On the structure of the set of bifurcation points of periodic solutions for multiparameter Hamiltonian systems

This paper deals with periodic solutions of the Hamilton equation with many parameters. Theorems on global bifurcation of solutions with periods $2\pi/j,$ $j\in\mathbb{N},$ from a stationary point are proved. The Hessian matrix of the Hamiltonian at the stationary point can be singular. However, it is assumed that the local topological degree of the gradient of the Hamiltonian at the stationary point is nonzero. It is shown that (global) bifurcation points of solutions with given periods can be identified with zeros of appropriate continuous functions on the space of parameters. Explicit formulae for such functions are given in the case when the Hessian matrix of the Hamiltonian at the stationary point is block-diagonal. Symmetry breaking results concerning bifurcation of solutions with different minimal periods are obtained. A geometric description of the set of bifurcation points is given. Examples of constructive application of the theorems proved to analytical and numerical investigation and visualization of the set of all bifurcation points in given domain are provided. This paper is based on a part of the author's thesis [W. Radzki, ``Branching points of periodic solutions of autonomous Hamiltonian systems'' (Polish), PhD thesis, Nicolaus Copernicus University, Faculty of Mathematics and Computer Science, Toru\'{n}, 2005].Comment: 35 pages, 4 figures, PDFLaTe

Multiplicity of periodic solutions for systems of weakly coupled parametrized second order differential equations

We prove a multiplicity result of periodic solutions for a system of second order differential equations having asymmetric nonlinearities. The proof is based on a recent generalization of the Poincar\ue9\u2013Birkhoff fixed point theorem provided by Fonda and Ure\uf1a

Syk activation in circulating and tissue innate immune cells in antineutrophil cytoplasmic antibody-associated vasculitis

OBJECTIVE: Syk is a cytoplasmic protein tyrosine kinase that plays a role in signaling via B cell and Fc receptors (FcR). FcR engagement and signaling via Syk is thought to be important in antineutrophil cytoplasm antibody (ANCA) IgG-mediated neutrophil activation. This study was undertaken to investigate the role of Syk in ANCA-induced myeloid cell activation and vasculitis pathogenesis. METHODS: Phosphorylation of Syk in myeloid cells from healthy controls and ANCA-associated vasculitis (AAV) patients was analyzed using flow cytometry. The effect of Syk inhibition on myeloperoxidase (MPO)-ANCA IgG activation of cells was investigated using functional assays (interleukin-8 and reactive oxygen species production) and targeted gene analysis with NanoString. Total and phosphorylated Syk at sites of tissue inflammation in patients with AAV was assessed using immunohistochemistry and RNAscope in situ hybridization. RESULTS: We identified increased phosphorylated Syk at critical activatory tyrosine residues in blood neutrophils and monocytes from patients with active AAV compared to patients with disease in remission or healthy controls. Syk was phosphorylated in vitro following MPO-ANCA IgG stimulation, and Syk inhibition was able to prevent ANCA-mediated cellular responses. Using targeted gene expression analysis, we identified up-regulation of FcR- and Syk-dependent signaling pathways following MPO-ANCA IgG stimulation. Finally, we showed that Syk is expressed and phosphorylated in tissue leukocytes at sites of organ inflammation in AAV. CONCLUSION: These findings indicate that Syk plays a critical role in MPO-ANCA IgG-induced myeloid cell responses and that Syk is activated in circulating immune cells and tissue immune cells in AAV; therefore, Syk inhibition may be a potential therapeutic option

Existence, regularity and boundary behaviour of bounded variation solutions of a one-dimensional capillarity equation

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Dynamics of generalized PT-symmetric dimers with time-periodic gain–loss

A parity-time (PT)-symmetric system with periodically varying-in-time gain and loss modeled by two coupled Schrödinger equations (dimer) is studied. It is shown that the problem can be reduced to a perturbed pendulum-like equation. This is done by finding two constants of motion. Firstly, a generalized problem using Melnikov-type analysis and topological degree arguments is studied for showing the existence of periodic (libration), shift- periodic (rotation), and chaotic solutions. Then these general results are applied to the PT-symmetric dimer. It is interestingly shown that if a sufficient condition is satisfied, then rotation modes, which do not exist in the dimer with constant gain–loss, will persist. An approximate threshold for PT-broken phase corresponding to the disappearance of bounded solutions is also presented. Numerical study is presented accompanying the analytical results

Longitudinal proteomic profiling of dialysis patients with COVID-19 reveals markers of severity and predictors of death

End-stage kidney disease (ESKD) patients are at high risk of severe COVID-19. We measured 436 circulating proteins in serial blood samples from hospitalised and non-hospitalised ESKD patients with COVID-19 (n=256 samples from 55 patients). Comparison to 51 non-infected patients revealed 221 differentially expressed proteins, with consistent results in a separate subcohort of 46 COVID-19 patients. 203 proteins were associated with clinical severity, including IL6, markers of monocyte recruitment (e.g. CCL2, CCL7), neutrophil activation (e.g. proteinase-3) and epithelial injury (e.g. KRT19). Machine learning identified predictors of severity including IL18BP, CTSD, GDF15, and KRT19. Survival analysis with joint models revealed 69 predictors of death. Longitudinal modelling with linear mixed models uncovered 32 proteins displaying different temporal profiles in severe versus non-severe disease, including integrins and adhesion molecules. These data implicate epithelial damage, innate immune activation, and leucocyte-endothelial interactions in the pathology of severe COVID-19 and provide a resource for identifying drug targets