Measuring the complete force field of an optical trap

The use of optical traps to measure or apply forces on the molecular level requires a precise knowledge of the trapping force field. Close to the trap center, this field is typically approximated as linear in the displacement of the trapped microsphere. However, applications demanding high forces at low laser intensities can probe the light-microsphere interaction beyond the linear regime. Here, we measured the full nonlinear force and displacement response of an optical trap in two dimensions using a dual-beam optical trap setup with back-focal-plane photodetection. We observed a substantial stiffening of the trap beyond the linear regime that depends on microsphere size, in agreement with Mie theory calculations. Surprisingly, we found that the linear detection range for forces exceeds the one for displacement by far. Our approach allows for a complete calibration of an optical trap.     

On cubic surfaces with a rational line

We report on our project to construct non-singular cubic surfaces over \mathbbQ{\mathbb{Q}} with a rational line. Our method is to start with degree 4 Del Pezzo surfaces in diagonal form. For these, we develop an explicit version of Galois descent.     

On the order three Brauer classes for cubic surfaces

We describe a method to compute the Brauer-Manin obstruction for smooth cubic surfaces over ℚ such that Br(S)/Br(ℚ) is a 3-group. Our approach is to associate a Brauer class with every ordered triplet of Galois invariant pairs of Steiner trihedra. We show that all order three Brauer classes may be obtained in this way. To show the effect of the obstruction, we give explicit examples.     

Organically bound sulfur in refractory organic substances

The sulfur compounds of refractory organic substances (ROS) of different origin have been characterized. Total organic sulfur was determined by elemental analysis. Sulfur-containing amino acids methionine and cystine were analyzed chromatographically after hydrolysis with HCl or by proteolytic digestion using enzymes. The results obtained from elemental analysis show that the total amount of sulfur is strongly dependent on the origin of the samples, because of different environmental factors during the formation of ROS. For naturally occurring samples isolated from soil seepage water, bog lake water and ground water the carbon-to-sulfur atomic ratios (C/S) decrease with the stage of humification, because of preferential loss of carbon. In humic acids (HA) isolated from secondary effluent the high value of the nitrogen-to-sulfur ratio (N/S) was indicative of a large amount of protein-derived nitrogen and sulfur compounds. In the solutions from acid hydrolysis the total amount of amino acid carbon related to the dissolved organic carbon (DOC) was generally less than 5%. Percentages of cystine related to all the amino acids detected were in the range 4 to 16%; methionine was below the detection limit for most samples. The results show that cystine is very important among the amino acids released. Enzymatic release generally resulted in smaller amounts of amino acids, indicating that these molecules are not only present in bioavailable protein-like structures. The data were compared with those from other approaches reported in the literature for the speciation of sulfur forms in ROS, including potentiometric titration, differential reduction methods, and spectroscopic investigations.     

Phase transitions for a model with uncountable spin space on the Cayley tree: the general case

Positivity - In this paper we complete the analysis of a statistical mechanics model on Cayley trees of any degree, started in Botirov (Positivity 21(3):955–961, 2017), Eshkabilov et al. (J...     

Cubic surfaces violating the Hasse principle are Zariski dense in the moduli scheme

We construct new examples of cubic surfaces, for which the Hasse principle fails. Thereby we show that, over every number field, the counterexamples to the Hasse principle are Zariski dense in the moduli scheme of non-singular cubic surfaces.     

On the computation of the Picard group for certain singular quartic surfaces

We test the methods for computing the Picard group of a K3 surface in a situation of high rank. The examples chosen are resolutions of quartics in P ³ having 14 singularities of type A ₁. Our computations show that the method of R. van Luijk works well when sufficiently large primes are used.     

Estimates for Tamagawa numbers of diagonal cubic surfaces

For diagonal cubic surfaces, we give an upper bound for E. Peyre's Tamagawa type number in terms of the coefficients of the defining equation. This bound shows that the reciprocal admits a fundamental finiteness property on the set of all diagonal cubic surfaces. As an application, we show that the infinite series of Tamagawa numbers related to the Fano cubic bundles considered by Batyrev and Tschinkel (1996) [BT] are indeed convergent.