Asymptotic Case Research Articles

Limits of multiplicative inhomogeneous random graphs and L\xe9vy trees: limit theorems

We consider a natural model of inhomogeneous random graphs that extends the classical Erdős–Rényi graphs and shares a close connection with the multiplicative coalescence, as pointed out by Aldous (Ann Probab 25:812–854, 1997). In this model, the vertices are assigned weights that govern their tendency to form edges. It is by looking at the asymptotic distributions of the masses (sum of the weights) of the connected components of these graphs that Aldous and Limic (Electron J Probab 3:1–59, 1998) have identified the entrance boundary of the multiplicative coalescence, which is intimately related to the excursion lengths of certain Lévy-type processes. We, instead, look at the metric structure of these components and prove their Gromov–Hausdorff–Prokhorov convergence to a class of (random) compact measured metric spaces that have been introduced in a companion paper (Broutin et al. in Limits of multiplicative inhomogeneous random graphs and Lévy trees: the continuum graphs. arXiv:1804.05871, 2020). Our asymptotic regimes relate directly to the general convergence condition appearing in the work of Aldous and Limic. Our techniques provide a unified approach for this general “critical” regime, and relies upon two key ingredients: an encoding of the graph by some Lévy process as well as an embedding of its connected components into Galton–Watson forests. This embedding transfers asymptotically into an embedding of the limit objects into a forest of Lévy trees, which allows us to give an explicit construction of the limit objects from the excursions of the Lévy-type process. The mains results combined with the ones in the other paper allow us to extend and complement several previous results that had been obtained via model- or regime-specific proofs, for instance: the case of Erdős–Rényi random graphs obtained by Addario-Berry et al. (Probab Theory Relat Fields 152:367–406, 2012), the asymptotic homogeneous case as studied by Bhamidi et al. (Probab Theory Relat Fields 169:565–641, 2017), or the power-law case as considered by Bhamidi et al. (Probab Theory Relat Fields 170:387–474, 2018).

Thermal effects on the Casimir energy of a Lorentz-violating scalar in magnetic field

In this work, I investigate the finite temperature Casimir effect due to a massive and charged scalar field that breaks Lorentz invariance in a CPT-even, aether-like way. I study the cases of Dirichlet and mixed (Dirichlet–Neumann) boundary conditions on a pair of parallel plates. I will not examine the case of Neumann boundary conditions since it produces the same results as Dirichlet boundary conditions. The main tool used in this investigation is the [Formula: see text]-function technique that allows me to obtain the Helmholtz free energy and Casimir pressure in the presence of a uniform magnetic field perpendicular to the plates. Three cases of Lorentz asymmetry are studied: timelike, spacelike and perpendicular to the magnetic field, spacelike and parallel to the magnetic field. Asymptotic cases of small plate distance, high temperature, strong magnetic field, and large mass will be considered for each of the three types of Lorentz asymmetry and each of the two types of boundary conditions examined. In all these cases, simple and very accurate analytic expressions of the thermal corrections to the Casimir energy and pressure are obtained and I discover that these corrections strongly depend on the direction of the unit vector that produces the breaking of the Lorentz symmetry.

Effective rate analysis of MISO over Beaulieu-Xie fading channel

This paper presents the effective rate (ER) analysis of a multiple input single output (MISO) system over the Beaulieu-Xie (BX) fading. The closed-form expressions were derived for the exact and asymptotic cases (for high and low signal-to-noise ratio (SNR)). The BX fading channel is extremely useful for characterizing small cell networks, hyper-dense networks, and femtocells, etc. It can also be used for modeling the fading channels for high-speed trains. Numerical results were presented using the derived closed-form expressions and compared with the results obtained from Monte Carlo simulations. Our findings illustrated that the analytical results have a close match with simulation results. In addition, the numerical findings also provide the important insights into the impact of delay constraint and fading parameters on the ER of considered wireless scenario. It is found that the performance of system improves as dominant specular power and/or number of transmitter antennas increases or the delay constraint parameter decreases.

Age of Information in Grant-Free Random Access With Massive MIMO

This letter aims to analyze and optimize the status update performance in a grant-free random access system with massive multiple-input multiple-output. We analyze the status update process, and derive a closed-form approximate expression of the average age of information (AoI). Then, in the asymptotic case where the large number of antennas at the base station causes the packet error rate to approach zero, we jointly optimize access control and pilot length to reduce the average AoI. Simulation and numerical results verify the correctness of the theoretical analysis, and show that the proposed scheme can achieve the same performance as the exhaustive search scheme and outperforms the benchmark ones.

An analytical solution to the multicommodity network flow problem with weighted random routing

We derive an analytical expression for the mean load at each node of an arbitrary undirected graph for the non-uniform multicommodity flow problem under weighted random routing. We show the mean load at each node, net of its demand and normalized by its (weighted) degree, is a constant equal to the trace of the product of two matrices: the Laplacian of the demand matrix and the generalized inverse of the graph Laplacian. For the case of uniform demand, this constant reduces to the sum of the inverses of the non-zero eigenvalues of the graph Laplacian. We note that such a closed-form expression for the network capacity for the general multicommodity network flow problem has not been reported before, and even though (weighted) random routing is not a practical procedure, it enables us to derive a (tight) upper bound for the capacity of the network under more standard routing policies. Using this new expression, we compute network capacity for a sample of demand matrices for some prototypical networks, including uniform demand (one unit between all node pairs) and broadcast demand (one unit between a source node and each other node as destination), and finally derive estimates of the mean load in some asymptotic cases.

Semi-infinite crack between two dissimilar orthotropic strips

The problem of a semi-infinite crack between two dissimilar orthotropic strips has been solved. Using the Fourier transform technique the problem has been converted to a standard Wiener-Hopf equation which has been solved for asymptotic cases only to deduce the expressions for the stress intensity factor and crack opening displacement. Finally the values of the stress intensity factor and crack opening displacement have been plotted graphically to show the effects of strip width and material orthotropy.

Non-Markovian process with variable memory functions

We present a treatment of the non-Markovian character of memory by incorporating different forms of Mittag-Leffler (ML) function, which generally arises in the solution of a fractional master equation, as different memory functions in the Generalized Kolmogorov-Feller Equation (GKFE). The cross-over from the short time (stretched exponential) to long time (inverse power law) approximations of the ML function incorporated in the GKFE is proven. We have found that the GKFE solutions are the same for negative exponential and upto first order expansion of the stretched exponential function for very small \(\tau \rightarrow 0\). A generalized integro-differential equation form of the GKFE along with an asymptotic case is provided.

Force on a moving liquid blister

Abstract

Improving the Discrete-Modulated Continuous-Variable Measurement-Device-Independent Quantum Key Distribution with Quantum Scissors

We propose a new scheme to strengthen the performance of the discrete-modulated continuous-variable measurement-device-independent quantum key distribution (DM-CV-MDI-QKD) protocol by introducing a quantum scissors (QS) operation to Bob’s side. According to the equivalent one-way scheme of the protocol, we further investigate the success probability of the QS operation, and analyze the security of the DM-CV-MDI-QKD equipped with this operation in the asymptotic case. Simulation results show that the QS operation can indeed effectively improve the performance of the existing protocol, and compared with the scheme without QS operation, our proposed scheme has a higher secret key rate and a longer secure transmission distance.

History and Spread of Viruses (COVID-19) and Associated Demographic and Clinical Parameters – A systematic review and meta-analysis

COVID-19 made a huge impact on the world due to its rapid transmission and no treatments being available for it. The virus affected more people and spread to various countries than what was predicted when COVID-19 initially began spreading. There have been numerous pandemics and epidemics in the 21st century yet COVID-19 has affected more people and spread widely. The primary objective of the study was to explore history, spread and associated parameters of existing viruses especially COVID-19. Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA) guideline was followed for a systematic search to identify eligible published articles. Clinical data, regarding COVID-19 patients, was obtained from previously published articles. The main cause of COVID-19 spreading rapidly was noted to be due to a high percentage of asymptomatic patients, transmission being air-borne, and the lack of knowledge and preventative measures being implemented when the virus began spreading. The common co-morbidity that found in patients was Diabetes Mellitus, Hypertension, and Coronary Heart Disease. The common symptoms, found through the Meta-analysis, that the patients faced included cough (55.4%), fever (68.4%), fatigue (20.3%), and shortness of breath (18.1%). The proportion of asymptotic positive cases was measured 58.3% (95%CI: 24.7% – 87.9%) while mortality proportion was found to be 6.7% (fixed-effect model) and 13.4% (random-effect model). The Meta-analysis indicated that a higher percentage of males were affected by COVID-19 than females and more patients are found to be asymptomatic. Moreover, the mortality rate of patients that have had COVID-19 was found to be low.

Ranking recovery from limited pairwise comparisons using low-rank matrix completion

This paper proposes a new methodology for solving the well-known rank aggregation problem from pairwise comparisons using low-rank matrix completion. Partial and noisy data of pairwise comparisons is first transformed into a matrix form. We then use tools from matrix completion, which has served as a major component in the low-rank based completion solution for the Netflix challenge, to construct the preference of different objects. In our approach, the data from multiple comparisons is used to create an estimate of the probability of object i winning (or be chosen) over object j, where only a partial set of comparisons between the N objects is known. These probabilities can be transformed to take the form of a rank-one matrix. An alternating minimization algorithm, in which the target matrix takes a bilinear form, is used in combination with maximum likelihood estimation for both factors. The reconstructed matrix is used to obtain the true underlying preference intensity. We start by exploring the asymptotic case of an infinitely large number of comparisons (“noiseless case”). We then extend our solution to the case of a finite number of comparisons for a subset of pairs (“noisy case”). This work demonstrates the improvement of our proposed algorithm over the state-of-the-art techniques in both simulated scenarios and real data.

COVID 19: A Strange Worldwide Pandemic

Corona virus disease (COVID -19) is a severe acute respiratory syndrome which cause respiratory tract illness. The pandemic caused by virus originated in Wuhan. Later declared as a pandemic by WHO. It causes symptom like common cold, fever, body ache, cough and loss of taste sensation which are commonly seen in most of the patient. Older and one with comorbidity are at high risk to get infected. The average time period from exposure to symptoms is 5 days. Many asymptotic cases where reported which shows no symptoms but show viral load on RT-PCR. Patients are considered covid-19 positive based on clinical features & radiological findings. On chest x-ray it shows ground-glass opacity. More than 75 percent of patients hospitalized require supplemental oxygen. Treatment for individual with COVID -19 include best practice for supportive management of acute hypoxic respiratory failure. Immediate hospitalization of positive patient if spo2 drops below 95 percent . Diagnosis is made by detection of SARS cov-2 via reverse transcription polymerase chain reaction testing, although false positive result may occur in up to 20 to 60 % patients. Suspected cases are quarantine for 14 days. Preventive measures include hand washing, social distancing etc. Recently COVISHEILD & COVAXIN is approved for emergency use. Reaction of vaccination include fever ,body ache, swelling at local site, pain, malaise, joint pains.

On Neyman–Pearson minimax detection of Poisson process intensity

The problem of the minimax testing of the Poisson process intensity $${\mathbf{s}}$$ is considered. For a given intensity $${\mathbf{p}}$$ and a set $$\mathcal{Q}$$ , the minimax testing of the simple hypothesis $$H_{0}: {\mathbf{s}} = {\mathbf{p}}$$ against the composite alternative $$H_{1}: {\mathbf{s}} = {\mathbf{q}},\,{\mathbf{q}} \in \mathcal{Q}$$ is investigated. The case, when the 1-st kind error probability $$\alpha $$ is fixed and we are interested in the minimal possible 2-nd kind error probability $$\beta ({\mathbf{p}},\mathcal{Q})$$ , is considered. What is the maximal set $$\mathcal{Q}$$ , which can be replaced by an intensity $${\mathbf{q}} \in \mathcal{Q}$$ without any loss of testing performance? In the asymptotic case ( $$T\rightarrow \infty $$ ) that maximal set $$\mathcal{Q}$$ is described.

Dynamics of inverted flags: Experiments and comparison with theory

The study of the dynamics of cantilevered thin flexible plates in reverse axial flow – also known as inverted flags – has become of significant interest, partly due to their energy harvesting potential. This paper presents fresh experimental results, aiming to enhance our understanding of the dynamics of inverted flags, particularly concerning stability and global dynamics sensitivity to various system parameters, such as the aspect ratio (i.e. height-to-length ratio) and flag material. This is achieved through testing flags of various dimensions, made from different materials. The dynamics of the system is presented in the form of bifurcation diagrams in which the flag tip angle is shown as a function of the dimensionless flow velocity. The frequency content of oscillatory motions is presented in the form of spectrograms. Also, time-histories, phase-plane portraits, as well as power spectral density and probability density function plots are presented for different flow regimes. Interesting dynamical features, such as small- and large-amplitude flapping, static divergence, large-amplitude buckling, and chaotic motions, have been observed experimentally, some for the first time. It is shown that for flags of very low aspect ratio the undeflected static equilibrium is stable prior to a subcritical pitchfork bifurcation. For flags of large aspect ratio, on the other hand, the undeflected static equilibrium becomes unstable via a supercritical pitchfork bifurcation leading to static divergence (buckling) at a sufficiently high flow velocity; at higher flow velocities, past the pitchfork bifurcation, a supercritical Hopf bifurcation materializes leading to flapping motion around the deflected static equilibrium; at even higher flow velocities, flapping motion becomes symmetric around the undeflected static equilibrium. Interestingly, it is found that heavy flags may exhibit large-amplitude flapping right after the initial static equilibrium, provided that they are subjected to a sufficiently large disturbance. In addition to experiments, nonlinear analytical models for the two asymptotic cases of zero and infinite aspect ratios are outlined, the results of which are used for comparison with the experimental results. Theory and experiments are found to be in reasonably good agreement.

Application of GSVD‐based precoding in MIMO‐NOMA relaying systems

In this study, a simple two-user multiple-input multiple-output (MIMO) relaying scenario is considered. To efficiently exploit the spectrum resources, a generalised singular value decomposition (GSVD) based precoding scheme with non-orthogonal multiple access (NOMA) is proposed, which converts the multiple channels into several parallel single channels. The performance of the proposed GSVD-MIMO relaying scheme is evaluated by using the distribution characteristics of squared generalised singular values of two users' channel matrices and the eigenvalues of the relay's channel matrix to obtain the expressions of the average data rate and outage probabilities. Both the asymptotic case and a practical scenario with nodes equipped with finite numbers of antennas are analysed because the distribution characteristics of squared generalised singular values in the two cases are different. Two special cases, single-input multiple-output and multiple-input single-output are considered. Simulation experiments are carried out to verify the effectiveness and accuracy of the obtained analytical results.

Practical decoy-state quantum random number generator with weak coherent sources

Considering imperfect experimental conditions, here we give a practical proposal on the decoy-state quantum random number generator with weak coherent sources. In the proposal, three different intensities of light sources are used to estimate the contribution of single-photon pulses. System randomness is characterized and simulated through the minimum entropy. Numerical simulation results show that our new scheme can extract out randomness close to the asymptotic case where the fraction of single-photon pulses are exactly known. Furthermore, a proof-of-principle experimental demonstration is presented, showing excellent agreement between experiment and theory.

Virtual zero-photon catalysis for improving continuous-variable quantum key distribution via Gaussian post-selection

Quantum catalysis is a feasible approach to increase the performance of continuous-variable quantum key distribution (CVQKD), involving the special zero-photon catalysis (ZPC) operation. However, in the practical point of view, the improvement effect of this operation will be limited by the imperfection of the photon detector. In this paper, we show that the ZPC operation at the sender can be simulated by a post-selection method without implementing it in practical devices. While performing this virtual version of ZPC in CVQKD, we can not only reach the ideal case of its practical implementation with minimal hardware requirement, but also keep the benefit of Gaussian security proofs. Based on Gaussian modulated coherent state protocols with achievable parameters, we enhance the security of the proposed scheme from the asymptotical case to the finite-size scenario and composable framework. Simulation results show that similar to the asymptotical case, both the maximal transmission distance and the tolerable excess noise of virtual ZPC-involved CVQKD outperform the original scheme and the scheme using virtual photon subtraction while considering finite-size effect and composable security. In addition, the virtual ZPC-involved CVQKD can tolerate a higher imperfection of the detector, enabling its practical implementation of the CVQKD system with state-of-the-art technology.

The 2-adic analysis of Stirling numbers of the second kind via higher order Bernoulli numbers and polynomials

Several new estimates for the [Formula: see text]-adic valuations of Stirling numbers of the second kind are proved. These estimates, together with criteria for when they are sharp, lead to improvements in several known theorems and their proofs, as well as to new theorems, including a long-standing open conjecture by Lengyel. The estimates and criteria all depend on our previous analysis of powers of [Formula: see text] in the denominators of coefficients of higher order Bernoulli polynomials. The corresponding estimates for Stirling numbers of the first kind are also proved. Some attention is given to asymptotic cases, which will be further explored in subsequent publications.

Сomparative Analysis of Polynomial Maximization and Maximum Likelihood Estimates for Data with Exponential Power Distribution

The work is devoted to the estimate accuracy comparative analysis of the experimental data parameters with exponential power distribution (EPD) using the classical Maximum Likelihood Estimation (MLE) and the original Polynomial Maximization Method (PMM). In contrast to the parametric approach of MLE, which uses the description in the form of probability density distribution, PMM is based on a partial description in the of higher-order statistics form and the mathematical apparatus of Kunchenko's stochastic polynomials. An algorithm for finding PMM estimates using 3rd order stochastic polynomials is presented. Analytical expressions allowing to determine the variance of PMM-estimates of the asymptotic case parameters and EPD parameters with a priori information are obtained. It is shown that the relative theoretical estimates accuracy of different methods significantly depends on the EPD shape parameter and matches only for a separate case of Gaussian distribution. The effectiveness of different approaches (including valuation of mean values estimates) both with and without a priori information on EPD properties was investigated by repeated statistical tests (through Monte Carlo Method). The greatest efficiency areas for each of methods depending on EPD shape parameter and sample data volume are constructed.

CONFIDENCE INTERVALS CONSTRUCTED BY MODEL AVERAGING AND BOOTSTRAP SMOOTHING

We assess the performance of confidence intervals constructed by model averaging and bootstrap smoothing, using a simple testbed situation of two nested linear regression models with either known error variance or unknown error variance. The assessment is carried out by deriving computationally convenient exact expressions for the coverage probability and scaled expected length where the scaling is with respect to the usual confidence interval, with the same minimum coverage, based on the full model. We assessed the properties of the confidence interval centered on the model averaged estimator proposed by Buckland et al. (1997) with half-width proportional to the standard error of the estimate put forward by Buckland et al. (1997) of their formula (9). We considered both the finite sample case and the asymptotic case. Our results show that this model averaged confidence interval cannot be generally recommended, although it performs well when the error degrees of freedom is 1. We also assessed the properties of the confidence intervals centered on the bootstrap smoothed estimator with half-width proportional to the estimate of (a) the standard deviation of this estimator and (b) the delta method approximation, derived by Efron (2014), to the standard deviation of this estimator. We show that, except for the case of unknown error variance and small error degrees of freedom, the confidence interval centered on the bootstrap smoothed estimator does not have scaled expected length that is substantially less than 1, when the simpler model is correct. For this reason, we sought a formula for the data-based width that leads to improved performance of the confidence interval centered on the bootstrap smoothed estimator. Since this search failed, it raises the following question. Is there any formula for the data-based width of such a confidence interval so that it performs well? To answer this question, we compute the performance bound due to Kabaila & Kong (2016).