For a t-norm T on a bounded lattice (L,≤), a partial order ≤T was recently defined and studied. In \cite{Karacal11}, it was pointed out that the binary relation ≤T is a partial order on L, but (L,≤T) may not be a lattice in general. In this paper, several sufficient conditions under which (L,≤T) is a lattice are given, as an answer to an open problem posed by the authors of \cite{Karacal11}. Furthermore, some examples of t-norms on L such that (L,≤T) is a lattice are presented.
Diabetes mellitus is associated with many complications including retinopathy, nephropathy, neuropathy and angiopathy. Increased cardiovascular risk is accompanied with diabetes-induced endothelial dysfunction. Pharmacological agents with endothelium-protective effects may decrease cardiovascular complications. In present study sulodexide (glycosaminoglycans composed from heparin-like and dermatan fractions) was chosen to evaluate its protective properties on endothelial dysfunction in diabetes. Effect of sulodexide treatment (SLX, 100 UI/kg/day, i.p.) in 5 and 10 weeks lasting streptozotocin-induced diabetes (30 mg/kg/day, i.p. administered for three consecutive days) was investigated. Animals were divided into four groups: control (injected with saline solution), control-treated with sulodexide (SLX), diabetic (DM) and diabetic-treated with sulodexide (DM+SLX). The pre-prandial and postprandial plasma glucose levels, number of circulating endothelial cells (EC) and acetylcholine-induced relaxation of isolated aorta and mesenteric artery were evaluated. Streptozotocin elicited hyperglycemia irrespective of SLX treatment. Streptozotocin-induced diabetes enhanced the number of circulating endothelial cells compared to controls. SLX treatment decreased the number of EC in 10-week diabetes. Acetylcholine-induced relaxation of mesenteric arteries was significantly impaired in 5 and 10-week diabetes. SLX administration improved relaxation to acetylcholine in 5 and 10-week diabetes. Diabetes impaired acetylcholine-induced relaxation of rat aorta irrespective of SLX treatment. Our results demonstrate that SLX treatment lowers the number of circulating endothelial cells and improves endothelium-dependent relaxation in small arteries. These findings suggest endothelium-protective effect of sulodexide in streptozotocin-induced diabetes., V. Kristová, S. Líšková, R. Sotníková, R. Vojtko, A. Kurtanský., and Obsahuje bibliografii a bibliografické odkazy
Let $n > m \geq 2$ be positive integers and $n=(m+1) \ell +r$, where $0 \leq r \leq m.$ Let $C$ be a subset of $\{0,1,\cdots ,n\}$. We prove that if $$ |C|>\begin {cases} \lfloor n/2 \rfloor +1 &\text {if $m$ is odd}, \\ m \ell /2 +\delta &\text {if $m$ is even},\\ \end {cases} $$ where $\lfloor x \rfloor $ denotes the largest integer less than or equal to $x$ and $\delta $ denotes the cardinality of even numbers in the interval $[0,\min \{r,m-2\}]$, then $C-C$ contains a power of $m$. We also show that these lower bounds are best possible.
An observer for a system with polynomial nonlinearities is designed. The system is assumed to exhibit a time delay whose value is supposed to be constant and known. The design is carried out using the sum-of-squares method. The key point is defining a suitable Lyapunov-Krasovskii functional. The resulting observer is in form of a polynomial in the observable variables. The results are illustrated by two examples.