Generalized Maps

2020 ◽  
pp. 157-170
Author(s):  
Daniel Canarutto
Keyword(s):  
Hilbert Space ◽  
Fourier Transforms ◽  
Tensor Products ◽  
Hermitian Structure ◽  
Quantum Geometry ◽  
Basic Notion ◽  
Rigged Hilbert Space ◽  
Fundamental Properties ◽  
Special Cases ◽  
Generalized Maps

Spaces of generalised sections (also called section-distributions) are introduced, and their fundamental properties are described. Several special cases are considered, with particular attention to the case of semi-densities; when a Hermitian structure on the underlying classical bundle is given, these determine a rigged Hilbert space, which can be regarded as a basic notion in quantum geometry. The essentials of tensor products in distributional spaces, kernels and Fourier transforms are exposed.

scholarly journals Thermal Disturbances in Twinned Orthotropic Thermoelastic Material

2018 ◽  
Vol 23 (4) ◽  
pp. 897-910 ◽  
Author(s):  
L. Rani ◽  
V. Singh
Keyword(s):  
Fourier Transforms ◽  
Crystallographic Axis ◽  
Physical Domain ◽  
Matrix Differential Equation ◽  
Special Cases ◽  
Eigen Value ◽  
Basic Equations ◽  
Matrix Differential ◽  
Thermally Conducting ◽  
Vector Matrix

Abstract This paper deals with deformation in homogeneous, thermally conducting, single-crystal orthotropic twins, bounded symmetrically along a plane containing only one common crystallographic axis. The Fourier transforms technique is applied to basic equations to form a vector matrix differential equation, which is then solved by the eigen value approach. The solution obtained is applied to specific problems of an orthotropic twin crystal subjected to triangular loading. The components of displacement, stresses and temperature distribution so obtained in the physical domain are computed numerically. A numerical inversion technique has been used to obtain the components in the physical domain. Particular cases as quasi-static thermo-elastic and static thermoelastic as well as special cases are also discussed in the context of the problem.

Extension of normal functionals on W*-tensor products

1975 ◽  
Vol 78 (2) ◽  
pp. 301-307 ◽  
Author(s):  
Simon Wassermann
Keyword(s):  
Representation Theory ◽  
Tensor Product ◽  
Hilbert Spaces ◽  
General Result ◽  
Von Neumann Algebras ◽  
Tensor Products ◽  
Von Neumann ◽  
Hilbert Algebras ◽  
Special Cases

A deep result in the theory of W*-tensor products, the Commutation theorem, states that if M and N are W*-algebras faithfully represented as von Neumann algebras on the Hilbert spaces H and K, respectively, then the commutant in L(H ⊗ K) of the W*-tensor product of M and N coincides with the W*-tensor product of M′ and N′. Although special cases of this theorem were established successively by Misonou (2) and Sakai (3), the validity of the general result remained conjectural until the advent of the Tomita-Takesaki theory of Modular Hilbert algebras (6). As formulated, the Commutation theorem is a spatial result; that is, the W*-algebras in its statement are taken to act on specific Hilbert spaces. Not surprisingly, therefore, known proofs rely heavily on techniques of representation theory.

Characterizations of Fully Constrained Poses of Parallel Cable-Driven Robots: A Review

2008 ◽  
Author(s):  
Marc Gouttefarde
Keyword(s):  
Linear Algebra ◽  
Convex Cones ◽  
Mobile Platform ◽  
Polyhedral Cones ◽  
Fundamental Properties ◽  
Special Cases ◽  
Force Closure

The pose of the mobile platform of a parallel cable-driven robot is said to be fully constrained if any wrench can be created at the platform by pulling on it with the cables. A fully constrained pose is also known as a force-closure pose. In this paper, a review of three useful characterizations of a force-closure pose is proposed. These characterizations are stated in the form of theorems for which proofs are presented. Tools from linear algebra allow to derive some of these proofs while the others are more difficult and can hardly be obtained in this manner. Therefore, polyhedral cones, which are special cases of convex cones, are introduced along with some of their well-known fundamental properties. Then, it is shown how the aforementioned difficult proofs can be obtained as direct consequences of these properties.

Gamow states in a rigged hilbert space

1998 ◽  
pp. 34-37
Author(s):  
C. G. Bollini ◽  
O. Civitarese ◽  
A. L. De Paoli ◽  
M. C. Rocca
Keyword(s):  
Hilbert Space ◽  
Rigged Hilbert Space ◽  
Gamow States

Diffraction by a half-plane perpendicular to the distinguished axis of a general gyrotropic medium

1977 ◽  
Vol 55 (4) ◽  
pp. 305-324 ◽  
Author(s):  
S. Przeździecki ◽  
R. A. Hurd
Keyword(s):  
Half Plane ◽  
Fourier Transforms ◽  
Diffraction Problem ◽  
Plane Waves ◽  
Closed Form Solution ◽  
Basic Feature ◽  
Form Solution ◽  
Electromagnetic Plane Wave ◽  
Edge Diffraction ◽  
Special Cases

An exact, closed-form solution is found for the following half-plane diffraction problem: (I) The medium surrounding the half-plane is both electrically and magnetically gyrotropic. (II) The scattering half-plane is perfectly conducting and oriented perpendicular to the distinguished axis of the medium. (III) The incident electromagnetic plane wave propagates in a direction normal to the edge of the half-plane.The formulation of the problem leads to a coupled pair of Wiener–Hopf equations. These had previously been thought insoluble by quadratures, but yield to a newly discovered technique : the Wiener–Hopf–Hilbert method. A basic feature of the problem is its two-mode character i.e. plane waves of both modes are necessary for the spectral representation of the solution. The coupling of these modes is purely due to edge diffraction, there being no reflection coupling. The solution obtained is simple in that the Fourier transforms of the field components are just algebraic functions. Properties of the solution are investigated in some special cases.

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