Solution to a parabolic differential equation in Hilbert space via Feynman formula - parts I and II

A parabolic partial differential equation $u'_t(t,x)=Lu(t,x)$ is considered, where $L$ is a linear second-order differential operator with time-independent coefficients, which may depend on $x$. We assume that the spatial coordinate $x$ belongs to a finite- or infinite-dimensional real separable Hilbert space $H$. Assuming the existence of a strongly continuous resolving semigroup for this equation, we construct a representation of this semigroup by a Feynman formula, i.e. we write it in the form of the limit of a multiple integral over $H$ as the multiplicity of the integral tends to infinity. This representation gives a unique solution to the Cauchy problem in the uniform closure of the set of smooth cylindrical functions on $H$. Moreover, this solution depends continuously on the initial condition. In the case where the coefficient of the first-derivative term in $L$ vanishes we prove that the strongly continuous resolving semigroup exists (this implies the existence of the unique solution to the Cauchy problem in the class mentioned above) and that the solution to the Cauchy problem depends continuously on the coefficients of the equation.Comment: This is a more or less stable version of the tex

Chernoff approximations of Feller semigroups in Riemannian manifolds

Chernoff approximations of Feller semigroups and the associated diffusion processes in Riemannian manifolds are studied. The manifolds are assumed to be of bounded geometry, thus including all compact manifolds and also a wide range of non-compact manifolds. Sufficient conditions are established for a class of second order elliptic operators to generate a Feller semigroup on a (generally non-compact) manifold of bounded geometry. A construction of Chernoff approximations is presented for these Feller semigroups in terms of shift operators. This provides approximations of solutions to initial value problems for parabolic equations with variable coefficients on the manifold. It also yields weak convergence of a sequence of random walks on the manifolds to the diffusion processes associated with the elliptic generator. For parallelizable manifolds this result is applied in particular to the representation of Brownian motion on the manifolds as limits of the corresponding random walks.Comment: 36 pages, no figures. Title and abstract changed. Minor corrections, some proposition generalise

The system of osteoprotegrin (OPG)/ligand of NF-kB receptor activator (RANKL) in patients with diabetes mellitus, mediacalcinosis and obliteratingatherosclerosis of lower leg arteries

Aim. To study the OPG/RANKL system in patients with diabetes mellitus, mediacalcinosis and obliterating atherosclerosis of lower leg arteries. Materials and methods. The study enrolled 70 patients including 20 with manifest diabetic neuropathy (DN) and mediacalcinosis of lower leg arteries(group 1). 29 patients with diabetes mellitus (DM) and clinical manifestations of obliterating atherosclerosis of lower leg arteries comprised group 2. Thecontrol group 3 consisted of 30 subjects without disturbances of carbohydrate metabolism. Immunoassays with Alkphase-B (Metra biosystems, USA),Osteoprotegrin (Biomedica, Austria), and sRANKL(Biomedica, Austria) kits were used to detect serum markers of bone formation (alkaline phosphatase(AP), OPG, and RANKL respectively). The patients underwent examination by digital X-ray of affected joints in frontal and lateral projections using anAxiom Iconos R 200 apparatus (Siemens, Germany). Bone mineral density (BMD) was measured by dual energy X-ray absorptiometry (Expert 1188,Lunar, USA). Ultrasound duplex scanning of lower leg arteries was performed with Sonoc-5500 (Agilent, USA). Results. OPG levels in diabetic patients were significantly higher than in controls (

Sakharnyy diabet 2 tipa u detey i podrostkov

До недавнего времени практически во всех случаях сахарного диабета (СД) у детей и подростков характеризовался как инсулиндефицитный. Полагали, что на других (редких) форм диабета приходится менее 1-2%. В настоящее время сообщается, что у 8-45% больных (в зависимости от популяции) имеют место неаутоиммунные формы СД. Среди неаутоиммунных форм заболевания наиболее часто встречаются СД 2 типа, MODY и атипичный СД молодых (АСД). В настоящей статье обобщены современные данные о распространенности и оптимальных подходах к дифференциальной диагностике и лечению СД 2 типа в детском возрасте