Note on vanishing power sums of roots of unity

For a given positive integers $m$ and $\ell$, we give a complete list of positive integers $n$ for which their exist $m$th roots of unity $x_1,\dots,x_n \in \mathbb{C}$ such that $x_1^{\ell} + \cdots + x_n^{\ell}=0$. This extends the earlier result of Lam and Leung on vanishing sums of roots of unity. Furthermore, we prove that for which integers $n$ with $2 \leq n \leq m$, there are distinct $m$th roots of unity $x_1,\dots,x_n \in \mathbb{C}$ such that $x_1^{\ell} + \cdots + x_n^{\ell}=0$.Comment: 5 pages, 1 table, to appear in Math. Studen

Techniques and Applications of Computation Slicing

Writing correct distributed programs is hard. In spite of extensive testing and debugging, software faults persist even in commercial grade software. Many distributed systems, especially those employed in safety-critical environments, should be able to operate properly even in the presence of software faults. Monitoring the execution of a distributed system, and, on detecting a fault, initiating the appropriate corrective action is an important way to tolerate such faults. This gives rise to the predicate detection problem which requires finding a consistent cut of a given computation that satisfies a given global predicate, if it exists. Detecting a predicate in a computation is, however, an NP-complete problem. To ameliorate the associated combinatorial explosion problem, we introduce the notion of computation slice. Formally, the slice of a computation with respect to a predicate is a (sub)computation with the least number of consistent cuts that contains all consistent cuts of the computation satisfying the predicate. To detect a predicate, rather than searching the state-space of the computation, it is much more efficient to search the state-space of the slice. We prove that the slice exists and is uniquely defined for all predicates. We present efficient slicing algorithms for several useful classes of predicates. We develop efficient heuristic algorithms for computing an approximate slice for predicates for which computing the slice is otherwise provably intractable. Our experimental results show that slicing can lead to an exponential improvement over existing techniques for predicate detection in terms of time and space.Comment: 50 pages, 14 figure

Diffusive Molecular Communication with Nanomachine Mobility

This work presents a performance analysis for diffusive molecular communication with mobile transmit and receive nanomachines. To begin with, the optimal test is obtained for symbol detection at the receiver nanomachine. Subsequently, closed-form expressions are derived for the probabilities of detection and false alarm, probability of error, and capacity considering also aberrations such as multi-source interference, inter-symbol interference, and counting errors. Simulation results are presented to corroborate the theoretical results derived and also, to yield various insights into the performance of the system. Interestingly, it is shown that the performance of the mobile diffusive molecular communication can be significantly enhanced by allocating large fraction of total available molecules for transmission as the slot interval increases.Comment: To be submitted in 52th Annual Conference on Information Sciences and Systems (CISS

Cognitive MIMO-RF/FSO Cooperative Relay Communication with Mobile Nodes and Imperfect Channel State Information

This work analyzes the performance of an underlay cognitive radio based decode-and-forward mixed multiple-input multiple-output (MIMO) radio frequency/free space optical (RF/FSO) cooperative relay system with multiple mobile secondary and primary user nodes. The effect of imperfect channel state information (CSI) arising due to channel estimation error is also considered at the secondary user transmitters (SU-TXs) and relay on the power control and symbol detection processes respectively. A unique aspect of this work is that both fixed and proportional interference power constraints are employed to limit the interference at the primary user receivers (PU-RXs). Analytical results are derived to characterize the exact and asymptotic outage and bit error probabilities of the above system under practical conditions of node mobility and imperfect CSI, together with impairments of the optical channel, such as path loss, atmospheric turbulence, and pointing errors, for orthogonal space-time block coded transmission between each SU-TX and relay. Finally, simulation results are presented to yield various interesting insights into the system performance such as the benefits of a midamble versus preamble for channel estimation.Comment: revision submitted to IEEE Transactions on Cognitive Communications and Networkin

Diffusion Based Cooperative Molecular Communication in Nano-Networks

This work presents a novel diffusion based dual-phase molecular communication system where the source leverages multiple cooperating nanomachines to improve the end-to-end reliability of communication. The Neyman-Pearson Likelihood Ratio Tests are derived for each of the cooperative as well as the destination nanomachines in the presence of multi-user interference. Further, to characterize the performance of the aforementioned system, closed form expressions are derived for the probabilities of detection, false alarm at the individual cooperative, destination nanomachines, as well as the overall end-to-end probability of error. Simulation results demonstrate a significant improvement in the end-to-end performance of the proposed cooperative framework in comparison to multiple-input single-output and single-input single-output molecular communication scenarios in the existing literature.Comment: Revised IEEE WCL Draft (in review process

On the discriminant of pure number fields

Let $K=\mathbb{Q}(\sqrt[n]{a})$ be an extension of degree $n$ of the field \Q of rational numbers, where the integer $a$ is such that for each prime $p$ dividing $n$ either $p\nmid a$ or the highest power of $p$ dividing $a$ is coprime to $p$; this condition is clearly satisfied when $a, n$ are coprime or $a$ is squarefree. The paper contains an explicit formula for the discriminant of $K$ involving only the prime powers dividing $a,n$

Design and Performance Analysis of Dual and Multi-hop Diffusive Molecular Communication Systems

This work presents a comprehensive performance analysis of diffusion based direct, dual-hop, and multi-hop molecular communication systems with Brownian motion and drift in the presence of various distortions such as inter-symbol interference (ISI), multi-source interference (MSI), and counting errors. Optimal decision rules are derived employing the likelihood ratio tests (LRTs) for symbol detection at each of the cooperative as well as the destination nanomachines. Further, closed-form expressions are also derived for the probabilities of detection, false alarm at the individual cooperative, destination nanomachines, as well as the overall end-to-end probability of error for source-destination communication. The results also characterize the impact of detection performance of the intermediate cooperative nanomachine(s) on the end-to-end performance of dual/multi hop diffusive molecular communication systems. In addition, capacity expressions are also derived for direct, dual-hop, and multi-hop molecular communication scenarios. Simulation results are presented to corroborate the theoretical results derived and also, to yield insights into system performance.Comment: in preparatio

On Hybrid MoSK-CSK Modulation based Molecular Communication: Error Rate Performance Analysis using Stochastic Geometry

Data transmission rate in molecular communication systems can be improved by using multiple transmitters and receivers. In molecular multiple-input multiple-output (MIMO) systems which use only single type of molecules, the performance at the destination is limited by inter-symbol interference (ISI), inter-link interference (ILI) and multi-user interference (MUI). This work proposes a new hybrid modulation for a system with multiple transmitters and receivers which uses different types of molecules to eliminate ILI. Further, to enhance the data rate of the proposed system under ISI and MUI, Mary CSK modulation scheme is used between each transmitter-receiver pair. In this paper, the random locations of transmitters present in the three dimensional (3-D) space are modeled as homogeneous Poisson point process (HPPP). Using stochastic geometry tools, analytical expression is derived for the probability of symbol error for the aforementioned scenario. Finally, the performance of the proposed system is compared using the different existing modulation schemes such as on-off keying (OOK), binary concentration shift keying (BCSK) and quadruple concentration shift keying (QCSK) to develop several important insights

Weakening of the spin density wave gap at low temperatures in SrFe2_2As2_2 single crystals

We report on temperature dependent scanning tunneling microscopy and spectroscopy studies of undoped SrFe$_{2}$As$_{2}$ single crystals from 6 K to 292 K. Resistivity data show spin density wave (SDW) transition at T$_{SDW}$ $\approx$ 205 K and the superconducting transition at $\sim$ 21 K while magnetic susceptibility does not show any superconductivity. Conductance maps and local tunneling spectra show an electronically homogeneous surface at all studied temperatures. Tunnel spectra correlate well with the SDW transition with a depression in the density of states near the Fermi energy below T$_{SDW}$. On further lowering the temperature, the width of this partial energy gap associated with the SDW decreases. Using the anti-correlation of superconducting phase with SDW, we interpret this gap weakening as a precursor to superconductivity. This may give rise to a facile nucleation of superconductivity near defects

Distributed Abstraction Algorithm for Online Predicate Detection

Analyzing a distributed computation is a hard problem in general due to the combinatorial explosion in the size of the state-space with the number of processes in the system. By abstracting the computation, unnecessary explorations can be avoided. Computation slicing is an approach for abstracting dis- tributed computations with respect to a given predicate. We focus on regular predicates, a family of predicates that covers a large number of commonly used predicates for runtime verification. The existing algorithms for computation slicing are centralized in nature in which a single process is responsible for computing the slice in either offline or online manner. In this paper, we present a distributed online algorithm for computing the slice of a distributed computation with respect to a regular predicate. Our algorithm distributes the work and storage requirements across the system, thus reducing the space and computation complexities per process. In addition, for conjunctive predicates, our algorithm also reduces the message load per process