Asymptotic pairs, stable sets and chaos in positive entropy systems

We consider positive entropy $G$-systems for certain countable, discrete, infinite left-orderable amenable groups $G$. By undertaking local analysis, the existence of asymptotic pairs and chaotic sets will be studied in connecting with the stable sets. Examples are given for the case of integer lattice groups, the Heisenberg group, and the groups of integral unipotent upper triangular matrices

A Connection Behind the Terwilliger Algebras of H(D,2)H(D,2) and 12H(D,2)\frac{1}{2} H(D,2)

The universal enveloping algebra $U(\mathfrak{sl}_2)$ of $\mathfrak{sl}_2$ is a unital associative algebra over $\mathbb C$ generated by $E,F,H$ subject to the relations \begin{align*} [H,E]=2E, \qquad [H,F]=-2F, \qquad [E,F]=H. \end{align*} The distinguished central element $$\Lambda=EF+FE+\frac{H^2}{2}$$ is called the Casimir element of $U(\mathfrak{sl}_2)$. The universal Hahn algebra $\mathcal H$ is a unital associative algebra over $\mathbb C$ with generators $A,B,C$ and the relations assert that $[A,B]=C$ and each of \begin{align*} \alpha=[C,A]+2A^2+B, \qquad \beta=[B,C]+4BA+2C \end{align*} is central in $\mathcal H$. The distinguished central element $$\Omega=4ABA+B^2-C^2-2\beta A+2(1-\alpha)B$$ is called the Casimir element of $\mathcal H$. By investigating the relationship between the Terwilliger algebras of the hypercube and its halved graph, we discover the algebra homomorphism $\natural:\mathcal H\rightarrow U(\mathfrak{sl}_2)$ that sends \begin{eqnarray*} A &\mapsto & \frac{H}{4}, \\ B & \mapsto & \frac{E^2+F^2+\Lambda-1}{4}-\frac{H^2}{8}, \\ C & \mapsto & \frac{E^2-F^2}{4}. \end{eqnarray*} We determine the image of $\natural$ and show that the kernel of $\natural$ is the two-sided ideal of $\mathcal H$ generated by $\beta$ and $16 \Omega-24 \alpha+3$. By pulling back via $\natural$ each $U(\mathfrak{sl}_2)$-module can be regarded as an $\mathcal H$-module. For each integer $n\geq 0$ there exists a unique $(n+1)$-dimensional irreducible $U(\mathfrak{sl}_2)$-module $L_n$ up to isomorphism. We show that the $\mathcal H$-module $L_n$ ($n\geq 1$) is a direct sum of two non-isomorphic irreducible $\mathcal H$-modules

Investigating the Relationship between Dietary Sodium Intake and Severity Levels of Fluid Overload Symptoms in Patients with Heart Failure

Aim: This study aimed to investigate dietary sodium intake levels and to explore the relationship between those levels and the severity of fluid overload symptoms.Background: The management of dietary sodium is an important nursing intervention in the care of patients with heart failure stemming from fluid overload. Recommendations for the intake of dietary sodium among heart failure patients were discussed. If a heart failure patient’s dietary sodium intake habits are understood, then the relationship between this intake and fluid overload can be elucidated. This knowledge would be beneficial for nursing intervention in cases of heart failure.Methods: A total of 98 patients selected from cardiology wards who had a diagnosis of heart failure were enrolled in this study. Their dietary sodium intake level was estimated from a 24-hour urinary sodium excretion analysis. The severity of fluid overload symptoms was assessed using the fluid volume overload symptoms scale. Results: This study showed that the mean dietary sodium intake for patients with heart failure was 2.49 g/day and that this intake had no correlation with the severity levels of fluid overload symptoms. Conclusions: Using the patients’ own perceptions of the severity of fluid overload symptoms as a reference, adopting more relaxed sodium dietary intake restrictions may lead patients to have better food consumption habits