Bi-Hamiltonian structures for integrable systems on regular time scales

A construction of the bi-Hamiltonian structures for integrable systems on regular time scales is presented. The trace functional on an algebra of $\delta$-pseudo-differential operators, valid on an arbitrary regular time scale, is introduced. The linear Poisson tensors and the related Hamiltonians are derived. The quadratic Poisson tensors is given by the use of the recursion operators of the Lax hierarchies. The theory is illustrated by $\Delta$-differential counterparts of Ablowitz-Kaup-Newell-Segur and Kaup-Broer hierarchies.Comment: 18 page

Ocena sprawności i jakości życia po paliatywnym zespoleniu złamań z powodu przerzutów nowotworowych do kości

BackgroundPathological fractures in the last phase of a neoplastic disease impair fitness and cause additional physical and mental suffering. Only surgical treatment ensures stabilization of a broken bone and thus lessens the pain and restores the function of the limb. The aim of this study is to evaluate fitness and quality of life after intramedullary fixation stabilizing of pathological fractures.Material and methods12 patients (8 femoral shaft fractures, 2 pertrochanteric fractures, 2 tibial shaft fractures) were treated with this method. In all patients a disseminated neoplastic disease was diagnosed. In 10 patients closed intramedullary fixation with a Kuntscher nail was performed (in 7 patients it was locked in a dynamic manner), in two patients fixation was made with Ender nails. Fitness was evaluated according to the ability to move, to load the operated limb, the range of motion in adjacent joints and muscle atrophy. One month after the operation quality of life was evaluated according to: the ability of self care, self dependence, subjective feeling of pain, necessity of analgesic intake and psychosomatic state.ResultsThree patients achieved preoperative fitness, despite worsened of fitness 6 patients were able to function independently and three patients required continuous care. All patients had to take analgesics, relief in severe pain typical of fracture was observed, but chronic pain typical of metastases was still present. The patients died within 2 to 16 months after the operation.ConclusionsPalliative intramedullary bone fixation in the last phase of a neoplastic disease makes possible stabilization of pathological fractures and is sufficient for the patient to function independently.WstępZłamania patologiczne w okresie terminalnym choroby nowotworowej upośledzają sprawność i powodują dodatkowe cierpienia fizyczne i psychiczne. Tylko leczenie operacyjne zapewnia stabilizację złamanej kości, a tym samym zmniejszenie bólu i przywrócenie funkcji kończyny. Celem pracy jest ocena sprawności i jakości życia po zespoleniu śródszpikowym stabilizującym złamanie patologiczne.Materiał i metodaTym sposobem leczono 12 chorych (8 – złamania trzonu kości udowej, 2 – złamania przezkrętarzowe, 2 – złamania trzonu kości piszczelowej). U wszystkich chorych stwierdzono rozsianą chorobę nowotworową. U 10 chorych zastosowano zamknięte zespolenie śródszpikowe gwoździem Kuntschera (w tym u 7 ryglowane sposobem dynamicznym), u 2 chorych zespolenie gwoździami Endera. Oceniono sprawność uwzględniając: zdolność poruszania się, obciążanie operowanej kończyny, zakres ruchów w sąsiednich stawach, zaniki mięśniowe. W miesiąc po operacji badano jakość życia oceniając: samodzielność, samoobsługę, subiektywną ocenę bólu, konieczność przyjmowania leków przeciwbólowych oraz stan psychosomatyczny.Wyniki3 chorych osiągnęło sprawność sprzed złamania, 6 pacjentów mimo pogorszenia sprawności było zdolnych do samodzielnego funkcjonowania, 3 chorych wymagało ciągłej opieki. Wszyscy chorzy musieli otrzymywać leki przeciwbólowe, obserwowano ustąpienie ostrych dolegliwości typowych dla złamania, natomiast utrzymywały się przewlekłe bóle charakterystyczne dla obecności przerzutów. Chorzy zmarli w 2 do 16 miesięcy od operacji.WnioskiPaliatywne zespolenie śródszpikowe w okresie schyłkowym choroby nowotworowej pozwala ustabilizować złamania patologiczne w sposób wystarczający dla samodzielnego funkcjonowania

The molecular pattern of histopathological progression to anaplastic meningioma – A case report

Meningiomas (MGs) are the most frequent primary tumours of the central nervous system (CNS) and exhibit a large spectrum of histological types and clinical phenotypes. The WHO classification of CNS tumours established strict diagnostic criteria of the benign (Grade 1), atypical (Grade 2) and anaplastic (Grade 3) subtypes. Combined with the resection rate, WHO grading has the most crucial role as the prognostic factor. Additionally, such biomarkers as Ki-67/MIB-1, progesterone receptors and phosphor-histone H3 were correlated with MG progression. Recently, it was suggested that the aggressive behaviour of some MGs is attributed to molecular alterations, regardless of their histopathology. The analysis of loss of heterozygosity (LOH) at chromosomes 1, 9, 10, 14 and 22 was performed. The presented case of WHO Grade 2 MG initially exhibited LOH at chromosomes 10, 14 and 22. In the first recurrence, the tumour genetic profiling revealed additional LOH at chromosome 1p and atypical histopathology. During the second recurrence, an aggressive phenotype was observed and tumour progressed to an anaplastic form. Considering the appearance of the tumour relapses, the set of molecular changes overtook the histopathological progression. The genetic and histopathological imbalance in the tumour progression in secondary anaplastic MGs has not been previously described. The evolution of genetic and histopathological changes was presented in the same patient. In the future, the individualised therapy of potentially more aggressive forms of MGs could be based on certain chromosome aberrations

R-matrix approach to integrable systems on time scales

A general unifying framework for integrable soliton-like systems on time scales is introduced. The $R$-matrix formalism is applied to the algebra of $\delta$-differential operators in terms of which one can construct infinite hierarchy of commuting vector fields. The theory is illustrated by two infinite-field integrable hierarchies on time scales which are difference counterparts of KP and mKP. The difference counterparts of AKNS and Kaup-Broer soliton systems are constructed as related finite-field restrictions.Comment: 21 page

Investigation of the cosmic ray population and magnetic field strength in the halo of NGC 891

Context. Cosmic rays and magnetic fields play an important role for the formation and dynamics of gaseous halos of galaxies. Aims. Low-frequency radio continuum observations of edge-on galaxies are ideal to study cosmic-ray electrons (CREs) in halos via radio synchrotron emission and to measure magnetic field strengths. Spectral information can be used to test models of CRE propagation. Free-free absorption by ionized gas at low frequencies allows us to investigate the properties of the warm ionized medium in the disk. Methods. We obtained new observations of the edge-on spiral galaxy NGC 891 at 129-163 MHz with the LOw Frequency ARray (LOFAR) and at 13-18 GHz with the Arcminute Microkelvin Imager (AMI) and combine them with recent high-resolution Very Large Array (VLA) observations at 1-2 GHz, enabling us to study the radio continuum emission over two orders of magnitude in frequency. Results. The spectrum of the integrated nonthermal flux density can be fitted by a power law with a spectral steepening towards higher frequencies or by a curved polynomial. Spectral flattening at low frequencies due to free-free absorption is detected in star-forming regions of the disk. The mean magnetic field strength in the halo is 7 +/- 2 mu G. The scale heights of the nonthermal halo emission at 146 MHz are larger than those at 1.5 GHz everywhere, with a mean ratio of 1.7 +/- 0.3, indicating that spectral ageing of CREs is important and that diffusive propagation dominates. The halo scale heights at 146 MHz decrease with increasing magnetic field strengths which is a signature of dominating synchrotron losses of CREs. On the other hand, the spectral index between 146 MHz and 1.5 GHz linearly steepens from the disk to the halo, indicating that advection rather than diffusion is the dominating CRE transport process. This issue calls for refined modelling of CRE propagation. Conclusions. Free-free absorption is probably important at and below about 150 MHz in the disks of edge-on galaxies. To reliably separate the thermal and nonthermal emission components, to investigate spectral steepening due to CRE energy losses, and to measure magnetic field strengths in the disk and halo, wide frequency coverage and high spatial resolution are indispensable

Classical R-matrix theory for bi-Hamiltonian field systems

The R-matrix formalism for the construction of integrable systems with infinitely many degrees of freedom is reviewed. Its application to Poisson, noncommutative and loop algebras as well as central extension procedure are presented. The theory is developed for (1+1)-dimensional case where the space variable belongs either to R or to various discrete sets. Then, the extension onto (2+1)-dimensional case is made, when the second space variable belongs to R. The formalism presented contains many proofs and important details to make it self-contained and complete. The general theory is applied to several infinite dimensional Lie algebras in order to construct both dispersionless and dispersive (soliton) integrable field systems.Comment: review article, 39 page

Satisfying positivity requirement in the Beyond Complex Langevin approach

The problem of finding a positive distribution, which corresponds to a given complex density, is studied. By the requirement that the moments of the positive distribution and of the complex density are equal, one can reduce the problem to solving the matching conditions. These conditions are a set of quadratic equations, thus Groebner basis method was used to find its solutions when it is restricted to a few lowest-order moments. For a Gaussian complex density, these approximate solutions are compared with the exact solution, that is known in this special case