Total Edge Irregularity Strength for Graphs

An edge irregular total $k$-labelling $f : V(G)\cup E(G)\rightarrow \{1,2,\dots,k\}$ of a graph $G$ is a labelling of the vertices and the edges of $G$ in such a way that any two different edges have distinct weights. The weight of an edge $e$, denoted by $wt(e)$, is defined as the sum of the label of $e$ and the labels of two vertices which incident with $e$, i.e. if $e=vw$, then $wt(e)=f(e)+f(v)+f(w)$. The minimum $k$ for which $G$ has an edge irregular total $k$-labelling is called the total edge irregularity strength of $G.$ In this paper, we determine total edge irregularity of connected and disconnected graphs

Weighted ℓ_1 minimization for sparse recovery with prior information

In this paper we study the compressed sensing problem of recovering a sparse signal from a system of underdetermined linear equations when we have prior information about the probability of each entry of the unknown signal being nonzero. In particular, we focus on a model where the entries of the unknown vector fall into two sets, each with a different probability of being nonzero. We propose a weighted ℓ_1 minimization recovery algorithm and analyze its performance using a Grassman angle approach. We compute explicitly the relationship between the system parameters (the weights, the number of measurements, the size of the two sets, the probabilities of being non-zero) so that an iid random Gaussian measurement matrix along with weighted ℓ_1 minimization recovers almost all such sparse signals with overwhelming probability as the problem dimension increases. This allows us to compute the optimal weights. We also provide simulations to demonstrate the advantages of the method over conventional ℓ_1 optimization

Divide-and-conquer: Approaching the capacity of the two-pair bidirectional Gaussian relay network

The capacity region of multi-pair bidirectional relay networks, in which a relay node facilitates the communication between multiple pairs of users, is studied. This problem is first examined in the context of the linear shift deterministic channel model. The capacity region of this network when the relay is operating at either full-duplex mode or half-duplex mode for arbitrary number of pairs is characterized. It is shown that the cut-set upper-bound is tight and the capacity region is achieved by a so called divide-and-conquer relaying strategy. The insights gained from the deterministic network are then used for the Gaussian bidirectional relay network. The strategy in the deterministic channel translates to a specific superposition of lattice codes and random Gaussian codes at the source nodes and successive interference cancelation at the receiving nodes for the Gaussian network. The achievable rate of this scheme with two pairs is analyzed and it is shown that for all channel gains it achieves to within 3 bits/sec/Hz per user of the cut-set upper-bound. Hence, the capacity region of the two-pair bidirectional Gaussian relay network to within 3 bits/sec/Hz per user is characterized.Comment: IEEE Trans. on Information Theory, accepte

Gerakan Salafiyah: Islam, Politik dan Rigiditas Interpretasi Hukum Islam

Salafiyah lately has become the scourge of modern states because the banality of their interpretation of Islamic law is considered rigid and not in line with the global world that is increasingly plural. This article attempts to discuss the various variants of salafism and the purpose behind its militant movement. The fundamental question in this article is about the purpose behind a non-compromising understanding of other understandings that are different from the Salafiyah. This article first describes some variant of Salafism, namely: al-Salafiyah al-Tārīkhiyah, al-Salafiyah al-Wahābiyah, al-Salafiyah al-Ishlāhiyah, al-Salafiyah al-Ta'sīliyah dan al-Salafiyah al-Jihādīyah al-Takfīriyah, and then analyze what the similarity of theological traits of these variant. This study confirms that Salafism is a doctrinal social movement that has political efforts to establish Islam by rejecting the differences that exist. The results of this study indicate that the Salafi initially is group that calls on Muslims to return to the original sources of the Qur'an and the Hadith leads to the resurrection of a new Islamic civilization against the Western world power hegemony (especially the United States), it becomes a group which is face to face with the Muslims themselves. In addition, a rigid interpretation of Islamic law has meaning to build political strength to slowly unify differences that do not correspond with the values of salafiyah