Two-frequency radiative transfer (2f-RT) theory is developed for geometrical optics in random media. The space-frequency correlation is described by the two-frequency Wigner distribution (2f-WD) which satisfies a closed-form equation, the two-frequency Wigner-Moyal equation. In the RT regime it is proved rigorously that 2f-WD satisfies a Fokker-Planck–like equation with complex-valued coefficie...

and Applied Analysis 3 for all x1, . . . , x2n ∈ V if and only if the odd mapping f : V → W is Cauchy additive, that is, f ( x y ) f x f ( y ) , 2.2 for all x, y ∈ V . Proof. Assume that f : V → W satisfies 2.1 . Letting x1 · · · xn x, xn 1 · · · x2n y in 2.1 , we get nf ( x − x y 2 ) nf ( y − x y 2 ) nf x nf ( y ) − 2nf ( x y 2 ) , 2.3 for all x, y ∈ V . Since f : V → W is odd, 0 nf x nf ( y )...

Discrimination studies suggest that two, and only two, channels encode relative spatial phase shifts in compound gratings (Bennett & Banks, 1991; Field & Nachmias, 1984). The more sensitive channel consists of even-symmetric filters and responds best to cosine phase shifts (e.g., 0 degree-180 degrees); the other consists of odd-symmetric filters and responds best to sine phase shifts (e.g., 90 ...

In 1933 Godel introduced a modal logic of provability (S4) and left open the problem of a formal provability semantics for this logic. Since then numerous attempts have been made to give an adequate provability semantics to Godel's provability logic with only partial success. In this paper we give the complete solution to this problem in the Logic of Proofs (LP). LP implements Godel's sugges...

The AIDS pandemic affects over 35 million people worldwide. Human immunodeficiency types 1 and 2 (HIV-1 and HIV-2), the etiologic agents of AIDS, originated from simian immunodeficiency viruses of chimpanzees (SIVcpz) in Cameroon and SIVsm from sooty mangabeys in West Africa. While the simian origin of HIV is well established, how the virus adapted to humans is poorly understood. The bulk of HI...

A multipole representation of the experimental electron density distribution for the high-pressure silica polymorph coesite, using Hirshfeld-type radial functions, has been generated with single-crystal X-ray diffraction data recorded to a sin θmax/λ value of 1.21 Å-1 at 100 K. Unlike an earlier modeling of the distribution, where a more limited data set was analyzed, deformation electron densi...

In this paper, we establish the generalized Hyers–Ulam–Rassias stability of C-ternary ring homomorphisms associated to the Trif functional equation d · C d−2f( x1 + · · ·+ xd d ) + C d−2 d ∑