Seller Inventory B More information about this seller Contact this seller. Add to Basket. Book Description American Mathematical Society, New Book. Shipped from UK. Established seller since Seller Inventory CE Language: English. Brand new Book. The first systematic theory of generalized functions also known as distributions was created in the early s, although some aspects were developed much earlier, most notably in the definition of the Green's function in mathematics and in the work of Paul Dirac on quantum electrodynamics in physics.
Seller Inventory AAN Book Description Chelsea Pub Co , Condition: As New. Books is in new condition. Seller Inventory DS Never used!. Seller Inventory P Condition: Very Good. Great condition with minimal wear, aging, or shelf wear. Book Description Chelsea Pub Co, For the set of hyperbolas, the maximal group of invariants is the affine group 6. Their density measure is equal to , where , , are the coefficients of the general equation of the hyperbola.
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Similarly, the maximal group of invariants of ellipses is measurable, but for parabolas it is non-measurable. For parabolas, only subgroups of it are measurable, such as the groups of unimodular affine and centro-affine transformations. The elementary kinematic measure of the group of projective transformations 4 is equal to , where is the determinant of the transformation. The set of lines of the centro-affine plane is measurable. Their density measure is equal to , where is the free term of the normal equation of the line. The kinematic measure of the group of transformations 5 of the non-centro-affine plane is equal to.
If is the width of an oval, then is its density measure for the affine unimodular transformations. Integral geometry in the projective space. The group of motions in projective space with a rectangular Cartesian coordinate system is measurable only for the set of quadruples of points. The density measure in this case is equal to , where is the volume of the tetrahedron whose vertices are the points. For pairs and triples of points, only the group of affine unimodular transformations is measurable. Its density measure is equal to the unit.
For triples of points, the group of centro-affine transformations is also measurable provided that the points do not lie on the same line. The set of straight lines in has as its maximal group of invariance the full group of motions, but it is non-measurable for them only a certain subgroup of it is measurable. The full group of transformations for pairs of straight lines is measurable. The set of planes does not admit a measure with respect to the full group of transformations in ; for the set of planes, only its subgroup of orthogonal transformations is measurable.
Pairs of planes admit a measure for the group of centro-affine unimodular transformations. Parallelopipeds admit a measure for the subgroup of affine transformations, the set of pairs of planes-points admits a measure for the full group of transformations in. The set of spheres in admits a measure for the group of similarity transformations, the density being equal to , where is the radius of the sphere. The set of second-order surfaces admits a measure for the full group of transformations in , the density being , where is the invariant of the surface. The set of circles in admits a measure for the group of similarity transformations, the density being equal to , where is the radius of the circle.
The kinematic measure in of the full group of transformations is equal to , where is its determinant. The density measure of a set of points in three-dimensional centro-affine unimodular space is equal to the unit. The set of planes in space is also measurable, with density , where is the parameter of the normal equation of the plane. Integral geometry on a surface of constant curvature. The family of curves in with constant positive curvature has as maximal group of invariance.
Families of special type three-, two- and one-parameter admit a density measure for the maximal group of invariance infinitesimal transformations of the group , and for in the one-parameter case. The same holds for with negative constant curvature. Three-parameter curves of special type admit a density measure for as maximal group of invariance; it is equal to the unit. Measures also exists for groups in the case of special type of two- and one-parameter families.
In both cases, the condition that the family of curves have a measure for as maximal group of invariance is that the adjoint group be spatially transitive measurable.
Generalizations of integral geometry. The above account relates to the traditional understanding of the content of integral geometry as a theory of invariant measures on sets of geometric objects in various spaces, mainly in homogeneous spaces. In the sense of integral geometry as a theory of transformation of functions given on a set of certain geometric objects in some space into functions defined on a set of other geometric objects of the same space, the problem converse to integrating some function of points of the space along some geometric objects of the same space is posed as the fundamental problem.
For example, if an integral transform of a function in -dimensional affine space a Radon transform is introduced as its integral over hypersurfaces, then the converse problem is to recover in terms of its integral over the hypersurfaces, that is, the problem of finding the inverse Radon transform. Similarly, problems have been posed and solved concerning recovering functions on ruled second-order surfaces in four-dimensional complex space for which the integrals over the straight lines forming this surface are known, and also the question of recovering a function in terms of its integral taken over horospheres in a real or imaginary Lobachevskii space.
Reference [a1] gives a fairly complete survey of classical integral geometry up to Part of the more recent development was essentially influenced by an important paper of H. Federer [a2] , who extended the classical kinematic and Crofton intersection formulas to curvature measures and sets of positive reach. Some of the later integral-geometric results involving curvature measures are described in the survey articles [a3] , [a4].
Integral geometry plays an essential role in the recent development of stochastic geometry , as in the work of R. Miles, e. Matheron [a6] , and others. The use of kinematic formulas for curvature measures in stochastic geometry can be seen in the articles [a7] , [a8]. Another new branch of integral geometry is the combinatorial integral geometry developed by R. Ambartzumian [a9]. This theory, in which combinatorial relations between measures of certain sets of geometric objects play a central role, and invariance properties are not necessarily assumed, has also applications to stochastic geometry and interesting connections to Hilbert's fourth problem.
An impression of the scope of the "generalizations of integral geometry" as it is called in the main article above, can be obtained from the contributions of the conference proceedings [a10] , and from [a11]. Log in. Namespaces Page Discussion. Views View View source History.
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Jump to: navigation , search. The latter can be found as a solution to the system of partial differential equations 1 where is the required integral invariant, is a point of the space having dimension , are the coefficients of the infinitesimal transformation of the group, and is the number of parameters of the transformation. The measure is given by the integral 2 where is a set of points in the parameter space of the Lie group and is an integral invariant of the group, defined by equation 1 , or the density measure.
In the case of a homogeneous multi-dimensional space, the measure of a set of manifolds for example, points, straight lines, hyperplanes, pairs of hyperplanes, hyperspheres, second-order hypersurfaces is uniquely defined up to a constant factor by the integral 3 where are the relative components of a given transitive Lie group. Outer common tangent lines Crossed common tangent lines Figure: ia The measure of the set of straight lines dividing two ovals is equal to the length of the crossed common tangent lines minus the sum of the lengths of the contours of the ovals.
The measure of a set of pairs of points is determined as where , are the parameters of the normal equation of the straight line passing through the points and and are the distances along this straight line from the points to the point on the line having minimal distance from the origin see Fig. Jantzen, Jens C.
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