Revisiting Generalized Bertand's Postulate and Prime Gaps

It is a well-known fact that for any natural number $n$, there always exists a prime in $[n, 2n]$. Our aim in this note is to generalize this result to $[n, kn]$. A lower as well as an upper bound on the number of primes in $[n, kn]$ were conjectured by Mitra et al. [Arxiv 2009]. In 2016, Christian Axler provided a proof of the lower bound which is valid only when $n$ is greater than a very large threshold. In this paper, after almost a decade, we for the first time provide a direct proof of the lower bound that holds for all $n \geq 2$. Further, we show that the upper bound is a consequence of Firoozbakht's conjecture. Finally, we also prove a stronger version of the bounded gaps between primes

Low Complexity Fair Scheduling in LTE/LTE-A Uplink Involving Multiple Traffic Classes

The bulk of the research on Long Term Evolution/Long Term Evolution-Advanced packet scheduling is concentrated in the downlink and the uplink is comparatively less explored. In up-link, channel aware scheduling with throughput maximization has been widely studied while considering an infinitely back-logged buffer model, which makes the investigations unrealistic. Therefore, we propose an optimal uplink packet scheduling pro-cedure with realistic traffic sources. Firstly, we advocate a joint channel and buffer aware algorithm, which maximizes the actual transmitted bit-count. Thereafter, we introduce delay constraints in our algorithm to support real-time traffic. We further enhance our algorithm by incorporating the varied delay and throughput requirements demanded by mixed traffic classes. Finally, we in-troduce priority flipping to minimize bandwidth starvation of lower priority traffic in presence of higher percentage of high priority traffic. We observe that a delay constraint may render the optimization-based proposals infeasible. Therefore, to avoid infeasibility, we replace the delay constraint with delay outage minimization (DOM). DOM aims at minimizing the packet drop due to delay violation. Moreover, DOM also helps in reducing the problems to a well-known assignment problem, which can be solved by applying the Hungarian algorithm. Hence, our approach delivers an optimal allocation with low computational complexity.Comment: in IEEE Systems Journa

Design of Energy-efficient EPON: a Novel Protocol Proposal and its Performance Analysis

Economical and environmental concerns necessitate network engineers to focus on energy-efficient access network design. The optical network units (ONUs), being predominantly responsible for the energy consumption of Ethernet Passive Optical Network (EPON), motivates us towards designing a novel protocol for saving energy at ONU. The proposed protocol exploits different low power modes (LPM) and opts for the suitable one using traffic prediction. This scheme provides a significant improvement of energy-efficiency especially at high load (~ 40%) over existing protocols. A better understanding of the performance and a deeper insight into several design aspects can only be addressed through a detailed mathematical analysis. The proposed protocol involves traffic prediction which infringes Markovian property. However, some pragmatic assumptions along with a proper selection of observation instances and state descriptions allow us to form a Discrete Time Markov Chain (DTMC) of the proposed algorithm. Thus, the primary objective of this paper is to propose a novel scheme for achieving energy-efficiency at ONU and mathematically analyze the performance of it with the help of a DTMC. The analysis reveals that the energy-efficiency is more sensitive to the power consumption of doze mode as compared to other LPM while the effect of sleep-to-wake-up time is minor

Neutral Triple Vector Boson Production in Randall-Sundrum Model at the LHC

In this paper, triple neutral electroweak gauge boson production processes, viz. \gamma\gamma\gamma, \gamma\gamma Z, \gamma ZZ and ZZZ productions merged to 1-jet have been studied at the leading order in QCD in the context of Randall-Sundrum model at the LHC with center of mass energy \sqrt{S}=13 TeV. Decay of Z bosons into lepton-pairs has been considered. We present a selection of kinematical distributions matched to parton shower and show their deviation from the SM results as a result of the RS model. The uncertainties as a result of the factorization and renormalization scales are also presented.Comment: 18 pages, 7 figures, version to appear in PR

Experimental test of Hardy's paradox on a five-qubit quantum computer

We test Hardy's paradox of non-locality experimentally on the IBM five-qubit quantum computer for the first time. The quantum circuit is constructed on superconducting qubits corresponding to the original Hardy's test of non-locality. Results confirmed the theory that any non-maximally entangled state of two qubits violates Hardy's Equations, whereas any maximally entangled state and product state of two qubits do not exhibit Hardy's non-locality. We also point out the difficulties associated with the practical implementation of any Hardy's paradox based quantum protocol and propose three performance measures for any two qubits of any quantum computer

Necessary and sufficient condition for the existence of Schmidt decomposition in multipartite Hilbert spaces

Pati (Physics Letters A, 2000) derived a sufficient condition for the existence of Schmidt decomposition in tripartite Hilbert spaces. In this paper, we show that the condition is erroneous by demonstrating some counter-examples. Moreover, we suitably modify the condition and provide a correctness proof. We also show for the first time how this can be generalized to $n$-partite Hilbert spaces for any $n\geq 2$. Finally, we prove that this condition is also a necessary condition

Modular Class of a Lie algebroid with a Nambu structure

In this paper, we introduce the notion of modular class of a Lie algebroid $A$ equipped with a Nambu structure satisfying some suitable hypothesis. We also introduce cohomology and homology theories for such Lie algebroids and prove that these theories are connected by a duality isomorphism when the modular class is null

Effect of Transmission Impairments in CO-OFDM Based Elastic Optical Network Design

Coherent Optical Orthogonal Frequency Division Multiplexing (CO-OFDM) based Elastic Optical Network (EON) is one of the emerging technologies being considered for next generation high data rate optical network systems. Routing and Spectrum Allocation (RSA) is an important aspect of EON. Apart from spectral fragmentation created due to spectrum continuity and contiguity constraints of RSA, transmission impairments such as shot noise, amplified spontaneous emission (ASE) beat noise due to coherent detection, crosstalk in cross-connect (XC), nonlinear interference, and filter narrowing, limit the transmission reach of optical signals in EON. This paper focuses on the cross-layer joint optimization of delay-bandwidth product, fragmentation and link congestion for RSA in CO-OFDM EON while considering the effect of physical layer impairments. First, we formulate an optimal Integer Linear Programming (ILP) that achieves load-balancing in presence of transmission impairments and minimizes delay-bandwidth product along with fragmentation. We next propose a heuristic algorithm for large networks with two different demand ordering techniques. We show the benefits of our algorithm compared to the existing load balancing algorithm

Constrained Receiver Scheduling in Flexible Time and Wavelength Division Multiplexed Optical

An increasing bandwidth demand has mandated a shift to the time and wavelength division multiplexing (TWDM) techniques in optical access networks (OAN). Typical TWDM scheduling schemes consider scheduling of the optical line terminal receiver only. In this paper we have identified an additional collision domain that is present in TWDM schemes that offer security, in addition to bandwidth flexibility. Scheduling of the identified collision domain is termed as group scheduling. We illustrate that consideration of receiver scheduling only (as done in typical TWDM schemes) severely affects their throughput when implemented on flexible and secure TWDM architectures. A novel media access control protocol has been proposed in this paper that considers the multiple collision domains. Through simulations, we are able to illustrate that the proposed scheme achieves a high throughput. A theoretical upper bound of throughput has also been derived to explain the simulation results. Complexity reduction of the proposed scheme has been illustrated, thereby making it an attractive proposal

Buffer-aided Resource Allocation for a Price Based Opportunistic Cognitive Radio Network

In this paper, a resource allocation problem for an opportunistic cooperative cognitive radio network is considered, where cognitive radio nodes send their hard decisions to the fusion center. The fusion center plays dual role, i.e., takes the global decision (i.e., decision about the primary user's activity) as well as allocates transmission time durations among cognitive radio nodes. Revenue based utility functions are considered at the fusion center and cognitive radio nodes. An optimization problem is formulated to maximize the fusion center's revenue while satisfying some well defined constraints. User selection among cognitive radio nodes is performed in order to make the optimization problem feasible.Comment: 32 pages, 8 figures, Journa