Attractors for damped quintic wave equations in bounded domains

The dissipative wave equation with a critical quintic nonlinearity in smooth bounded three dimensional domain is considered. Based on the recent extension of the Strichartz estimates to the case of bounded domains, the existence of a compact global attractor for the solution semigroup of this equation is established. Moreover, the smoothness of the obtained attractor is also shown

Global well-posedness and attractors for the hyperbolic Cahn-Hilliard-Oono equation in the whole space

We prove the global well-posedness of the so-called hyperbolic relaxation of the Cahn-Hilliard-Oono equation in the whole space R^3 with the non-linearity of the sub-quintic growth rate. Moreover, the dissipativity and the existence of a smooth global attractor in the naturally defined energy space is also verified. The result is crucially based on the Strichartz estimates for the linear Scroedinger equation in R^3

Quantifying the structural stability of simplicial homology

The homology groups of a simplicial complex reveal fundamental properties of the topology of the data or the system and the notion of topological stability naturally poses an important yet not fully investigated question. In the current work, we study the stability in terms of the smallest perturbation sufficient to change the dimensionality of the corresponding homology group. Such definition requires an appropriate weighting and normalizing procedure for the boundary operators acting on the Hodge algebra's homology groups. Using the resulting boundary operators, we then formulate the question of structural stability as a spectral matrix nearness problem for the corresponding higher-order graph Laplacian. We develop a bilevel optimization procedure suitable for the formulated matrix nearness problem and illustrate the method's performance on a variety of synthetic quasi-triangulation datasets and transportation networks.Comment: 25 pages, 9 figure

Error Analysis of Explicit Partitioned Runge-Kutta Schemes for Conservation Laws

An error analysis is presented for explicit partitioned Runge-Kutta methods and multirate methods applied to conservation laws. The interfaces, across which different methods or time steps are used, lead to order reduction of the schemes. Along with cell-based decompositions, also flux-based decompositions are studied. In the latter case mass conservation is guaranteed, but it will be seen that the accuracy may deteriorate

CLINICAL RECOMMENDATION TO THE DIAGNOSTICS AND TREATMENT OF HYPOPHOSPHATASIA IN CHILDREN

Hypophosphatasia is a rare genetic disorder caused by deficiency of tissue-specific  alkaline phosphatase as a result of mutations in the ALPL gene. Depending on the form and severity of the disease, pathology may spawn in utero, in childhood or in adult age. Given functions of alkaline phosphatase, patients experience multisystem disorders: primarily changes in bone (osteoporosis, rachitic deformations,  fractures),  lung  disease  (hypoplasia  with respiratory  failure)  and  central  nervous  system  (seizures),  hypercalcemia with development of nephrocalcinosis. Without timely treatment, the disease may be harmful to life in most cases. Patients required observation of a multidisciplinary team of physicians. The only effective treatment is enzyme replacement therapy with asfotase alpha. It is also necessary to carry out symptomatic treatment and rehabilitation of patients with the use of physiotherapy and therapeutic physical training complexes of exercises

Federal Clinical Guidelines on Rendering Help to Children with Hemolytic Uremic Syndrome

The hemolytic uremic syndrome (HUS) is a serious therapeutic problem in pediatrics and pediatric nephrology. HUS is one of the leading causes of acute renal failure with the potential of transforming into terminal chronic renal failure at various periods from the disease onset. The typical form of HUS with a diarrheal prodrome associated with Shiga toxin (STEC) is the most common form. Despite this fact, it requires careful confirmation of infectious etiology to exclude atypical HUS and HUS associated with pneumococcal infection in time. In respect of STEC-HUS it is recommended to conduct adequate symptomatic therapy with a timely dialysis connection if needed. The prognosis here will depend on the anuretic period duration and on the related central nervous system injuries. Atypical HUS is often based on genetic mutations leading to the complement cascade disfunction with uncontrolled activation of the alternative pathway. The overall prognosis for this prone to recurrence form is unfavourable, however, it is recommended to conduct aeculizumabum treatment which will block the terminal components of the complement cascade