Unbounded C<SUP>&#x002A;</SUP>-seminorms and unbounded C<SUP>&#x002A;</SUP>-spectral algebras

Several &#x002A;-algebras A&#167; carry with them unbounded C&#x002A;-seminorms in the s ense that they are C&#x002A;-seminorms defined on &#x002A;-subalgebras. Unbounded operator representations of A are constructed from such unbounded C&#x002A;-seminorms and they are investigated. The notions of spectrality and sta bility of unbounded C&#x002A;-seminorms are defined and studied

Conditional Expectations for Unbounded Operator Algebras

Two conditional expectations in unbounded operator algebras (O∗-algebras) are discussed. One is a vector conditional expectation defined by a linear map of an O∗-algebra into the Hilbert space on which the O∗-algebra acts. This has the usual properties of conditional expectations. This was defined by Gudder and Hudson. Another is an unbounded conditional expectation which is a positive linear map ℰ of an O∗-algebra ℳ onto a given O∗-subalgebra of ℳ. Here the domain D(ℰ) of ℰ does not equal to ℳ in general, and so such a conditional expectation is called unbounded

Ultrasonic diffraction from a transducer with arbitrary geometry and strength distribution

The exact solution to the Helmholtz equation with a Dirichlet boundary condition is obtained to study three-dimensional ultrasonic diffraction phenomena and derive the numerical data of amplitude loss and the phase shift for correcting induced errors. Calculation is made for near-field diffraction, for the rectangular transducers, and for the transducers with strength distribution on the radiating area. In the near field, where the wavelength and the propagation distance are comparable with each other, the longitudinal and shear waves undergo different diffraction. For transducers having a noncircular shape and a strength distribution on the area, both the amplitude loss and the phase shift experience different tendencies from the classical work on the circular piston source. Use of diffraction data specific to each measurement condition is then necessary to correct the errors. The calculated results are verified for pulse-echo measurements using a shear-wave electromagnetic acoustic transducer.Hirotsugu Ogi, Masahiko Hirao, and Takashi Honda. Ultrasonic diffraction from a transducer with arbitrary geometry and strength distribution. Journal of the Acoustical Society of America, 1995, 98(2), 1191. https://doi.org/10.1121/1.413617