Fluorescence tomographic imaging in turbid media using early-arriving photons and Laplace transforms

We present a multichannel tomographic technique to detect fluorescent objects embedded in thick (6.4 cm) tissue-like turbid media using early-arriving photons. The experiments use picosecond laser pulses and a streak camera with single photon counting capability to provide short time resolution and high signal-to-noise ratio. The tomographic algorithm is based on the Laplace transform of an analytical diffusion approximation of the photon migration process and provides excellent agreement between the actual positions of the fluorescent objects and the experimental estimates. Submillimeter localization accuracy and 4- to 5-mm resolution are demonstrated. Moreover, objects can be accurately localized when fluorescence background is present. The results show the feasibility of using early-arriving photons to image fluorescent objects embedded in a turbid medium and its potential in clinical applications such as breast tumor detection.

Fibroblast Growth Factor (FGF) Soluble Receptor 1 Acts as a Natural Inhibitor of FGF2 Neurotrophic Activity during Retinal Degeneration

Fibroblast growth factors (FGF) 1 and 2 and their tyrosine kinase receptor (FGFR) are present throughout the adult retina. FGFs are potential mitogens, but adult retinal cells are maintained in a nonproliferative state unless the retina is damaged. Our work aims to find a modulator of FGF signaling in normal and pathological retina. We identified and sequenced a truncated FGFR1 form from rat retina generated by the use of selective polyadenylation sites. This 70-kDa form of soluble extracellular FGFR1 (SR1) was distributed mainly localized in the inner nuclear layer of the retina, whereas the full-length FGFR1 form was detected in the retinal Muller glial cells. FGF2 and FGFR1 mRNA levels greatly increased in light-induced retinal degeneration. FGFR1 was detected in the radial fibers of activated retinal Muller glial cells. In contrast, SR1 mRNA synthesis followed a biphasic pattern of down- and up-regulation, and anti-SR1 staining was intense in retinal pigmented epithelial cells. The synthesis of SR1 and FGFR1 specifically and independently regulated in normal and degenerating retina suggests that changes in the proportion of various FGFR forms may control the bioavailability of FGFs and thus their potential as neurotrophic factors. This was demonstrated in vivo during retinal degeneration when recombinant SR1 inhibited the neurotrophic activity of exogenous FGF2 and increased damaging effects of light by inhibiting endogenous FGF. This study highlights the significance of the generation of SR1 in normal and pathological conditions.

New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan’s tau function

In this paper, we give two infinite families of explicit exact formulas that generalize Jacobi’s (1829) 4 and 8 squares identities to 4n2 or 4n(n + 1) squares, respectively, without using cusp forms. Our 24 squares identity leads to a different formula for Ramanujan’s tau function τ(n), when n is odd. These results arise in the setting of Jacobi elliptic functions, Jacobi continued fractions, Hankel or Turánian determinants, Fourier series, Lambert series, inclusion/exclusion, Laplace expansion formula for determinants, and Schur functions. We have also obtained many additional infinite families of identities in this same setting that are analogous to the η-function identities in appendix I of Macdonald’s work [Macdonald, I. G. (1972) Invent. Math. 15, 91–143]. A special case of our methods yields a proof of the two conjectured [Kac, V. G. and Wakimoto, M. (1994) in Progress in Mathematics, eds. Brylinski, J.-L., Brylinski, R., Guillemin, V. & Kac, V. (Birkhäuser Boston, Boston, MA), Vol. 123, pp. 415–456] identities involving representing a positive integer by sums of 4n2 or 4n(n + 1) triangular numbers, respectively. Our 16 and 24 squares identities were originally obtained via multiple basic hypergeometric series, Gustafson’s Cℓ nonterminating 6φ5 summation theorem, and Andrews’ basic hypergeometric series proof of Jacobi’s 4 and 8 squares identities. We have (elsewhere) applied symmetry and Schur function techniques to this original approach to prove the existence of similar infinite families of sums of squares identities for n2 or n(n + 1) squares, respectively. Our sums of more than 8 squares identities are not the same as the formulas of Mathews (1895), Glaisher (1907), Ramanujan (1916), Mordell (1917, 1919), Hardy (1918, 1920), Kac and Wakimoto, and many others.