Search

We introduce and study a strict monotonicity property of the norm in solid Banach lattices of real functions that prevents such spaces from having the local diameter two property. Then we show that any strictly convex 1-symmetric norm on $\ell_\infty(\mathbb N)$ possesses this strict monotonicity property. In the opposite direction, we show that any Banach space which is strictly convex renormable and contains a complemented copy of $c_0,$ admits an equivalent strictly convex norm for which the space has the local diameter two property. In particular, this enables us to construct a strictly convex norm on $c_0(\Gamma),$ where $\Gamma$ is uncountable, for which the space has a 1-unconditional basis and the local diameter two property., Comment: Some text revision. Fixed some misprints

In this study, we propose a robust set of features derived from a thorough research of contemporary practices in voice pathology detection. The feature set is based on the combination of acoustic handcrafted features. Additionally, we introduce pitch difference as a novel feature. We combine this feature set, containing data from the publicly available Saarbr\"ucken Voice Database (SVD), with preprocessing using the K-Means Synthetic Minority Over-Sampling Technique algorithm to address class imbalance. Moreover, we applied multiple ML models as binary classifiers. We utilized support vector machine, k-nearest neighbors, naive Bayes, decision tree, random forest and AdaBoost classifiers. To determine the best classification approach, we performed grid search on feasible hyperparameters of respective classifiers and subsections of features. Our approach has achieved the state-of-the-art performance, measured by unweighted average recall in voice pathology detection on SVD database. We intentionally omit accuracy as it is highly biased metric in case of unbalanced data compared to aforementioned metrics. The results are further enhanced by eliminating the potential overestimation of the results with repeated stratified cross-validation. This advancement demonstrates significant potential for the clinical deployment of ML methods, offering a valuable tool for an objective examination of voice pathologies. To support our claims, we provide a publicly available GitHub repository with DOI 10.5281/zenodo.13771573. Finally, we provide REFORMS checklist., Comment: 33 pages, 8 figures, code repository: https://github.com/aailab-uct/Automated-Robust-and-Reproducible-Voice-Pathology-Detection

We consider a general reaction--nonlinear-diffusion equation with a region of negative diffusivity, and show how a nonlinear regularisation selects a shock position. Negative diffusivity can model population aggregation, but leads to shock-fronted solutions for population density. In general the shock position is non-unique. Previous studies have defined shock selection criteria such as the equal area rule, and shown how these arise from specific regularisations to the reaction--diffusion equation. In this work, we show that a nonlinear regularisation leads to travelling wave solutions where the shock is selected according to a modified equal area rule. Adjusting the nonlinearity in the regularisation moves the shock location. We focus on attaining shocks that conserve diffusivity across the shock, and demonstrate that this condition yields the longest possible shock length. Using geometric singular perturbation theory, we prove the existence of shock-fronted travelling wave solutions with continuous diffusivity, show how to construct them, and demonstrate that they correspond to a unique wave speed. Numerical solutions align with theoretical predictions for shock position and wave speed, confirming that a single regularisation term can vary the shock position and attain shocks with continuous diffusivity., Comment: 31 pages, 14 figures

Background: Patient-reported survey data are used to train prognostic models aimed at improving healthcare. However, such data are typically available multi-centric and, for privacy reasons, cannot easily be centralized in one data repository. Models trained locally are less accurate, robust, and generalizable. We present and apply privacy-preserving federated machine learning techniques for prognostic model building, where local survey data never leaves the legally safe harbors of the medical centers. Methods: We used centralized, local, and federated learning techniques on two healthcare datasets (GLA:D data from the five health regions of Denmark and international SHARE data of 27 countries) to predict two different health outcomes. We compared linear regression, random forest regression, and random forest classification models trained on local data with those trained on the entire data in a centralized and in a federated fashion. Results: In GLA:D data, federated linear regression (R2 0.34, RMSE 18.2) and federated random forest regression (R2 0.34, RMSE 18.3) models outperform their local counterparts (i.e., R2 0.32, RMSE 18.6, R2 0.30, RMSE 18.8) with statistical significance. We also found that centralized models (R2 0.34, RMSE 18.2, R2 0.32, RMSE 18.5, respectively) did not perform significantly better than the federated models. In SHARE, the federated model (AC 0.78, AUROC: 0.71) and centralized model (AC 0.84, AUROC: 0.66) perform significantly better than the local models (AC: 0.74, AUROC: 0.69). Conclusion: Federated learning enables the training of prognostic models from multi-center surveys without compromising privacy and with only minimal or no compromise regarding model performance.

We present annular dark field scanning transmission electron microscopy (ADF-STEM) as an efficient, fast, and non-destructive nanoscale tool for monitoring solid-state phase transition. Using metal-insulator transition in vanadium dioxide nanoparticles as an example, we characterize lattice and electronic signatures of the phase transition using analytical transmission electron microscopy including diffraction and electron energy-loss spectroscopy. We demonstrate that ADF-STEM shows a clear contrast across the transition, interpreted with the help of convergent electron beam diffraction as stemming from the crystal-lattice modification accompanying the transition. In addition, ADF-STEM utilizes 3--6 orders of magnitude lower electron dose when compared to electron microscopy techniques able to reveal the phase transition with the same spatial resolution and universality. The benefits of ADF-STEM are emphasized by recording a full hysteresis loop for the metal-insulator transition of a single vanadium dioxide nanoparticle. Our study opens the prospect for fast, non-destructive, large-area and nanoscale characterization of solid-state phase transitions.

We prove that every Banach space admitting a Gateaux smooth norm and containing a complemented copy of $\ell_1$ has an equivalent renorming which is simultaneously G\^ateaux smooth and octahedral. This is a partial solution to a problem from the early nineties.

Fast emitting polymeric scintillators are requested in advanced applications where high-speed detectors with large signal-to-noise ratio are needed. However, their low density implies a weak stopping power of high energy radiations, thus a limited light output and sensitivity. To enhance their performances, polymeric scintillators can be loaded with dense nanoparticles (NPs). We investigate the properties of a series of polymeric scintillators by means of photoluminescence and scintillation spectroscopy, comparing standard scintillators with a composite system loaded with dense hafnium dioxide (HfO2) NPs. The nanocomposite shows a scintillation yield enhancement of +100% with unchanged time response. We provide for the first time an interpretation of this effect, pointing out the local effect of NPs in the generation of emissive states upon interaction with the ionizing radiation. The obtained results indicate that coupling of fast conjugated emitters with optically inert dense NPs could allow to surpass the actual limits of pure polymeric scintillators., Comment: 14 pages, 4 figures

In this paper, we present some sufficient conditions on a metric space $M$ for which every molecule is a strongly subdifferentiable (SSD, for short) point in the Lipschitz-free space $\mathcal{F}(M)$ over $M$. Our main result reads as follows: if $(M,d)$ is a metric space and $\gamma > 0$, then there exists a (not necessarily equivalent) metric $d_{\gamma}$ in $M$ such that every finitely supported element in $\mathcal{F}(M, d_{\gamma})$ is an SSD point. As an application of the main result, it follows that if $M$ is uniformly discrete and $\varepsilon > 0$ is given, there exists a metric space $N$ and a $(1+\varepsilon)$-bi-Lipschitz map $\phi: M \rightarrow N$ such that the set of all SSD points in $\mathcal{F}(N)$ is dense., Comment: 20 pages, Some typos were fixed and a couple of remarks were added

Reliable and profound studies of actual magnetic domain structure in rare-earth A2Ir2O7 pyrochlore iridates are frequently limited by insufficient sample quality or lack of single crystals. We report the magnetic properties of the for-the-first-time synthesised Lu2Ir2O7 and Er2Ir2O7 single-crystals. The paper is focused on the robust ferromagnetic component of magnetisation present in the material with the antiferromagnetically ordered state of the all-in-all-out (AIAO) type, and is discussed in the framework of AIAO and AOAI domains and interfaces on the geometrically frustrated lattice., Comment: 6 pages, 3 figures in main text, supplementary material attached

This work studies fundamental limits for recovering the underlying correspondence among multiple correlated random graphs. We identify a necessary condition for any algorithm to correctly match all nodes across all graphs, and propose two algorithms for which the same condition is also sufficient. The first algorithm employs global information to simultaneously match all the graphs, whereas the second algorithm first partially matches the graphs pairwise and then combines the partial matchings by transitivity. Both algorithms work down to the information theoretic threshold. Our analysis reveals a scenario where exact matching between two graphs alone is impossible, but leveraging more than two graphs allows exact matching among all the graphs. Along the way, we derive independent results about the k-core of Erdos-Renyi graphs., Comment: 20 pages, 3 figures

Stablecoins - crypto tokens whose value is pegged to a real-world asset such as the US Dollar - are an important component of the DeFi ecosystem as they mitigate the impact of token price volatility. In crypto-backed stablecoins, the peg is founded on the guarantee that in case of system shutdown, each stablecoin can be exchanged for a basket of other crypto tokens worth approximately its nominal value. However, price fluctuations that affect the collateral tokens may cause this guarantee to be invalidated. In this work, we investigate the impact of the collateral portfolio's composition on the resilience to this type of catastrophic event. For stablecoins whose developers maintain a significant portion of the collateral (e.g., MakerDAO's Dai), we propose two portfolio optimization methods, based on convex optimization and (semi)variance minimization, that account for the correlation between the various token prices. We compare the optimal portfolios to the historical evolution of Dai's collateral portfolio, and to aid reproducibility, we have made our data and code publicly available., Comment: Accepted for presentation at MARBLE 2024

We survey several results concerning norming Markushevich bases (M-bases, for short), focusing in particular on two recent examples of a weakly compactly generated Banach space with no norming M-basis and of an Asplund space with norming M-basis that is not weakly compactly generated. We highlight the context for these problems and state several open problems in different directions that arise from these results.

We give a brief survey of the results on coarse or uniform embeddings of Banach spaces into $c_0(\Ga)$ and the point character of Banach spaces. In the process we prove several new results in this direction (for example we determine the point character of the spaces $L_p(\mu)$, $1\le p\le2$) solving open problems posed by C.~Avart, P.~Komjath, and V.~Roedl and by G.~Godefroy, G.~Lancien, and V.~Zizler. In particular, we show that $X=L_p(\mu)$, $1\le p<\infty$, bi-Lipschitz embeds into $c_0(\Ga)$ if and only if $\dens X<\om_\om$.

Two models are introduced to investigate graph matching in the presence of corrupt nodes. The weak model, inspired by biological networks, allows one or both networks to have a positive fraction of molecular entities interact randomly with their network. For this model, it is shown that no estimator can correctly recover a positive fraction of the corrupt nodes. Necessary conditions for any estimator to correctly identify and match all the uncorrupt nodes are derived, and it is shown that these conditions are also sufficient for the k-core estimator. The strong model, inspired by social networks, permits one or both networks to have a positive fraction of users connect arbitrarily. For this model, detection of corrupt nodes is impossible. Even so, we show that if only one of the networks is compromised, then under appropriate conditions, the maximum overlap estimator can correctly match a positive fraction of nodes albeit without explicitly identifying them., Comment: 31 pages, 1 figure

Reaction-diffusion equations (RDEs) model the spatiotemporal evolution of a density field $u(\vec{x},t)$ according to diffusion and net local changes. Usually, the diffusivity is positive for all values of $u,$ which causes the density to disperse. However, RDEs with partially negative diffusivity can model aggregation, which is the preferred behaviour in some circumstances. In this paper, we consider a nonlinear RDE with quadratic diffusivity $D(u) = (u - a)(u - b)$ that is negative for $u\in(a,b)$. We use a nonclassical symmetry to construct analytic receding time-dependent, colliding wave, and receding travelling wave solutions. These solutions are multi-valued, and we convert them to single-valued solutions by inserting a shock. We examine properties of these analytic solutions including their Stefan-like boundary condition, and perform a phase plane analysis. We also investigate the spectral stability of the $u = 0$ and $u = 1$ constant solutions, and prove for certain $a$ and $b$ that receding travelling waves are spectrally stable. Additionally, we introduce a new shock condition where the diffusivity and flux are continuous across the shock. For diffusivity symmetric about the midpoint of its zeros, this condition recovers the well-known equal-area rule, but for non-symmetric diffusivity it results in a different shock position., Comment: 36 pages, 10 figures

Reaction-nonlinear diffusion (RND) partial differential equations are a fruitful playground to model the formation of sharp travelling fronts, a fundamental pattern in nature. In this work, we demonstrate the utility and scope of regularisation as a technique to investigate shock-fronted solutions of RND PDEs, using geometric singular perturbation theory (GSPT) as the mathematical framework. In particular, we show that composite regularisations can be used to construct families of monotone shock-fronted travelling waves sweeping out distinct generalised area rules, which interpolate between the equal area and extremal area (i.e. algebraic decay) rules that are well-known in the shockwave literature. We further demonstrate that our RND PDE supports other kinds of shock-fronted solutions, namely, nonmonotone shockwaves as well as shockwaves containing slow tails in the aggregation (negative diffusion) regime. Our analysis blends Melnikov methods -- in both smooth and piecewise-smooth settings -- with GSPT techniques applied to the PDE over distinct spatiotemporal scales. We also consider the spectral stability of these new interpolated shockwaves. Using techniques from geometric spectral stability theory, we determine that our RND PDE admits spectrally stable shock-fronted travelling waves. The multiple-scale nature of the regularised RND PDE continues to play an important role in the analysis of the spatial eigenvalue problem., Comment: 43 pages, 18 figures

Van der Waals heterostructures of two-dimensional materials have opened up new frontiers in condensed matter physics, unlocking unexplored possibilities in electronic and photonic device applications. However, the investigation of wide-gap high-$\kappa$ layered dielectrics for devices based on van der Waals structures has been relatively limited. In this work, we demonstrate an easily reproducible synthesis method for the rare earth oxyhalide LaOBr, and we exfoliate it as a 2D layered material with a measured static dielectric constant of $\epsilon_{0, \perp} \simeq 9$ and a wide bandgap of 5.3 eV. Furthermore, our research demonstrates that LaOBr can be used as a high-$\kappa$ dielectric in van der Waals field-effect transistors with high performance and low interface defect concentrations. Additionally, it proves to be an attractive choice for electrical gating in excitonic devices based on 2D materials. Our work demonstrates the versatile realization and functionality of 2D systems with wide-gap and high-$\kappa$ van der Waals dielectric environments.

Decentralized Finance (DeFi), in which digital assets are exchanged without trusted intermediaries, has grown rapidly in value in recent years. The global DeFi ecosystem is fragmented into multiple blockchains, fueling the demand for cross-chain commerce. Existing approaches for cross-chain transactions, e.g., bridges and cross-chain deals, achieve atomicity by locking assets in escrow. However, locking up assets increases the financial risks for the participants, especially due to price fluctuations and the long latency of cross-chain transactions. Stablecoins, which are pegged to a non-volatile asset such as the US dollar, help mitigate the risk associated with price fluctuations. However, existing stablecoin designs are tied to individual blockchain platforms, and trusted parties or complex protocols are needed to exchange stablecoin tokens between blockchains. Our goal is to design a practical stablecoin for cross-chain commerce. Realizing this goal requires addressing two challenges. The first challenge is to support a large and growing number of blockchains efficiently. The second challenge is to be resilient to price fluctuations and blockchain platform failures. We present CroCoDai to address these challenges. We also present three prototype implementations of our stablecoin system, and show that it incurs small execution overhead., Comment: Accepted for publication in ACM Distributed Ledger Technologies: Research and Practice

The field of audio captioning has seen significant advancements in recent years, driven by the availability of large-scale audio datasets and advancements in deep learning techniques. In this technical report, we present our approach to audio captioning, focusing on the use of a pretrained speech-to-text Whisper model and pretraining on synthetic captions. We discuss our training procedures and present our experiments' results, which include model size variations, dataset mixtures, and other hyperparameters. Our findings demonstrate the impact of different training strategies on the performance of the audio captioning model. Our code and trained models are publicly available on GitHub and Hugging Face Hub.

Identification of experimentally acquired mass spectra of unknown compounds presents a~particular challenge because reliable spectral databases do not cover the potential chemical space with sufficient density. Therefore machine learning based \emph{de-novo} methods, which derive molecular structure directly from its mass spectrum gained attention recently. We present a~novel method in this family, addressing a~specific usecase of GC-EI-MS spectra, which is particularly hard due to lack of additional information from the first stage of MS/MS experiments, on which the previously published methods rely. We analyze strengths and drawbacks or our approach and discuss future directions.

We study several classical concepts in the topic of strict convexity of norms in infinite dimensional Banach spaces. Specifically, and in descending order of strength, we deal with Uniform Rotundity (UR), Weak Uniform Rotundity (WUR) and Uniform Rotundity in Every Direction (URED). Our first three results show that we may distinguish between all of these three properties in every Banach space where such renormings are possible. Specifically, we show that in every infinite dimensional Banach space which admits a WUR (resp. URED) renorming, we can find a norm with the same condition and which moreover fails to be UR (resp. WUR). We prove that these norms can be constructed to be Locally Uniformly Rotund (LUR) in Banach spaces admitting such renormings. Additionally, we obtain that in every Banach space with a LUR norm we can find a LUR renorming which is not URED. These results solve three open problems posed by A.J. Guirao, V. Montesinos and V. Zizler. The norms we construct in this first part are dense. In the last part of this note, we solve a fourth question posed by the same three authors by constructing a $C^\infty$-smooth norm in $c_0$ whose dual norm is not strictly convex., Comment: 19 pages, 1 figure

Formalizing surgical activities as triplets of the used instruments, actions performed, and target anatomies is becoming a gold standard approach for surgical activity modeling. The benefit is that this formalization helps to obtain a more detailed understanding of tool-tissue interaction which can be used to develop better Artificial Intelligence assistance for image-guided surgery. Earlier efforts and the CholecTriplet challenge introduced in 2021 have put together techniques aimed at recognizing these triplets from surgical footage. Estimating also the spatial locations of the triplets would offer a more precise intraoperative context-aware decision support for computer-assisted intervention. This paper presents the CholecTriplet2022 challenge, which extends surgical action triplet modeling from recognition to detection. It includes weakly-supervised bounding box localization of every visible surgical instrument (or tool), as the key actors, and the modeling of each tool-activity in the form of triplet. The paper describes a baseline method and 10 new deep learning algorithms presented at the challenge to solve the task. It also provides thorough methodological comparisons of the methods, an in-depth analysis of the obtained results across multiple metrics, visual and procedural challenges; their significance, and useful insights for future research directions and applications in surgery., Comment: MICCAI EndoVis CholecTriplet2022 challenge report. Published at Elsevier journal of Medical Image Analysis. 25 pages, 15 figures, 8 tables

This paper introduces the ``SurgT: Surgical Tracking" challenge which was organised in conjunction with MICCAI 2022. There were two purposes for the creation of this challenge: (1) the establishment of the first standardised benchmark for the research community to assess soft-tissue trackers; and (2) to encourage the development of unsupervised deep learning methods, given the lack of annotated data in surgery. A dataset of 157 stereo endoscopic videos from 20 clinical cases, along with stereo camera calibration parameters, have been provided. Participants were assigned the task of developing algorithms to track the movement of soft tissues, represented by bounding boxes, in stereo endoscopic videos. At the end of the challenge, the developed methods were assessed on a previously hidden test subset. This assessment uses benchmarking metrics that were purposely developed for this challenge, to verify the efficacy of unsupervised deep learning algorithms in tracking soft-tissue. The metric used for ranking the methods was the Expected Average Overlap (EAO) score, which measures the average overlap between a tracker's and the ground truth bounding boxes. Coming first in the challenge was the deep learning submission by ICVS-2Ai with a superior EAO score of 0.617. This method employs ARFlow to estimate unsupervised dense optical flow from cropped images, using photometric and regularization losses. Second, Jmees with an EAO of 0.583, uses deep learning for surgical tool segmentation on top of a non-deep learning baseline method: CSRT. CSRT by itself scores a similar EAO of 0.563. The results from this challenge show that currently, non-deep learning methods are still competitive. The dataset and benchmarking tool created for this challenge have been made publicly available at https://surgt.grand-challenge.org/.

We construct a complete metric space $M$ of cardinality continuum such that every non-singleton closed separable subset of $M$ fails to be a Lipschitz retract of $M$. This provides a metric analogue to the various classical and recent examples of Banach spaces failing to have linearly complemented subspaces of prescribed smaller density character., Comment: 42 pages, 9 figures

For $1\leq p<\infty$, we prove that the dense subspace $\mathcal{Y}_p$ of $\ell_p(\Gamma)$ comprising all elements $y$ such that $y \in \ell_q(\Gamma)$ for some $q \in (0,p)$ admits a $C^{\infty}$-smooth norm which locally depends on finitely many coordinates. Moreover, such a norm can be chosen as to approximate the $\left\Vert\cdot \right\Vert_p $-norm. This provides examples of dense subspaces of $\ell_p(\Gamma)$ with a smooth norm which have the maximal possible linear dimension and are not obtained as the linear span of a biorthogonal system. Moreover, when $p>1$ or $\Gamma$ is countable, such subspaces additionally contain dense operator ranges; on the other hand, no non-separable operator range in $\ell_1(\Gamma)$ admits a $C^1$-smooth norm.

This paper proposes regenerative particle Thompson sampling (RPTS), a flexible variation of Thompson sampling. Thompson sampling itself is a Bayesian heuristic for solving stochastic bandit problems, but it is hard to implement in practice due to the intractability of maintaining a continuous posterior distribution. Particle Thompson sampling (PTS) is an approximation of Thompson sampling obtained by simply replacing the continuous distribution by a discrete distribution supported at a set of weighted static particles. We observe that in PTS, the weights of all but a few fit particles converge to zero. RPTS is based on the heuristic: delete the decaying unfit particles and regenerate new particles in the vicinity of fit surviving particles. Empirical evidence shows uniform improvement from PTS to RPTS and flexibility and efficacy of RPTS across a set of representative bandit problems, including an application to 5G network slicing., Comment: Mainbody 14 pages, appendix 32 pages, 16 figures

We consider a family of exact solutions to a nonlinear reaction-diffusion model, constructed using nonclassical symmetry analysis. In a particular limit, the mathematical model approaches the well-known Fisher-KPP model, which means that it is related to various applications including cancer progression, wound healing and ecological invasion. The exact solution is mathematically interesting since exact solutions of the Fisher-KPP model are rare, and often restricted to long-time travelling wave solutions for special values of the travelling wave speed., Comment: 9 pages, 2 figures

In the present paper we prove that a necessary condition for a Banach space $X$ to admit a generating compact Lipschitz retract $K$, which satisfies an additional mild assumption on its shape, is that $X$ enjoys the Bounded Approximation Property. This is a partial solution to a question raised by Godefroy and Ozawa., Comment: arXiv admin note: text overlap with arXiv:2106.00331

In the present note we give a construction (based on a retractional argument) of a Schauder basis for the Lipschitz free space $\mathcal{F}(N)$, over a net $N$ in any separable infinite dimensional $\mathcal{L}_\infty$-space $X$. In particular, this yields the first example of an infinite dimensional Banach space $X$ not containing $c_0$ with such a property.

Security analyses for consensus protocols in blockchain research have primarily focused on the synchronous model, where point-to-point communication delays are upper bounded by a known finite constant. These models are unrealistic in noisy settings, where messages may be lost (i.e. incur infinite delay). In this work, we study the impact of message losses on the security of the proof-of-work longest-chain protocol. We introduce a new communication model to capture the impact of message loss called the $0-\infty$ model, and derive a region of tolerable adversarial power under which the consensus protocol is secure. The guarantees are derived as a simple bound for the probability that a transaction violates desired security properties. Specifically, we show that this violation probability decays almost exponentially in the security parameter. Our approach involves constructing combinatorial objects from blocktrees, and identifying random variables associated with them that are amenable to analysis. This approach improves existing bounds and extends the known regime for tolerable adversarial threshold in settings where messages may be lost., Comment: 27 pages, 5 figures

We prove that on 2-connected closed oriented manifolds, the analytic and algebraic constructions of an IBL$_\infty$ structure associated to a closed oriented manifold coincide. The corresponding structure is invariant under orientation preserving homotopy equivalences and induces on homology the involutive Lie bialgebra structure of Chas and Sullivan., Comment: Title changed from "Chain-level equivariant string topology for simply connected manifolds". 45p

The optimal receiver operating characteristic (ROC) curve, giving the maximum probability of detection as a function of the probability of false alarm, is a key information-theoretic indicator of the difficulty of a binary hypothesis testing problem (BHT). It is well known that the optimal ROC curve for a given BHT, corresponding to the likelihood ratio test, is theoretically determined by the probability distribution of the observed data under each of the two hypotheses. In some cases, these two distributions may be unknown or computationally intractable, but independent samples of the likelihood ratio can be observed. This raises the problem of estimating the optimal ROC for a BHT from such samples. The maximum likelihood estimator of the optimal ROC curve is derived, and it is shown to converge to the true optimal ROC curve in the \levy\ metric, as the number of observations tends to infinity. A classical empirical estimator, based on estimating the two types of error probabilities from two separate sets of samples, is also considered. The maximum likelihood estimator is observed in simulation experiments to be considerably more accurate than the empirical estimator, especially when the number of samples obtained under one of the two hypotheses is small. The area under the maximum likelihood estimator is derived; it is a consistent estimator of the true area under the optimal ROC curve.

Let $\mathcal{X}$ be a Banach space with a fundamental biorthogonal system and let $\mathcal{Y}$ be the dense subspace spanned by the vectors of the system. We prove that $\mathcal{Y}$ admits a $C^\infty$-smooth norm that locally depends on finitely many coordinates (LFC, for short), as well as a polyhedral norm that locally depends on finitely many coordinates. As a consequence, we also prove that $\mathcal{Y}$ admits locally finite, $\sigma$-uniformly discrete $C^\infty$-smooth and LFC partitions of unity and a $C^1$-smooth LUR norm. This theorem substantially generalises several results present in the literature and gives a complete picture concerning smoothness in such dense subspaces. Our result covers, for instance, every WLD Banach space (hence, all reflexive ones), $L_1(\mu)$ for every measure $\mu$, $\ell_\infty(\Gamma)$ spaces for every set $\Gamma$, $C(K)$ spaces where $K$ is a Valdivia compactum or a compact Abelian group, duals of Asplund spaces, or preduals of Von Neumann algebras. Additionally, under Martin Maximum {\sf MM}, all Banach spaces of density $\omega_1$ are covered by our result., Comment: Int. Math. Res. Not. IMRN (online first)

In the present note we give two explicit constructions (based on a retractional argument) of a Schauder basis for the Lipschitz free space $\mathcal{F}(N)$, over certain uniformly discrete metric spaces $N$. The first one applies to every net $N$ in a finite dimensional Banach space, leading to the basis constant independent of the dimension. The second one applies to grids in Banach spaces with an FDD. As a corollary, we obtain a retractional Schauder basis for the Lipschitz free space $\mathcal{F}(N)$ over a net $N$ in every Banach space $X$ with a Schauder basis containing a copy of $c_0$, as well as in every Banach space with a $c_0$-like FDD.

In the present paper we introduce and study the Lipschitz retractional structure of metric spaces. This topic was motivated by the analogous projectional structure of Banach spaces, a topic that has been thoroughly investigated. The more general metric setting fits well with the currently active theory of Lipschitz free spaces and spaces of Lipschitz functions. Among our applications we show that the Lipschitz free space $\mathcal{F}(X)$ is a Plichko space whenever $X$ is a Plichko Banach space. Our main results include two examples of metric spaces. The first one $M$ contains two points $\{0,1\}$ such that no separable subset of $M$ containing these points is a Lipschitz retract of $M$. The second example fails the analogous property for arbitrary infinite density. Finally, we introduce the metric version of the concept of locally complemented Banach subspace, and prove some metric analogues to the linear theory., Comment: 35 pages, 1 figure

In our note we show the very close connection between the existence of a Finite Dimensional Decomposition (FDD for short) for a separable Banach space $X$ and the existence of a Lipschitz retraction of $X$ onto a small (in a certain precise sense) generating convex and compact subset $K$ of $X$. In one direction, if $X$ admits an FDD then we construct a Lipschitz retraction onto a small generating convex and compact set $K$. On the other hand, we prove that if $X$ admits a small generating compact Lipschitz retract then $X$ has the $\pi$-property. We note that it is still unknown if the $\pi$-property is isomorphically equivalent to the existence of an FDD. For dual Banach spaces this is true, so our results lead in particular to a characterization of the FDD property for dual Banach spaces $X$ in terms of the existence of Lipschitz retractions onto small generating convex and compact subsets of $X$. It is conceivable that our results will find applications in the area of Lipschitz isomorphisms of Banach spaces. Our arguments make critical use of the Lipschitzization of coarse Lipschitz mappings due to J. Bourgain, and of an unpublished complementability result of V. Milman. We give an example of a small generating convex compact set which is not a Lipschitz retract of $C[0,1]$, although it is contained in a small convex Lipschitz retract and contains another one. In the last part of our note we characterize isomorphically Hilbertian spaces as those Banach spaces $X$ for which every convex and compact subset is a Lipschitz retract of $X$. Finally, we prove that a convex and compact set $K$ in any Banach space with a Uniformly Rotund in Every Direction norm is a uniform retract, of every bounded set containing it, via the nearest point map., Comment: 36 pages

The Banach space $\mathcal{P}({}^2X)$ of $2$-homogeneous polynomials on the Banach space $X$ can be naturally embedded in the Banach space ${{\rm Lip}_0}(B_X)$ of real-valued Lipschitz functions on $B_X$ that vanish at $0$. We investigate whether $\mathcal{P}({}^2X)$ is a complemented subspace of ${{\rm Lip}_0}(B_X)$. This line of research can be considered as a polynomial counterpart to a classical result by Joram Lindenstrauss, asserting that $\mathcal{P}({}^1X)=X^*$ is complemented in ${{\rm Lip}_0}(B_X)$ for every Banach space $X$. Our main result asserts that $\mathcal{P}({}^2X)$ is not complemented in ${{\rm Lip}_0}(B_X)$ for every Banach space $X$ with non-trivial type.

In the field of distributed consensus and blockchains, the synchronous communication model assumes that all messages between honest parties are delayed at most by a known constant $\Delta$. Recent literature establishes that the longest-chain blockchain protocol is secure under the synchronous model. However, for a fixed mining rate, the security guarantees degrade with $\Delta$. We analyze the performance of the longest-chain protocol under the assumption that the communication delays are random, independent, and identically distributed. This communication model allows for distributions with unbounded support and is a strict generalization of the synchronous model. We provide safety and liveness guarantees with simple, explicit bounds on the failure probabilities. These bounds hold for infinite-horizon executions and decay exponentially with the security parameter. In particular, we show that the longest-chain protocol has good security guarantees when delays are sporadically large and possibly unbounded, which is reflective of real-world network conditions., Comment: 30 pages, 1 figure

Recovery of the causal structure of dynamic networks from noisy measurements has long been a problem of intense interest across many areas of science and engineering. Many algorithms have been proposed, but there is no work that compares the performance of the algorithms to converse bounds in a non-asymptotic setting. As a step to address this problem, this paper gives lower bounds on the error probability for causal network support recovery in a linear Gaussian setting. The bounds are based on the use of the Bhattacharyya coefficient for binary hypothesis testing problems with mixture probability distributions. Comparison of the bounds and the performance achieved by two representative recovery algorithms are given for sparse random networks based on the Erd\H{o}s-R\'enyi model., Comment: Full version of the ISIT paper

We construct an Asplund Banach space $\mathcal{X}$ with a norming Marku\v{s}evi\v{c} basis such that $\mathcal{X}$ is not weakly compactly generated. This solves a long-standing open problem from the early nineties, originally due to Gilles Godefroy. En route to the proof, we construct a peculiar example of scattered compact space, that also solves a question due to Wies{\l}aw Kubi\'s and Arkady Leiderman.

In this paper we explore the following question: how weak can a logic be for Rosser's essential undecidability result to be provable for a weak arithmetical theory? It is well known that Robinson's Q is essentially undecidable in intuitionistic logic, and P. Hajek proved it in the fuzzy logic BL for Grzegorczyk's variant of Q which interprets the arithmetic operations as non-total non-functional relations. We present a proof of essential undecidability in a much weaker substructural logic and for a much weaker arithmetic theory, a version of Robinson's R (with arithmetic operations also interpreted as mere relations). Our result is based on a structural version of the undecidability argument introduced by Kleene and we show that it goes well beyond the scope of the Boolean, intuitionistic, or fuzzy logic.

We extend a CDGA $V$ with a perfect pairing of degree $n$ on cohomology to a CDGA $\hat V$ with a pairing of degree $n$ on chain level such that $\hat V$ admits a Hodge decomposition and retracts onto $V$ preserving the pairing on cohomology; here we suppose that $V$ is either 1-connected, or that $V$ is connected, of finite type, and $n$ is odd. We show that a Hodge decomposition of $\hat V$ induces a differential Poincar\'e duality model of $V$ in a natural way. Assuming that $H(V)$ is 1-connected, we apply our extension to a Sullivan model of $V$ in the proof of the existence and "uniqueness" of a 1-connected differential Poincar\'e duality model of $V$ by Lambrechts & Stanley; we eliminate their extra assumptions in the uniqueness statement, including $H^2(V)=0$ if $n$ is odd., Comment: Corrected and streamlined version. 30p

Following Cieliebak, Fukaya, Latschev and Volkov, we construct an IBL-infinity chain model for equivariant string topology on cyclic Hochschild cochains of de Rham cohomology. We study its properties and perform explicit computations., Comment: PHD thesis. Part I is a slightly corrected version of arXiv:1811.05281

We show that finite dimensional Banach spaces fail to be uniformly non locally almost square. Moreover, we construct an equivalent almost square bidual norm on $\ell_\infty.$ As a consequence we get that every dual Banach space containing $c_0$ has an equivalent almost square dual norm. Finally we characterize separable real almost square spaces in terms of their position in their fourth duals., Comment: Added a new result (Theorem 3.7) that there exists an equivalent renorming of c_0 which is M-embedded and whose bidual is ASQ. Added a couple of new paragraphs and fixed some typos

In the first part of our paper, we show that $\ell_\infty$ has a dense linear subspace which admits an equivalent real analytic norm. As a corollary, every separable Banach space, as well as $\ell_1(\mathfrak{c})$, also has a dense linear subspace which admits an analytic renorming. By contrast, no dense subspace of $c_0(\omega_1)$ admits an analytic norm. In the second part, we prove (solving in particular an open problem of Guirao, Montesinos, and Zizler) that every Banach space with a long unconditional Schauder basis contains a dense subspace that admits a $C^{\infty}$-smooth norm. Finally, we prove that there is a proper dense subspace of $\ell_{\infty}^{c}(\omega_1)$ that admits no G\^ateaux smooth norm. (Here, $\ell_{\infty}^{c} (\omega_1)$ denotes the Banach space of real-valued, bounded, and countably supported functions on $\omega_1$.), Comment: 18 pp

In the first part of our note we prove that every Weakly Lindel\"of Determined (WLD) (in particular, every reflexive) non-separable Banach $X$ space contains two dense linear subspaces $Y$ and $Z$ that are not densely isomorphic. This means that there are no further dense linear subspaces $Y_0$ and $Z_0$ of $Y$ and $Z$ which are linearly isomorphic. Our main result (Theorem B) concerns the existence of biorthogonal systems in normed spaces. In particular, we prove under the Continuum Hypothesis (CH) that there exists a dense linear subspace of $\ell_2(\omega_1)$ (or more generally every WLD space of density $\omega_1$) which contains no uncountable biorthogonal system. This result lies between two fundamental results concerning biorthogonal systems, namely the construction of Kunen (under CH) of a non-separable Banach space which contains no uncountable biorthogonal system, and the construction of Todor\u{c}evi\'c (under Martin Maximum) of an uncountable biorthogonal system in every non-separable Banach space., Comment: 21 pp

The theory of mean field games is a tool to understand noncooperative dynamic stochastic games with a large number of players. Much of the theory has evolved under conditions ensuring uniqueness of the mean field game Nash equilibrium. However, in some situations, typically involving symmetry breaking, non-uniqueness of solutions is an essential feature. To investigate the nature of non-unique solutions, this paper focuses on the technically simple setting where players have one of two states, with continuous time dynamics, and the game is symmetric in the players, and players are restricted to using Markov strategies. All the mean field game Nash equilibria are identified for a symmetric follow the crowd game. Such equilibria correspond to symmetric $\epsilon$-Nash Markov equilibria for $N$ players with $\epsilon$ converging to zero as $N$ goes to infinity. In contrast to the mean field game, there is a unique Nash equilibrium for finite $N.$ It is shown that fluid limits arising from the Nash equilibria for finite $N$ as $N$ goes to infinity are mean field game Nash equilibria, and evidence is given supporting the conjecture that such limits, among all mean field game Nash equilibria, are the ones that are stable fixed points of the mean field best response mapping.

The way how recurrently connected networks of spiking neurons in the brain acquire powerful information processing capabilities through learning has remained a mystery. This lack of understanding is linked to a lack of learning algorithms for recurrent networks of spiking neurons (RSNNs) that are both functionally powerful and can be implemented by known biological mechanisms. Since RSNNs are simultaneously a primary target for implementations of brain-inspired circuits in neuromorphic hardware, this lack of algorithmic insight also hinders technological progress in that area. The gold standard for learning in recurrent neural networks in machine learning is back-propagation through time (BPTT), which implements stochastic gradient descent with regard to a given loss function. But BPTT is unrealistic from a biological perspective, since it requires a transmission of error signals backwards in time and in space, i.e., from post- to presynaptic neurons. We show that an online merging of locally available information during a computation with suitable top-down learning signals in real-time provides highly capable approximations to BPTT. For tasks where information on errors arises only late during a network computation, we enrich locally available information through feedforward eligibility traces of synapses that can easily be computed in an online manner. The resulting new generation of learning algorithms for recurrent neural networks provides a new understanding of network learning in the brain that can be tested experimentally. In addition, these algorithms provide efficient methods for on-chip training of RSNNs in neuromorphic hardware., Comment: We changed in this version 2 of the paper the name of the new learning algorithms to e-prop, corrected minor errors, added details -- especially for resulting new rules for synaptic plasticity, edited the notation, and included new results for TIMIT

We study the application of IBL-infinity-algebras to string topology and explicitly compute the case of spheres. This involves finding a Green kernel and computing integrals associated to trivalent ribbon graphs, which are similar to integrals from perturbative Chern-Simons theory. We generalize the computation and show that the twisted IBL-infinity-structure in question is in many cases trivial., Comment: New version from January, 2019: a general vanishing result for the Maurer-Cartan element due to a tricky Green kernel modification from "A. Cattaneo, P. Mnev, Remarks on Chern-Simons theory"; more precise formulation of the string topology conjecture; other minor changes

Interventional C-arm imaging is crucial to percutaneous orthopedic procedures as it enables the surgeon to monitor the progress of surgery on the anatomy level. Minimally invasive interventions require repeated acquisition of X-ray images from different anatomical views to verify tool placement. Achieving and reproducing these views often comes at the cost of increased surgical time and radiation dose to both patient and staff. This work proposes a marker-free "technician-in-the-loop" Augmented Reality (AR) solution for C-arm repositioning. The X-ray technician operating the C-arm interventionally is equipped with a head-mounted display capable of recording desired C-arm poses in 3D via an integrated infrared sensor. For C-arm repositioning to a particular target view, the recorded C-arm pose is restored as a virtual object and visualized in an AR environment, serving as a perceptual reference for the technician. We conduct experiments in a setting simulating orthopedic trauma surgery. Our proof-of-principle findings indicate that the proposed system can decrease the 2.76 X-ray images required per desired view down to zero, suggesting substantial reductions of radiation dose during C-arm repositioning. The proposed AR solution is a first step towards facilitating communication between the surgeon and the surgical staff, improving the quality of surgical image acquisition, and enabling context-aware guidance for surgery rooms of the future. The concept of technician-in-the-loop design will become relevant to various interventions considering the expected advancements of sensing and wearable computing in the near future.