Rhythm

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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

Multi-Hop Routing Mechanism for Reliable Sensor Computing

Current research on routing in wireless sensor computing concentrates on increasing the service lifetime, enabling scalability for large number of sensors and supporting fault tolerance for battery exhaustion and broken nodes. A sensor node is naturally exposed to various sources of unreliable communication channels and node failures. Sensor nodes have many failure modes, and each failure degrades the network performance. This work develops a novel mechanism, called Reliable Routing Mechanism (RRM), based on a hybrid cluster-based routing protocol to specify the best reliable routing path for sensor computing. Table-driven intra-cluster routing and on-demand inter-cluster routing are combined by changing the relationship between clusters for sensor computing. Applying a reliable routing mechanism in sensor computing can improve routing reliability, maintain low packet loss, minimize management overhead and save energy consumption. Simulation results indicate that the reliability of the proposed RRM mechanism is around 25% higher than that of the Dynamic Source Routing (DSR) and ad hoc On-demand Distance Vector routing (AODV) mechanisms

EFFECTS OF BACKPACK LOADS ON NECK-TRUNK MUSCLE ACTIVATION AMONG OFFICE WORKERS

The main purposes of this study were to investigate the effect of weight carriage on necktrunk muscle activation during standing and walking among office workers and to compare electromyography activation between healthy and symptomatic office workers. Twenty-one participants were recruited. Three load trials (0%, 10%, and 15% BW) and two conditions (standing and walking) were encountered. Repeated measure ANOVA was used to test main effect of load and condition on kinetic data. There was a significant condition*load interaction on right trapezius. Significantly increasing activation of right abdominis was found as carrying 15% BW. There was a significant decrease on activation of left erector spinae while carrying 10% BW. Considering to electromyography data, we suggest the backpack load under 10% BW was suitable for office workers