Tractable Expressions Research Articles

Modelling and analysis of repulsive cellular networks using Matérn hard‐core point processes

AbstractThe repulsive nature of Matérn hard‐core point process makes it well‐suited for modelling base station placements, where proximity constraints are critical. This letter reexamines the spatial properties of the model and proposes an empirical Weibull approximation for the contact distance distribution, where the shape and scale parameters can be reliably estimated through tractable regression models. Additionally, the probability generating functional is approximated using an exponential upper bound, simplifying the computation of the Laplace transform of interference. From these results, analytically tractable expressions are derived for the probability of coverage and average data rate. Simulations validate the accuracy of these analytical estimates.

Approximation for probability of coverage in softly inhibitive cellular networks

AbstractThe Strauss point process is a very popular model for describing the random cellular networks, yet several key statistical properties such as intensity, empty space function, and probability generating functional have remained elusive. This article addresses these issues by first leveraging the Poisson saddle point method to approximate the distance‐conditioned intensity for Strauss point processes. Subsequently, the author derives an analytically tractable expression for the distribution of empty space distance based on a conditional thinning mechanism. Additionally, the author establishes an upper bound for the probability generating functional in Strauss point processes, which is crucial for evaluating the Laplace transform of cumulative interference in relevant cellular networks. These findings facilitate the systematic derivation of spatially averaged probability of coverage, and the accuracy of analytic results is validated through simulations.

Global knowledge and trade flows: Theory and measurement

We present a model of trade, global innovation, and diffusion, inspired by Eaton and Kortum (1999). The specific structure for innovation and diffusion we propose, which leverages general results developed in our previous work (Lind and Ramondo, 2023a), allows us to measure the flow of ideas across countries and over time. By deriving tractable expressions for productivity and expenditure, we can use easily-available international trade data to estimate both innovation and diffusion rates across countries and over time. We find that, although innovation is correlated with economic growth, there are many high income countries that primarily produce using diffused ideas from foreign sources.

Profiling expression strategies for a type III polyketide synthase in a lysate-based, cell-free system

Some of the most metabolically diverse species of bacteria (e.g., Actinobacteria) have higher GC content in their DNA, differ substantially in codon usage, and have distinct protein folding environments compared to tractable expression hosts like Escherichia coli. Consequentially, expressing biosynthetic gene clusters (BGCs) from these bacteria in E. coli often results in a myriad of unpredictable issues with regard to protein expression and folding, delaying the biochemical characterization of new natural products. Current strategies to achieve soluble, active expression of these enzymes in tractable hosts can be a lengthy trial-and-error process. Cell-free expression (CFE) has emerged as a valuable expression platform as a testbed for rapid prototyping expression parameters. Here, we use a type III polyketide synthase from Streptomyces griseus, RppA, which catalyzes the formation of the red pigment flaviolin, as a reporter to investigate BGC refactoring techniques. We applied a library of constructs with different combinations of promoters and rppA coding sequences to investigate the synergies between promoter and codon usage. Subsequently, we assess the utility of cell-free systems for prototyping these refactoring tactics prior to their implementation in cells. Overall, codon harmonization improves natural product synthesis more than traditional codon optimization across cell-free and cellular environments. More importantly, the choice of coding sequences and promoters impact protein expression synergistically, which should be considered for future efforts to use CFE for high-yield protein expression. The promoter strategy when applied to RppA was not completely correlated with that observed with GFP, indicating that different promoter strategies should be applied for different proteins. In vivo experiments suggest that there is correlation, but not complete alignment between expressing in cell free and in vivo. Refactoring promoters and/or coding sequences via CFE can be a valuable strategy to rapidly screen for catalytically functional production of enzymes from BCGs, which advances CFE as a tool for natural product research.

Age of Information in NOMA-IoT Networks: A Temporal Queuing Model Perspective

The Internet of Things (IoT) with non-orthogonal multiple access (NOMA) has been anticipated to offer diverse real-time applications, wherein the crux is to guarantee the age of information (AoI) for dynamic traffic. However, the traffic temporal variation provokes the interdependence between queue status and interference, in which context the AoI performance remains to be further explored. In this paper, an analytical framework is established to characterize the AoI performance in NOMA-IoT networks with random Bernoulli and deterministic periodic arrivals. Particularly, a numerical algorithm is devised to obtain the queue service rate, and tractable expressions for AoI violation probability and average AoI under both the first-come first-served (FCFS) and the preemptive last-come first-served (LCFS-PR) service disciplines are derived. Simulations are conducted to validate the proposed analysis. The results unveil that LCFS-PR conduces to better AoI performance than FCFS, and yet the gain diverges for each device with different traffic arrival configurations. In addition, the result shows that with sporadic traffic arrival, the periodic pattern outperforms the Bernoulli pattern, whereas this advantage gradually diminishes with more frequent packet arrival.

Computationally Tractable Gaussian Process-based Stochastic Predictive Control Using Backoffs

Current stochastic nonlinear model predictive control (SNMPC) hinges on the lack of high-fidelity models that describe the system behavior and the lack of tractable solution methods that handle chance constraints. Motivated by this, a model-and data-driven predictive control approach using Gaussian processes (GP-MDPC) is synthesized in this paper. It exploits GPs to learn the unknown dynamics and apply Taylor expansion for uncertainty propagation through probabilistic modeling. A backoff approximation method is explored to reformulate the chance constraints into tractable expressions. Finally, a finite-horizon stochastic optimal control problem (FH-SOCP) is formulated.

Random Caching Design for Multi-User Multi-Antenna HetNets With Interference Nulling

The strong interference suffered by users can be a severe problem in cache-enabled networks (CENs) due to the content-centric user association mechanism. To tackle this issue, multi-antenna technology may be employed for interference management. In this paper, we consider a user-centric interference nulling (IN) scheme in two-tier multi-user multi-antenna CEN, with a hybrid most-popular and random caching policy at macro base stations (MBSs) and small base stations (SBSs) to provide file diversity. All the interfering SBSs within the IN range of a user are requested to suppress the interference at this user using zero-forcing beamforming. Using stochastic geometry analysis techniques, we derive a tractable expression for the area spectral efficiency (ASE). A lower bound on the ASE is also obtained, with which we then consider ASE maximization, by optimizing the caching policy and IN coefficient. To solve the resultant mixed integer programming problem, we design an alternating optimization algorithm to minimize the lower bound of the ASE. Our numerical results demonstrate that the proposed caching policy yields performance that is close to the optimum, and it outperforms several existing baselines.

Entropy and Energy Profiles of Chemical Reactions.

The description of chemical processes at the molecular level is often facilitated by the use of reaction coordinates or collective variables (CVs). The CV measures the progress of the reaction and allows the construction of profiles that track how specific properties evolve as the reaction progresses. Whereas CVs are routinely used, especially alongside enhanced sampling techniques, the links among reaction profiles, thermodynamic state functions, and reaction rate constants are not rigorously exploited. Here, we report a unified treatment of such reaction profiles. Tractable expressions are derived for the free-energy, internal-energy, and entropy profiles as functions of only the CV. We demonstrate the ability of this treatment to extract quantitative insight from the entropy and internal-energy profiles of various real-world physicochemical processes, including intramolecular organic reactions, ionic transport in superionic electrolytes, and molecular transport in nanoporous materials.

Coexisting Terahertz and RF Finite Wireless Networks: Coverage and Rate Analysis

Wireless communications over Terahertz (THz)-band frequencies are vital enablers of ultra-high rate applications and services in sixth-generation (6G) networks. However, THz communications suffer from poor coverage because of inherent THz features such as high penetration losses, significant molecular absorption, and severe path loss. To surmount these critical challenges and fully exploit the THz band, we explore a coexisting radio frequency (RF) and THz finite indoor network in which THz small cells are deployed to provide high data rates, and RF macrocells are deployed to satisfy coverage requirements. Using stochastic geometry tools, we assess the performance of coexisting RF and THz networks and derive tractable analytical expressions for the coverage probability and average achievable rate. The analytical results are validated with Monte-Carlo simulations. Several insights are devised for accurate tuning and optimization of THz system parameters, including the THz bias, and the fraction of THz access points (APs) to deploy. The obtained results recognize a clear coverage/rate trade-off where a high fraction of THz AP improves the rate significantly but may degrade the coverage performance. Furthermore, the location of a user in the finite area highly affects the fraction of THz APs that optimizes its quality of service.

Analysis and Optimization of a Double-IRS Cooperatively Assisted System With a Quasi-Static Phase Shift Design

The analysis and optimization of single intelligent reflecting surface (IRS)-assisted systems have been extensively studied, whereas little is known regarding multiple-IRS-assisted systems. This paper investigates the analysis and optimization of a double-IRS cooperatively assisted downlink system (D-IRS-C), where a multi-antenna base station (BS) serves a single-antenna user with the help of two multi-element IRSs, connected by an inter-IRS channel. The channel between any two nodes is modeled with Rician fading. The BS adopts the instantaneous CSI-adaptive maximum-ratio transmission (MRT) beamformer, and the two IRSs adopt a cooperative quasi-static phase shift design. The goal is to maximize the average achievable rate, which can be reflected by the average channel power of the equivalent channel between the BS and user at low channel estimation and phase adjustment costs and computational complexity. First, we obtain tractable expressions of the average channel power of the equivalent channel in the general (Rician factor), pure line of sight (LoS), and pure non-line of sight (NLoS) regimes, respectively. Then, we jointly optimize the phase shifts of the two IRSs to maximize the average channel power of the equivalent channel in these regimes. The optimization problems are challenging non-convex problems. We obtain globally optimal closed-form solutions for some cases and propose computationally efficient iterative algorithms to obtain stationary points for the other cases. Next, we compare the computational complexity for optimizing the phase shifts and the optimal average channel power of D-IRS-C with those of a counterpart double-IRS non-cooperatively assisted system (D-IRS-NC) and a counterpart single-IRS-assisted system (S-IRS) at a large number of reflecting elements in the three regimes. Finally, we numerically demonstrate notable gains of the proposed solutions over the existing solutions at different system parameters. To our knowledge, this is the first work that optimizes the quasi-static phase shift design of D-IRS-C and characterizes its advantages over the optimal quasi-static phase shift design of the counterpart D-IRS-NC and S-IRS.

Underwater Propulsion Using Forced Excitation of a Flexible Beam

Abstract Aquatic animals commonly oscillate their fins, tails, or other structures to propel and control themselves in water. These elements are not perfectly rigid, so the interplay between their stiffness and the fluid loading dictates their dynamics. We examine the propulsive qualities of a tail-like flexible beam actuated by a dynamic moment over a range of frequencies and flow speeds. This is accomplished using the equations of fluid-immersed beams in combination with a set of tractable expressions for thrust and efficiency. We solve these expressions over the velocity–frequency plane and show that the flexible propulsor has regions of both positive and negative thrust. We also show the behavior of a sample underwater vehicle with fixed drag characteristics as an illustration of a realizable system.

To Filter or Not to Filter? Impact on Stability of Delay-Difference and Neutral Equations With Multiple Delays

For control systems where the closed-loop system description is governed by linear delay-differential equations of neutral type, it is known that stability may be fragile, in the sense of sensitive to infinitesimal perturbations to parameters in the system model or arbitrarily small errors in the implementation of the controller. A natural approach to resolve this problem of ill-posedness and to break down the underlying instability mechanisms, rooted in characteristic roots moving from the left plane to the right one via the point at infinity, consists of including a low-pass filter in the control loop, provided the inclusion preserves stability. Independently of the particular control problem, the addition of a low-pass filter essentially boils down to a “regularization” of delay-difference equations and delay equations of neutral type in terms of parametrized delay equations of retarded type, where the parameter can be interpreted as the inverse of the filter's cut-off frequency. In this paper, the stability properties of these parametrized delay equations are analyzed in a general, multi-delay setting, with focus on the transition to the original delay-difference or neutral equations. It is illustrated that the spectral abscissa may not be continuous at the transition, which may impact stability. Hence, conditions for preservation of stability in terms of a robustified stability indicator called filtered spectral abscissa are presented, for which mathematical characterizations and a computationally tractable expression are provided. The application of a PD controller to a time-delay system with relative degree one is used to motivate the structure of the equations studied throughout the paper, and to explicate the implications of the presented results on control design, discussed in the last section.

Killing Horizons and Surface Gravities for a Well-Behaved Three-Function Generalization of the Kerr Spacetime

Thanks to the recent advent of the event horizon telescope (EHT), we now have the opportunity to test the physical ramifications of the strong-field near-horizon regime for astrophysical black holes. Herein, emphasizing the trade-off between tractability and generality, the authors discuss a particularly powerful three-function distortion of the Kerr spacetime, depending on three arbitrary functions of the radial coordinate r, which on the one hand can be fit to future observational data, and on the other hand is sufficiently general so as to encompass an extremely wide class of theoretical models. In all of these spacetimes, both the timelike Hamilton–Jacobi (geodesic) and massive Klein–Gordon (wave) equations separate, and the spacetime geometry is asymptotically Kerr; hence, these spacetimes are well-suited to modeling real astrophysical black holes. The authors then prove the existence of Killing horizons for this entire class of spacetimes, and give tractable expressions for the angular velocities, areas, and surface gravities of these horizons. We emphasize the validity of rigidity results and zeroth laws for these horizons.

Two-Timescale Design for Reconfigurable Intelligent Surface-Aided Massive MIMO Systems With Imperfect CSI

This paper investigates the two-timescale transmission scheme for reconfigurable intelligent surface (RIS)-aided massive multiple-input multiple-output (MIMO) systems, where the beamforming at the base station (BS) is adapted to the rapidly-changing instantaneous channel state information (CSI), while the nearly-passive beamforming at the RIS is adapted to the slowly-changing statistical CSI. Specifically, we first consider a system model with spatially independent Rician fading channels, which leads to tractable expressions and offers analytical insights on the power scaling laws and on the impact of various system parameters. Then, we analyze a more general system model with spatially correlated Rician fading channels and consider the impact of electromagnetic interference (EMI) caused by any uncontrollable sources present in the considered environment. For both case studies, we apply the linear minimum mean square error (LMMSE) estimator to estimate the aggregated channel from the users to the BS, utilize the low-complexity maximal ratio combining (MRC) detector, and derive a closed-form expression for a lower bound of the achievable rate. Besides, an accelerated gradient ascent-based algorithm is proposed for solving the minimum user rate maximization problem. Numerical results show that, in the considered setup, the spatially independent model without EMI is sufficiently accurate when the inter-distance of the RIS elements is sufficiently large and the EMI is mild. In the presence of spatial correlation, we show that an RIS can better tailor the wireless environment. Furthermore, it is shown that deploying an RIS in a massive MIMO network brings significant gains when the RIS is deployed close to the cell-edge users. On the other hand, the gains obtained by the users distributed over a large area are shown to be modest.

Coverage Analysis for Reconfigurable Intelligent Surface-Assisted Cellular Networks Over Nakagami-m Fading Channels

Unlike prior research works that focus on cell-level coverage analysis and optimization, this letter develops a system-level analysis framework based on stochastic geometry for RIS-assisted cellular networks experiencing Nakagami- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${m}$ </tex-math></inline-formula> fading. First, we reveal that the passive beamforming gain of RISs for the signal approachs a deterministic value due to the channel hardening effect, while it is approximated as a Gamma distribution for the interference. Then, the tractable closed-form expressions for the signal-to-interference ratio (SIR) coverage probability of each tier and the entire network are derived. The simulation results verify the correctness of the theoretical analysis. Finally, it is shown that given RIS element density, for scenarios with the path loss exponent greater than 4, densely deploying small-size RISs achieves better coverage performance, while for scenarios with the path loss exponent less than 4, sparse deployment of large-size RISs is a better option.

A Tractable Approach to Coverage Analysis in Downlink Satellite Networks

Satellite networks are promising to provide ubiquitous and high-capacity global wireless connectivity. Traditionally, satellite networks are modeled by placing satellites on a grid of multiple circular orbit geometries. Such a network model, however, requires intricate system-level simulations to evaluate coverage performance, and analytical understanding of the satellite network is limited. Continuing the success of stochastic geometry in a tractable analysis for terrestrial networks, in this paper, we develop novel models that are tractable for the coverage analysis of satellite networks using stochastic geometry. By modeling the locations of satellites and users using Poisson point processes on the surfaces of concentric spheres, we characterize analytical expressions for the coverage probability of a typical downlink user as a function of relevant parameters, including path-loss exponent, satellite height, density, and Nakagami fading parameter. Then, we also derive a tight lower bound of the coverage probability in tractable expression while keeping full generality. Leveraging the derived expression, we identify the optimal density of satellites in terms of the height and the path-loss exponent. Our key finding is that the optimal average number of satellites decreases logarithmically with the satellite height to maximize the coverage performance. Simulation results verify the exactness of the derived expressions.

MIMO Device-to-Device Communications via Cooperative Dual-Polarized Intelligent Surfaces

By making the propagation environment a programmable entity, reconfigurable intelligent surfaces (RIS) constitute a key enabler to fulfill the quality-of-service requirements of device-to-device (D2D) communications. In practice, however, a single RIS may not be able to establish a line-of-sight path between the devices, e.g., in environments with rich scattering or when devices are far apart. This can hinder the communications especially in applications that require the transfer of different data sets from a source to multiple destinations. Considering such type of applications, this letter proposes a dual-polarized cooperative RIS-assisted MIMO D2D communication scheme, where the beam routing path via multiple RISs is selected based on the maximum received signal-to-noise ratio and path constraints. Looking into system operation in indoor environments, and utilizing a realistic channel model, tractable expressions are obtained for the system’s error rate and outage probability. An asymptotic analysis is also conducted to evaluate the diversity gain. The findings reveal significant advantages of the dual-polarized cooperative RIS scheme over other RIS-based schemes.

Discovering explicit Reynolds-averaged turbulence closures for turbulent separated flows through deep learning-based symbolic regression with non-linear corrections

This work introduces a novel data-driven framework to formulate explicit algebraic Reynolds-averaged Navier–Stokes (RANS) turbulence closures. Recent years have witnessed a blossom in applying machine learning (ML) methods to revolutionize the paradigm of turbulence modeling. However, due to the black-box essence of most ML methods, it is currently hard to extract interpretable information and knowledge from data-driven models. To address this critical limitation, this work leverages deep learning with symbolic regression methods to discover hidden governing equations of Reynolds stress models. Specifically, the Reynolds stress tensor is decomposed into linear and non-linear parts. While the linear part is taken as the regular linear eddy viscosity model, a long short-term memory neural network is employed to generate symbolic terms on which tractable mathematical expressions for the non-linear counterpart are built. A novel reinforcement learning algorithm is employed to train the neural network to produce best-fitted symbolic expressions. Within the proposed framework, the Reynolds stress closure is explicitly expressed in algebraic forms, thus allowing for direct functional inference. On the other hand, the Galilean and rotational invariance are craftily respected by constructing the training feature space with independent invariants and tensor basis functions. The performance of the present methodology is validated through numerical simulations of three different canonical flows that deviate in geometrical configurations. The results demonstrate promising accuracy improvements over traditional RANS models, showing the generalization ability of the proposed method. Moreover, with the given explicit model equations, it can be easier to interpret the influence of input features on generated models.

The Energy Efficiency of Heterogeneous Cellular Networks Based on the Poisson Hole Process

In order to decrease energy consumption caused by the dense deployment of pico base stations (PBSs) in heterogeneous cellular networks (HetNets), this paper first analyzes the energy efficiency (EE) of two-tier HetNets and then proposes a method to maximize the network EE by adjusting the PBS transmit power. The two-tier HetNets are modeled by the Poisson point process (PPP) and the Poisson hole process (PHP), and then the coverage probability of the macro base station (MBS) and the PBS in the two-tier HetNets is derived based on the mean interference to signal ratio (MISR). According to the user association probability, the coverage probability of the PPP-PHP HetNets is obtained. Then, the tractable expression of the average achievable rate is deduced on the basis of the relationship between the coverage probability and the average achievable rate. Finally, the expression of EE is derived and the EE optimization algorithm is proposed based on the PBS transmit power. The simulation results show that the PPP-PHP network is superior to the PPP-PPP network in terms of coverage probability and EE, and the network EE can be effectively improved by setting an appropriate PBS transmit power.

Modeling and Analysis of UAV-Assisted Mobile Network with Imperfect Beam Alignment

With the rapid development of emerging 5G and beyond (B5G), Unmanned Aerial Vehicles (UAVs) are increasingly important to improve the performance of dense cellular networks. As a conventional metric, coverage probability has been widely studied in communication systems due to the increasing density of users and complexity of the heterogeneous environment. In recent years, stochastic geometry has attracted more attention as a mathematical tool for modeling mobile network systems. In this paper, an analytical approach to the coverage probability analysis of UAV-assisted cellular networks with imperfect beam alignment has been proposed. An assumption was considered that all users are distributed according to Poisson Cluster Process (PCP) around base stations, in particular, Thomas Cluster Process (TCP). Using this model, the impact of beam alignment errors on the coverage probability was investigated. Initially, the Probability Density Function (PDF) of directional antenna gain between the user and its serving base station was obtained. Then, association probability with each tier was achieved. A tractable expression was derived for coverage probability in both Line-of-Sight (LoS) and Non-Line-of-Sight (NLoS) condition links. Numerical results demonstrated that at low UAVs altitude, beam alignment errors significantly degrade coverage performance. Moreover, for a small cluster size, alignment errors do not necessarily affect the coverage performance.