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In this paper we study the deterministic and stochastic homogenisation of free-discontinuity functionals under \emph{linear} growth and coercivity conditions. The main novelty of our deterministic result is that we work under very general assumptions on the integrands which, in particular, are not required to be periodic in the space variable. Combining this result with the pointwise Subadditive Ergodic Theorem by Akcoglu and Krengel, we prove a stochastic homogenisation result, in the case of stationary random integrands. In particular, we characterise the limit integrands in terms of asymptotic cell formulas, as in the classical case of periodic homogenisation.

In fifth-generation (5G) cellular networks, users feed back to the base station the index of the precoder (from a codebook) to be used for downlink transmission. The precoder is strongly related to the user channel and in turn to the user position within the cell. We propose a method by which an external attacker determines the user position by passively overhearing this unencrypted layer-2 feedback signal. The attacker first builds a map of fed back precoder indices in the cell. Then, by overhearing the precoder index fed back by the victim user, the attacker finds its position on the map. We focus on the type-I single-panel codebook, which today is the only mandatory solution in the 3GPP standard. We analyze the attack and assess the obtained localization accuracy against various parameters. We analyze the localization error of a simplified precoder feedback model and describe its asymptotic localization precision. We also propose a mitigation against our attack, wherein the user randomly selects the precoder among those providing the highest rate. Simulations confirm that the attack can achieve a high localization accuracy, which is significantly reduced when the mitigation solution is adopted, at the cost of a negligible rate degradation.

We put forward a novel approach to study the evolution of an arbitrary open quantum system under a resetting process. Using the framework of renewal equations, we find a universal behavior for the mean first return time that goes beyond unitary dynamics and Markovian measurements. Our results show a non-trivial behavior of the mean switching times with the mean measurement time $\tau$, which permits tuning $\tau$ for minimizing the mean transition time between states. We complement our results with a numerical analysis and we benchmark the results against the corresponding analytical study for low dimensional systems under unitary \textit{and} open system dynamics.

We investigate the effects of markovian resseting events on continuous time random walks where the waiting times and the jump lengths are random variables distributed according to power law probability density functions. We prove the existence of a non-equilibrium stationary state and finite mean first arrival time. However, the existence of an optimum reset rate is conditioned to a specific relationship between the exponents of both power law tails. We also investigate the search efficiency by finding the optimal random walk which minimizes the mean first arrival time in terms of the reset rate, the distance of the initial position to the target and the characteristic transport exponents., Comment: Submitted to PRE

We study a diffusion process on a three-dimensional comb under stochastic resetting. We consider three different types of resetting: global resetting from any point in the comb to the initial position, resetting from a finger to the corresponding backbone and resetting from secondary fingers to the main fingers. The transient dynamics along the backbone in all three cases is different due to the different resetting mechanisms, finding a wide range of dynamics for the mean squared displacement. For the particular geometry studied herein, we compute the stationary solution and the mean square displacement and find that the global resetting breaks the transport in the three directions. Regarding the resetting to the backbone, the transport is broken in two directions but it is enhanced in the main axis. Finally, the resetting to the fingers enhances the transport in the backbone and the main fingers but reaches a steady value for the mean squared displacement in the secondary fingers., Comment: Submitted to Physical Review Research (12/12/19)

We present a numerical implementation of a model of quasi-static crack growth in linearly elastic-perfectly plastic materials. We assume that the displacement is antiplane, and that the cracks and the plastic slips are localized on a prescribed path. We provide numerical evidence of the fact that the crack growth is intermittent, with jump characteristics that depend on the material properties., Comment: 13 pages, 7 figures

This chapter focuses on the performance enhancement brought by the addition of caching capabilities to full-duplex (FD) radios in the context of ultra-dense networks (UDNs). More specifically, we aim at showing that the interference footprint of such networks, i.e., the major bottleneck to overcome to observe the theoretical FD throughput doubling at the network level, can be significantly reduced thanks to edge caching. Fundamental results show that most of the gain, as compared to their half-duplex (HD) counterparts, can be achieved by such networks only if costly modifications to their infrastructure are performed and/or if high-rate signaling is exchanged between user equipments (UEs) over suitable control links. Therefore, we aim at proposing a viable and cost-effective alternative to these solutions based on pre-fetching locally popular contents at the network edge. We start by considering an interference-rich scenario such as an ultra-dense FD small-cell network, in which several non-cooperative FD base stations (BSs) serve their associated UEs while communicating with a wireless backhaul node (BN) to retrieve the content to deliver. We then describe a geographical caching policy aiming at capturing local files popularity and compute the corresponding cache-hit probability. Thereupon, we calculate the probability of successful transmission of a file requested by a UE, either directly by its serving small-cell base station (SBS) or by the corresponding BN: this quantity is then used to lower-bound the throughput of the considered network. Our approach leverages tools from stochastic geometry in order to guarantee both analytical tractability of the problem and generality of the results. Our numerical simulations show that shifting from cache-free to cache-aided FD small-cell networks yields a remarkable performance improvement., Comment: To appear in "Wireless Edge Caching: Modelling, Analysis and Optimization", Cambridge University Press

Random walks with stochastic resetting provides a treatable framework to study interesting features about central-place motion. In this work, we introduce non-instantaneous resetting as a two-state model being a combination of an exploring state where the walker moves randomly according to a propagator and a returning state where the walker performs a ballistic motion with constant velocity towards the origin. We study the emerging transport properties for two types of reset time probability density functions (PDFs): exponential and Pareto. In the first case, we find the stationary distribution and a general expression for the stationary mean square displacement (MSD) in terms of the propagator. We find that the stationary MSD may increase, decrease or remain constant with the returning velocity. This depends on the moments of the propagator. Regarding the Pareto resetting PDF we also study the stationary distribution and the asymptotic scaling of the MSD for diffusive motion. In this case, we see that the resetting modifies the transport regime, making the overall transport sub-diffusive and even reaching a stationary MSD., i.e., a stochastic localization. This phenomena is also observed in diffusion under instantaneous Pareto resetting. We check the main results with stochastic simulations of the process., Comment: 4 Figures. Published in https://doi.org/10.1103/PhysRevE.100.042104

In this minireview we present the main results regarding the transport properties of stochastic movement with relocations to known positions. To do so, we formulate the problem in a general manner to see several cases extensively studied during the last years as particular situations within a framework of random walks with memory. We focus on (i) stochastic motion with resets to its initial position followed by a waiting period, and (ii) diffusive motion with memory-driven relocations to previously visited positions. For both of them we show how the overall transport regime may be actively modified by the details of the relocation mechanism., Comment: 1 Figure. Published on https://doi.org/10.3389/fphy.2019.00112

The majority of modern communication systems adopts quadrature amplitude modulation (QAM) constellations as transmission schemes. Due to their square structure, however, QAM do not provide satisfying protection to phase noise effects as the number of constellation points grows, increasing at the same time their peak to average power ratio (PAPR). This requires an expensive power amplifier and oscillator at the transmitter to guarantee low distortion, complicating the adoption of dense transmission schemes in practical high-data rate systems. In this paper, we construct a coded modulation scheme based on regular amplitude and phase shift keying (RAPSK) modulations. We propose a novel multilevel coding (MLC) labeling for the constellation points separating amplitude and phase domains. We provide a novel multistage decoding (MSD) scheme allowing for a low-complexity log-likelihood ratio (LLR) calculation for soft-input decoding of component codes, along with a suitable rate design. Finally, we compare the proposed scheme with state-of-the-art QAM constellations and optimized constellations in the presence of phase noise.

In this work we consider a stochastic movement process with random resets to the origin followed by a random residence time there before the walker restarts its motion. First, we study the transport properties of the walker, we derive an expression for the mean square displacement of the overall process and study its dependence with the statistical properties of the resets, the residence and the movement. From this general formula, we see that the inclusion of the residence after the resets is able to induce super-diffusive to sub-diffusive (or diffusive) regimes and it can also make a sub-diffusive walker reach a constant MSD or even collapse. Second, we study how the reset-and-residence mechanism affects the first arrival time of different search processes to a given position, showing that the long time behavior of the reset and residence time distributions determine the existence of the mean first arrival time., Comment: 11 pages, 8 figures. Submitted to Journal of Statistical Mechanics: Theory and Experiment

Stochastic resets have lately emerged as a mechanism able to generate finite equilibrium mean square displacement (MSD) when they are applied to diffusive motion. Furthermore, walkers with an infinite mean first arrival time (MFAT) to a given position $x$, may reach it in a finite time when they reset their position. In this work we study these emerging phenomena from a unified perspective. On one hand we study the existence of a finite equilibrium MSD when resets are applied to random motion with $\langle x^2(t)\rangle _m\sim t^p$ for $0

We study simple stochastic scenarios, based on birth-and-death Markovian processes, that describe populations with Allee effect, to account for the role of demographic stochasticity. In the mean-field deterministic limit we recover well-known deterministic evolution equations widely employed in population ecology. The mean-time to extinction is in general obtained by the Wentzel-Kramers-Brillouin (WKB) approximation for populations with strong and weak Allee effects. An exact solution for the mean time to extinction can be found via a recursive equation for special cases of the stochastic dynamics. We study the conditions for the validity of the WKB solution and analyze the boundary between the weak and strong Allee effect by comparing exact solutions with numerical simulations., Comment: 12 pages, 7 figures, to appear in Phys. Rev. E (2019)

Neuroimaging data analysis often involves \emph{a-priori} selection of data features to study the underlying neural activity. Since this could lead to sub-optimal feature selection and thereby prevent the detection of subtle patterns in neural activity, data-driven methods have recently gained popularity for optimizing neuroimaging data analysis pipelines and thereby, improving our understanding of neural mechanisms. In this context, we developed a deep convolutional architecture that can identify discriminating patterns in neuroimaging data and applied it to electroencephalography (EEG) recordings collected from 25 subjects performing a hand motor task before and after a rest period or a bout of exercise. The deep network was trained to classify subjects into exercise and control groups based on differences in their EEG signals. Subsequently, we developed a novel method termed the cue-combination for Class Activation Map (ccCAM), which enabled us to identify discriminating spatio-temporal features within definite frequency bands (23--33 Hz) and assess the effects of exercise on the brain. Additionally, the proposed architecture allowed the visualization of the differences in the propagation of underlying neural activity across the cortex between the two groups, for the first time in our knowledge. Our results demonstrate the feasibility of using deep network architectures for neuroimaging analysis in different contexts such as, for the identification of robust brain biomarkers to better characterize and potentially treat neurological disorders.

In this work we propose to validate the predictive capabilities of one-dimensional (1D) blood flow models with full three-dimensional (3D) models in the context of patient-specific coronary hemodynamics in hyperemic conditions. Such conditions mimic the state of coronary circulation during the acquisition of the Fractional Flow Reserve (FFR) index. Demonstrating that 1D models accurately reproduce FFR estimates obtained with 3D models has implications in the approach to computationally estimate FFR. To this end, a sample of 20 patients was employed from which 29 3D geometries of arterial trees were constructed, 9 obtained from coronary computed tomography angiography (CCTA) and 20 from intra-vascular ultrasound (IVUS). For each 3D arterial model, a 1D counterpart was generated. The same outflow and inlet pressure boundary conditions were applied to both (3D and 1D) models. In the 1D setting, pressure losses at stenoses and bifurcations were accounted for through specific lumped models. Comparisons between 1D models ($\text{FFR}_{\text{1D}}$) and 3D models ($\text{FFR}_{\text{3D}}$) were performed in terms of predicted $\text{FFR}$ value. Compared to $\text{FFR}_{\text{3D}}$, $\text{FFR}_{\text{1D}}$ resulted with a difference of 0.00$\pm$0.03 and overall predictive capability AUC, Acc, Spe, Sen, PPV and NPV of 0.97, 0.98, 0.90, 0.99, 0.82, and 0.99, with an FFR threshold of 0.8. We conclude that inexpensive $\text{FFR}_{\text{1D}}$ simulations can be reliably used as a surrogate of demanding $\text{FFR}_{\text{3D}}$ computations., Comment: 11 pages, 2 figures, 2 tables, submited to Scientific Reports

Given a bounded open set in $\mathbb{R}^n$, $n\ge 2$, and a sequence $(K_j)$ of compact sets converging to an $(n-1)$-dimensional manifold $M$, we study the asymptotic behaviour of the solutions to some minimum problems for integral functionals on $\Omega\setminus K_j$, with Neumann boundary conditions on $\partial(\Omega\setminus K_j)$. We prove that the limit of these solutions is a minimiser of the same functional on $\Omega\setminus M$ subjected to a transmission condition on $M$, which can be expressed through a measure $\mu$ supported on $M$. The class of all measures that can be obtained in this way is characterised, and the link between the measure $\mu$ and the sequence $(K_j)$ is expressed by means of suitable local minimum problems.

In this paper we study the stochastic homogenisation of free-discontinuity functionals. Assuming stationarity for the random volume and surface integrands, we prove the existence of a homogenised random free-discontinuity functional, which is deterministic in the ergodic case. Moreover, by establishing a connection between the deterministic convergence of the functionals at any fixed realisation and the pointwise Subadditive Ergodic Theorem by Akcoglou and Krengel, we characterise the limit volume and surface integrands in terms of asymptotic cell formulas., Comment: 23 pages

We study the $\Gamma$-convergence of sequences of free-discontinuity functionals depending on vector-valued functions $u$ which can be discontinuous across hypersurfaces whose shape and location are not known a priori. The main novelty of our result is that we work under very general assumptions on the integrands which, in particular, are not required to be periodic in the space variable. Further, we consider the case of surface integrands which are not bounded from below by the amplitude of the jump of $u$. We obtain three main results: compactness with respect to $\Gamma$-convergence, representation of the $\Gamma$-limit in an integral form and identification of its integrands, and homogenisation formulas without periodicity assumptions. In particular, the classical case of periodic homogenisation follows as a by-product of our analysis. Moreover, our result covers also the case of stochastic homogenisation, as we will show in a forthcoming paper., Comment: 37 pages

In this paper, we consider a stochastic process that may experience random reset events which relocate the system to its starting position. We focus our attention on a one-dimensional, monotonic continuous-time random walk with a constant drift: the process moves in a fixed direction between the reset events, either by the effect of the random jumps, or by the action of a deterministic bias. However, the orientation of its motion is randomly determined after each restart. As a result of these alternating dynamics, interesting properties do emerge. General formulas for the propagator as well as for two extreme statistics, the survival probability and the mean first-passage time, are also derived. The rigor of these analytical results is verified by numerical estimations, for particular but illuminating examples., Comment: 11 pages, 5 figures

Caching at the edge is a promising technique to cope with the increasing data demand in wireless networks. This paper analyzes the performance of cellular networks consisting of a tier macro-cell wireless backhaul nodes overlaid with a tier of cache-aided small cells. We consider both static and dynamic association policies for content delivery to the user terminals and analyze their performance. In particular, we derive closed-form expressions for the area spectral efficiency and the energy efficiency, which are used to optimize relevant design parameters such as the density of cache-aided small cells and the storage size. By means of this approach, we are able to draw useful design insights for the deployment of highly performing cache-aided tiered networks., Comment: 5 pages, 5 figures, to be presented at 18th IEEE International Workshop on Signal Processing Advances in Wireless Communications (SPAWC'2017), Sapporo, Japan, 2017

Caching popular contents at the edge of the network can positively impact the performance and future sustainability of wireless networks in several ways, e.g., end-to-end access delay reduction and peak rate increase. In this paper, we aim at showing that non-negligible performance enhancements can be observed in terms of network interference footprint as well. To this end, we consider a full-duplex small-cell network consisting of non-cooperative cache-aided base stations, which communicate simultaneously with both downlink users and wireless backhaul nodes. We propose a novel caching model seeking to mimic a geographical policy based on local files popularity and calculate the corresponding cache hit probability. Subsequently we study the performance of the considered network in terms of throughput gain with respect to its cache-free half-duplex counterpart. Numerical results corroborate our theoretical findings and highlight remarkable performance gains when moving from cache-free to cache-aided full-duplex small-cell networks., Comment: To be presented at Content Caching and Delivery in Wireless Networks Workshop - CCDWN (WiOpt)

In this paper the study of a nonlocal second order Cahn-Hilliard-type singularly perturbed family of functions is undertaken. The kernels considered include those leading to Gagliardo fractional seminorms for gradients. Using Gamma convergence the integral representation of the limit energy is characterized leading to an anisotropic surface energy on interfaces separating different phases.

In this work, a novel energy-recycling single-antenna full-duplex (FD) radio is designed, in which a new 3-port element including a power divider and an energy harvester is added between the circulator and the receiver (RX) chain. The presence of this new element brings advantages over the state of the art in terms of both spectral efficiency and energy consumption. In particular, it provides the means of performing both an arbitrary attenuation of the incoming signal, which in turn increases the effectiveness of the state-of-the-art self-interference cancellation strategies subsequently adopted in the RX chain, and the recycling of a non-negligible portion of the energy leaked through the non-ideal circulator. The performance of this architecture is analyzed in a practically relevant 4-node scenario in which 2 nodes operate in FD and 2 nodes in half-duplex (HD). Analytical approximations are derived for both the achievable rates of the transmissions performed by the FD and HD radios and the energy recycled by the FD radios. The accuracy of these derivations is confirmed by numerical simulations. Quantitatively, achievable rate gains up to 40% over the state-of-the-art alternatives, in the considered scenario, are highlighted. Furthermore, up to 50% of the leaked energy at the circulator, i.e., 5% of the energy of the transmitted signal, can be recycled.

In this work, we consider a multiple-input single-output system in which an access point (AP) performs a simultaneous wireless information and power transfer (SWIPT) to serve a user terminal (UT) that is not equipped with external power supply. In order to assess the efficacy of the SWIPT, we target a practically relevant scenario characterized by imperfect channel state information (CSI) at the transmitter, the presence of penalties associated to the CSI acquisition procedures, and non-zero power consumption for the operations performed by the UT, such as CSI estimation, uplink signaling and data decoding. We analyze three different cases for the CSI knowledge at the AP: no CSI, and imperfect CSI in case of time-division duplexing and frequency-division duplexing communications. Closed-form representations of the ergodic downlink rate and both the energy shortage and data outage probability are derived for the three cases. Additionally, analytic expressions for the ergodically optimal duration of power transfer and channel estimation/feedback phases are provided. Our numerical findings verify the correctness of our derivations, and also show the importance and benefits of CSI knowledge at the AP in SWIPT systems, albeit imperfect and acquired at the expense of the time available for the information transfer.

We study a variational model for the quasistatic growth of cracks with fractional di- mension in brittle materials. We give a minimal set of properties of the collection of admissible cracks which ensure the existence of a quasistatic evolution. Both the an- tiplane and the planar cases are treated., Comment: 17 pages

Self-interference (SI) significantly limits the performance of full-duplex (FD) radio devices if not properly cancelled. State-of-the-art SI cancellation (SIC) techniques at the receive chain implicitly set an upper bound on the transmit power of the device. This paper starts from this observation and proposes a transmit beamforming design for FD multiple-antenna radios that: i) leverages the inherent SIC capabilities at the receiver and the channel state information; and ii) exploits the potential of multiple antennas in terms of spatial SIC. The proposed solution not only maximizes the throughput while complying with the SIC requirements of the FD device, but also enjoys a very low complexity that allows it to outperform state-of-the-art counterparts in terms of processing time and power requirements. Numerical results show that our transmit beamforming design achieves significant gains with respect to applying zero-forcing to the SI channel when the number of transmit antennas is small to moderate, which makes it particularly appealing for FD small-cell base stations.