Unreferenced date January 2009 In general relativity , the tidal tensor or gravitoelectric tensor is one of the pieces in the Bel decomposition of the Riemann tensor . It is physically interpreted as giving the tidal stresses on small bits of a material object which may also be acted upon by other physical forces , or the tidal accelerations of a small cloud of test particle s in a vacuum solution or electrovacuum solution . Category Tensors in general relativity relativity stub ... more details
other uses Dablink Note that in common usage, the term tensor is also used to refer to a tensor field . File Components stress tensor.svg right thumb 300px Stress, a second order tensor. The tensor s components ... themselves are also tensors. A tensor can be represented as a Array data structure Multidimensional arrays multi dimensional array of numerical values. The order also degree or rank of a tensor is the dimensionality ... dimensional array, and therefore is a 2nd order tensor. A vector can be represented as a 1 dimensional array and is a 1st order tensor. Scalars are single numbers and are thus zeroth order tensors. Tensors ..., the Stress mechanics Cauchy s stress theorem stress tensor stress tensor T takes a direction ... system . Taking a coordinate basis of a vector space basis or frame of reference and applying the tensor to it results in an organized multidimensional array representing the tensor in that basis, or as it looks from that frame of reference. The coordinate independence of a tensor then takes the form ... in to the notion of a tensor in a geometrical or physical setting, and the precise form of the transformation law determines the type or valence of the tensor. Tensors are important in physics because ... curvature tensor . ref name Kline cite book title Mathematical thought from ancient to modern ... year 1972 ref History The concepts of later tensor analysis arose from the work of Carl Gauss in differential ... books?id O6lixBzbc0gC ref The word tensor itself was introduced in 1846 by William Rowan ... ref to describe something different from what is now meant by a tensor. ref group Note Namely, the norm ... publisher Von Veit year 1898 ref Tensor calculus was developed around 1890 by Gregorio Ricci ... to be known as tensor analysis , and achieved broader acceptance with the introduction of Albert Einstein ... to correct mistakes Einstein had made in his use of tensor analysis. The correspondence lasted ... tensor s, and the Riemann curvature tensor . The exterior algebra of Hermann Grassmann , from ... more details
In mathematics , the nonmetricity tensor in differential geometry is the covariant derivative of the metric tensor . It is therefore a tensor field of TensorTensor rank rank three. It vanishes for the case of Riemannian geometry . Category Differential geometry Geometry stub ... more details
, the tidal tensor is identified with the electrogravitictensor , which is one piece of the Bel decomposition of the Riemann tensor . DEFAULTSORT Tidal Tensor Category Tensors Category Gravitation ... r sin theta , partial phi math We will directly compute the tidal tensor, expressed in this frame, by elementary ... are math Phi 22 Phi 33 m r 3 math . Combining this information, we find that the tidal tensor is diagonal ... more details
Stress tensor may refer to Stress mechanics , in classical physics Stress energy tensor , in relativistic theories Maxwell stress tensor , in electromagnetism See also Stress disambiguation Tensor disambiguation disambig de Spannungstensor he ... more details
In Riemannian geometry , the Schouten tensor is a second order tensor which is introduced by Jan Arnoldus Schouten . It is defined by, for n 3 dimensions, math P frac 1 n 2 left Ric frac R 2 n 1 g right , math where Ric is the Ricci tensor , R is the scalar curvature , g is the Riemannian metric and n is the dimension of the manifold. The Weyl tensor equals the Riemann curvature tensor minus the Kulkarni&ndash Nomizu product of the Schouten tensor with the metric. See also Weyl Schouten theorem Cotton tensor Category Riemannian geometry Category Tensors differential geometry stub ru ... more details
Unreferenced date August 2008 In mathematics , the tensor bundle of a manifold is the direct sum of vector bundles direct sum of all tensor products of the tangent bundle and the cotangent bundle of that manifold. To do calculus on the tensor bundle a connection mathematics connection is needed. Category Vector bundles geometry stub ru ... more details
Unreferenced stub auto yes date December 2009 In general relativity , the topogravitic tensor is one of the three pieces of the Bel decomposition of the Riemann tensor . The topogravitic tensor can be interpreted as representing the sectional curvatures for the spatial part of a frame fields in general relativity frame field . DEFAULTSORT Topogravitic Tensor Category Tensors in general relativity Relativity stub ... more details
A Killing tensor , named after Wilhelm Killing , is a tensor , known in the theory of general relativity , math K math that satisfies math nabla alpha K beta gamma 0 , math where the parentheses on the indices refer to the symmetric tensor symmetric part . This is a generalization of a Killing vector. While Killing vectors are associated with continuous symmetries more precisely, differentiable , and hence very common, the concept of Killing tensor arises much less frequently. The Kerr metric Kerr solution is the most famous example of a semi Riemannian manifold manifold possessing a Killing tensor. See also Killing form Killing vector field Wilhelm Killing Category Riemannian geometry ... more details
In mathematics and theoretical physics , a tensor is antisymmetric on two indices i and j if it flips sign when the two indices are interchanged math T ijk dots T jik dots math An antisymmetric tensor is a tensor for which there are two indices on which it is antisymmetric. If a tensor changes sign under the exchange of any pair of indices, then the tensor is completely antisymmetric and it is also referred to as a differential form . A tensor A which is antisymmetric on indices i and j has the property that the Tensor contraction contraction with a tensor B , which is symmetric on indices i and j , is identically 0. For a general tensor U with components math U ijk dots math and a pair of indices i and j , U has symmetric and antisymmetric parts defined as math U ij k dots frac 1 2 U ijk dots U jik dots math symmetric part math U ij k dots frac 1 2 U ijk dots U jik dots math antisymmetric part Similar definitions can be given for other pairs of indices. As the term part suggests, a tensor is the sum of its symmetric part and antisymmetric part for a given pair of indices, as in math U ijk dots U ij k dots U ij k dots math An important antisymmetric tensor in physics is the electromagnetic tensor F in electromagnetism . See also antisymmetric matrix Exterior algebra symmetric tensor Levi Civita symbol Category Tensors geometry stub eo Malsimetria tensoro fr Tenseur antisym trique ko he nl Anti symmetrische tensor pt Tensor antissim trico ru ... more details
In differential geometry , the Weyl curvature tensor , named after Hermann Weyl , is a measure of the curvature ... tensor , the Weyl tensor expresses the tidal force that a body feels when moving along a geodesic . The Weyl tensor differs from the Riemann curvature tensor in that it does not convey information on how ... force. The Ricci curvature , or trace linear algebra trace component of the Riemann tensor contains ... tensor is the traceless component of the Riemann tensor. It is a tensor that has the same symmetries as the Riemann tensor with the extra condition that it be trace free Tensor contraction Metric contraction ... tensor vanishes identically. In dimensions 4, the Weyl curvature is generally nonzero. If the Weyl tensor ... system in which the metric tensor is proportional to a constant tensor. This fact was a key component ... . Definition The Weyl tensor can be obtained from the full curvature tensor by subtracting out various traces. This is most easily done by writing the Riemann tensor as a 0,4 valence tensor by contracting with the metric . The 0,4 valence Weyl tensor is then Harv Petersen 2006 p 92 NumBlk math W R ... n is the dimension of the manifold, g is the metric, R is the Riemann tensor, Ric is the Ricci tensor , s is the scalar curvature , and h small O small k denotes the Kulkarni&ndash Nomizu product ... 1,3 valent Weyl tensor is then given by contracting the above with the inverse of the metric. The decomposition EquationNote 1 expresses the Riemann tensor as an orthogonal direct sum of vector bundles ... decomposition , expresses the Riemann curvature tensor into its irreducible representation irreducible ... tensor further decomposes into invariant factors for the action of the special orthogonal group , the self dual and antiself dual parts W sup sup and W sup &minus sup . The Weyl tensor can also be expressed using the Schouten tensor , which is a trace adjusted multiple of the Ricci tensor, math ... math R abcd math is the Riemann tensor, math R ab math is the Ricci tensor, math R math is the Ricci ... more details
Unreferenced date December 2009 In tensor analysis , a mixed tensor is a tensor which is neither strictly Covariance and contravariance of vectors covariant nor strictly Covariance and contravariance of vectors contravariant at least one of the indices of a mixed tensor will be a subscript covariant and at least one of the indices will be a superscript contravariant . A mixed tensor of type math begin pmatrix M N end pmatrix math , also written type M , N , with both M 0 and N 0, is a tensor which has M contravariant indices and N covariant indices. Such tensor can be defined as a linear operator linear function which maps an M N tuple of M one form s and N Vector geometry vector s to a scalar ... other by the covariance contravariance of their indices. A given contravariant index of a tensor can be lowered using the metric tensor g sub sub , and a given covariant index can be raised using the inverse metric tensor g sup sup . Thus, g sub sub could be called the index lowering operator and g sup sup the index raising operator . Generally, the covariant metric tensor, contracted with a tensor of type M , N , yields a tensor of type math M 1,N 1 math , whereas its contravariant inverse, contracted with a tensor of type math M,N math , yields a tensor of type math M 1,N 1 math . Examples As an example, a mixed tensor of type 1, 2 can be obtained by raising an index of a covariant tensor of type 0, 3 , math T alpha beta tau T alpha beta gamma , g gamma tau math , where math T alpha beta tau math is the same tensor as math T alpha beta gamma math , because math T alpha beta ... Raising an index of the metric tensor is equivalent to contracting it with its inverse, yielding ... of the metric tensor will be equal to the Kronecker delta, which will also be mixed. See also Covariance and contravariance of vectors Tensor intrinsic definition Two point tensor DEFAULTSORT Mixed Tensor Category Tensors ... more details
In differential geometry and general relativity , the Bach tensor is a tensor of rank 2 which is conformally invariant in dimension n 4. It is the only known conformally invariant tensor that is algebraically independent of the Weyl tensor . ref P. Szekeres, Conformal Tensors. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences Vol. 304, No. 1476 Apr. 2, 1968 , pp. http www.jstor.org pss 2416002 113 122 ref In abstract index notation abstract indices the Bach tensor is given by math B ab P cd W a c b d nabla c nabla aP bc nabla c nabla cP ab math where math W math is the Weyl tensor , and math P math the Schouten tensor given in terms of the Ricci tensor math R ab math and scalar curvature math R math by math P ab frac 1 n 2 left R ab frac R 2 n 1 g ab right math . References Reflist Category Tensors Category Tensors in general relativity geometry stub relativity stub ru ... more details
Expert subject Physics date December 2008 Unreferenced date December 2008 In general relativity , the magnetogravitic tensor is one of the three pieces appearing in the Bel decomposition of the Riemann tensor . The magnetogravitic tensor can be interpreted physically as a specifying possible spin spin force s on spinning bits of matter, such as spinning test particle s. See also Papapetrou Dixon equations Curvature invariant s References reflist Category Tensors in general relativity relativity stub ... more details
In differential geometry , the term curvature tensor may refer to the Riemann curvature tensor of a Riemannian manifold &mdash see also Curvature of Riemannian manifolds the curvature of an affine connection or covariant derivative on tensors the curvature form of an Ehresmann connection see Ehresmann connection , connection principal bundle or connection vector bundle . See also Tensor disambiguation mathdab zh ... more details
In differential geometry , the Cotton tensor on a pseudo Riemannian manifold of dimension n is a third order tensor field tensor concomitant of the metric tensor metric , like the Weyl curvature Weyl tensor . The concept is named after mile Cotton . Just as the vanishing of the Weyl tensor for n 4 is a necessary condition necessary and sufficient condition for the manifold to be conformally flat , the same is true for the Cotton tensor for n 3, while for n 3 it is identically zero. In coordinates, and denoting the Ricci tensor by R sub ij sub and the scalar curvature by R , the components of the Cotton tensor are math C ijk nabla k R ij nabla j R ik frac 1 2 n 1 left nabla j Rg ik nabla k Rg ij right . math The Cotton tensor can be regarded as a vector valued Differential form 2 form , and for n 3 one can use the Hodge star operator to convert this in to a second order trace free tensor density math C i j nabla k left R li frac 1 4 Rg li right epsilon klj , math sometimes called the Cotton York tensor . The proof of the classical result that for n 3 the vanishing of the Cotton tensor is equivalent the metric being conformally flat is given by Luther P. Eisenhart Eisenhart using a standard integrability condition integrability argument. This tensor density is uniquely characterized by its conformal properties coupled with the demand that it be differentiable for arbitrary metrics, as shown by Stephen Aldersley Aldersley . Properties Conformal rescaling Under conformal rescaling of the metric math tilde g e 2 omega g math for some scalar function math omega math the Cotton York tensor transforms as math tilde C C operatorname ... part of the Weyl tensor W . Symmetries The Cotton tensor has the following symmetries math ... tensor for can be rewritten as math delta W 3 n C, , math where math delta math is the positive divergence ... on certain divergence free tensor densities in a 3 space journal Journal of Mathematical Physics ... more details
External links date October 2009 Tensor software is a class of mathematical software designed for manipulation and calculation with tensor s. Standalone open source software http www.aei.mpg.de peekas ... encountered in field theory. It has extensive functionality for tensor polynomial simplification including ... Maxima is a free open source computer algebra system which can be used for tensor algebra calculations ... without defining all components of the tensor explicitly . It comes with three tensor packages itensor for abstract indicial tensor manipulation, ctensor for component defined tensors, and atensor for algebraic tensor manipulation. http maxima.sourceforge.net docs manual en maxima 27.html SEC90 ... is a tensor analysis system written for the Mathematica system. It provides more than 250 ... is a tensor package written for the Mathematica system. It provides many functions relevant for General ... lee Ricci Ricci is a system for Mathematica 2.x and later for doing basic tensor analysis, available for free. http baldufa.upc.es xjaen ttc index.htm TTC Tools of Tensor Calculus is a Mathematica package for doing tensor and exterior calculus on differentiable manifolds. http www.inp.demokritos.gr sbonano RGTC EDC and RGTC Exterior Differential Calculus and Riemannian Geometry & Tensor Calculus are free Mathematica packages for tensor calculus especially designed but not only for general relativity ... tensor calculus package for Mathematica. http www.xact.es xAct Efficient Tensor Computer Algebra for Mathematica. xAct is a collection of packages for fast manipulation of tensor expressions. http library.wolfram.com ... connection and the basic tensors of General Relativity from a given metric tensor. http ... Tensor Toolbox Multilinear algebra MATLAB software. http www.mathworks.com matlabcentral fileexchange ... analysis . Libraries http www.oonumerics.org FTensor FTensor is a high performance tensor library ... threaded tensor library implemented in C used in Dynare . The library allows for folded unfolded ... more details
In differential geometry , the Einstein tensor also trace reversed Ricci tensor , named after Albert ... tensor occurs in the Einstein field equations for gravitation describing spacetime curvature in a manner consistent with energy considerations. Definition The Einstein tensor math mathbf G math is a rank 2 tensor defined over Riemannian manifold s. In index free notation it is defined as math mathbf G mathbf R frac 1 2 mathbf g R, math where math mathbf R math is the Ricci tensor , math mathbf g math is the metric tensor and math R math is the scalar curvature . In component form, the previous equation reads as math G mu nu R mu nu 1 over2 g mu nu R. math The Einstein tensor is symmetric math G mu nu G nu mu , math and, like the stress energy tensor , divergenceless math G mu nu nu 0 ,. math Explicit form The Ricci tensor depends only on the metric tensor , so the Einstein tensor can be defined directly with just the metric tensor. However, this expression is complex and rarely quoted in textbooks. The complexity of this expression can be shown using the formula for the Ricci tensor ... align math where math delta alpha beta math is the Kronecker tensor and the Christoffel symbol math ... derivatives of the metric tensor vanish and the component form of the Einstein tensor is considerably ... denote Antisymmetric tensor antisymmetrization over bracketed indices, i.e. math g alpha beta, gamma ... linear algebra trace of the Einstein tensor can be computed by Tensor contraction contract ing the equation in the Definition definition with the metric tensor math g mu nu math . In math n math dimensions ... 3 space, 1 time gives math G , math , the trace of the Einstein tensor, as the negative of math R , math , the Ricci tensor s trace. Thus another name for the Einstein tensor is the trace reversed Ricci tensor . Use in general relativity The Einstein tensor allows the Einstein field equations without ... explicit form of the Einstein tensor , the Einstein tensor is a nonlinear function of the metric tensor ... more details
Multiple issues notability August 2009 refimprove August 2009 Image Tensor logo.png thumb right Tensor Trucks logo Tensor Trucks is a skateboarding truck company founded and designed by professional skateboarder Rodney Mullen in 2000 . Tensor s parent company is Dwindle Distribution . They offer trucks in three different heights lo, mid, hi tailored for differing wheel diameters the hi is designed for 58mm wheels and smaller, the mid for 54mm or smaller, and the lo for 52 and smaller. Mullen holds US patent no. 6,443,471B1 ref http patft.uspto.gov netacgi nph Parser?Sect1 PTO1&Sect2 HITOFF&d PALL&p 1&u 2Fnetahtml 2FPTO 2Fsrchnum.htm&r 1&f G&l 50&s1 6443471.PN.&OS PN 6443471&RS PN 6443471 United States Patent 6,443,471 ref for design features implemented in Tensors. Tensor trucks are manufactured in China. Tensor has also teamed up with Oust Skateboard Bearings to put out a co branded Oust Tensor line of skateboard bearings, as well as working on experimental Tensor trucks with enhanced parts. Tensor released a new truck called the Response in March 2007. It is an all metal design, forgoing the plastic baseplate sliders of the original design, and is touted as the lightest truck, 11 ... Dwindle.com ref ref http www.tensortrucks.com response Tensor Trucks dead link date January 2011 ref In 2008 Tensor released their lightest design yet using magnesium and touted to be 25 lighter than the industry average truck. Key design features Baseplate sliders The most visible feature on a Tensor ... standard aluminum baseplates. Interlocking bushings Tensor bushings or cushions use a patented design .... Baseplate nibs Tensor features four fangs on each baseplate designed to dig into the board to prevent the truck from shifting when mounting hardware loosens. Buttonhead kingpin Tensor implemented ... Song . ref http www.tensortrucks.com Tensor Team ref Notable ex team members Citation needed date ... Oust Bearings Category Skateboarding companies nl Tensor Trucks ... more details
Unreferenced date August 2008 In mathematics , physics and engineering , a tensor field assigns a tensor to each point of a mathematical space typically a Euclidean space or manifold . Tensor fields are used ... physics stress and strain tensor strain in materials, and in numerous applications in the physical sciences and engineering. As a tensor is a generalization of a scalar physics scalar a pure number representing a value, like length and a Euclidean vector vector a geometrical arrow in space , a tensor ... tensor fields , that is, fields defined over a manifold which define a tensor at every point of the manifold. An example is the Riemann curvature tensor . Geometric introduction Intuitively, a vector ... velocity at each point of the Earth s surface. The general idea of tensor field combines the requirement ... tensor &mdash with the idea that we don t want our notion to depend on the particular method ... explanation The contemporary mathematical expression of the idea of tensor field breaks it down into a two ... the tensor product concept is independent of any choice of basis, taking the tensor product of two ... of tensor field , namely as a section fiber bundle section of some tensor bundle . There are vector bundles which are not tensor bundles the M bius band for instance. This is then guaranteed geometric content, since everything has been done in an intrinsic way. More precisely, a tensor field assigns to any given point of the manifold a tensor in the space math V otimes cdots otimes V otimes ... . See also tangent bundle and cotangent bundle . Given two tensor bundles E M and F M , a map A E F from the space of sections of E to sections of F can be considered itself as a tensor section ..., where f is a smooth function on M . Thus a tensor is not only a linear map on the vector space ... tensor torsion and Affine connection curvature tensors built from them are. Notation The notation for tensor fields can sometimes be confusingly similar to the notation for tensor spaces. Thus, the tangent ... more details
General relativity In differential geometry , a tensor density or relative tensor is a generalization of the tensor concept. A tensor density transforms as a tensor when passing from one coordinate system .... A distinction is made among authentic tensor densities, pseudotensor densities, even tensor densities and odd tensor densities. A tensor density can also be regarded as a section of the tensor product of a tensor bundle with a density bundle . Definition Unreferenced section date June 2011 Some authors classify tensor densities into the two types called authentic tensor densities and pseudotensor ... tensor densities and odd tensor densities. When a tensor density weight is an integer there is an equivalence ... elucidate the different ways that tensor densities transform under the somewhat ... into these types, there is only one way that tensor densities transform under orientation preserving ... than 2 to the determinant of the metric tensor expressed with Covariance and contravariance of vectors covariant indices. cn date June 2011 Tensor and pseudotensor densities For example, a mixed rank 2 authentic tensor density of weight W transforms as math mathfrak T alpha beta left det left frac ... partial bar x epsilon partial x beta , bar mathfrak T delta epsilon ,, quad text authentic tensor density of integer weight , W math where math bar mathfrak T math is the order 2 tensor density in the math bar x math coordinate system, math mathfrak T math is the transformed tensor density in the math ... only when W is an integer. However, see even and odd tensor densities below. We say that a tensor ... or 1 when its argument is negative. Even and odd tensor densities The transformations for even and odd tensor densities have the benefit of being well defined even when W is not an integer. Thus one can speak of, say, an odd tensor density of weight 2 or an even tensor density of weight 1 2. When W is an even integer the above formula for an authentic tensor density can be rewritten as math ... more details
multiple issues context September 2011 no footnotes September 2011 one source September 2011 orphan September 2011 technical September 2011 In Scientific Visualization a Tensor Glyph is an object that can visualize all or most of the nine degrees of freedom, such as acceleration , twist or shear of a math 3 times 3 math matrix. It is used for tensor field visualization, where a data matrix is available at every point in the grid. There are certain types of glyphs that can be used ellipsoid cuboid cylindrical superquadrics References reflist http www.cs.utah.edu gk papers vissym04 Superquadric Tensor Glyphs Images and Examples Category Scientific modeling cat improve date September 2011 ... more details
In multilinear algebra , a tensor contraction is an operation on one or more tensor s that arises from ... vector space dual . In components, it is expressed as a sum of products of scalar components of the tensor ... to each other in an expression. The contraction of a single mixed tensor occurs when a pair of literal indices one a subscript, the other a superscript of the tensor are set equal to each other and summed ... tensor with TensorTensor rank rank or order reduced by 2. Tensor contraction can be seen as a generalization ... from the Component free treatment of tensors Definition Tensor Product of Vector Spaces tensor product of these two spaces to the field k math C V otimes V rightarrow k math corresponding ... the contraction operation on a tensor of type 1,1 , which is an element of math V otimes V math ... obtains a basis free definition of the trace linear algebra trace . In general, a tensor of type m , n ... operation, which is a linear map which yields a tensor of type m &minus 1, n ... dual basis . Since a general mixed dyadic tensor is a linear combination of decomposable tensors ... T T i j mathbf e i e j math be a mixed dyadic tensor. Then its contraction is math T i j mathbf ... being implied by the summation convention . The resulting contracted tensor inherits the remaining indices of the original tensor. For example, contracting a tensor T of type 2,2 on the second and third indices to create a new tensor U of type 1,1 is written as math T ab bc sum b T ab bc T a1 ... dyadic tensor. This tensor does not contract if its base vectors are dotted the result is the contravariant metric mathematics metric tensor , math g ij mathbf e i cdot mathbf e j math , whose rank ... product also known as a Metric tensor metric g , such contractions are possible. One uses the metric ... to tensor fields Contraction is often applied to tensor fields over spaces e.g. Euclidean space ... be applied pointwise to a tensor field, e.g. if T is a 1,1 tensor field on Euclidean space, then in any ... more details
Expert subject Physics date November 2008 There are two different tensor s sometimes referred to as the Lanczos tensor both named after Cornelius Lanczos A tensor in the theory of quadratic Lagrangian s, which vanishes in Four dimensional space four dimensions . The potential tensor H for the Weyl tensor C , this can be expressed as math C abcd H abc d H abd c H cda b H cdb a , math math g ac H bd H db g ad H bc H cb g bd H ac H ca g bc H ad H da 2 , math math 2H ef e f g ac g bd g ad g bc 3, , math where the Lanczos tensor has the symmetries math H abc H bac 0, , math math H abc H bca H cab 0, , math and where math H bd math is defined by math H bd stackrel mathrm def H e b d e H e b e d . math Thus, the Lanczos potential tensor is a gravitational field analog of the vector potential A for the electromagnetic field . See also Introduction to 2 spinors in general relativity World Scientific, 2003 by Peter O Donnell for a more detailed discussion of the Lanczos tensor and spinor. External links http www.worldscibooks.com physics 5222.html Introduction to 2 spinors in general relativity http www.arXiv.org abs gr qc 9904006 gr qc 9904006 Category Tensors Category Differential geometry Category Tensors in general relativity relativity stub ... more details
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