Unreferenced date March 2007 For a surface in three dimension the focal surface , surface of centers or evolute is formed by taking the centers of the curvature sphere s, which are the tangent ial sphere s whose radii are the Multiplicative inverse reciprocals of one of the principal curvature s at the point of tangency. Equivalently it is the surface formed by the centers of the circles which osculate the curvature line s. As the principal curvatures are the eigenvalues of the second fundamental form, there are two at each point, and these give rise to two points of the focal surface on each surface normal normal direction to the surface. Away from umbilical point s, these two points of the focal surface are distinct at umbilical points the two sheets come together. At points where the Gaussian curvature is zero, one sheet of the focal surface will have a point at infinity corresponding to the zero principal curvature. Special cases The sphere is the only surface where both sheets of the focal surface degenerate to a single point. Both sheets of the focal surface of a Dupin cyclide degenerate to curves. These curves form a pair of anticonics, i.e., an ellipse and a hyperbola in perpendicular planes, each passing through the respective foci of the other curve. ref Harvnb Chandru Dutta Hoffmann 1988 ref For the torus , a special Dupin cyclide, the focal ellipse becomes a circle and the focal hyperbola degenerates to the axis of the circle. One sheet of the focal surface of a channel surface will form a degenerate curve. Such surfaces includes all surface of revolution surfaces of revolution , where the degenerate curve is the axis of revolution. See also Focus optics Evolute Notes reflist References citation title On the Geometry of Dupin Cyclides first1 V. last1 Chandru first2 D. last2 Dutta first3 C.M. last3 Hoffmann year 1988 publisher Purdue University e Pubs . DEFAULTSORT Focal Surface Category Surfaces geometry stub ... more details
About the financial math concept the company with the name Efficient Frontier Efficient Frontier company Image markowitz frontier.jpg right frame Efficient Frontier. The hyperbola is sometimes referred to as the Markowitz Bullet , and its upward sloped portion is the efficient frontier if no risk free asset is available. With a risk free asset, the straight line is the efficient frontier. The efficient frontier is a concept in modern portfolio theory introduced by Harry Markowitz and others. A combination of assets, i.e. a portfolio finance portfolio , is referred to as efficient if it has the best possible expected value expected level of return finance return for its level of risk measure risk usually proxied by the standard deviation of the portfolio s return . ref cite book title Investments and Portfolio Performance author Edwin J. Elton and Martin J. Gruber publisher World Scientific year 2011 isbn 9789814335393 pages 382 383 url http books.google.com books?id aJ7Cp5ZwZ9kC&pg PA382 ref Here, every possible combination of risky assets, without including any holdings of the Risk free interest rate risk free asset , can be plotted in risk expected return space, and the collection of all such possible portfolios defines a region in this space. The upward sloped part of the left boundary of this region, a hyperbola , is then called the efficient frontier . For more information see Modern portfolio theory The efficient frontier with no risk free asset modern portfolio theory . The efficient frontier is the positively sloped portion of the opportunity set that offers the highest expected return for a given level of risk. The efficient frontier lies at the top of the opportunity set or the feasible set. References reflist Econ theory stub Category Financial economics Category Finance theories Category Mathematical finance Category Portfolio theories ... more details
In mathematics , a projective range is a set of points in projective geometry considered in a unified fashion. A projective range may be a real projective line projective line or a conic section conic . A projective range is the projective duality dual of a pencil mathematics pencil of lines on a given point. For instance, a correlation projective geometry correlation interchanges the points of a projective range with the lines of a pencil. A projectivity is said to act from one range to another, though the two ranges may coincide as sets. A projective range expresses projective invariance of the relation of projective harmonic conjugate s. Indeed, three points on a projective line determine a fourth by this relation. Application of a projectivity to this quadruple results in four points likewise in the harmonic relation. Such a quadruple of points is termed a harmonic range . When a conic is chosen for a projective range, and a particular point E is selected as origin on the conic, then addition of points may be defined as follows Let A and B be in the range conic and AB the line connecting them. Let L be the line through E and parallel to AB . The sum of points A and B , A B , is the intersection of L with the range. The circle and hyperbola are instances of a conic and the summation of angles on either can be generated by the method of sum of points , provided points are associated with angle s on the circle and hyperbolic angle s on the hyperbola. References Reflist H. S. M. Coxeter 1955 The Real Projective Plane, University of Toronto Press , p 20 for line, p 101 for conic. Viktor Prasolov & Yuri Solovyev 1997 Elliptic Functions and Elliptic Integrals , page one, Translations of Mathematical Monographs volume 170, American Mathematical Society . Category Projective geometry ... more details
the area of a circular sector of radians is . The area of the whole circle is 2 . As the hyperbola ... can also be applied to the hyperbola. If P sub 0 sub is taken to be the point 1,1 , P sub 1 sub the point ... traces out a hyperbola. In Euclidean space, the multiple of a given angle traces equal distances ... quadrature of the hyperbola is the evaluation of the area swept out by a radial segment from the origin as the terminus moves along the hyperbola, just the topic of hyperbolic angle. The quadrature of the hyperbola was first accomplished by Gregoire de Saint Vincent in 1647 in his momentous Opus ... of a hyperbola to its asymptotes, and showed that as the area increased in arithmetic series ... to the hyperbola in Augustus De Morgan s 1849 textbook Trigonometry and Double Algebra see references for link . In 1878 W.K. Clifford used hyperbolic angle to parametrize a unit hyperbola , describing ... more details
jesuit Alphonse Antonio de Sarasa was a Jesuit mathematician who contributed to the understanding of logarithm s, particularly as area s under a hyperbola . Alphonse de Sarasa was born in 1618, in Nieuwpoort Belgium Nieuwpoort in Flanders. In 1632 he was admitted as a novice in Ghent . It was there that he worked alongside Gregoire de Saint Vincent whose ideas he developed, exploited, and promulgated. According to Sommervogel 1896 , Alphonse de Sarasa also held academic positions in Antwerp and Brussels. In 1649 Alphonse de Sarasa published Solutio problematis a R.P. Marino Mersenne Minimo propositi . This book was in response to Marin Mersenne s pamphlet Reflexiones Physico mathematicae which reviewed Saint Vincent s Opus Geometricum and posed this challenge Given three arbitrary magnitudes, rational or irrational, and given the logarithms of the two, to find the logarithm of the third geometrically. R.P. Burn 2001 explains that the term logarithm was used differently in the seventeenth century. Logarithms were any arithmetic progression which corresponded to a geometric progression . Burn says, in reviewing de Sarasa s popularization of de Saint Vincent, and concurring with Moritz Cantor , that the relationship between logarithms and the hyperbola was found by Saint Vincent in all but name . Burn quotes de Sarasa on this point the foundation of the teaching embracing logarithms are contained in Saint Vincent s Opus Geometricum , part 4 of Book 6, de Hyperbola . Alphonse Antonio de Sarasa died in Brussels in 1667. See also List of Roman Catholic scientist clerics References R. P. Burn 2001 Alphonse Antonio de Sarasa and Logarithms , Historia Mathematica 28 1 17. C.H. Edwards, Jr. 1979 The Historical Development of the Calculus , pp. 154 8, Springer Verlag, ISBN 0 387 90436 0 . C. Sommervogel 1896 Biblioth que de la Compagnie de J sus , vol. VII, pp. 621 7. Use dmy dates date January 2011 Persondata Metadata see Wikipedia Persondata . NAME Sarasa, Alphonse Anto ... more details
hyperbola if and only if x cos A y cos B z cos C 0. The general inconic reduces to a parabola if and only ... rp p.136 Examples Circumconics circumcircle Steiner circumellipse Kiepert hyperbola Je bek hyperbola Feuerbach hyperbola Inconics incircle Steiner inellipse Mandart inellipse Kiepert parabola Yff ... more details
thumb Types of conic sections br 1. Parabola br 2. Circle and ellipse br 3. Hyperbola Image Table of Conics ... section are the hyperbola , the parabola , and the ellipse . The circle is a special case of the ellipse ... is right then the conic is a parabola and if the angle is obtuse then the conic is a hyperbola ... are the ellipse , parabola , and hyperbola . The order ellipse parabola hyperbola is important in discussions ... is unbounded and is called a parabola. In the remaining case, the figure is a hyperbola. In this case ... case, interchange the symbols a and b . For the hyperbola the east west opening case is given. In all ... y 2 4ax , math math 1 , math math a , math math 2a , math math 2a , math hyperbola math frac x 2 a 2 ... side but with the inside of the hyperbola the direction of curvature on the other side left vs ... . These correspond respectively to degeneration of an ellipse, parabola, and a hyperbola, which ... e 1 2 font , font color 00cc00 parabola e 1 font and font color 0000ff hyperbola e 2 font with fixed ... 0 e 1 math we obtain an ellipse, for math e 1 math a parabola, and for math e 1 math a hyperbola. For an ellipse and a hyperbola, two focus directrix combinations can be taken, each giving the same full ellipse or hyperbola. The distance from the center to the directrix is math a e math , where math a math is the semi major axis of the ellipse, or the distance from the center to the tops of the hyperbola ... difference between 1 and &minus 1 precisely, the ellipse math x 2 y 2 1 math becomes a hyperbola under ... a hyperbola is simply an ellipse with an imaginary axis length. Thus there is a 2 way classification ellipse hyperbola and parabola. Geometrically, this corresponds to intersecting the line at infinity ... of a parabola , and thus the real hyperbola is a more suggestive image for the complex ellipse hyperbola, as it also has 2 real intersections with the line at infinity. In projective space , over ... to an ellipse or a hyperbola. In other areas of mathematics The classification into elliptic, parabolic ... more details
the curves that were later known as the ellipse, the parabola, and the hyperbola. ... Yet the first ... it was the parabola and hyperbola that proffered the properties needed in the solution of the Delian ... could be achieved also by the use of a rectangular hyperbola and a parabola. ref There are few direct ... more details
quantity , by Noether s theorem . In the study of spacetime the use of the unit hyperbola to calibrate .... The principle of relativity was reduced to arbitrariness of which diameter of the unit hyperbola was used to determine a world line . Using the parametrization of the hyperbola with hyperbolic angle ... more details
Year nav topic 1668 science The year 1668 in science and technology involved some significant events. Astronomy Isaac Newton invents the reflecting telescope. Biology Francesco Redi publishes Esperienze Intorno alla Generazione degl Insetti Experiments on the Generation of Insects , disproving theories of the spontaneous generation of maggot s in putrefying matter. Mathematics Nicholas Mercator and William Brouncker discover an infinite series for the logarithm while attempting to calculate the area under a hyperbola hyperbolic segment . Medicine Fran ois Mauriceau publishes Trait des Maladies des Femmes Grosses et Accouch es in Paris , a key text in scientific obstetrics . ref cite book title A Medical Bibliography a check list of texts illustrating the history of the medical sciences first Leslie T. last Morton location London publisher Grafton year 1943 ref Births December 31 Hermann Boerhaave , Netherlands Dutch physician and chemist who makes Leiden a Europe an centre of medical knowledge d. 1738 in science 1738 Deaths References reflist Category 1668 in science fr 1668 en science mk 1668 pt 1668 na ci ncia ... more details
Wiktionary angle Angle An angle is a geometrical figure that divides a circle. A hyperbolic angle is a geometrical figure that divides a hyperbola. Angle may also refer to Angle astrology , a cardinal point of an astrological chart Angle rib , an anatomical characteristic Angle, Pembrokeshire , Wales The Angle , in the American Civil War, an area of the Gettysburg battlefield Northwest Angle , known by locals as The Angle, the only place in the United States outside Alaska that is north of the 49th parallel Angles , a Germanic tribe that settled in Britain Angling , a fishing technique Angle, in Glossary of professional wrestling terms Angle professional wrestling terminology , a character s motivating story People Jared Angle , New York City Ballet principal dancer Eric Angle , brother of Kurt Angle, professional wrestler Kurt Angle , Olympic gold medalist in amateur wrestling, and professional wrestler Sharron Angle , Nevada politician Tyler Angle , New York City Ballet soloist See also Angle of attack Angle of incidence Angle of parallelism Angle of repose Angle of view Angle Man Angles disambiguation Angel disambiguation Bloody Angle disambiguation Team Angle disambiguation Disambig de Angle es ngulo desambiguaci n eu Angelu fr Angle homonymie pt Angle ru zh yue zh ... more details
In geometry , a trisectrix is a curve which can be used to angle trisection trisect an arbitrary angle. Such a method falls outside those allowed by compass and straightedge constructions , so they do not contradict the well known theorem which states that an arbitrary angle cannot be trisected with that type of construction. There are a variety of such curves and the methods used to construct an angle trisector differ according to the curve. Examples include Lima on trisectrix some sources refer to this curve as simply the trisectrix. Trisectrix of Maclaurin Equilateral trefoil aka Longchamps Trisectrix Tschirnhausen cubic aka Catalan s trisectrix and L hospital s cubic Durer s folium Cubic parabola Hyperbola with eccentricity 2 Rose mathematics Rose with 3 petals Parabola A related concept is a sectrix , which is a curve which can be used to divide an arbitrary angle by any integer. Examples include Quadratrix of Hippias Sectrix of Maclaurin Sectrix of Ceva Sectrix of Delanges See also Angle trisection Neusis construction Quadratrix Doubling the cube References http www.jimloy.com geometry trisect.htm curves Loy, Jim Trisection of an Angle , Part VI MathWorld title Trisectrix urlname Trisectrix http www.mathcurve.com courbes2d sectrice sectrice.shtml Sectrix curve at Encyclop die des Formes Math matiques Remarquables In French 1911 Category Curves de Trisektrix ru sl Trisektrisa ... more details
Perseus c. 150 BC was an Greece ancient Greek geometer , who invented the concept of spiric section s, in analogy to the conic section s studied by Apollonius of Perga . Few details of Perseus life are known, as he is mentioned only by Proclus and Geminus none of his own works have survived. The spiric sections result from the intersection of a torus with a Plane mathematics plane that is parallel to the rotational symmetry axis of the torus. Consequently, spiric sections are fourth order quartic curve quartic plane curve s, whereas the conic section s are second order quadratic function quadratic plane curve s. Spiric sections are a special case of a toric section , and were the first toric sections to be described. The most famous spiric section is the Cassini oval , which is the locus of points having a constant product of distances to two foci. For comparison, an ellipse has a constant sum of focal distances, a hyperbola has a constant difference of focal distances and a circle has a constant ratio of focal distances. References Tannery P. 1884 Pour l histoire des lignes et de surfaces courbes dans l antiquit , Bull. des sciences math matique et astronomique , 8 , 19 30. Heath TL. 1931 A history of Greek mathematics , vols. I & II, Oxford. Greek mathematics DEFAULTSORT Perseus Category Ancient Greek mathematicians Category Geometers Category 2nd century BC Greek people it Perseo matematico ru ... more details
Probability distribution name hyperbolic type density pdf image cdf image parameters math mu math location parameter location real number real br math alpha math to do real br math beta math asymmetry parameter real br math delta math scale parameter real br math gamma sqrt alpha 2 beta 2 math support math x in infty infty math pdf math frac gamma 2 alpha delta K 1 delta gamma e alpha sqrt delta 2 x mu 2 beta x mu math br br math K lambda math denotes a modified Bessel function of the second kind cdf to do mean math mu frac delta beta K 2 delta gamma gamma K 1 delta gamma math median to do mode math mu frac delta beta gamma math variance math frac delta K 2 delta gamma gamma K 1 delta gamma frac beta 2 delta 2 gamma 2 left frac K 3 delta gamma K 1 delta gamma frac K 2 2 delta gamma K 1 2 delta gamma right math skewness to do kurtosis to do entropy to do mgf math frac e mu z gamma K 1 delta alpha 2 beta z 2 alpha 2 beta z 2 K 1 delta gamma math char to do The hyperbolic distribution is a continuous probability distribution that is characterized by the fact that the logarithm of the probability density function is a hyperbola . Thus the distribution decreases exponentially, which is more slowly than the normal distribution . It is therefore suitable to model phenomena where numerically large values are more probable than is the case for the normal distribution. Examples are returns from financial assets and turbulent wind speeds. The hyperbolic distributions form a subclass of the generalised hyperbolic distribution s. The origin of the distribution is the observation by Ralph Alger Bagnold in his book The Physics of Blown Sand and Desert Dunes 1941 that the logarithm of the histogram of the empirical size distribution of sand deposits tends to form a hyperbola. This observation was formalised mathematically by Ole Barndorff Nielsen in a paper in 1977, ref cite journal doi 10.1098 rspa.1977.0041 last Barndorff Nielsen first Ole year 1977 title Exponentially decreasing ... more details
math 1 math math a math hyperbola math frac x 2 a 2 frac y 2 b 2 1 math math sqrt 1 frac b 2 a 2 ... is not a parabola which has eccentricity equal to 1 , not a degenerate hyperbola or degenerate ellipse ... r min r max r min frac r max r min 2a . math Hyperbolas The eccentricity of a hyperbola can be any real number greater than 1, with no upper bound. The eccentricity of a rectangular hyperbola is math ... more details
, and the denominator 17 p 3 is the 3rd prime of form 4k 1. The unit hyperbola s group of rational points There is a close connection between this group on the unit hyperbola and the group discussed above ... fractions, then c a,b a is a rational point on the unit hyperbola, since math c a 2 b a 2 1, math satisfying the equation for the unit hyperbola. The group operation here is math x,y times u,v xu yv ,xv ... identity element is 0,1,0, 1 . The unit hyperbola group corresponds to points of form 0,1, y,z , with math ... more details
The same process of finding smaller roots is used instead to find lower lattice points on a hyperbola ... y x math . Assume there is some lattice point math scriptstyle x, , y math on some hyperbola and without ... lattice point with the same y coordinate on the other branch of the hyperbola, and by reflection through math scriptstyle y x math a new point on the original branch of the hyperbola is obtained. It is shown ... of the hyperbola, the desired conclusion will be proven. Example This method can be applied ... ab , , 1 q math , then we have the hyperbola math scriptstyle a 2 , , b 2 , , qab , , q 0 math . Call this hyperbola math scriptstyle H math . If math scriptstyle a b math then we find math scriptstyle ... more details
Following is a list of some mathematics mathematically well defined shape s. See also list of geometric shapes , list of polygons, polyhedra and polytopes , and list of curves . 0D with no surface Point geometry point 1D with 0D surface interval mathematics interval Line geometry line 2D with 1D surface B zier curve As Bt sup n sup 0 s 1 0 t 1 s t 1, A sup n sup , A sup n &minus 1 sup B , ..., B sup n sup are vectors circle x sup 2 sup y sup 2 sup r sup 2 sup ellipse parabola hyperbola Plane mathematics plane polygon chiliagon decagon enneagon googolgon hectagon heptagon hendecagon hexagon myriagon octagon pentagon quadrilateral triangle trapezium 3D with 1D surface helix x sin z y cos z 3D with 2D surface B zier triangle As Bt Cu sup n sup 0 s 1 0 t 1 0 u 1 s t u 1, A sup p sup B sup q sup C sup r sup vectors if p q r n and p , q , r are nonnegative integers cylinder geometry cylinder hyperplane m bius strip platonic solid dodecahedron hexahedron cube icosahedron octahedron tetrahedron torus doughnut quadric cone geometry cone cylinder geometry cylinder ellipsoid spheroid sphere hyperboloid paraboloid sphere 4D with 3D surface polychoron hecatonicosachoron hexacosichoron hexadecachoron icositetrachoron pentachoron simplex tesseract spherical cone 5D with 4D surfaces regular 5 polytopes 5 dimensional simplex 5 dimensional cross polytope 5 dimensional hypercube 5 measure polytope Fractal s Apollonian gasket Cantor set Dragon curve Koch snowflake L vy C curve Lyapunov fractal Mandelbrot set Sierpinski carpet Peano curve Sierpinski triangle See also List of mathematical topics Periodic table of shapes The Periodic table of mathematical shapes Category Mathematics related lists Shapes ... more details
catenary ref name 2dcurves roulettede Line mathematics Line Hyperbola Focus geometry Focus of the hyperbola Hyperbolic catenary ref name 2dcurves roulettede Line mathematics Line Hyperbola Centre geometry Center of the hyperbola Rectangular elastica ref name sturm Failed verification date August 2008 ... more details
In the differential geometry of surfaces , an asymptotic curve is a curve always tangent to an asymptotic direction of the surface where they exist . It is sometimes called an asymptotic line , although it need not be a line mathematics line . An asymptotic direction is one in which the normal curvature is zero. Which is to say for a point on an asymptotic curve, take the plane mathematics plane which bears both the curve s tangent and the surface s surface normal normal at that point. The curve of intersection of the plane and the surface will have zero curvature at that point. Asymptotic directions can only occur when the Gaussian curvature is negative or zero . There will be two asymptotic directions through every point with negative Gaussian curvature, these directions are bisected by the principal curvature principal directions . The direction of the asymptotic direction are the same as the asymptote s of the hyperbola of the Dupin indicatrix . ref cite book title Geometry and Imagination author David Hilbert authorlink David Hilbert coauthors Stephan Cohn Vossen Cohn Vossen, S. year 1999 publisher American Mathematical Society isbn 0 8218 1998 4 ref A related notion is a Monstar curvature line , which is a curve always tangent to a principal direction. References MathWorld urlname AsymptoticCurve title Asymptotic Curve http www.seas.upenn.edu cis70005 cis700sl10pdf.pdf Lines of Curvature, Geodesic Torsion, Asymptotic Lines http www.mathcurve.com surfaces asymptotic asymptotic.shtml Asymptotic line of a surface at Encyclop die des Formes Math matiques Remarquables in French language French references Category Curves Category Differential geometry of surfaces Category Surfaces differential geometry stub ar cs Asymptotick k ivka eo Asimptota kurbo fr Branche parabolique ru ... more details
In astrodynamics a characteristic energy math C 3 , math , a form of specific energy , is a measure of the energy required for an interplanetary mission that requires attaining an excess orbital velocity over an escape velocity required for additional orbital maneuver s. The unit of the characteristic energy is kilometre km sup 2 sup second s sup 2 sup . Characteristic energy can be computed as math C 3 v infty 2 , math where math v infty math is the orbital velocity when the orbital distance tends to infinity. Note that, since the kinetic energy is one half m math v 2, math C sub 3 sub is in fact equal to twice the magnitude of the specific orbital energy math epsilon math of the escaping object. Parabolic trajectory For a spacecraft that is leaving the central body e.g. earth on a parabolic trajectory math C 3 0 , math Hyperbolic trajectory For a spacecraft that is leaving the central body on a hyperbolic trajectory math C 3 mu over a , math where math mu , math is the standard gravitational parameter , math a , math is length of semi major axis of orbit s hyperbola . See also Specific orbital energy Orbit Parabolic trajectory Hyperbolic trajectory References cite book last Wie first Bong title Space Vehicle Dynamics and Control publisher American Institute of Aeronautics and Astronautics location Reston, Virginia date 1998 series AIAA Education Series chapter Orbital Dynamics isbn 1563472619 accessdate 2009 07 05 Category Astrodynamics Category Celestial mechanics Category Energy in physics ... more details
The Scaled Composites SpaceShipThree SS3 is a proposed spaceplane to be developed by Virgin Galactic and Scaled Composites if SpaceShipTwo is successful. The originally proposed vehicle mission was orbital spaceflight . ref cite web last title SpaceShipThree poised to follow if SS2 succeeds work publisher date August 23, 2005 url http www.flightglobal.com articles 2005 08 23 201097 spaceshipthree poised to follow if ss2 succeeds.html accessdate 2008 01 25 ref As of 2008 , the company has scaled back those plans and articulated a design that would be a point to point vehicle traveling outside the Earth s atmosphere atmosphere . ref name fg200802 http www.flightglobal.com blogs hyperbola 2008 02 spaceshipthree revealed.html SpaceShipThree revealed? , FlightGlobal Hyperbola, Rob Coppinger, 29 Feb 2008 ref asof 2008 , the SpaceShipThree concept spacecraft will be used for transportation through point to point suborbital spaceflight . This service could provide, for example, a rapid trans atlantic trip, or a two hour trip from London to Sydney . Kangaroo Route ref name fg200802 References reflist External links http www.newscientist.com article dn7897 space tourism companies aiming for orbit.html Space tourism companies aiming for orbit New Scientist Space, 8 24 2005 http www.space tourism.ws spaceshipthree.htm SpaceShipThree Goals See also StratoLaunch Systems Suborbital transport Space bomber disambiguation Suborbital bomber The Spaceship Company Space tourism Reusable launch systems DEFAULTSORT Spaceship 3 Category Spaceplanes Category Space tourism Category Air launch to orbit Category Proposed spacecraft Category Proposed aircraft of the United States Category Space access Category Scaled Composites aero 2010s stub US spacecraft stub cs SpaceShipThree de SpaceShipThree eo SpaceShipThree ... more details
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