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Validating smartphone sensor-based tests to study gait and balance against reference measurement systems in a laboratory setting poses several technical challenges related to data quality and data processing. One challenge is to guarantee the correct annotation of the data, which is required to ensure that only data collected during the same test execution are compared across measurement systems in subsequent analyses. A second challenge is to accurately synchronize the data across the different systems. Here, we propose innovative solutions for both challenges and illustrate their use in the example of comparing smartphone sensor data collected with the Floodlight technology with data collected with a motion capture system. These solutions form important tools for guaranteeing the data quality and data integrity required for the validation of gait and balance characteristics measured by digital health technology tools such as the Floodlight technology., Comment: 16 pages, 4 figures, 1 table

In this paper, we present a novel approach for learning bimanual manipulation actions from human demonstration by extracting spatial constraints between affordance regions, termed affordance constraints, of the objects involved. Affordance regions are defined as object parts that provide interaction possibilities to an agent. For example, the bottom of a bottle affords the object to be placed on a surface, while its spout affords the contained liquid to be poured. We propose a novel approach to learn changes of affordance constraints in human demonstration to construct spatial bimanual action models representing object interactions. To exploit the information encoded in these spatial bimanual action models, we formulate an optimization problem to determine optimal object configurations across multiple execution keypoints while taking into account the initial scene, the learned affordance constraints, and the robot's kinematics. We evaluate the approach in simulation with two example tasks (pouring drinks and rolling dough) and compare three different definitions of affordance constraints: (i) component-wise distances between affordance regions in Cartesian space, (ii) component-wise distances between affordance regions in cylindrical space, and (iii) degrees of satisfaction of manually defined symbolic spatial affordance constraints., Comment: 8 pages, accepted for publication at Humanoids 2024 - Copyright IEEE

We propose a new fast algorithm for simultaneous recovery of the coil sensitivities and of the magnetization image from incomplete Fourier measurements in parallel MRI. Our approach is based on a parameter model for the coil sensitivities using bivariate trigonometric polynomials of small degree. The derived MOCCA algorithm has low computational complexity of $O(N_c N^2 \log N)$ for $N \times N$ images and $N_c$ coils and achieves very good performance for incomplete MRI data. We present a complete mathematical analysis of the proposed reconstruction method. Further, we show that MOCCA achieves similarly good reconstruction results as ESPIRiT with a considerably smaller numerical effort which is due to the employed parameter model. Our numerical examples show that MOCCA can outperform several other reconstruction methods., Comment: 32 pages

In this paper we propose a method to approximate the Gaussian function on ${\mathbb R}$ by a short cosine sum. We extend the differential approximation method proposed in [4,39] to approximate $\mathrm{e}^{-t^{2}/2\sigma}$ in the weighted space $L_2({\mathbb R}, \mathrm{e}^{-t^{2}/2\rho})$ where $\sigma, \, \rho >0$. We prove that the optimal frequency parameters $\lambda_1, \ldots , \lambda_{N}$ for this method in the approximation problem $ \min\limits_{\lambda_{1},\ldots, \lambda_{N}, \gamma_{1} \ldots \gamma_{N}}\|\mathrm{e}^{-\cdot^{2}/2\sigma} - \sum\limits_{j=1}^{N} \gamma_{j} \, {\mathrm e}^{\lambda_{j} \cdot}\|_{L_{2}({\mathbb R}, \mathrm{e}^{-t^{2}/2\rho})}$, are zeros of a scaled Hermite polynomial. This observation leads us to a numerically stable approximation method with low computational cost of $\mathit{O}(N^{3})$ operations. Furthermore, we derive a direct algorithm to solve this approximation problem based on a matrix pencil method for a special structured matrix. The entries of this matrix are determined by hypergeometric functions. For the weighted $L_{2}$-norm, we prove that the approximation error decays exponentially with respect to the length $N$ of the sum. An exponentially decaying error in the (unweighted) $L^{2}$-norm is achieved using a truncated cosine sum., Comment: 30 pages, 12 figures

We analyze the structure of a one-dimensional deep ReLU neural network (ReLU DNN) in comparison to the model of continuous piecewise linear (CPL) spline functions with arbitrary knots. In particular, we give a recursive algorithm to transfer the parameter set determining the ReLU DNN into the parameter set of a CPL spline function. Using this representation, we show that after removing the well-known parameter redundancies of the ReLU DNN, which are caused by the positive scaling property, all remaining parameters are independent. Moreover, we show that the ReLU DNN with one, two or three hidden layers can represent CPL spline functions with $K$ arbitrarily prescribed knots (breakpoints), where $K$ is the number of real parameters determining the normalized ReLU DNN (up to the output layer parameters). Our findings are useful to fix a priori conditions on the ReLU DNN to achieve an output with prescribed breakpoints and function values., Comment: 31 pages, 5 figurs

In this paper we introduce two new algorithms for stable approximation with and recovery of short cosine sums. The used signal model contains cosine terms with arbitrary real positive frequency parameters and therefore strongly generalizes usual Fourier sums. The proposed methods both employ a set of equidistant signal values as input data. The ESPRIT method for cosine sums is a Prony-like method and applies matrix pencils of Toeplitz+Hankel matrices while the ESPIRA method is based on rational approximation of DCT data and can be understood as a matrix pencil method for special Loewner matrices. Compared to known numerical methods for recovery of exponential sums, the design of the considered new algorithms directly exploits the special real structure of the signal model and therefore usually provides real parameter estimates for noisy input data, while the known general recovery algorithms for complex exponential sums tend to yield complex parameters in this case., Comment: 29 pages, 6 figures

We introduce a new method for Estimation of Signal Parameters based on Iterative Rational Approximation (ESPIRA) for sparse exponential sums. Our algorithm uses the AAA algorithm for rational approximation of the discrete Fourier transform of the given equidistant signal values. We show that ESPIRA can be interpreted as a matrix pencil method applied to Loewner matrices. These Loewner matrices are closely connected with the Hankel matrices which are usually employed for signal recovery. Due to the construction of the Loewner matrices via an adaptive selection of index sets, the matrix pencil method is stabilized. ESPIRA achieves similar recovery results for exact data as ESPRIT and the matrix pencil method but with less computational effort. Moreover, ESPIRA strongly outperforms ESPRIT and the matrix pencil method for noisy data and for signal approximation by short exponential sums., Comment: 36 pages, 18 figures

In this paper we derive a new recovery procedure for the reconstruction of extended exponential sums of the form $y(t) = \sum_{j=1}^{M} \left( \sum_{m=0}^{n_j} \, \gamma_{j,m} \, t^{m} \right) {\mathrm e}^{2\pi \lambda_j t}$, where the frequency parameters $\lambda_{j} \in {\mathbb C}$ are pairwise distinct. For the reconstruction we employ a finite set of classical Fourier coefficients of $y$ with regard to a finite interval $[0,P] \subset {\mathbb R}$ with $P>0$. Our method requires at most $2N+2$ Fourier coefficients $c_{k}(y)$ to recover all parameters of $y$, where $N:=\sum_{j=1}^{M} (1+n_{j})$ denotes the order of $y$. The recovery is based on the observation that for $\lambda_{j} \not\in \frac{{\mathrm i}}{P} {\mathbb Z}$ the terms of $y$ possess Fourier coefficients with rational structure. We employ a recently proposed stable iterative rational approximation algorithm in [12]. If a sufficiently large set of $L$ Fourier coefficients of $y$ is available (i.e., $L > 2N+2$), then our recovery method automatically detects the number $M$ of terms of $y$, the multiplicities $n_{j}$ for $j=1, \ldots , M$, as well as all parameters $\lambda_{j}$, $j=1, \ldots , M$ and $ \gamma_{j,m}$ $j=1, \ldots , M$, $m=0, \ldots , n_{j}$, determining $y$. Therefore our method provides a new stable alternative to the known numerical approaches for the recovery of exponential sums that are based on Prony's method., Comment: 29 pages, 7 figures

In this paper, we derive a new reconstruction method for real non-harmonic Fourier sums, i.e., real signals which can be represented as sparse exponential sums of the form $f(t) = \sum_{j=1}^{K} \gamma_{j} \, \cos(2\pi a_{j} t + b_{j})$, where the frequency parameters $a_{j} \in {\mathbb R}$ (or $a_{j} \in {\mathrm i} {\mathbb R}$) are pairwise different. Our method is based on the recently proposed stable iterative rational approximation algorithm in \cite{NST18}. For signal reconstruction we use a set of classical Fourier coefficients of $f$ with regard to a fixed interval $(0, P)$ with $P>0$. Even though all terms of $f$ may be non-$P$-periodic, our reconstruction method requires at most $2K+2$ Fourier coefficients $c_{n}(f)$ to recover all parameters of $f$. We show that in the case of exact data, the proposed iterative algorithm terminates after at most $K+1$ steps. The algorithm can also detect the number $K$ of terms of $f$, if $K$ is a priori unknown and $L>2K+2$ Fourier coefficients are available. Therefore our method provides a new stable alternative to the known numerical approaches for the recovery of exponential sums that are based on Prony's method. Keywords: sparse exponential sums, non-harmonic Fourier sums, reconstruction of sparse non-periodic signals, rational approximation, AAA algorithm, barycentric representation, Fourier coefficients, Comment: 28 pages, 2 figures

Internet users and service providers do not often know when traffic is being watched but desire a way to determine when, where, and by whom. We present NOISE, the Nonce Observatory for Inverse Surveillance of Eavesdroppers, a method and system that detects monitoring by disseminating nonces - unique, pseudorandom values - in traffic and seeing if they are acted upon unexpectedly, indicating that the nonce-laden traffic is being monitored. Specifically, we embed 64-bit nonces innocuously into IPv6 addresses and disseminate these nonces Internet-wide using a modified traceroute-like tool that makes each outbound probe's source address unique. We continually monitor for subsequent nonce propagation, i.e., activity or interest involving these nonces, e.g., via packet capture on our system's infrastructure. Across three experiments and four months, NOISE detects monitoring more than 200k times, ostensibly in 268 networks, for probes destined for 437 networks. Our results reveal: (a) data collection for security incident handling, (b) traffic information being shared with third parties, and (c) eavesdropping in or near a large commercial peering exchange.

In this paper we extend the deterministic sublinear FFT algorithm in Plonka et al. (2018) for fast reconstruction of $M$-sparse vectors ${\mathbf x}$ of length $N= 2^J$, where we assume that all components of the discrete Fourier transform $\hat{\mathbf x}= {\mathbf F}_{N} {\mathbf x}$ are available. The sparsity of ${\mathbf x}$ needs not to be known a priori, but is determined by the algorithm. If the sparsity $M$ is larger than $2^{J/2}$, then the algorithm turns into a usual FFT algorithm with runtime ${\mathcal O}(N \log N)$. For $M^{2} < N$, the runtime of the algorithm is ${\mathcal O}(M^2 \, \log N)$. The proposed modifications of the approach in Plonka et al. (2018) lead to a significant improvement of the condition numbers of the Vandermonde matrices which are employed in the iterative reconstruction. Our numerical experiments show that our modification has a huge impact on the stability of the algorithm. While the algorithm in Plonka et al. (2018) starts to be unreliable for $M>20$ because of numerical instabilities, the modified algorithm is still numerically stable for $M=200$., Comment: 15 pages, 3 figures

We characterize optimal rank-1 matrix approximations with Hankel or Toeplitz structure with regard to two different norms, the Frobenius norm and the spectral norm, in a new way. More precisely, we show that these rank-1 matrix approximation problems can be solved by maximizing special rational functions. Our approach enables us to show that the optimal solutions with respect to these two norms have completely different structure and only coincide in the trivial case when the singular value decomposition already provides an optimal rank-1 approximation with the desired Hankel or Toeplitz structure. We also prove that the Cadzow algorithm for structured low-rank approximations always converges to a fixed point in the rank-1 case. However, it usually does not converge to the optimal solution, neither with regard to the Frobenius norm nor the spectral norm., Comment: 34 pages

We present the result of an experiment to measure the electric dipole moment (EDM) of the neutron at the Paul Scherrer Institute using Ramsey's method of separated oscillating magnetic fields with ultracold neutrons (UCN). Our measurement stands in the long history of EDM experiments probing physics violating time reversal invariance. The salient features of this experiment were the use of a Hg-199 co-magnetometer and an array of optically pumped cesium vapor magnetometers to cancel and correct for magnetic field changes. The statistical analysis was performed on blinded datasets by two separate groups while the estimation of systematic effects profited from an unprecedented knowledge of the magnetic field. The measured value of the neutron EDM is $d_{\rm n} = (0.0\pm1.1_{\rm stat}\pm0.2_{\rm sys})\times10^{-26}e\,{\rm cm}$., Comment: 5 pages, 4 figures, submitted to PRL on 18.12.2019

In this survey we describe some modifications of Prony's method. In particular, we consider the recovery of general expansions into eigenfunctions of linear differential operators of first order and show, how these expansions can be recovered from function samples using generalized shift operators. We derive an ESPRIT-like algorithm for the generalized recovery method and show, that this approach can be directly used to reconstruct classical exponential sums from non-equispaced data. Furthermore, we derive a modification of Prony's method for sparse approximation with exponential sums which leads to a non-linear least-squares problem., Comment: 27 pages, 2 figures

The aim of this paper is twofold. First, we show that a certain concatenation of a proximity operator with an affine operator is again a proximity operator on a suitable Hilbert space. Second, we use our findings to establish so-called proximal neural networks (PNNs) and stable tight frame proximal neural networks. Let $\mathcal H$ and $\mathcal K$ be real Hilbert spaces, $b\in\mathcal K$ and $T\in\mathcal{B}(\mathcal H,\mathcal K)$ have closed range and Moore-Penrose inverse $T^\dagger$. Based on the well-known characterization of proximity operators by Moreau, we prove that for any proximity operator $\text{Prox}\colon\mathcal K\to\mathcal K$ the operator $T^\dagger\,\text{Prox} (T\cdot +b)$ is a proximity operator on $\mathcal H$ equipped with a suitable norm. In particular, it follows for the frequently applied soft shrinkage operator $\text{Prox} = S_{\lambda}\colon\ell_2 \rightarrow\ell_2$ and any frame analysis operator $T\colon\mathcal H\to\ell_2$ that the frame shrinkage operator $T^\dagger\, S_\lambda\,T$ is a proximity operator on a suitable Hilbert space. The concatenation of proximity operators on $\mathbb R^d$ equipped with different norms establishes a PNN. If the network arises from tight frame analysis or synthesis operators, then it forms an averaged operator. Hence, it has Lipschitz constant 1 and belongs to the class of so-called Lipschitz networks, which were recently applied to defend against adversarial attacks. Moreover, due to its averaging property, PNNs can be used within so-called Plug-and-Play algorithms with convergence guarantee. In case of Parseval frames, we call the networks Parseval proximal neural networks (PPNNs). Then, the involved linear operators are in a Stiefel manifold and corresponding minimization methods can be applied for training. Finally, some proof-of-the concept examples demonstrate the performance of PPNNs., Comment: arXiv admin note: text overlap with arXiv:1910.02843

Let $\mathcal H$ and $\mathcal K$ be real Hilbert spaces and $T \in \mathcal{B} (\mathcal H,\mathcal K)$ an injective operator with closed range and Moore-Penrose inverse $T^\dagger$. Based on the well-known characterization of proximity operators by Moreau, we prove that for any proximity operator $\text{Prox} \colon \mathcal K \to \mathcal K$ the operator $T^\dagger \, \text{Prox} \, T$ is a proximity operator on the linear space $\mathcal H$ equipped with a suitable norm. In particular, it follows for the frequently applied soft shrinkage operator $\text{Prox} = S_{\lambda}\colon \ell_2 \rightarrow \ell_2$ and any frame analysis operator $T\colon \mathcal H \to \ell_2$, that the frame shrinkage operator $T^\dagger\, S_\lambda\, T$ is a proximity operator in a suitable Hilbert space.

In this paper, we show that the commonly used frame soft shrinkage operator, that maps a given vector ${\mathbf x} \in {\mathbb R}^{N}$ onto the vector ${\mathbf T}^{\dagger} S_{\gamma} {\mathbf T} {\mathbf x}$, is already a proximity operator, which can therefore be directly used in corresponding splitting algorithms. In our setting, the frame transform matrix ${\mathbf T} \in {\mathbb R}^{L \times N}$ with $L \ge N$ has full rank $N$, ${\mathbf T}^{\dagger}$ denotes the Moore-Penrose inverse of ${\mathbf T}$, and $S_{\gamma}$ is the usual soft shrinkage operator with threshold parameter $\gamma >0$. Our result generalizes the known assertion that ${\mathbf T}^{*} S_{\gamma} {\mathbf T}$ is the proximity operator of $\| {\mathbf T} \cdot \|_{1}$ if ${\mathbf T}$ is an orthogonal (square) matrix. It is well-known that for rectangular frame matrices ${\mathbf T}$ with $L > N$, the proximity operator of $\| {\mathbf T} \cdot \|_{1}$ does not have a closed representation and needs to be computed iteratively. We show that the frame soft shrinkage operator {${\mathbf T}^{\dagger} S_{\gamma} {\mathbf T}$} is a proximity operator as well, thereby motivating its application as a replacement of the exact proximity operator of $\| {\mathbf T} \cdot \|_{1}$. We further give an explanation, why the usage of the frame soft shrinkage operator still provides good results in various applications. In particular, we provide some properties of the subdifferential of the convex functional $\Phi$ which leads to the proximity operator ${\mathbf T}^{\dagger} S_{\gamma} {\mathbf T}$ and show that ${\mathbf T}^{\dagger} S_{\gamma} {\mathbf T}$ approximates $\textrm{prox}_{\|{\mathbf T} \cdot\|_{1}}$., Comment: 16 pages, 1 figure

We propose a new outline for adaptive dictionary learning methods for sparse encoding based on a hierarchical clustering of the training data. Through recursive application of a clustering method, the data is organized into a binary partition tree representing a multiscale structure. The dictionary atoms are defined adaptively based on the data clusters in the partition tree. This approach can be interpreted as a generalization of a discrete Haar wavelet transform. Furthermore, any prior knowledge on the wanted structure of the dictionary elements can be simply incorporated. The computational complexity of our proposed algorithm depends on the employed clustering method and on the chosen similarity measure between data points. Thanks to the multiscale properties of the partition tree, our dictionary is structured: when using Orthogonal Matching Pursuit to reconstruct patches from a natural image, dictionary atoms corresponding to nodes being closer to the root node in the tree have a tendency to be used with greater coefficients.

The generalized Prony method introduced by Peter & Plonka (2013) is a reconstruction technique for a large variety of sparse signal models that can be represented as sparse expansions into eigenfunctions of a linear operator $A$. However, this procedure requires the evaluation of higher powers of the linear operator $A$ that are often expensive to provide. In this paper we propose two important extensions of the generalized Prony method that simplify the acquisition of the needed samples essentially and at the same time can improve the numerical stability of the method. The first extension regards the change of operators from $A$ to $\varphi(A)$, where $\varphi$ is an analytic function, while $A$ and $\varphi(A)$ possess the same set of eigenfunctions. The goal is now to choose $\varphi$ such that the powers of $\varphi(A)$ are much simpler to evaluate than the powers of $A$. The second extension concerns the choice of the sampling functionals. We show, how new sets of different sampling functionals $F_{k}$ can be applied with the goal to reduce the needed number of powers of the operator $A$ (resp. $\varphi(A)$) in the sampling scheme and to simplify the acquisition process for the recovery method., Comment: 31 pages, 2 figures

The experimental setup for examining the low-molecular-weight fluoropolymer CF$_{3}$(CF$_{2})_{3}$-O-CF$_{2}$-O-(CF$_{2})_{3}$CF$_{3}$, which is a promising coating material for the walls of storage chambers for ultracold neutrons, is described. The results are detailed. The measurement data are interpreted in the model of a multilayer complex quantum-mechanical potential of the chamber walls., Comment: Corrected version of the paper published in Phys. of Part. and Nucl Lett., 15 (No.5) (2018) 541

In this paper we present a new fast and deterministic algorithm for the inverse discrete cosine transform of type II for reconstructing the input vector $\mathbf x\in\mathbb R^N$, $N=2^J$, with short support of length $m$ from its discrete cosine transform $\mathbf x^{\widehat{\mathrm{II}}}=C^{\mathrm{II}}_N\mathbf x$ if an upper bound $M\geq m$ is known. The resulting algorithm only uses real arithmetic, has a runtime of $\mathcal{O}\left(M\log M+m\log_2\frac{N}{M}\right)$ and requires $\mathcal{O}\left(M+m\log_2\frac{N}{M}\right)$ samples of $\mathbf x^{\widehat{\mathrm{II}}}$. For $m,M\rightarrow N$ the runtime and sampling requirements approach those of a regular IDCT-II for vectors with full support. The algorithm presented hereafter does not employ inverse FFT algorithms to recover $\mathbf x$., Comment: 28 pages, 5 figures

Existing methods for active topology discovery within the IPv6 Internet largely mirror those of IPv4. In light of the large and sparsely populated address space, in conjunction with aggressive ICMPv6 rate limiting by routers, this work develops a different approach to Internet-wide IPv6 topology mapping. We adopt randomized probing techniques in order to distribute probing load, minimize the effects of rate limiting, and probe at higher rates. Second, we extensively analyze the efficiency and efficacy of various IPv6 hitlists and target generation methods when used for topology discovery, and synthesize new target lists based on our empirical results to provide both breadth (coverage across networks) and depth (to find potential subnetting). Employing our probing strategy, we discover more than 1.3M IPv6 router interface addresses from a single vantage point. Finally, we share our prober implementation, synthesized target lists, and discovered IPv6 topology results.

We consider Perron-Frobenius and Koopman operators associated to time-inhomogeneous ordinary stochastic differential equations, and establish their Fr\'{e}chet differentiability with respect to the drift. This result relies on a similar differentiability result for pathwise expectations of path functionals of the solution of the stochastic differential equation, which we establish using Girsanov's formula. We demonstrate the significance of our result in the context of dynamical systems and operator theory, by proving continuously differentiable drift dependence of the simple eigen- and singular values and the corresponding eigen- and singular functions of the stochastic Perron-Frobenius and Koopman operators.

In this paper we present a new fast and deterministic algorithm for the inverse discrete cosine transform of type II that reconstructs the input vector $\mathbf{x}\in\mathbb{R}^{N}$, $N=2^{J-1}$, with short support of length $m$ from its discrete cosine transform $\mathbf{x}^{\widehat{\mathrm{II}}}=\mathbf{C}_N^{\mathrm{II}}\mathbf{x}$. The resulting algorithm has a runtime of $\mathcal{O}\left(m\log m\log \frac{2N}{m}\right)$ and requires $\mathcal{O}\left(m\log \frac{2N}{m}\right)$ samples of $\mathbf{x}^{\widehat{\mathrm{II}}}$. In order to derive this algorithm we also develop a new fast and deterministic inverse FFT algorithm that constructs the input vector $\mathbf{y}\in\mathbb{R}^{2N}$ with reflected block support of block length $m$ from $\widehat{\mathbf{y}}$ with the same runtime and sampling complexities as our DCT algorithm., Comment: 27 pages, 6 figures

Privacy-minded Internet service operators anonymize IPv6 addresses by truncating them to a fixed length, perhaps due to long-standing use of this technique with IPv4 and a belief that it's "good enough." We claim that simple anonymization by truncation is suspect since it does not entail privacy guarantees nor does it take into account some common address assignment practices observed today. To investigate, with standard activity logs as input, we develop a counting method to determine a lower bound on the number of active IPv6 addresses that are simultaneously assigned, such as those of clients that access World-Wide Web services. In many instances, we find that these empirical measurements offer no evidence that truncating IPv6 addresses to a fixed number of bits, e.g., 48 in common practice, protects individuals' privacy. To remedy this problem, we propose kIP anonymization, an aggregation method that ensures a certain level of address privacy. Our method adaptively determines variable truncation lengths using parameter k, the desired number of active (rather than merely potential) addresses, e.g., 32 or 256, that can not be distinguished from each other once anonymized. We describe our implementation and present first results of its application to millions of real IPv6 client addresses active over a week's time, demonstrating both feasibility at large scale and ability to automatically adapt to each network's address assignment practice and synthesize a set of anonymous aggregates (prefixes), each of which is guaranteed to cover (contain) at least k of the active addresses. Each address is anonymized by truncating it to the length of its longest matching prefix in that set.

In this paper, we show that sparse signals f representable as a linear combination of a finite number N of spikes at arbitrary real locations or as a finite linear combination of B-splines of order m with arbitrary real knots can be almost surely recovered from O(N^2) Fourier intensity measurements up to trivial ambiguities. The constructive proof consists of two steps, where in the first step the Prony method is applied to recover all parameters of the autocorrelation function and in the second step the parameters of f are derived. Moreover, we present an algorithm to evaluate f from its Fourier intensities and illustrate it at different numerical examples.

In this paper, we derive a new method for optimal $\ell^{1}$- and $\ell^2$-approximation of discrete signals on ${\mathbb N}_{0}$ whose entries can be represented as an exponential sum of finite length. Our approach employs Prony's method in a first step to recover the exponential sum that is determined by the signal. In the second step we use the AAK-theory to derive an algorithm for computing a shorter exponential sum that approximates the original signal in the $\ell^{p}$-norm well. AAK-theory originally determines best approximations of bounded periodic functions in Hardy-subspaces. We rewrite these ideas for our purposes and give a proof of the used AAK theorem based only on basic tools from linear algebra and Fourier analysis. The new algorithm is tested numerically in different examples., Comment: 24 pages, 1 figure

In this paper, we introduce Entropy/IP: a system that discovers Internet address structure based on analyses of a subset of IPv6 addresses known to be active, i.e., training data, gleaned by readily available passive and active means. The system is completely automated and employs a combination of information-theoretic and machine learning techniques to probabilistically model IPv6 addresses. We present results showing that our system is effective in exposing structural characteristics of portions of the IPv6 Internet address space populated by active client, service, and router addresses. In addition to visualizing the address structure for exploration, the system uses its models to generate candidate target addresses for scanning. For each of 15 evaluated datasets, we train on 1K addresses and generate 1M candidates for scanning. We achieve some success in 14 datasets, finding up to 40% of the generated addresses to be active. In 11 of these datasets, we find active network identifiers (e.g., /64 prefixes or `subnets') not seen in training. Thus, we provide the first evidence that it is practical to discover subnets and hosts by scanning probabilistically selected areas of the IPv6 address space not known to contain active hosts a priori., Comment: Paper presented at the ACM IMC 2016 in Santa Monica, USA (https://dl.acm.org/citation.cfm?id=2987445). Live Demo site available at http://www.entropy-ip.com/

In this study, we report on techniques and analyses that enable us to capture Internet-wide activity at individual IP address-level granularity by relying on server logs of a large commercial content delivery network (CDN) that serves close to 3 trillion HTTP requests on a daily basis. Across the whole of 2015, these logs recorded client activity involving 1.2 billion unique IPv4 addresses, the highest ever measured, in agreement with recent estimates. Monthly client IPv4 address counts showed constant growth for years prior, but since 2014, the IPv4 count has stagnated while IPv6 counts have grown. Thus, it seems we have entered an era marked by increased complexity, one in which the sole enumeration of active IPv4 addresses is of little use to characterize recent growth of the Internet as a whole. With this observation in mind, we consider new points of view in the study of global IPv4 address activity. Our analysis shows significant churn in active IPv4 addresses: the set of active IPv4 addresses varies by as much as 25% over the course of a year. Second, by looking across the active addresses in a prefix, we are able to identify and attribute activity patterns to network restructurings, user behaviors, and, in particular, various address assignment practices. Third, by combining spatio-temporal measures of address utilization with measures of traffic volume, and sampling-based estimates of relative host counts, we present novel perspectives on worldwide IPv4 address activity, including empirical observation of under-utilization in some areas, and complete utilization, or exhaustion, in others., Comment: in Proceedings of ACM IMC 2016

Considering the ambiguousness of the discrete-time phase retrieval problem to recover a signal from its Fourier intensities, one can ask the question: what additional information about the unknown signal do we need to select the correct solution within the large solution set? Based on a characterization of the occurring ambiguities, we investigate different a priori conditions in order to reduce the number of ambiguities or even to receive a unique solution. Particularly, if we have access to additional magnitudes of the unknown signal in the time domain, we can show that almost all signals with finite support can be uniquely recovered. Moreover, we prove that an analogous result can be obtained by exploiting additional phase information.

In this paper we propose a new fast Fourier transform to recover a real nonnegative signal ${\bf x}$ from its discrete Fourier transform. If the signal ${\mathbf x}$ appears to have a short support, i.e., vanishes outside a support interval of length $m < N$, then the algorithm has an arithmetical complexity of only ${\cal O}(m \log m \log (N/m)) $ and requires ${\cal O}(m \log (N/m))$ Fourier samples for this computation. In contrast to other approaches there is no a priori knowledge needed about sparsity or support bounds for the vector ${\bf x}$. The algorithm automatically recognizes and exploits a possible short support of the vector and falls back to a usual radix-2 FFT algorithm if ${\bf x}$ has (almost) full support. The numerical stability of the proposed algorithm ist shown by numerical examples.

There is striking volume of World-Wide Web activity on IPv6 today. In early 2015, one large Content Distribution Network handles 50 billion IPv6 requests per day from hundreds of millions of IPv6 client addresses; billions of unique client addresses are observed per month. Address counts, however, obscure the number of hosts with IPv6 connectivity to the global Internet. There are numerous address assignment and subnetting options in use; privacy addresses and dynamic subnet pools significantly inflate the number of active IPv6 addresses. As the IPv6 address space is vast, it is infeasible to comprehensively probe every possible unicast IPv6 address. Thus, to survey the characteristics of IPv6 addressing, we perform a year-long passive measurement study, analyzing the IPv6 addresses gleaned from activity logs for all clients accessing a global CDN. The goal of our work is to develop flexible classification and measurement methods for IPv6, motivated by the fact that its addresses are not merely more numerous; they are different in kind. We introduce the notion of classifying addresses and prefixes in two ways: (1) temporally, according to their instances of activity to discern which addresses can be considered stable; (2) spatially, according to the density or sparsity of aggregates in which active addresses reside. We present measurement and classification results numerically and visually that: provide details on IPv6 address use and structure in global operation across the past year; establish the efficacy of our classification methods; and demonstrate that such classification can clarify dimensions of the Internet that otherwise appear quite blurred by current IPv6 addressing practices.

We compare the expected effects of so-called gravitationally enhanced depolarization of ultracold neutrons to measurements carried out in a spin-precession chamber exposed to a variety of vertical magnetic-field gradients. In particular, we have investigated the dependence upon these field gradients of spin depolarization rates and also of shifts in the measured neutron Larmor precession frequency. We find excellent qualitative agreement, with gravitationally enhanced depolarization accounting for several previously unexplained features in the data., Comment: 10 pages, 6 figures. Updated: section added about implications for current nEDM limit

We describe a spin-echo method for ultracold neutrons (UCNs) confined in a precession chamber and exposed to a $|B_0|=1~\text{\mu T}$ magnetic field. We have demonstrated that the analysis of UCN spin-echo resonance signals in combination with knowledge of the ambient magnetic field provides an excellent method by which to reconstruct the energy spectrum of a confined ensemble of neutrons. The method takes advantage of the relative dephasing of spins arising from a gravitationally induced striation of stored UCN of different energies, and also permits an improved determination of the vertical magnetic-field gradient with an exceptional accuracy of $1.1~\text{pT/cm}$. This novel combination of a well-known nuclear resonance method and gravitationally induced vertical striation is unique in the realm of nuclear and particle physics and should prove to be invaluable for the assessment of systematic effects in precision experiments such as searches for an electric dipole moment of the neutron or the measurement of the neutron lifetime., Comment: 7 pages 5 figures, accepted by PRL, September, 08 2015

The Hubbard model with local on-site repulsion is generally thought to possess a superconducting ground-state for appropriate parameters, but the effects of more realistic long-range Coulomb interactions have not been studied extensively. We study the influence of these interactions on superconductivity by including nearest and next-nearest neighbor extended Hubbard interactions in addition to the usual on-site terms. Utilizing numerical exact diagonalization, we analyze the signatures of superconductivity in the ground states through the fidelity metric of quantum information theory. We find that nearest and next-nearest neighbor interactions have thresholds above which they destabilize superconductivity regardless of whether they are attractive or repulsive, seemingly due to competing charge fluctuations., Comment: 5 pages, 3 figures

We derive new explicit expressions for the components of Moore-Penrose inverses of symmetric difference matrices. These generalized inverses are applied in a new regularization approach for scattered data interpolation based on partial differential equations. The columns of the Moore-Penrose inverse then serve as elements of a dictionary that allow a sparse signal approximation. In order to find a set of suitable data points for signal representation we apply the orthogonal patching pursuit (OMP) method., Comment: 16 pages

In this paper we consider the special case where a discrete signal ${\bf x}$ of length N is known to vanish outside a support interval of length $m < N$. If the support length $m$ of ${\bf x}$ or a good bound of it is a-priori known we derive a sublinear deterministic algorithm to compute ${\bf x}$ from its discrete Fourier transform. In case of exact Fourier measurements we require only ${\cal O}(m \log m)$ arithmetical operations. For noisy measurements, we propose a stable ${\cal O}(m \log N)$ algorithm., Comment: 16 pages

Ultrafast light pulses can modify the electronic properties of quantum materials by perturbing the underlying, intertwined degrees of freedom. In particular, iron-based superconductors exhibit a strong coupling among electronic nematic fluctuations, spins, and the lattice, serving as a playground for ultrafast manipulation. Here we use time-resolved x-ray scattering to measure the lattice dynamics of photo-excited BaFe2As2. Upon optical excitation, no signature of an ultrafast change of the crystal symmetry is observed, but the lattice oscillates rapidly in time due to the coherent excitation of an A1g mode that modulates the Fe-As-Fe bond angle. We directly quantify the coherent lattice dynamics and show that even a small photo-induced lattice distortion can induce notable changes in the electronic and magnetic properties. Our analysis implies that transient structural modification can generally be an effective tool for manipulating the electronic properties of multi-orbital systems, where electronic instabilities are sensitive to the orbital character of bands near the Fermi level.

This work presents selected results from the first round of the DFG Priority Programme SPP 1491 "precision experiments in particle and astroparticle physics with cold and ultra-cold neutrons"., Comment: 8 pages, 4 figures, JPS Conference Proceedings of the 2nd International Symposium on Science at J-PARC (J-PARC 2014), Tsukuba, Japan, July 12-15, 2014

The Surrounding Field Compensation (SFC) system described in this work is installed around the four-layer Mu-metal magnetic shield of the neutron electric dipole moment spectrometer located at the Paul Scherrer Institute. The SFC system reduces the DC component of the external magnetic field by a factor of about 20. Within a control volume of approximately 2.5m x 2.5m x 3m disturbances of the magnetic field are attenuated by factors of 5 to 50 at a bandwidth from $10^{-3}$ Hz up to 0.5 Hz, which corresponds to integration times longer than several hundreds of seconds and represent the important timescale for the nEDM measurement. These shielding factors apply to random environmental noise from arbitrary sources. This is achieved via a proportional-integral feedback stabilization system that includes a regularized pseudoinverse matrix of proportionality factors which correlates magnetic field changes at all sensor positions to current changes in the SFC coils., Comment: 33 pages, 18 figures

An intriguing aspect of unconventional superconductivity is that it always appears in the vicinity of other competing phases, whose suppression brings the full emergence of superconductivity. In the iron-pnictides, these competing phases are marked by a tetragonal-to-orthorhombic structural transition and a collinear spin-density-wave (SDW) transition. There has been macroscopic evidence for competition between these phases and superconductivity as the magnitude of both the orthorhombicity and magnetic moment are suppressed in the superconducting state. Here, using angle-resolved photoemission spectroscopy on detwinned underdoped $Ba_{1-x}K_xFe_2As_2$, we observe a coexistence of both the SDW gap and superconducting gap in the same electronic structure. Furthermore, our data reveal that following the onset of superconductivity, the SDW gap decreases in magnitude and shifts in a direction consistent with a reduction of the orbital anisotropy. This observation provides direct spectroscopic evidence for the dynamic competition between superconductivity and both SDW and electronic nematic orders in these materials.

Using realistic multi-orbital tight-binding Hamiltonians and the T-matrix formalism, we explore the effects of a non-magnetic impurity on the local density of states in Fe-based compounds. We show that scanning tunneling spectroscopy (STS) has very specific anisotropic signatures that track the evolution of orbital splitting (OS) and antiferromagnetic gaps. Both anisotropies exhibit two patterns that split in energy with decreasing temperature, but for OS these two patterns map onto each other under 90 degree rotation. STS experiments that observe these signatures should expose the underlying magnetic and orbital order as a function of temperature across various phase transitions., Comment: 12 pages, 9 figures, replacement with minor changes suggested by referees

A permanent electric dipole moment of fundamental spin-1/2 particles violates both parity (P) and time re- versal (T) symmetry, and hence, also charge-parity (CP) symmetry since there is no sign of CPT-violation. The search for a neutron electric dipole moment (nEDM) probes CP violation within and beyond the Stan- dard Model. The experiment, set up at the Paul Scherrer Institute (PSI), an improved, upgraded version of the apparatus which provided the current best experimental limit, dn < 2.9E-26 ecm (90% C.L.), by the RAL/Sussex/ILL collaboration: Baker et al., Phys. Rev. Lett. 97, 131801 (2006). In the next two years we aim to improve the sensitivity of the apparatus to sigma(dn) = 2.6E-27 ecm corresponding to an upper limit of dn < 5E-27 ecm (95% C.L.), in case for a null result. In parallel the collaboration works on the design of a new apparatus to further increase the sensitivity to sigma(dn) = 2.6E-28 ecm., Comment: APS Division for particles and fields, Conference Proceedings, Two figures

Diffraction experiments with holographic gratings recorded in SiO$_2$ nanoparticle-polymer composites have been carried out with slow neutrons. The influence of parameters such as nanoparticle concentration, grating thickness and grating spacing on the neutron-optical properties of such materials has been tested. Decay of the grating structure along the sample depth due to disturbance of the recording process becomes an issue at grating thicknesses of about 100 microns and larger. This limits the achievable diffraction efficiency for neutrons. As a solution to this problem, the Pendell\"{o}sung interference effect in holographic gratings has been exploited to reach a diffraction efficiency of 83% for very cold neutrons., Comment: 7 pages, 3 figures, submitted to Phys. Rev. A

We developed an optical device for ultra-cold neutrons and investigated the influence of a tilt of its guiding components. A measurement of the time-of-flight of the neutrons through the device by means of a dedicated chopper system was performed and a light-optical method for the alignment of the guiding components is demonstrated. A comparative analysis of former experiments with our results shows the potential of such a device to test the electrical neutrality of the free neutron on the $10^{-22} q_{\rm e}$ level and to investigate the interaction of neutrons with gravity.

We performed ultracold neutron (UCN) storage measurements to search for additional losses due to neutron (n) to mirror-neutron (n') oscillations as a function of an applied magnetic field B. In the presence of a mirror magnetic field B', UCN losses would be maximal for B = B'. We did not observe any indication for nn' oscillations and placed a lower limit on the oscillation time of tau_{nn'} > 12.0 s at 95% C.L. for any B' between 0 and 12.5 uT., Comment: 4 pages, 2 figures. Accepted for publication in Phys. Rev. D

For the first time, the neutron optical wall-potential of natural cubic boron nitride (cBN) was measured at the ultracold neutron (UCN) source of the research reactor TRIGA Mainz using the time-of-flight method (TOF). The samples investigated had a wall-potential of (305 +/- 15) neV. This value is in good agreement with the result extracted from neutron reflectometry data and theoretical expectations. Because of its high critical velocity for UCN and its good dielectric characteristics, cubic boron nitride coatings (isotopically enriched) will be useful for a number of applications in UCN experiments.

The geometric phase has been proposed as a candidate for noise resilient coherent manipulation of fragile quantum systems. Since it is determined only by the path of the quantum state, the presence of noise fluctuations affects the geometric phase in a different way than the dynamical phase. We have experimentally tested the robustness of Berry's geometric phase for spin-1/2 particles in a cyclically varying magnetic field. Using trapped polarized ultra-cold neutrons it is demonstrated that the geometric phase contributions to dephasing due to adiabatic field fluctuations vanish for long evolution times., Comment: 4 pages, 4 figures

A new measurement of the neutron EDM, using Ramsey's method of separated oscillatory fields, is in preparation at the new high intensity source of ultra-cold neutrons (UCN) at the Paul Scherrer Institute, Villigen, Switzerland (PSI). The existence of a non-zero nEDM would violate both parity and time reversal symmetry and, given the CPT theorem, might lead to a discovery of new CP violating mechanisms. Already the current upper limit for the nEDM (|d_n|<2.9E-26 e.cm) constrains some extensions of the Standard Model. The new experiment aims at a two orders of magnitude reduction of the experimental uncertainty, to be achieved mainly by (1) the higher UCN flux provided by the new PSI source, (2) better magnetic field control with improved magnetometry and (3) a double chamber configuration with opposite electric field directions. The first stage of the experiment will use an upgrade of the RAL/Sussex/ILL group's apparatus (which has produced the current best result) moved from Institut Laue-Langevin to PSI. The final accuracy will be achieved in a further step with a new spectrometer, presently in the design phase., Comment: Flavor Physics & CP Violation Conference, Taipei, 2008

We present the status of the development of a dedicated high density ultra-cold neutron (UCN) source dedicated to the gravitational spectrometer GRANIT. The source employs superthermal conversion of cold neutrons to UCN in superfluid helium. Tests have shown that UCN produced inside the liquid can be extracted into vacuum. Furthermore a dedicated neutron selection channel was tested to maintain high initial density and extract only neutrons with a vertical velocity component 20 cm/s for the spectrometer. This new source would have a phase-space density of 0.18 cm-3(m/s)-3 for the spectrometer., Comment: ISINN 15 conference proceeding Article has been replaced due to minor orthographic changes