Narrow Results

In this paper we provide a mathematical reconstruction of what might have been Gauss' own derivation of the linking number of 1833, providing also an alternative, explicit proof of its modern interpretation in terms of degree, signed crossings and intersection number. The reconstruction presented here is entirely based on an accurate study of Gauss' own work on terrestrial magnetism. A brief discussion of a possibly independent derivation made by Maxwell in 1867 completes this reconstruction. Since the linking number interpretations in terms of degree, signed crossings and intersection index play such an important role in modern mathematical physics, we offer a direct proof of their equivalence. Explicit examples of its interpretation in terms of oriented area are also provided.

Let $K$ be a number field, and let ${\alpha }_1,\dots ,{\alpha }_r$ be elements of $K^{\times }$ which generate a subgroup of $K^{\times }$ of rank $r$. Consider the cyclotomic-Kummer extensions of $K$ given by $K({\zeta }_n,\sqrt[n_1]{{\alpha }_1},\dots ,\sqrt[n_r]{{\alpha }_r})$, where $n_i$ divides $n$ for all $i$. There is an integer $x$ such that these extensions have maximal degree over $K({\zeta }_g,\sqrt[g_1]{{\alpha }_1},\dots ,\sqrt[g_r]{{\alpha }_r})$, where $g=gcd(n,x)$ and $g_i=gcd(n_i,x)$. We prove that the constant $x$ is computable. This result reduces to finitely many cases the computation of the degrees of the extensions $K({\zeta }_n,\sqrt[n_1]{{\alpha }_1},\dots ,\sqrt[n_r]{{\alpha }_r})$ over $K$.

The third leap Zagreb index of a graph $G$ is denoted as ${LM}_3(G)$ and is defined as ${LM}_3(G)={\sum }_{b\in V(G)}d(b/G)d_2(b/G)$, where $d_2(b/G)$ and $d(b/G)$ are the 2-distance degree and the degree of the vertex $b$ in $G$, respectively. The first, second and third leap Zagreb indices were introduced by Naji et al. [A. M. Naji, N. D. Soner and I. Gutman, On leap Zagreb indices of graphs, Commun. Combin. Optim. 2(2) (2017) 99–117] in 2017. In this paper, the behavior of the third leap Zagreb index under several graph operations like the Cartesian product, Corona product, neighborhood Corona product, lexicographic product, strong product, tensor product, symmetric difference and disjunction of two graphs is studied.

We study the minimum distance function of a complete intersection graded ideal in a polynomial ring with coefficients in a field. For graded ideals of dimension one, whose initial ideal is a complete intersection, we use the footprint function to give a sharp lower bound for the minimum distance function. Then we show some applications to coding theory.

Let $G$ be a finitely generated multiplicative subgroup of ${\mathbb{Q}}^{\times }$ having rank $r$. The ratio between $n^r$ and the Kummer degree $[\mathbb{Q}({\zeta }_m,\sqrt[n]{G}):\mathbb{Q}({\zeta }_m)]$, where $n$ divides $m$, is bounded independently of $n$ and $m$. We prove that there exist integers $m_0,n_0$ such that the above ratio depends only on $G$, $gcd(m,m_0)$, and $gcd(n,n_0)$. Our results are very explicit and they yield an algorithm that provides formulas for all the above Kummer degrees (the formulas involve a finite case distinction).

Terrorist network may be defined as collection of suspected terrorist nodes which may function in disguise towards accomplishing a terrorist activity. They use extensive communication channel for sharing crucial information. Terrorist network analysis is highly efficacious for intelligence analysis and deriving useful conclusions from available data. Computer Science and Network analysis act as pertinent fields for the study and graphical interpretation of these networks. In this paper, we examine the 26/11 Mumbai attack terrorist network dataset and employ the ELECTRE method for identification of key node in the terrorist network. ELECTRE is an effective multi-criteria decision-making model. It provides a framework for structuring a decision problem integrates the quantitative and qualitative factors of the problem and facilitates easy computation. From the 26/11 Mumbai attack dataset of terrorist network, we have determined that out of several terrorists in the network “Wassi” was the momentous and mastermind of all. The proposed work also demonstrates improvement of result in terms of concurrence, generalization accuracy and genuineness. Based on the solution of ELECTRE framework, it is resolved that the obtained (terrorist) nodes will step up the work of law enforcement agencies and enable them to confine their focus on important members of the terrorist network. Identification of key terrorist is highly important for developing long-term strategies to counter forthcoming terrorist attacks. It can be better implemented during the development of smart city especially for India.

Link prediction of complex network intends to estimate the probability of existence of links between two nodes. In order to improve link prediction accuracy and fully exploit the potentialities of nodes, many studies focus more on the influence of degree on nodes but less on the hybrid influence of degree and H-index. The nodes with a larger degree have more neighbors, and the nodes with larger H-index have more neighbors of neighbors. Meanwhile, weak ties consisting of neighbors with a small degree have powerful strength of intermediary ability and a high probability of passing similarity. A novel link prediction model is proposed considering the hybrid influence of degree and H-index and weak ties, which is called Hybrid Weak Influence, marked as HWI. After experimenting with nine real datasets, the results show that this method can significantly improve the link prediction accuracy, compared with the empirical methods: Common Neighbors (CN), Resource-Allocation (RA) and Adamic/Adar (AA). Meanwhile, the computation complexity is less than the long path algorithm of LP, SRW, PCEN.

Traditional derivations of general relativity (GR) from the graviton degrees of freedom assume spacetime Lorentz covariance as an axiom. In this paper, we survey recent evidence that GR is the unique spatially-covariant effective field theory of the transverse, traceless graviton degrees of freedom. The Lorentz covariance of GR, having not been assumed in our analysis, is thus plausibly interpreted as an accidental or emergent symmetry of the gravitational sector. From this point of view, Lorentz covariance is a necessary feature of low-energy graviton dynamics, not a property of spacetime. This result has revolutionary implications for fundamental physics.

Hydrogen bond is a key factor in the determination of structures and properties of room-temperature ionic liquids. Connections of these hydrogen bonds form a network. In this work, we analyzed the hydrogen bond network of 1-alkyl-3-methylimidazolium ionic liquids using network theory. A two-dimensional view of the hydrogen bond network has been generated, the connection pattern shown that the average length of line shape connection is 2.44 to 2.77 for six 1-alkyl-3-methylimidazolium ionic liquids, and the connection patterns are different for short and long alkyl side chain length. The degree of each ion was calculated and analyzed. The nodes with zero degree were adopted to detect the boundary of the clusters in the ionic liquids, which have no hydrogen bond connected with neighbor ions. This work indicates that the network analysis method is useful for understanding and predicting the structure and function of RTILs.

There are growing interests for studying collective behavior including the dynamics of markets, the emergence of social norms and conventions and collective phenomena in daily life such as traffic congestion. In our previous work [Iwanaga and Namatame, Collective behavior and diverse social network, International Journal of Advancements in Computing Technology4(22) (2012) 321–320], we showed that collective behavior in cooperative relationships is affected in the structure of the social network, the initial collective behavior and diversity of payoff parameter. In this paper, we focus on scale-free network and investigate the effect of number of interactions on collective behavior. And we found that choices of hub agents determine collective behavior.

A smart grid consists of two networks: the power network and the communication network, which are interconnected by edges spanning across the networks. We model smart grids as complex interdependent networks, and study targeted and adaptive attacks on smart grids for the first time. Due to an attack on one network, nodes in the other network might get isolated, which in turn will disconnect nodes in the first network. Such cascading failures can result in disintegration of either or both of the networks. Earlier works considered only random failures. In real life, an attacker is more likely to compromise nodes selectively.

We study cascading failures in smart grids, where an attacker selectively compromises the nodes with probabilities proportional to their degrees, betweenness, or clustering coefficient. We mathematically and experimentally analyze the sizes of the giant components of the networks under different types of targeted attacks, and compare the results with the corresponding sizes under random attacks. We show that networks disintegrate faster for targeted attacks compared to random attacks. We next study adaptive attacks, where an attacker compromises nodes in rounds. Here, some nodes are compromised in each round based on their degree, betweenness or clustering coefficients, instead of compromising all nodes together. We show experimentally that an adversary has an advantage in this adaptive approach, compared to compromising the same number of nodes all at once.