Arbitrary Trace Research Articles

Kernels of trace class operators

Let X ⊂ R n X \subset {{\mathbf {R}}^n} and let K K be a trace class operator on L 2 ( X ) {L^2}(X) with corresponding kernel K ( x , y ) ∈ L 2 ( X × X ) K(x,y) \in {L^2}(X \times X) . An integral formula for tr K K , proven by Duflo for continuous kernels, is generalized for arbitrary trace class kernels. This formula is shown to be equivalent to one involving the factorization of K K into a product of Hilbert-Schmidt operators. The formula and its derivation yield two new necessary conditions for traceability of a Hilbert-Schmidt kernel, and these conditions are also shown to be sufficient for positive operators. The proofs make use of the boundedness of the Hardy-Littlewood maximal function on L 2 ( R n ) {L^2}({{\mathbf {R}}^n}) .

A THEORY OF ACOUSTIC DIFFRACTORS APPLIED TO 2‐D MODELS*

AbstractFinite‐offset seismic reflection modeling of acoustic waves, propagating in a two‐dimensional depth section of arbitrary complexity, is discussed. The procedure developed employs the principles of simplified (far‐field) diffractor theory and ray tracing. Each reflector is represented by a set of discrete secondary sources or diffractors and the wavefield associated with each diffractor is calculated directly in the time domain by ray tracing. Reflections and diffractions are subsequently built up by the numerical superposition of these wavefields. This superposition is nondispersive for all frequencies for which the Fresnel zones are large compared with the diffractor separation.All primary travel paths connecting the shot to diffractor and diffractor to geophone are accounted for together with phase changes induced by focal events. The method allows the modeling of arbitrary trace gathers for energy originating from selected reflectors. The nonsequential nature of the algorithm makes it suited to machines capable of carrying out many similar operations in parallel or concurrently. Diffractor theory also provides physical insight into wave scattering and focusing. In particular, the half‐differential waveform associated with a line diffractor leads to an explanation of the 90° phase lead induced by a cylindrical focus and, similarly, the full differential waveform of a point diffractor can be used to explain the 180° phase shift induced by a point focus.

Duffin-Kemmer-Petiau subalgebras: Representations and applications

A reduction of the Duffin-Kemmer-Petiau algebra to a direct sum of irreducible subalgebras for spin-0 and spin-1 bosons is presented. The subalgebras are defined by multiplication rules for the linearly independent basis elements. In the representations discussed the spin projection operators are independent basis elements of the subalgebras. The formal utility of these representations is demonstrated by obtaining the reduction of arbitrary operator products and trace theorems. The practical utility is demonstrated by application to the analysis of free and interacting boson field currents. Most importantly, one can understand the differences between DKP nonconserved currents and those obtained from second-order wave equations.