Counting systems and the First Hilbert problem

The First Hilbert problem is studied in this paper by applying two instruments: a new methodology distinguishing between mathematical objects and mathematical languages used to describe these objects; and a new numeral system allowing one to express different infinite numbers and to use these numbers for measuring infinite sets. Several counting systems are taken into consideration. It is emphasized in the paper that different mathematical languages can describe mathematical objects (in particular, sets and the number of their elements) with different accuracies. The traditional and the new approaches are compared and discussed.     

Switchable solvent based liquid phase microextraction of palladium coupled with determination by flame atomic absorption spectrometry

A switchable solvent-based micro-extraction method for pre-concentration and separation of ultratrace palladium was developed prior to its flame atomic absorption spectrometric detection. Reverse change of hydrophilicity of N,N-dimethylcyclohexylamine (DMA) was achieved by reaction with carbonated water. The hydrophilic bicarbonate salt of the protonated DMA was used as extractant for palladium complexed with 2-(5-bromo-2-pyridylazo)-5-diethylaminophenol (5-Br-PADAP). Formation of the hydrophobic form of switchable solvent from hydrophilic form of switchable solvent phase was achieved by addition of sodium hydroxide into the extractant-sample solution. The effect of key parameters affected extraction recovery was studied and optimised by using Plackett–Burman design, central composite design and three dimension (3D) surfaces response. The calibration plot was linear in the range 0.015–1.6 mg L^?1 of palladium with a correlation coefficient of 0.999. The limit of detections values of palladium for liquid and solid samples were 4.28 μg L^?1 and 0.54 µg g^?1, respectively. The pre-concentration factor was 37.5. The accuracy was confirmed by determination of palladium in certified reference material. The procedure was also applied for determination of palladium content of real samples as automotive catalytic converter, roadside dust, sea water and river water.     

Calibration function for the orbitrap FTMS accounting for the space charge effect

Ion storage in an electrostatic trap has been implemented with the introduction of the Orbitrap Fourier transform mass spectrometer (FTMS), which demonstrates performance similar to high-field ion cyclotron resonance MS. High mass spectral characteristics resulted in rapid acceptance of the Orbitrap FTMS for Life Sciences applications. The basics of Orbitrap operation are well documented; however, like in any ion trap MS technology, its performance is limited by interactions between the ion clouds. These interactions result in ion cloud couplings, systematic errors in measured masses, interference between ion clouds of different size yet with close m/z ratios, etc. In this work, we have characterized the space-charge effect on the measured frequency for the Orbitrap FTMS, looking for the possibility to achieve sub-ppm levels of mass measurement accuracy (MMA) for peptides in a wide range of total ion population. As a result of this characterization, we proposed an m/z calibration law for the Orbitrap FTMS that accounts for the total ion population present in the trap during a data acquisition event. Using this law, we were able to achieve a zero-space charge MMA limit of 80 ppb for the commercial Orbitrap FTMS system and sub-ppm level of MMA over a wide range of total ion populations with the automatic gain control values varying from 10 to 10⁷.     

Solitons in geometric potentials

We show that the geometrically induced potential existing in undulated slab waveguides dramatically affects the properties of solitons. In particular, whereas solitons residing in the potential maxima do not feature power thresholds and are stable, their counterparts residing in the potential minima are unstable and may exhibit a power threshold for their existence. Additionally, the geometric potential is shown to support stable multipole solitons that cannot be supported by straight waveguides. Finally, the geometric potential results in the appearance of the effective barriers that prevent transverse soliton motion.     

Solitons in a medium with linear dissipation and localized gain

We present a variety of dissipative solitons and breathing modes in a medium with localized gain and homogeneous linear dissipation. The system possesses a number of unusual properties, like exponentially localized modes in both focusing and defocusing media, existence of modes in focusing media at negative propagation constant values, simultaneous existence of stable symmetric and antisymmetric localized modes when the gain landscape possesses two local maxima, as well as the existence of stable breathing solutions.     

Rotating vortex solitons supported by localized gain

We show that ringlike localized gain landscapes imprinted in focusing cubic (Kerr) nonlinear media with strong two-photon absorption support new types of stable higher-order vortex solitons containing multiple phase singularities nested inside a single core. The phase singularities are found to rotate around the center of the gain landscape, with the rotation period being determined by the strength of the gain and the nonlinear absorption.     

Stable radially symmetric and azimuthally modulated vortex solitons supported by localized gain

We discover that a spatially localized gain supports stable vortex solitons in media with cubic nonlinearity and two-photon absorption. The interplay between nonlinear losses and gain in amplifying rings results in the suppression of otherwise ubiquitous azimuthal modulation instabilities of radially symmetric vortex solitons. We find that the topology of the gain profile imposes restrictions on the maximal possible charge of vortex solitons. Symmetry breaking occurs at high gain levels, resulting in the formation of necklace vortex solitons composed of asymmetric bright spots.     

The complexity of tropical matrix factorization

The tropical arithmetic operations on R are defined by a⊕b=min{a,b}$a\oplus b=\min \{a,b\}$ and a⊗b=a+b$a\otimes b=a+b$. Let A be a tropical matrix and k   a positive integer, the problem of Tropical Matrix Factorization (TMF) asks whether there exist tropical matrices B∈R^m×k$B\in {\mathbf{R}}^{m\times k}$ and C∈R^k×n$C\in {\mathbf{R}}^{k\times n}$ satisfying B⊗C=A$B\otimes C=A$. We show that the TMF problem is NP-hard for every k≥7$k\ge 7$ fixed in advance, thus resolving a problem proposed by Barvinok in 1993.