Boxicity of Leaf Powers

The boxicity of a graph G, denoted as boxi(G), is defined as the minimum integer t such that G is an intersection graph of axis-parallel t-dimensional boxes. A graph G is a k-leaf power if there exists a tree T such that the leaves of the tree correspond to the vertices of G and two vertices in G are adjacent if and only if their corresponding leaves in T are at a distance of at most k. Leaf powers are used in the construction of phylogenetic trees in evolutionary biology and have been studied in many recent papers. We show that for a k-leaf power G, boxi(G) a parts per thousand currency sign k-1. We also show the tightness of this bound by constructing a k-leaf power with boxicity equal to k-1. This result implies that there exist strongly chordal graphs with arbitrarily high boxicity which is somewhat counterintuitive.

Bipartite Powers of k-chordal Graphs

Let k be an integer and k >= 3. A graph G is k-chordal if G does not have an induced cycle of length greater than k. From the definition it is clear that 3-chordal graphs are precisely the class of chordal graphs. Duchet proved that, for every positive integer m, if G m is chordal then so is G(m+2). Brandst `` adt et al. in Andreas Brandsadt, Van Bang Le, and Thomas Szymczak. Duchet- type theorems for powers of HHD- free graphs. Discrete Mathematics, 177(1- 3): 9- 16, 1997.] showed that if G m is k - chordal, then so is G(m+2). Powering a bipartite graph does not preserve its bipartitedness. In order to preserve the bipartitedness of a bipartite graph while powering Chandran et al. introduced the notion of bipartite powering. This notion was introduced to aid their study of boxicity of chordal bipartite graphs. The m - th bipartite power G(m]) of a bipartite graph G is the bipartite graph obtained from G by adding edges (u; v) where d G (u; v) is odd and less than or equal to m. Note that G(m]) = G(m+1]) for each odd m. In this paper we show that, given a bipartite graph G, if G is k-chordal then so is G m], where k, m are positive integers with k >= 4

On interference alignment for symmetrically connected interference networks with line of sight channels at finite powers

In this work, interference alignment for a class of Gaussian interference networks with general message demands, having line of sight (LOS) channels, at finite powers is considered. We assume that each transmitter has one independent message to be transmitted and the propagation delays are uniformly distributed between 0 and (L - 1) (L >; 0). If receiver-j, j ∈{1,2,..., J}, requires the message of transmitter-i, i ∈ {1, 2, ..., K}, we say (i, j) belongs to a connection. A class of interference networks called the symmetrically connected interference network is defined as a network where, the number of connections required at each transmitter-i is equal to ct for all i and the number of connections required at each receiver-j is equal to cr for all j, for some fixed positive integers ct and cr. For such networks with a LOS channel between every transmitter and every receiver, we show that an expected sum-spectral efficiency (in bits/sec/Hz) of at least K/(e+c1-1)(ct+1) (ct/ct+1)ct log2 (1+min(i, j)∈c|hi, j|2 P/WN0) can be achieved as the number of transmitters and receivers tend to infinity, i.e., K, J →∞ where, C denotes the set of all connections, hij is the channel gain between transmitter-i and receiver-j, P is the average power constraint at each transmitter, W is the bandwidth and N0 W is the variance of Gaussian noise at each receiver. This means that, for an LOS symmetrically connected interference network, at any finite power, the total spectral efficiency can grow linearly with K as K, J →∞. This is achieved by extending the time domain interference alignment scheme proposed by Grokop et al. for the k-user Gaussian interference channel to interference networks.

Analysis of OSNR Requirements for Higher Modulation Schemes for Various Local Oscillator Powers

To meet the growing demands of data traffic in long haul communication, it is necessary to efficiently use the low-loss region(C-band) of the optical spectrum, by increasing the no. of optical channels and increasing the bit rate on each channel But narrow pulses occupy higher spectral bandwidth. To circumvent this problem, higher order modulation schemes such as QPSK and QAM can be used to modulate the bits, which increases the spectral efficiency without demanding any extra spectral bandwidth. On the receiver side, to meet a satisfy, a given BER, the received optical signal requires to have minimum OSNR. In our study in this paper, we analyses for different modulation schemes, the OSNR required with and without preamplifier. The theoretical limit of OSNR requirement for a modulation scheme is compared for a given link length by varying the local oscillator (LO) power. Our analysis shows that as we increase the local oscillator (LO) power, the OSNR requirement decreases for a given BER. Also a combination of preamplifier and local oscillator (LO) gives the OSNR closest to theoretical limit.

Lower Bounds for Sums of Powers of Low Degree Univariates

We consider the problem of representing a univariate polynomial f(x) as a sum of powers of low degree polynomials. We prove a lower bound of Omega(root d/t) for writing an explicit univariate degree-d polynomial f(x) as a sum of powers of degree-t polynomials.

Nonlinear X-wave formation and conical emission at different powers of a femtosecond laser pulse in water

Nonlinear X-wave formation at different pulse powers in water is simulated using the standard model of nonlinear Schrodinger equation (NLSE). It is shown that in near field X-shape originally emerges from the interplay between radial diffraction and optical Kerr effect. At relatively low power group-velocity dispersion (GVD) arrests the collapse and leads to pulse splitting on axis. With high enough power, multi-photon ionization (NIPI) and multi-photon absorption (MPA) play great importance in arresting the collapse. The tailing part of pulse is first defocused by MPI and then refocuses. Pulse splitting on axis is a manifestation of this process. Double X-wave forms when the split sub-pulses are self-focusing. In the far field, the character of the central X structure of conical emission (CE) is directly related to the single or double X-shape in the near field. (c) 2007 Elsevier B.V. All rights reserved.

Multi-Party Negotiation When Agents Have Subjective Estimates of Bargaining Powers

This paper presents a method for generating Pareto-optimal solutions in multi-party negotiations. In this iterative method, decision makers (DMs) formulate proposals that yield a minimum payoff to their opponents. Each proposal belongs to the efficient frontier, DMs try to adjust to a common one. In this setting, each DM is supposed to have a given bargaining power. More precisely each DM is supposed to have a subjective estimate of the power of the different parties. We study the convergence of the method, and provide examples where there is no possible agreement resulting from it.

Political Posturing or Real Powers: Control of Genetically Modified Crops by the Member States and their Regions

The cultivation of genetically modified (GM) crops in the EU is highly harmonised, involving a central authorisation procedure that aims to ensure a high level of environmental and human health protection. However conflicts over authority persist and the Commission has responded to a combination of internal and external pressures with a more flexible approach to coexistence, a proposed opt-out clause and recently a promise by the head of the Commission to review the existing EU GM legislative regime, providing an opportunity to consider and suggest paths of development. In light of the significance of multilevel governance and subsidiarity for GM cultivation, this paper considers the policy-making powers of the Member States and subnational regions in this regime, focussing upon post-authorisation options in particular. A number of core mechanisms exist, including voluntary measures, safeguard clauses, coexistence measures, a proposed express opt-out and Article 4(2) TEU on ‘national identity. These mechanisms are examined in light of the goals and challenges of multilevel governance, in order to consider whether the relevant powers are located at the appropriate level. Overall, it is apparent that the developments occurring at the EU level are strengthening multilevel governance, but with significant opportunities to improve it further through focussing on the supporting roles and the regional levels in particular.