Search Results (4)

Compared with scalar linear network coding (LNC) formulated over the finite field GF($2^L$), vector LNC offers enhanced flexibility in the code design by enabling linear operations over the vector space $\mathrm{GF}{\left(2\right)}^L$ and demonstrates a number of advantages over scalar LNC. While random LNC (RLNC) has shown significant potential to improve the completion delay performance in wireless broadcasts, most prior studies focus on scalar RLNC. In particular, it is well known that, with increasing L, primitive scalar RLNC over GF($2^L$) asymptotically achieves the optimal completion delay. However, the completion delay performance of primitive vector RLNC remains unexplored. This work aims to fill in this blank. We derive closed-form expressions for the probability distribution and the expected value of both the completion delay at a single receiver and the system completion delay. We further unveil a fundamental limitation that is different from scalar RLNC: even for large enough L, primitive vector RLNC over $\mathrm{GF}{\left(2\right)}^L$ inherently fails to reach optimal completion delay. In spite of this, the gap between the expected completion delay at a receiver and the optimal one is shown to be a constant smaller than $0.714$, which implies that the expected completion delay normalized by the number P of original packets is asymptotically optimal with increasing P. We also validate our theoretical characterization through numerical simulations. Our theoretical characterization establishes primitive vector RLNC as a performance baseline for the future design of practical vector RLNC schemes with different design goals. Full article

We propose fundamental inequalities for contact pseudo-slant submanifolds of $(ϵ)$-para Sasakian space form employing generalized normalized $\delta$-Casorati curvature. We characterize submanifolds for which equality cases hold and illustrate the main result with some applications. Further, we have considered a certain type of submanifold for a Ricci soliton and after computing its scalar curvature, developed an inequality to find correlations between intrinsic or extrinsic invariants. Full article

In this article, we obtain certain bounds for statistical curvatures of submanifolds with any codimension of Kenmotsu-like statistical manifolds. In this context, we construct a class of optimum inequalities for submanifolds in Kenmotsu-like statistical manifolds containing the normalized scalar curvature and the generalized normalized Casorati curvatures. We also define the second fundamental form of those submanifolds that satisfy the equality condition. On Legendrian submanifolds of Kenmotsu-like statistical manifolds, we discuss a conjecture for Wintgen inequality. At the end, some immediate geometric consequences are stated. Full article

The 2 + 1 + 1 decomposition of space-time is useful in monitoring the temporal evolution of gravitational perturbations/waves in space-times with a spatial direction singled-out by symmetries. Such an approach based on a perpendicular double foliation has been employed in the framework of dark matter and dark energy-motivated scalar-tensor gravitational theories for the discussion of the odd sector perturbations of spherically-symmetric gravity. For the even sector, however, the perpendicularity has to be suppressed in order to allow for suitable gauge freedom, recovering the 10th metric variable. The 2 + 1 + 1 decomposition of the Einstein–Hilbert action leads to the identification of the canonical pairs, the Hamiltonian and momentum constraints. Hamiltonian dynamics is then derived via Poisson brackets. Full article