Correction of diffraction effects in confocal raman microspectroscopy

A mathematical approach developed to correct depth profiles of wet-chemically modified polymer films obtained by confocal Raman microscopy is presented which takes into account scattered contributions originated from a diffraction-limited laser focal volume. It is demonstrated that the problem can be described using a linear Fredholm integral equation of the first kind which correlates apparent and true Raman intensities with the depth resolution curve of the instrument. The calculations of the corrected depth profiles show that considerable differences between apparent and corrected depth profiles exist at the surface, especially when profiles with strong concentration gradients are dealt with or an instrument with poor depth resolution is used. Degrees of modification at the surface obtained by calculation of the corrected depth profiles are compared with those measured by FTIR-ATR and show an excellent concordance.</p

Artin-Schreier, Erdős, and Kurepa\u27s conjecture

We discuss possible generalizations of Erdŏs\u27s problem about factorials in Fp to the Artin-Schreier extension Fpp of Fp. The generalizations are related to Bell numbers in Fp and to Kurepa\u27s conjecture

Fixed points of the sum of divisors function on ({{mathbb{F}}}_2[x])

We work on an analogue of a classical arithmetic problem over polynomials. More precisely, we study the fixed points (F) of the sum of divisors function (sigma : {mathbb{F}}_2[x] mapsto {mathbb{F}}_2[x]) (defined mutatis mutandi like the usual sum of divisors over the integers) of the form (F := A^2 cdot S), (S) square-free, with (omega(S) leq 3), coprime with (A), for (A) even, of whatever degree, under some conditions. This gives a characterization of (5) of the (11) known fixed points of (sigma) in ({mathbb{F}}_2[x])

Under a mild condition, Ryser\u27s Conjecture holds for every ( n:= 4h^2) with h>1 odd and non square-free

We prove, under a mild condition, that there is no circulant Hadamard matrix ( H) with (n >4) rows when (sqrt{n/4}) is not square-free. The proof introduces a new method to attack Ryser\u27s Conjecture, that is a long standing difficult conjecture