singapore, the first plasma focus to be commissioned in a US university in recent times, the KSU Plasma Focus at Kansas State University which recorded its first fusion neutron emitting pinch on New Year's Eve 2009 and the IR-MPF-100 plasma focus (115kJ) in Iran.
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The code below uses a function ccw: ccw > 0 if three points make a counter-clockwise turn, clockwise if ccw < 0, and collinear if ccw = 0. (In real applications, if the coordinates are arbitrary real numbers, the function requires exact comparison of floating- point numbers, and one has to beware of numeric singularities for "nearly" collinear points.) Then let the result be stored in the `stack`. let points be the list of points let stack = empty_stack() find the lowest y-coordinate and leftmost point, called P0 sort points by polar angle with P0, if several points have the same polar angle then only keep the farthest for point in points: # pop the last point from the stack if we turn clockwise to reach this point while count stack > 1 and ccw(next_to_top(stack), top(stack), point) <= 0: pop stack push point to stack end Now the stack contains the convex hull, where the points are oriented counter-clockwise and P0 is the first point. Here, `next_to_top()` is a function for returning the item one entry below the top of stack, without changing the stack, and similarly, `top()` for returning the topmost element.
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Automorphic forms are a generalization of modular forms to more general analytic functions, perhaps of several complex variables, with similar transformation properties.. The generalization involves replacing the modular group PSL2 (R) and a chosen congruence subgroup by a semisimple Lie group G and a discrete subgroup Γ. Just as modular forms can be viewed as differential forms on a quotient of the upper half space H = PSL2 (R)/SO(2), automorphic forms can be viewed as differential forms (or similar objects) on Γ\G/K, where K is (typically) a maximal compact subgroup of G. Some care is required, however, as the quotient typically has singularities. The quotient of a semisimple Lie group by a compact subgroup is a symmetric space and so the theory of automorphic forms is intimately related to harmonic analysis on symmetric spaces. Before the development of the general theory, many important special cases were worked out in detail, including the Hilbert modular forms and Siegel modular forms. Important results in the theory include the Selberg trace formula and the realization by Robert Langlands that the Riemann-Roch theorem could be applied to calculate the dimension of the space of automorphic forms.
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"A Mathematical Theory of Stochastic Microlensing I. Random Time Delay Functions and Lensing Maps," A. O. Petters, B. Rider, and A. M. Teguia, J. Math Phys., 50, 072503 (2009); astro-ph arXiv:0807.0232v2 "A Mathematical Theory of Stochastic Microlensing II. Random Images, Shear, and the Kac-Rice Formula," A. O. Petters, B. Rider, and A. M. Teguia (2008); astro-ph arXiv:0807.4984 The work forms a concrete framework from which extensions to more general random maps can be made. In two additional papers, he and Aazami found geometric universal magnification invariants of higher- order caustics occurring in lensing and caustics produced by generic general maps up to codimension five."A Universal Magnification Theorem for Higher- Order Caustic Singularities," A. B. Aazami and A. O. Petters, J. Math. Phys. 50, 032501 (2009); astro-ph arXiv:0811.3447v2 "A Universal Magnification Theorem II. Caustics up to Codimension Five," A. B. Aazami and A. O. Petters, 50, 082501 (2009); math-ph arXiv:0904.2236v4 National Science Foundation, Science360 News Server, April 16, 2009 The invariants hold with a probability of 1 for random lenses and thereby form important consistency checks for research on random image magnifications of sources near stable caustics.
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Physics beyond the Standard Model (BSM) refers to the theoretical developments needed to explain the deficiencies of the Standard Model, such as the strong CP problem, neutrino oscillations, matter–antimatter asymmetry, and the nature of dark matter and dark energy. Another problem lies within the mathematical framework of the Standard Model itself: the Standard Model is inconsistent with that of general relativity, to the point where one or both theories break down under certain conditions (for example within known spacetime singularities like the Big Bang and black hole event horizons). Theories that lie beyond the Standard Model include various extensions of the standard model through supersymmetry, such as the Minimal Supersymmetric Standard Model (MSSM) and Next-to-Minimal Supersymmetric Standard Model (NMSSM), and entirely novel explanations, such as string theory, M-theory, and extra dimensions. As these theories tend to reproduce the entirety of current phenomena, the question of which theory is the right one, or at least the "best step" towards a Theory of Everything, can only be settled via experiments, and is one of the most active areas of research in both theoretical and experimental physics.
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Despite its own singularities and concepts, the "Great Replacement" is encompassed in a larger and older "white genocide" conspiracy theory, popularized in the US by neo- Nazi David Lane in his 1995 White Genocide Manifesto, where he asserted that governments in Western countries were intending to turn white people into "extinct species". The idea of a "replacement" of indigenous white people under the guidance of a hostile elite can be further traced back to pre-WWII antisemitic conspiracy theories which posited the existence of a Jewish plot to destroy Europe through miscegenation, especially in Édouard Drumont's antisemitic bestseller La France juive (1886). Commenting on this resemblance, historian Nicolas Lebourg and political scientist Jean-Yves Camus suggest that Camus's contribution was to replace the antisemitic elements with a clash of civilizations between Muslims and Europeans. Maurice Barrès's nationalist writings of that period have also been noted in the ideological genealogy of the "Great Replacement", Barrès contending both in 1889 and in 1900 that a replacement of the native population under the combined effect of immigration and a decline in the birth rate was happening in France.
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The theory of generalized Cauchy-Riemann systems with a singular point of the 1st and above 1st order in the coefficients, as well as with the 1st order singularity in the coefficients on the boundary circle, which was a natural generalization of the classical analytical apparatus of I. N. Vekua, developed to study generalized analytic functions. Based on the fundamental achievements in the development of the theory of generalized Cauchy-Riemann systems with singularities, in-depth studies have been carried out on the effect of an isolated flattening point on infinitesimal and exact bends of surfaces of positive curvature. Some progress has been made in solving the generalized Christoffel problem of determining convex surfaces from a predetermined sum of conditional radii of curvature defined on a convex surface with an isolated flattening point (together with A. Khakimov). For a wide class of natural processes described by ordinary differential equations and partial differential equations, natural metrics such as spatio-temporal Minkowski metrics are constructed, based on which a definition of the concept of the intrinsic time of a process and constructive methods for measuring it are proposed.
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