Asymptotic Expressions Research Articles

Phenomenological analysis of simple ion channel block in large populations of uncoupled cardiomyocytes.

Current understanding of arrhythmia mechanisms and design of anti-arrhythmic drug therapies hinges on the assumption that myocytes from the same region of a single heart have similar, if not identical, action potential waveforms and drug responses. On the contrary, recent experiments reveal significant heterogeneity in uncoupled healthy myocytes both from different hearts as well as from identical regions within a single heart. In this work, a methodology is developed for quantifying the individual electrophysiological properties of large numbers of uncoupled cardiomyocytes under ion channel block in terms of the parameters values of a conceptual fast-slow model of electrical excitability. The approach is applied to a population of nearly 500 rabbit ventricular myocytes for which action potential duration (APD) before and after the application of the drug nifedipine was experimentally measured (Lachaud et al., 2022, Cardiovasc. Res.). To this end, drug action is represented by a multiplicative factor to an effective ion conductance, a closed form asymptotic expression for APD is derived and inverted to determine model parameters as functions of APD and $\varDelta $APD (drug-induced change in APD) for each myocyte. Two free protocol-related quantities are calibrated to experiment using an adaptive-domain procedure based on an original assumption of optimal excitability. The explicit APD expression and the resulting set of model parameter values allow (a) direct evaluation of conditions necessary to maintain fixed APD or $\varDelta $APD, (b) predictions of the proportion of cells remaining excitable after drug application, (c) predictions of stimulus period dependency and (d) predictions of dose-response curves, the latter being in agreement with additional experimental data.

Learning curves for the multi-class teacher–student perceptron

One of the most classical results in high-dimensional learning theory provides a closed-form expression for the generalisation error of binary classification with a single-layer teacher–student perceptron on i.i.d. Gaussian inputs. Both Bayes-optimal (BO) estimation and empirical risk minimisation (ERM) were extensively analysed in this setting. At the same time, a considerable part of modern machine learning practice concerns multi-class classification. Yet, an analogous analysis for the multi-class teacher–student perceptron was missing. In this manuscript we fill this gap by deriving and evaluating asymptotic expressions for the BO and ERM generalisation errors in the high-dimensional regime. For Gaussian teacher, we investigate the performance of ERM with both cross-entropy and square losses, and explore the role of ridge regularisation in approaching Bayes-optimality. In particular, we observe that regularised cross-entropy minimisation yields close-to-optimal accuracy. Instead, for Rademacher teacher we show that a first-order phase transition arises in the BO performance.

Spin-foams as semiclassical vertices: Gluing constraints and a hybrid algorithm

Numerical methods in spin-foam models have significantly advanced in the last few years, yet challenges remain in efficiently extracting results for amplitudes with many quantum degrees of freedom. In this paper we sketch a proposal for a ``hybrid algorithm'' that would use both the full quantum amplitude and its asymptotic approximation in the relevant regimes. As a first step towards the algorithm, we derive a new representation of the partition function where each spin-foam vertex possesses its own coherent data, such that it can be individually asymptotically approximated. We do this through the implementation of gluing constraints between vertices, which we study numerically. We further derive an asymptotic expression for the constraints for arbitrary boundary data, including data for which there are no critical points. From this new representation we conjecture an intermediate quasi-geometric spin-foam regime describing a superposition of semi-classical vertices glued in a non-matching way via the gluing constraints.

Mathematical analysis of electromagnetic radiations diffracted by symmetric strip with Leontovich conditions in an-isotropic medium

The current article explores the meticulous analysis of the an-isotropic medium affecting the high-frequency signals diffracted by a conductible finite symmetric strip. A symmetric strip of finite length is considered here with impedance characteristics. The non-thermal plasma in the ionosphere contains a geomagnetic field which makes the surrounding medium as an an-isotropic medium. This idea is devised, mathematically, to analyze the propagation and diffraction pattern of EM-waves in the an-isotropic medium. Maxwell's equations modeled for non-thermal plasma are used to devise the EM-wave equation. Time is executed harmonically throughout the investigation. A coupled system of Wiener–Hopf equations is formulated and kernel functions are put to product decomposition to get the integral form of desired result. This integral form is then coped by employing the modified stationary phase method which yields an asymptotic expression of the separated field in the an-isotropic medium. The model can be turned out into a model for the isotropic medium by allocating the specific numeric values to the parameters controlling the medium. Physical interpretation of the separated field (diffraction by finite symmetric strip) is described for both the an-isotropic as well as the isotropic medium.

On Censoring Time in Statistical Monitoring of Lifetime Data

Life tests for highly reliable products often take a long time even using accelerated life testing with censoring. When the production process is monitored by control charts with the product lifetime as the key quality characteristic, the time spent on life testing could incur significant delays for practitioners to make decisions after sampling. However, shortening the test duration, that results in excessive right-censored observations, inevitably degrades the test power for anomaly detection. This study pays close attention to the determination of censoring time in life tests when monitoring lifetime data with the likelihood-based control charts. To interpret the optimal censoring time, the performance metric—out-of-control average time to signal (OC ATS), is deconstructed into two parts: the original OC ATS and the delay caused by life testing. Finite-sample analytical and large-sample asymptotic expressions of ATS metrics are derived for Type-I censored exponential lifetimes. Similar analytical expressions are also derived for the Weibull case. For general distributions, a Monte Carlo simulation procedure is developed for obtaining approximate results. Our numerical investigation uncovers the 2-fold impact of censoring time on the actual performance of control charts under various scenarios and provides useful references for practitioners to set more sensible censoring times in life testing.

Security-reliability tradeoff of MIMO TAS/SC networks using harvest-to-jam cooperative jamming methods with random jammer location

This paper evaluates outage probability (OP) and intercept probability (IP) of physical-layer security based MIMO networks adopting cooperative jamming (Coop-Jam). In the considered scenario, a multi-antenna source communicates with a multi-antenna destination employing transmit antenna selection (TAS)/ selection combining (SC), in presence of a multi-antenna eavesdropper using SC. One of jammers appearing near the destination is selected for generating jamming noises on the eavesdropper. Moreover, the destination supports the wireless energy for the chosen jammer, and cooperates with it to remove the jamming noises. We consider two jammer selection approaches, named RAND and SHORT. In RAND, the destination randomly selects the jammer, and in SHORT, the jammer, which is nearest to the destination, is chosen. We derive exact and asymptotic expressions of OP and IP over Rayleigh fading, and perform Monte-Carlo simulations to verify the correction of our derivation. The results present advantages of the proposed RAND and SHORT methods, as compared with the corresponding one without using Coop-Jam.

Joint Effects of Co-Channel Interferences and Pointing Errors on Dual-Hop Mixed RF/FSO Fixed-Gain and Variable-Gain Relaying Systems

The performance of a dual-hop mixed radio frequency (RF)/free space optical (FSO) system with co-channel interferences (CCIs) is studied for both fixed-gain and variable-gain amplify-and-forward(AF) relaying scenarios. The RF link and interference link undergo the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\kappa$</tex-math></inline-formula> - <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mu$</tex-math></inline-formula> shadowed fading and the FSO link experiences the Fisher-Snedecor <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${\cal F}$</tex-math></inline-formula> distribution with pointing errors. For the considered system, we derive new closed-form expressions for the probability density function (PDF), cumulative distribution function (CDF), and average bit error rate (BER) of the end-to-end signal-to-interference-plus-noise ratio (SINR). In addition, efficient numerical expressions for the ergodic capacity are also obtained. Moveover, the asymptotic expressions for the CDF and average BER are derived. Our results can be regarded as a generalization of the previous results. Finally, numerical and simulation results are presented to verify the correctness of the proposed expressions.

Performance Analysis of Full-Duplex Two-Way RIS-Based Systems With Imperfect CSI and Discrete Phase-Shift Design

This letter investigates the effects of imperfect channel state information (CSI) for incident channels and discrete phase-shift design on the performance of full-duplex two-way communication-based reconfigurable intelligent surface (RIS) systems. In order to characterize the system performance, the approximate and asymptotic expressions for the outage probability (OP) and ergodic capacity (EC) are derived. Relying on these analyses, the diversity order, the ergodic slope, and some valuable technical insights regarding performance trends are also deduced. Extensive simulations are provided to corroborate the theoretical analysis. Numerical results show that: 1) the derived analysis can approach well with the simulation results; 2) imperfect CSI leads to the performance floor at high transmit power regimes; 3) better CSI estimation provides significant system performance improvement; and 4) by using 3-bits for DPD’s quantization and/or increasing the number of RIS’s elements, the system performance with imperfect CSI can be significantly enhanced.

Performance analysis of FBMC/OQAM systems in frequency-selective channels

Filter bank multicarrier (FBMC) systems based on offset QAM modulation (FBMC/OQAM) represent an interesting alternative to the widely used OFDM scheme for machine type communications or cognitive radio. In this paper the performance of FBMC/OQAM systems exploiting a single-tap receiver in frequency-selective Rayleigh fading channels, is analyzed. In particular, the SER and the asymptotic SER (i.e., for a large number of subcarriers) for any prototype filter, is derived. Moreover, a simple closed-form approximate expression of the asymptotic SER is obtained by taking into account, for each active subcarrier, only intersymbol interference terms due to the considered subcarrier and the adjacent subacarriers, and intercarrier interference terms due to the adjacent subcarriers. The numerical results show that the obtained SER expression can predict the actual performance of the system while the asymptotic SER expression provides a useful upper-bound. Moreover, the curves obtained by exploiting the derived simple closed-form approximate expression can be used to predict the asymptotic upper-bound when the considered prototype filter is adopted.

On randomized sketching algorithms and the Tracy–Widom law

There is an increasing body of work exploring the integration of random projection into algorithms for numerical linear algebra. The primary motivation is to reduce the overall computational cost of processing large datasets. A suitably chosen random projection can be used to embed the original dataset in a lower-dimensional space such that key properties of the original dataset are retained. These algorithms are often referred to as sketching algorithms, as the projected dataset can be used as a compressed representation of the full dataset. We show that random matrix theory, in particular the Tracy–Widom law, is useful for describing the operating characteristics of sketching algorithms in the tall-data regime when the sample size n is much greater than the number of variables d. Asymptotic large sample results are of particular interest as this is the regime where sketching is most useful for data compression. In particular, we develop asymptotic approximations for the success rate in generating random subspace embeddings and the convergence probability of iterative sketching algorithms. We test a number of sketching algorithms on real large high-dimensional datasets and find that the asymptotic expressions give accurate predictions of the empirical performance.

Data Transmission of Digital Grid Assisted by Intelligent Relaying

In this paper, we study the relaying and cache aided digital grid data transmission, where the relaying may be equipped by caching or not, depending on specific applications. For both cases, we evaluate the impact of relaying and caching on the system performance of digital grid data transmission through theoretical derivation. To this end, an analytical expression on the outage probability is firstly derived for the data transmission. We then provide an asymptotic expression on the system outage probability. Finally, some simulation results are provided to verify the correctness of the derived analysis on the system performance, and show the impact of relaying and caching on the data transmission of digital grid system. In particular, the usage of caching at the relaying can help strengthen the data transmission performance of the considered system effectively. The results in this paper could provide some reference to the development of wireless transmission and scalable information systems.

Entropy of rigid k-mers on a square lattice.

Using the transfer matrix technique, we estimate the entropy for a gas of rods of sizes equal to k (named k-mers), which cover completely a square lattice. Our calculations were made considering three different constructions, using periodical and helical boundary conditions. One of those constructions, which we call profile method, was based on the calculations performed by Dhar and Rajesh to obtain a lower limit to the entropy of very large chains placed on the square lattice. This method, so far as we know, was never used before to define the transfer matrix, but turned out to be very useful, since it produces matrices with smaller dimensions than those obtained using the usual approach. Our results were obtained for chain sizes ranging from k=2 to k=10 and they are compared with results already available in the literature. In the case of dimers (k=2) our results are compatible with the exact result. For trimers (k=3), recently investigated by Ghosh etal., also our results were compatible, with the same happening for the simulational estimates obtained by Pasinetti etal. in the whole range of rod sizes. Our results are also consistent with the asymptotic expression for the behavior of the entropy as a function of the size k, proposed by Dhar and Rajesh for very large rods (k≫1).

Outage probability and ergodic capacity analysis of satellite–terrestrial NOMA system with mixed RF/mmWave relaying

This paper investigates the performance of hybrid radio frequency/millimeter wave (RF/mmWave) satellite–terrestrial relay network with non-orthogonal multiple access (NOMA) scheme. In particular, the satellite network operates at the Ka-band while the mmWave band is adopted at decode-and-forward (DF) relay–destination link. The blockage effect in mmWave communication is considered to deduce the ordered probability density function (PDF) of the user equipment (UE) distance. However, the randomness of the UEs’ location and the random beamforming design cause the difficulty of the performance analysis. Herein, we provide a closed-form approximation for the outage probability. Moreover, the system diversity order is derived from the asymptotic expressions of the outage probability at high signal-to-noise ratio (SNR) slopes. Besides, approximate ergodic capacity performances of the multi-UEs are evaluated with numerical integration. Simulations are applied to demonstrate the accuracy of the theoretical analysis and show that the performance of the ordered selection NOMA scheme outperforms the orthogonal multiple access scheme and random selection scheme.

Free Dynamics of Feature Learning Processes

Regression models usually tend to recover a noisy signal in the form of a combination of regressors, also called features in machine learning, themselves being the result of a learning process. The alignment of the prior covariance feature matrix with the signal is known to play a key role in the generalization properties of the model, i.e. its ability to make predictions on unseen data during training. We present a statistical physics picture of the learning process. First we revisit the ridge regression to obtain compact asymptotic expressions for train and test errors, rendering manifest the conditions under which efficient generalization occurs. It is established thanks to an exact test-train sample error ratio combined with random matrix properties. Along the way in the form of a self-energy emerges an effective ridge penalty—precisely the train to test error ratio—which offers a very simple parameterization of the problem. This formulation appears convenient to tackle the learning process of the feature matrix itself. We derive an autonomous dynamical system in terms of elementary degrees of freedom of the problem determining the evolution of the relative alignment between the population matrix and the signal. A macroscopic counterpart of these equations is also obtained and various dynamical mechanisms are unveiled, allowing one to interpret the dynamics of simulated learning processes and to reproduce trajectories of single experimental run with high precision.

Outage performance for full-duplex relaying satellite-terrestrial multi-antenna NOMA systems with imperfect SIC

Due to the masking effect, a reliable line-of-sight transmission between a satellite and terrestrial nodes is not always guaranteed, and a half-duplex (HD) relaying is commonly performed in literature to assist such transmission. However, the system efficiency could be seriously degraded with the HD relaying as a two-time slot is required for a dual-hop information transmission. Therefore, in this paper, a full-duplex (FD) relaying, which can enable signal receiving and transmitting simultaneously, is considered for a satellite-terrestrial non-orthogonal multiple access (NOMA) system consisting of a multi-antenna satellite source, a FD relay, and two multi-antenna destination nodes. Specifically, the satellite source adopts a maximum ratio transmission scheme to transmit a NOMA signal to the relay, and the relay deploys a decode-and-forward protocol to forward the information to two destination nodes simultaneously, which employ the maximal ratio combining strategy to process the received signals. Assuming the satellite link experiences Shadowed-Rician fading and the terrestrial links undergo Nakagami-m fading distribution, both analytical and asymptotic expressions of outage probability (OP) for two destination nodes are presented in the presence of residual self-interference and imperfect successive interference cancellation. In addition, an adaptive power allocation (APA) scheme is performed, and the system throughput under the delay-limited transmission mode is also analyzed. Finally, Monte-Carlo simulations are adopted to verify the accuracy of those derived analytical expressions. Numerical results show that the OP performance with the FD mode outperforms that with the HD one, and adopting the APA scheme can result in the best OP performance.

Mechanisms of mass transfer to small spheres sinking in turbulence

Using laboratory experiments and numerical simulations, we examine the transfer of soluble material from small, spherical particles sinking in homogeneous turbulence at large Péclet number. A theoretical analysis predicts two distinct mechanisms of convective mass transfer: strain due to turbulence and slip due to gravitational settling. Their relative strength is parametrised by the sinking ratio, ${Sr} = w_0 \tau _\eta /a$ , where $w_0$ is the quiescent settling velocity, $a$ is the particle radius and $\tau _\eta$ is the Kolmogorov time scale. This analysis predicts that the topology of the concentration wake changes from a symmetric topology at ${Sr} \ll 1$ to an asymmetric topology at ${Sr} \gg 1$ as the dominant mechanism of mass transfer changes. Particle tracking flow visualisations of small spheres releasing dye in turbulence confirm the existence of this change in mechanism at ${Sr} = O(1)$ . We complement these experiments with numerical simulations of the mass transfer from sinking particles. The transfer rate predicted by the simulation is found to be in good agreement with literature data for mass transfer to turbulent suspensions of solid particles and is consistent with asymptotic expressions for mass transfer in uniform flow when ${Sr} \gg 1$ . A decomposition of the convective fluxes confirms the transition in the transfer mechanism. At ${Sr} = O(1)$ , both mechanisms provide comparable contributions to the transfer rate. Cross-correlation analysis reveals that particle-scale knowledge of both the recent strain and velocity history is required to predict the instantaneous transfer rate. Turbulence-induced particle rotation has a modest suppression effect upon convective transfer by sinking.

Hybrid SAG-FSO/SH-FSO/RF Transmission for Next-Generation Satellite Communication Systems

Recent advances demonstrate satellite communication (SatCom) as a potent enabler for future Terabit/s global connectivity. Existing SatCom systems, however, rely mostly on radio frequency (RF) transmission, whose limited available bandwidth is the main bottleneck for further data rate increases. Free-space optical (FSO) communication links, with huge license-free bandwidth, have emerged as an attractive alternative. Despite their ability to deliver high-throughput transmissions, FSO communications are weather-dependent and susceptible to atmospheric turbulence. Hybrid FSO/RF and space-air-ground (SAG) FSO transmissions are possible solutions to increase the reliability of FSO-based transmission systems. A strategically deployed unmanned-aerial vehicle, acting as a relay, can successfully mitigate the adverse effect of atmospheric turbulence, while the hybrid FSO/RF transmission can address weather-dependent effects. These solutions can also be integrated to create a system with significantly improved performance and reliability. To evaluate the performance of the resulting integrated transmission system, we analyze the outage probability and average symbol error probability in this work. Asymptotic expressions are also derived to get further insight into the system behavior and calculate the overall diversity gain. Furthermore, we consider the optimal design of switching thresholds. The numerical results show that the integrated transmission system achieves about 10 dB performance gain over existing solutions for both downlink and uplink scenarios.

Performance Analysis of RIS Assisted NOMA Networks over Rician Fading Channels

In this paper, we consider a downlink non-orthogonal multiple access (NOMA) network assisted by two reconfigurable intelligent surfaces (RISs) over Rician fading channels, in which each user communicates with the base station by the virtue of a RIS to enhance the reliability of the received signal. To evaluate the system performance of our proposed RIS-NOMA network, we first derive the exact and asymptotic expressions for the outage probability and ergodic rate of two users. Then, we derive the exact and asymptotic upper bound expressions for the ergodic rate of the nearby user. Based on asymptotic analytical results, the diversity orders for the outage probability and the high signal-to-noise ratio (SNR) slopes for the ergodic rate of the two users are obtained in the high SNR regime. Moreover, we derive the system throughputs of the proposed RIS-NOMA network in delay-limited and delay-tolerant transmission modes. Numerical results confirm our analysis and demonstrate that: 1) The outage probability and ergodic rate of RIS-NOMA networks are superior to that of RIS-assisted orthogonal multiple access (OMA) networks; 2) The RIS-NOMA networks have ability to achieve a larger system throughput compared to RIS-OMA networks; and 3) The system performance of RIS-NOMA networks can be significantly improved as the number of reflecting elements and Rician factor increases.

TRAPPING PROBLEM OF HONEYPOTS ON FRACTAL NETWORKS WITH THE STURMIAN STRUCTURE

This paper studies the average trapping time of honeypots on some evolving networks. We propose a simple algorithmic framework for generating networks with Sturmian structure. From the balance property and the recurrence property of Sturmian words, we estimate the average trapping time of our proposed networks with an asymptotic expression [Formula: see text], where [Formula: see text] is a bounded expression related to word [Formula: see text]. We next consider networks with multi-honeypots and generalize our basic models. Additionally, we give an symmetrical method to create a series of networks with the Sturmian structure, and the average trapping time satisfies [Formula: see text], which is independent of any word [Formula: see text]. The generalized methods may have some illuminating effects on the study of networks with randomness.

A Unified Framework Analysis for Reconfigurable Intelligent Surface-Aided Coordinated NOMA Systems

This paper presents a unified framework analysis for the performance of reconfigurable intelligent surface (RIS)-aided coordinated non-orthogonal multiple access (C-NOMA) systems where the discrete phase-shift alignment and imperfect successive interference cancellation (SIC) are investigated. This framework yields double opportunities to enhance users' performance via RIS and dedicated relay transmission. To assess the system characteristic, the users' outage probability (OP) and ergodic capacity (EC) expressions are evaluated and derived in terms of approximate closed-form and asymptotic expressions. Numerical results validate our derivations and show that: i) expanding the number of RISs element implementations can significantly improve the users' OP, and ii) the RIS-aided C-NOMA system achieves a superior EC improvement compared to RIS-orthogonal multiple access (OMA) and conventional C-NOMA ones.