In this paper described numerical expansion of natural-valued power function xⁿ, in point x = x_0, where n; x₀ - positive integers. Applying numerical methods, the calculus of nite dierences, particular pattern, that is sequence A287326 in OEIS, which shows the expansion of perfect cube n as row sum over k; 0 ≤ k ≤ n − 1 is generalized, obtained results are applied to show expansion of monomial n^2m+1; m = 0; 1; 2; ..., N. Additionally, relation between Faulhaber's sum ∑_n^m and nite dierences of power are shown in section 4.

Here, we propose a novel scheme based on advanced techniques of digital modulation in optical communications to achieve a single-channel transmission rate above 100 Gb/s. We utilize a hybrid scheme amplitude/phase/frequency/dual polarization, combined with multidimensional dual lattice and a low-density parity-check-coded modulation. The Stokes parameters are applied to the proposed scheme to map the four-dimensional classical polarization \({I_X},{Q_X},{I_Y},{Q_Y}\) in a three-dimensional space. In addition, in the proposed system, the packing theory is applied to the bit interleaver process, three wavelengths are packaged before being transmitted over a wavelength-division multiplexing optical channel. This modulation process is carried out using symmetrical geometric shapes, such as a hypercube or a polyhedron, based on the molecular links theory using a grouping of 12 and 13/15 bits for the cubic and spherical lattices, respectively. The proposed technique is evaluated in the context of long distance communications over distances up to 100 km. The bit error rate results showed that the optical signal-to-noise ratio was approximately 4 dB over a distance of 50 km. In addition, the power spectral efficiency was found to be 3 lambda, which is considered good performance considering the effects of distance and the non-linear effects influencing the number of lambdas. Also, we use an optical time division multiplexing scheme (OTDM) in order to achieve a transmission rate beyond 1Tbit/s, where the speed effect is evaluated, taking into consideration that the power spectral efficiency is degraded. Furthermore, the average noise floor is -60dBm for all transmission distances. These findings are expected to contribute to the future implementation of robust long-distance optical communication infrastructure.

Multi-level direct networks fueled by the evolving technology of active optical cables and increasing pin-bandwidth achieve reduced diameters and cost, with high-radix switches. These networks, like Dragonfly, are becoming the preferred aspirants for extreme high-performance parallel machines such as exa-scale computing. In this paper, we introduce a Hyper Z-Tree topology, which deploys the Z-Tree, a variant of fat-tree, as a computing node of a Generalized Hypercube (GHC) configuration. The resulting configuration provides higher bisection bandwidth, lower latency for some applications and higher throughput. Furthermore, the levels of the fat tree offer several path diversities across the GHC dimensions, hence conceding a more fault tolerant architecture. To profit from these path diversities, we propose two adaptive routing algorithms, which are extensions to the routing algorithm suggested in HyperX topology. These two algorithms exhibit better latencies and throughput than the HyperX.

The $k$-set tree connectivity, as a natural extension of classical connectivity, is a very important index to evaluate the fault-tolerance of interconnection networks. Let $G=(V, E)$ be a connected graph and a subset $S\subseteq V$, an $S$-tree of graph $G$ is a tree $T=(V',E')$ that contains all the vertices of $S$. Two $S$-trees $T$ and $T'$ are internally disjoint if and only if $E(T)\cap E(T')=\varnothing$ and $V(T)\cap V(T')=S$. The cardinality of maximum internally disjoint $S$-trees is defined as $\kappa_{G}(S)$, and the $k$-set tree connectivity is defined by $\kappa_{k}(G)=\min\{\kappa_{G}(S)|S\subseteq V(G)\ \text{and} \ |S|=k\}$. In this paper, we show that the $k$-set tree connectivity of hierarchical folded hypercube when $k=4$, that is, $\kappa_{4}(HFQ_{n})=n+1$, where $HFQ_{n}$ is hierarchical folded hypercube for $n\geq 7$.