Delay Distribution Research Articles

Strong-lensing cosmography using third-generation gravitational-wave detectors

Abstract We present a detailed exposition of a statistical method for estimating cosmological parameters from the observation of a large number of strongly lensed binary-black-hole (BBH) mergers observable by next (third) generation (XG) gravitational-wave (GW) detectors. This method, first presented in [1], compares the observed number of strongly lensed GW events and their time delay distribution (between lensed images) with observed events to infer cosmological parameters. We show that the precision of the estimation of the cosmological parameters does not have a strong dependance on the assumed BBH redshift distribution model. Using the large number of unlensed mergers, XG detectors are expected to measure the BBH redshift distribution with sufficient precision for the cosmological inference. However, a biased inference of the BBH redshift distribution will bias the estimation of cosmological parameters. An incorrect model for the distribution of lens properties can also lead to a biased cosmological inference. However, Bayesian model selection can assist in selecting the right model from a set of available parametric models for the lens distribution. We also present a way to incorporate the effect of contamination in the data due to the limited efficiency of lensing identification methods, so that it will not bias the cosmological inference.

Best practices for estimating and reporting epidemiological delay distributions of infectious diseases.

Epidemiological delays are key quantities that inform public health policy and clinical practice. They are used as inputs for mathematical and statistical models, which in turn can guide control strategies. In recent work, we found that censoring, right truncation, and dynamical bias were rarely addressed correctly when estimating delays and that these biases were large enough to have knock-on impacts across a large number of use cases. Here, we formulate a checklist of best practices for estimating and reporting epidemiological delays. We also provide a flowchart to guide practitioners based on their data. Our examples are focused on the incubation period and serial interval due to their importance in outbreak response and modeling, but our recommendations are applicable to other delays. The recommendations, which are based on the literature and our experience estimating epidemiological delay distributions during outbreak responses, can help improve the robustness and utility of reported estimates and provide guidance for the evaluation of estimates for downstream use in transmission models or other analyses.

Image formation near hyperbolic umbilic in strong gravitational lensing

Hyperbolic umbilic (HU) is a point singularity of the gravitational lens equation, giving rise to a ring-shaped image formation made of four highly magnified images, off-centred from the lens centre. Recent observations have revealed new strongly lensed image formations near HU singularities, and many more are expected in ongoing and future observations. Like fold/cusp, image formations near HU also satisfy magnification relation (), i.e., the signed magnification sum of the four images equals zero. Here, we study how deviates from zero as a function of area () covered by the image formation near HU and the distance ( d) of the central maxima image (which is part of the HU image formation) from the lens centre for ideal single- and double-component cluster-scale lenses. For lens ellipticity values ≥0.3, the central maxima image will form sufficiently far from the lens centre ( d≳5″), similar to the observed HU image formations with . We also find that, in some cases, double-component and actual cluster-scale lenses can lead to large cross-sections for HU image formations for sources at z≳5, effectively increasing the chances to observe HU image formation at high redshifts. Finally, we study the time delay distribution in the observed HU image formation, finding that not only are these images highly magnified, but the relative time delay between various pairs of HU characteristic image formation has a typical value of ∼100 days, an order of magnitude smaller than generic five-image formations in cluster lenses, making such image formations optimal targets for time delay cosmography studies.

Dynamics of a Price Adjustment Model with Distributed Delay

This paper deals with the stability and occurrence of Hopf bifurcation of a distributed delay differential cobweb model using the chain trick technique. This is a generalized form of the fixed delay cobweb model to which it is compared using the same parameter values. The results from the delay distribution showed that whenever less weight (γ=0.146) is put on past prices, the current equilibrium price is adjusted upwards while the reverse is observed when a higher weight (γ=0.186) is put on the previous price. It is also observed that if the initial price is set below/above the equilibrium price, the price adjustment either affects the consumers or benefits the suppliers. However, the fixed delay cobweb model does not display the consumers or suppliers benefits of the price dynamics in either direction. These are unique, underlying patterns in price dynamics discovered when using a distributed delay model compared to traditional fixed delay cobweb models. Furthermore, our model challenges the traditional cobweb model’s requirement for divergence, as it is based on the weight assigned to past prices rather than the relationship between the elasticities of supply and demand, which is the determining factor in the classical model. Based on these insights, we recommend that future price adjustment models incorporate distributed delays, as they reveal more intricate price dynamics and provide a more comprehensive understanding of market behavior than fixed delay models.

On the stability of a single-species model with a generic delay distribution kernel

The delayed logistic equation, also known as Hutchinson’s equation, is a simple and elegant model commonly used to capture critical features of complex phenomena in biology, medicine, and economics. This paper studies the stability of a single-species logistic model with a general delay distribution and a constant inflow of nutritional resources. We provide conditions for the linear stability of the positive equilibrium and the occurrence of Hopf bifurcation. The findings complement existing literature and are applied to specific delay distributions: Uniform, Dirac-delta, and gamma distributions. Without resource inflow, we find that the positive equilibrium is stable for short delays but loses stability through Hopf bifurcation as the mean delay increases. The model’s dynamics vary with resource inflow based on the delay distribution: in uniform and Dirac-delta distributions, the dynamics are similar to the no-inflow case, whereas for the gamma distribution, stability depends on the delay order p=1,2,3.

Secure Patient Data Monitoring and Efficient Routing Optimization using a Hyperelliptic Curve Cryptography with Fuzzy-based Priority in WBSN

Aims and Background: Wireless Body Sensor Network (WBSN) technology is one of the major research areas in the medical and entertainment industries. A wireless sensor network (WSN) is a dense sensor network that senses environmental conditions, processes, and outgoing data at the sink node. A WBSN develops patient monitoring systems that provide the flexibility and mobility needed to monitor patient health. In data communications, it is difficult to find flexible optical routing paths, switching capabilities, and packet processing in the composition of optical networks. Information-centric networks (ICNs) are a new network model and are different from information- centric models. The priority of the information-centric model is the communication network. Objectives: In the existing literature, such methods are typically developed using computationally expensive procedures, such as bilinear pairing, elliptic curve operations, etc., which are unsuitable for biomedical devices with limited resources. Using the concept of hyperelliptic curve cryptography (HECC), we propose a new solution: a smart card-based two-factor mutual authentication scheme. In this new scheme, HECC’s finest properties, such as compact parameters and key sizes, are utilized to enhance the real-time performance of an IoT-based TMIS system. Methodology: A fuzzy–based Priority Aware Data Sharing (FPADS) method is introduced to schedule the priority data and monitor the transmission length. The child node adjusts the transmission speed of the cluster head with the help of a fuzzy logic controller (FLC). Results: The proposed model estimated the traffic load of the child node and the priority of the different amounts of data to be transmitted. The principle of scheduling data packets to be developed is based on the precedence of the data with the lowest transmit length in the network. Conclusion: The proposed FPADS performance increases in terms of scheduling time utilisation, traffic distribution, and mean delay. Simulations have been done using NS2, and the outcomes have shown that the proposed methodology is efficient and improves the overall QoS of the system.

Cosmic Type Ia supernova rate and constraints on supernova Ia progenitors

Type Ia supernovae play a key role in the evolution of galaxies by polluting the interstellar medium with a fraction of iron peak elements larger than that released in the core-collapse supernova events. Their light curve, moreover, is widely used in cosmological studies as it constitutes a reliable distance indicator on extragalactic scales. Among the mechanisms proposed to explain the Type Ia supernovae (SNe), the single- and double-degenerate channels are thought to be the dominant ones, which implies a different distribution of time delays between the progenitor formation and the explosion. In this paper, we aim to determine the dominant mechanism by comparing a compilation of Type Ia SN rates with those computed from various cosmic star-formation histories coupled with different delay-time distribution functions. We also evaluate the relative contributions of both channels. By using a least-squares fitting procedure, we modeled the observations of Type Ia SN rates assuming different combinations of three recent cosmic star-formation rates and seven delay-time distributions. The goodness of these fits are statistically quantified by the $ test. For two of the three cosmic star-formation rates, the single degenerate scenario provides the most accurate explanation for the observations, while a combination of 34<!PCT!> single-degenerate- and 66<!PCT!> double-degenerate delay-time distributions is more plausible for the remaining tested cosmic star-formation rates. Though dependent on the assumed cosmic star-formation rate, we find arguments in favor of the single-degenerate model. From the theoretic point of view, at least sim 34<!PCT!> of the Type Ia SN must have been produced through the single-degenerate channel to account for the observations. The wide, double-degenerate mechanism slightly under-predicts the observations at redshift $z 1$, unless the cosmic SFR flattens in that regime. On the contrary, although the purely close double-degenerate scenario can be ruled out, we cannot rule out a mixed scenario with single- and double-degenerate progenitors.

Novel mirror-flipped mode permutation technique for long-haul 6-mode transmission

Mode-division multiplexing based on few-mode fibers is expected to overcome capacity limits of current single-mode fiber communications. Mode permutation as an approach for extending transmission reach has attracted great attention. In this paper, we propose a mirror-flipped mode permutation (MFMP) scheme based on newly developed fibers with symmetric differential mode delay (DMD). We demonstrate a 28-GBaud transmission over a 1542-km 6-mode fiber link, which suppresses modal-dispersion impact by 36% and mitigates mode-dependent-loss effect with 3.1-dB Q-factor improvement, compared with the traditional cyclic mode permutation at 1233 km. Then, we experimentally verify that the proposed MFMP works well even using a fiber with a not exactly symmetric group delay distribution. Benefiting from a low DMD in this fiber, we demonstrate a 6-mode PM-QPSK transmission at 25 GBaud with 16-λ channels, extending the transmission distance to 3000 km.

Combustion instability characteristics via fuel nozzle modification in a hydrogen and natural gas Co-firing gas turbine combustor

In this study, three premixed swirl gas turbine nozzles with different fuel injection hole sizes were designed to optimize combustion characteristics influenced by changes in the hydrogen mixing ratio. Experiments involved increasing the hydrogen volume fraction of the fuel from 0 to 60%. The amount of fuel supplied through the pilot flow was increased to suppress combustion instability. The OH* chemiluminescence technique was introduced to analyze flame structure. To understand combustion instability, time delay and time-delay distribution were derived from flame structure analysis and applied to the 1D thermoacoustic model. The results indicated that the amount of fuel required for pilot injection with increasing hydrogen ratio varied with fuel hole size. It was also observed that changes in fuel hole size significantly affected flame characteristic length. Findings from the 1D thermoacoustic analysis confirmed that fuel hole size variation altered combustion instability characteristics by influencing time delay and time-delay distribution variation.

Transit fares integrating alternative modes as a delay insurance

Public transport (PT) fare policy remains subject to innovations, not least evident in the Mobility as a Service discussion. Mode integration and related fare strategies can be used to increase the attractiveness of PT by compensating for potential delays. This study proposes “premium fares” as a novel pricing tool that can evaluate and improve the travel time reliability on a multimodal transportation network. The premium fare is higher than the standard fare but allows passengers to use an alternative service free of charge if waiting for the delayed public transport service is anticipated to be longer than a certain qualification threshold. Properties that guarantee monotonicity of the premium fare with respect to distance traveled are developed. The operator aims to find the premium fare price and qualification threshold that can maximize its profit, based on the probability distributions of delay and passengers’ value of time (VOT). We model this optimization problem for a railway line given a limited capacity of alternative mode services, e.g., the number of taxis. A two-stage approach using nonlinear and dynamic programming is developed to obtain the optimal decision variables and associated capacity allocation plan. Our results show that the introduction of the premium fare can benefit both operators and travelers with increased profits and improved travel time reliability. The interplay between fares and the qualification threshold is illustrated using various delay and VOT distributions. To attract enough customers the premium fare has to be set below a specific level dependent on the VOT distribution. Meanwhile, the operator adjusts the qualification threshold to control the cost paid to the alternative service provider.

A large-aperture, high-power ultrawideband radiation system with beam broadening capacity.

Due to the limited maximum output power of the pulsers based on avalanche transistors, high-power ultrawideband (UWB) radiation systems usually synthesize plenty of modules simultaneously to achieve a high peak effective potential (rEp). However, this would lead to an increased aperture size as well as a narrower beam, which would limit their applications in intentional electromagnetic interference fields. In this paper, a high-power UWB radiation system with beam broadening capacity is developed. To achieve beam broadening in the time domain, a power-law time delay distribution method is proposed and studied by simulation, and then the relative excitation time delays of the modules are optimized to achieve higher rEp and avoid beam splitting in the beam broadening mode. In order to avoid false triggering of the pulser elements when implementing the beam broadening, the mutual coupling effect in the system is analyzed and suppressed by employing onboard high-pass filters, since the mutual coupling effect is much more severe in the low-frequency range. Finally, a radiation system with 36 modules is developed. Measuring results indicate that in the high-rEp mode, the developed system could achieve a maximum effective potential rEp of 313.6kV and a maximum pulse-repetition-rate of 20kHz. In the beam broadening mode, its half-peak-power beam width in the H-plane is broadened from the original value of 3.9° to 7.9°, with a maximum rEp of 272.9kV. The polarization direction of the system could be flexibly adjusted by a built-in motor.

Analysis of potential 5G transmission methods concerning Bit Error Rate

Fifth-generation wireless (5G) significantly impacts individuals' lives and work and is expected to increase. Although OFDM has been used in previous generation technologies (4G), it has limitations in meeting specific criteria such as data. Issues like better power consumption and Bit Error Rate make this technology unsuitable for meeting current needs such as the Internet of Things and user-based processing. This research aims to test the BER success of the transmission technique proposed as a candidate for the 5G communication system. Performance analysis is carried out considering the Tapped Delay Line (TDL-A) channel model, which is recommended for 5G research. Various scenarios, including situations where the transmitter/receiver is fixed or mobile, are considered to provide a more comprehensive evaluation. Evaluations were also carried out on various channel delay distribution profiles to expand the scope of analysis. The research results show that in channel conditions with high delay distribution and mobility, the communication system that uses the FBMC transmission technique shows better and more stable performance compared to OFMD, F-OFMD, WOLA, UFMC systems, it has an SNR of about 36 dB to achieve a BER of approx 10-3 under 10 ns delay spreading conditions, demonstrating the superiority of FBMC in difficult channel conditions These findings emphasize the importance of considering BER and system complexity in the design of future communications systems. Thus, this research provides a significant original contribution to the understanding and developing 5G communication systems.

DNS root server resolution anomaly detection

The DNS root server is at the top of the hierarchical structure of the DNS system and is the initial node that bootstraps all DNS queries. If the root server resolves abnormally, all domain name resolutions will fail, and many users cannot access the Internet. For this reason, this paper detects the root server itself resolution anomalies and non-self resolution anomalies by constructing high-confidence root zone file and anomaly judgment rules. First, we use the weighted voting statistics method to build high-confidence root zone file by calculating the confidence of multi-source root zone files. Based on high-confidence root zone files, we construct three types of anomaly detection judgment rules: (1) root-side resolution anomaly judgment rules based on feature value matching, (2) response hijacking judgment rules by correlating response anomaly features and resolution routing information, (3) root zone file synchronization anomaly judgment rules by calculating the relative synchronization delay of multi-source root zone files. Finally, using three anomaly judgment rules, we perform anomaly detection on the root resolution data obtained by active measurement. Our detection results show that root zone file synchronization delay distributions of different root server instances vary greatly. Some instances even show minute-level convergence, resulting in incorrect resolution for some TLDs. We also detect one response hijacking incident for 2 TLDs resolution, caused by the domain takeover mechanism adopted by the ISP to reduce inter-domain traffic settlement and decrease resolution latency. Except for the unresponsive exception caused by network packet loss, no root-side resolution anomaly is found, indicating that there is no artificial manipulation of TLD resolution on the root server and reflecting the responsibility of each root server operator aiming to maintain global Internet interconnection. The detection results show that the detection rules proposed in this paper can effectively achieve the anomaly detection of root server resolution and help to maintain the health and stability of the DNS system.

Characteristics of electric breakdown in repeated frequency pulse with microcavity effect

The electric breakdown characteristics in microcavity structure under repeated frequency pulse (RFP) were studied, and the physical mechanism was investigated quantitatively based on the full statistical distribution of breakdown time delay obtained in step rectangular pulse (SRP). Experimentally, microcavity heights of 300, 800, and 2000 μm were used. In RFP, the occurrence of breakdown becomes probabilistic when the time delay t d and pulse width t PW satisfy the condition t s-min < t PW < t s-max. The breakdown probability increases with pulse width, and the probability distributions are roughly exponential and Gaussian at pulse frequencies of 3 and 1000 Hz, respectively. We found the results are attributed to the similar distributions of time delay in RFP and SRP with similar afterglow time and pulse voltage, and the equal distributions of breakdown probability (with pulse width) and cumulative probability of t d in RFP. The microcavity effect will decrease the breakdown probability under given pulse width and voltage. Additionally, it is found that in RFP the increase of pulse width from 1 to 1000 μ s will decrease the threshold voltages at 0% and 100% breakdown probabilities, and the threshold voltage difference will decrease simultaneously to around 0, which results in the transition of breakdown feature from probability to certainty. This phenomenon is due to that the reduction of pulse voltage will increase the time delay significantly and meanwhile the variation rate of time delay with pulse voltage Δt d/ΔU w decreases sharply. The microcavity effect will cause the increase of threshold breakdown voltages at a given pulse width and frequency. Finally, it is found that in RFP the breakdown voltage will decrease with the rise of pulse frequency from 10° to 104 Hz, which is consistent with the variation of time delay with afterglow time (from 10−1 to 103 ms) in the memory curve measured in SRP under similar afterglow time. Overall, the microcavity effect will enhance the adsorption of charged and excited species by dielectric walls during afterglow period and enlarge the time delay in the following pulse breakdown, and then influence the RFP breakdown characteristics.

Time Delay Characterization in Wireless Sensor Networks for Distributed Measurement Applications

This paper investigates the critical aspect of synchronization in wireless sensor networks (WSNs) across diverse industrial applications. The low-cost sensor network topologies are analyzed. The communication delay measurements and quantitative jitter analysis are performed under different conditions, and dependencies of the propagation time delay on the data bitrate and modulation type for different hardware implementations of the WSNs are presented. The time delay distribution influence on the time synchronization error propagation over WSN layers was assessed from the experimental probability density functions. The network synchronization based on the controlled propagation delay jitter approach has been proposed. This research contributes quantitative insights into the complexities of synchronization in WSNs, offering a foundation for optimizing network configurations and parameters to extend the operational life of low-power sensor nodes.

Boosting reservoir computer performance with multiple delays.

Time delays play a significant role in dynamical systems, as they affect their transient behavior and the dimensionality of their attractors. The number, values, and spacing of these time delays influences the eigenvalues of a nonlinear delay-differential system at its fixed point. Here we explore a multidelay system as the core computational element of a reservoir computer making predictions on its input in the usual regime close to fixed point instability. Variations in the number and separation of time delays are first examined to determine the effect of such parameters of the delay distribution on the effectiveness of time-delay reservoirs for nonlinear time series prediction. We demonstrate computationally that an optoelectronic device with multiple different delays can improve the mapping of scalar input into higher-dimensional dynamics, and thus its memory and prediction capabilities for input time series generated by low- and high-dimensional dynamical systems. In particular, this enhances the suitability of such reservoir computers for predicting input data with temporal correlations. Additionally, we highlight the pronounced harmful resonance condition for reservoir computing when using an electro-optic oscillator model with multiple delays. We illustrate that the resonance point may shift depending on the task at hand, such as cross prediction or multistep ahead prediction, in both single delay and multiple delay cases.

Modelling of Timestamp Processing within Transport Network Equipment

The modern telecommunication networks are forming on the basis of different transmission and data processing technologies. High load on the transport layer nodes demands quality synchronization of equipment that works on it. The transport network is an important element which delivers the time scale to slave nodes. Using the time-division multiplexing for the high-speed channel processing with the use of wavelength division multiplexing (OTN, Optical Transport Network) and classic Ethernet networks as a transport layer systems needs the precision time synchronization which can be achieved with the use of Precision Time Protocol (PTP). The modeling apparatus of PTP messages exchanging within OTN multiplexers and IP-network based on the routers are suggested. The packet transferring through the network are given as sequence of simple processing steps. The impact of a network load on PTP information processing capacity on the routers and efficiency of the Forward Error Correction algorithm for OTN equipment are analyzed. The experimental distribution of variable delays on that network elements are given.

Phase Delays between Mouse Globus Pallidus Neurons Entrained by Common Oscillatory Drive Arise from Their Intrinsic Properties, Not Their Coupling.

A hallmark of Parkinson's disease is the appearance of correlated oscillatory discharge throughout the cortico-basal ganglia (BG) circuits. In the primate globus pallidus (GP), where the discharge of GP neurons is normally uncorrelated, pairs of GP neurons exhibit oscillatory spike correlations with a broad distribution of pairwise phase delays in experimental parkinsonism. The transition to oscillatory correlations is thought to indicate the collapse of the normally segregated information channels traversing the BG. The large phase delays are thought to reflect pathological changes in synaptic connectivity in the BG. Here we study the structure and phase delays of spike correlations measured from neurons in the mouse external GP (GPe) subjected to identical 1-100 Hz sinusoidal drive but recorded in separate experiments. First, we found that spectral modes of a GPe neuron's empirical instantaneous phase response curve (iPRC) elucidate at what phases of the oscillatory drive the GPe neuron locks when it is entrained and the distribution of phases at which it spikes when it is not. Then, we show that in this case the pairwise spike cross-correlation equals the cross-correlation function of these spike phase distributions. Finally, we show that the distribution of GPe phase delays arises from the diversity of iPRCs and is broadened when the neurons become entrained. Modeling GPe networks with realistic intranuclear connectivity demonstrates that the connectivity decorrelates GPe neurons without affecting phase delays. Thus, common oscillatory input gives rise to GPe correlations whose structure and pairwise phase delays reflect their intrinsic properties captured by their iPRCs.

Quantifying the impact of hospital catchment area definitions on hospital admissions forecasts: COVID-19 in England, September 2020–April 2021

BackgroundDefining healthcare facility catchment areas is a key step in predicting future healthcare demand in epidemic settings. Forecasts of hospitalisations can be informed by leading indicators measured at the community level. However, this relies on the definition of so-called catchment areas or the geographies whose populations make up the patients admitted to a given hospital, which are often not well-defined. Little work has been done to quantify the impact of hospital catchment area definitions on healthcare demand forecasting.MethodsWe made forecasts of local-level hospital admissions using a scaled convolution of local cases (as defined by the hospital catchment area) and delay distribution. Hospital catchment area definitions were derived from either simple heuristics (in which people are admitted to their nearest hospital or any nearby hospital) or historical admissions data (all emergency or elective admissions in 2019, or COVID-19 admissions), plus a marginal baseline definition based on the distribution of all hospital admissions. We evaluated predictive performance using each hospital catchment area definition using the weighted interval score and considered how this changed by the length of the predictive horizon, the date on which the forecast was made, and by location. We also considered the change, if any, in the relative performance of each definition in retrospective vs. real-time settings, or at different spatial scales.ResultsThe choice of hospital catchment area definition affected the accuracy of hospital admission forecasts. The definition based on COVID-19 admissions data resulted in the most accurate forecasts at both a 7- and 14-day horizon and was one of the top two best-performing definitions across forecast dates and locations. The “nearby” heuristic also performed well, but less consistently than the COVID-19 data definition. The marginal distribution baseline, which did not include any spatial information, was the lowest-ranked definition. The relative performance of the definitions was larger when using case forecasts compared to future observed cases. All results were consistent across spatial scales of the catchment area definitions.ConclusionsUsing catchment area definitions derived from context-specific data can improve local-level hospital admission forecasts. Where context-specific data is not available, using catchment areas defined by carefully chosen heuristics is a sufficiently good substitute. There is clear value in understanding what drives local admissions patterns, and further research is needed to understand the impact of different catchment area definitions on forecast performance where case trends are more heterogeneous.

Switching-like event-triggered control of uncertain NCSs under delay distribution and DoS attacks

The paper addresses the switching-like event-triggered control for uncertain networked control systems with time-varying delay under DoS attacks. First of all, to reduce the communication burden, a switching-like event-triggered mechanism is designed to automatically select the trigger condition according to whether the system is under DoS attacks, which have the advantage of reducing the number of data packets transmitted. Secondly, unlike the traditional assumption of time-varying delay, here it satisfies the condition that the probability is known, and combines the networked control systems to propose a novel time-delay system model, which can obtain a larger upper bound on the delay. Then, by using both the Lyapunov functional method and linear matrix inequality technique, we obtain sufficient conditions of uncertain networked control systems to achieve exponentially stable in the mean square sense. Furthermore, under the common limitations of the maximum continuous packet losses caused by the DoS attacks and delay, the stability criterion is derived, which can be used to estimate the communication parameters and security controller gain. Finally, through two simulation examples, the larger upper bound of time delay, less trigger times, faster convergence rate are obtained, which verify the validity of our theoretical analysis.