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We prove that the Riemannian metric on a compact manifold of dimension $n\geq 3$ with smooth boundary can be uniquely determined, up to an isometry fixing the boundary, by the Dirichlet-to-Neumann map associated to the Laplace-Beltrami operator., Comment: The argument is incorrect

We consider an inverse problem of recovering a parameter appearing in all levels in a second-order hyperbolic equation from a single boundary measurement. The model is motivated from applications in photoacoustic tomography when one seeks to recover both the wave speed and the initial ultrasound pressure from a single ultrasound signal. In particular, our result shows that the ratio of the initial ultrasound pressure and the wave speed squared uniquely determines both of them respectively., Comment: 19 pages. arXiv admin note: text overlap with arXiv:2210.03865

Recently, it has been observed that the Floquet-Bloch transform with real quasiperiodicities fails to capture the spectral properties of non-reciprocal systems. The aim of this paper is to introduce the notion of a generalised Brillouin zone by allowing the quasiperiodicities to be complex in order to rectify this. It is proved that this shift of the Brillouin zone into the complex plane accounts for the unidirectional spatial decay of the eigenmodes and leads to correct spectral convergence properties. The results in this paper clarify and prove rigorously how the spectral properties of a finite structure are associated with those of the corresponding semi-infinitely or infinitely periodic lattices and give explicit characterisations of how to extend the Hermitian theory to non-reciprocal settings. Based on our theory, we characterise the generalised Brillouin zone for both open boundary conditions and periodic boundary conditions. Our results are consistent with the physical literature and give explicit generalisations to the $k$-Toeplitz matrix cases.

We study an analog of the anisotropic Calder\'on problem for fractional Schr\"odinger operators $(-\Delta_g)^\alpha + V$ with $\alpha \in (0,1)$ on closed Riemannian manifolds of dimensions two and higher. We prove that the knowledge of a Cauchy data set of solutions of the fractional Schr\"odinger equation, given on an open nonempty a priori known subset of the manifold determines both the Riemannian manifold up to an isometry and the potential up to the corresponding gauge transformation, under certain geometric assumptions on the manifold as well as the observation set. Our method of proof is based on: (i) studying a new variant of the Gel'fand inverse spectral problem without the normalization assumption on the energy of eigenfunctions, and (ii) the discovery of an entanglement principle for nonlocal equations involving two or more compactly supported functions. Our solution to (i) makes connections to antipodal sets as well as local control for eigenfunctions and quantum chaos, while (ii) requires sharp interpolation results for holomorphic functions. We believe that both of these results can find applications in other areas of inverse problems., Comment: 49 pages

A small number of pulsars are known to emit giant pulses, single pulses much brighter than average. Among these is PSR J0534+2200, also known as the Crab pulsar, a young pulsar with high giant pulse rates. Long-term monitoring of the Crab pulsar presents an excellent opportunity to perform statistical studies of its giant pulses and the processes affecting them, potentially providing insight into the behavior of other neutron stars that emit bright single pulses. Here, we present an analysis of a set of 24,985 Crab giant pulses obtained from 88 hours of daily observations at a center frequency of 1.55 GHz by the 20-meter telescope at the Green Bank Observatory, spread over 461 days. We study the effects of refractive scintillation at higher frequencies than previous studies and compare methods of correcting for this effect. We also search for deterministic patterns seen in other single-pulse sources, possible periodicities seen in several rotating radio transients and fast radio bursts, and clustering of giant pulses like that seen in the repeating fast radio burst FRB121102., Comment: 21 pages, 14 figures, accepted to ApJ

We study partial data inverse problems for linear and nonlinear parabolic equations with unknown time-dependent coefficients. In particular, we prove uniqueness results for partial data inverse problems for semilinear reaction-diffusion equations where Dirichlet boundary data and Neumann measurements of solutions are restricted to any open subset of the boundary. We also prove injectivity of the Fr\'{e}chet derivative of the partial Dirichlet-to-Neumann map associated to heat equations. Our proof consists of two crucial ingredients; (i) we introduce an asymptotic family of spherical quasimodes that approximately solve heat equations modulo an exponentially decaying remainder term and (ii) the asymptotic study of a weighted Laplace transform of the unknown coefficient along a straight line segment in the domain where the weight may be viewed as a semiclassical symbol that itself depends on the complex-valued frequency. The latter analysis will rely on Phragm\'{e}n-Lindel\"{o}f principle and Gr\"{o}nwall inequality., Comment: 39 pages. Comments are welcome!

The aim of this paper is to illustrate both analytically and numerically the interplay of two fundamentally distinct non-Hermitian mechanisms in a deep subwavelength regime. Considering a parity-time symmetric system of one-dimensional subwavelength resonators equipped with two kinds of non-Hermiticity - an imaginary gauge potential and on-site gain and loss - we prove that all but two eigenmodes of the system decouple when going through an exceptional point. By tuning the gain-to-loss ratio, the system changes from a phase with unbroken parity-time symmetry to a phase with broken parity-time symmetry. At the macroscopic level, this is observed as a transition from symmetrical eigenmodes to condensated eigenmodes at one edge of the structure. Mathematically, it arises from a topological state change. The results of this paper open the door to the justification of a variety of phenomena arising from the interplay between non-Hermitian reciprocal and non-reciprocal mechanisms not only in subwavelength wave physics but also in quantum mechanics where the tight binding model coupled with the nearest neighbour approximation can be analysed with the same tools as those developed here.

With our work, we contribute towards a qualitative analysis of the discourse on controversies in online news media. For this, we employ Formal Concept Analysis and the economics of conventions to derive conceptual controversy maps. In our experiments, we analyze two maps from different news journals with methods from ordinal data science. We show how these methods can be used to assess the diversity, complexity and potential bias of controversies. In addition to that, we discuss how the diagrams of concept lattices can be used to navigate between news articles.

One of the most common misconceptions made about the Kalman filter when applied to linear systems is that it requires an assumption that all error and noise processes are Gaussian. This misconception has frequently led to the Kalman filter being dismissed in favor of complicated and/or purely heuristic approaches that are supposedly "more general" in that they can be applied to problems involving non-Gaussian noise. The fact is that the Kalman filter provides rigorous and optimal performance guarantees that do not rely on any distribution assumptions beyond mean and error covariance information. These guarantees even apply to use of the Kalman update formula when applied with nonlinear models, as long as its other required assumptions are satisfied. Here we discuss misconceptions about its generality that are often found and reinforced in the literature, especially outside the traditional fields of estimation and control.

In this paper we argue that conventional unitary-invariant measures of recommender system (RS) performance based on measuring differences between predicted ratings and actual user ratings fail to assess fundamental RS properties. More specifically, posing the optimization problem as one of predicting exact user ratings provides only an indirect suboptimal approximation for what RS applications typically need, which is an ability to accurately predict user preferences. We argue that scalar measures such as RMSE and MAE with respect to differences between actual and predicted ratings are only proxies for measuring RS ability to accurately estimate user preferences. We propose what we consider to be a measure that is more fundamentally appropriate for assessing RS performance, rank-preference consistency, which simply counts the number of prediction pairs that are inconsistent with the user's expressed product preferences. For example, if an RS predicts the user will prefer product A over product B, but the user's withheld ratings indicate s/he prefers product B over A, then rank-preference consistency has been violated. Our test results conclusively demonstrate that methods tailored to optimize arbitrary measures such as RMSE are not generally effective at accurately predicting user preferences. Thus, we conclude that conventional methods used for assessing RS performance are arbitrary and misleading.

Soft targets combined with the cross-entropy loss have shown to improve generalization performance of deep neural networks on supervised classification tasks. The standard cross-entropy loss however assumes data to be categorically distributed, which may often not be the case in practice. In contrast, InfoNCE does not rely on such an explicit assumption but instead implicitly estimates the true conditional through negative sampling. Unfortunately, it cannot be combined with soft targets in its standard formulation, hindering its use in combination with sophisticated training strategies. In this paper, we address this limitation by proposing a loss function that is compatible with probabilistic targets. Our new soft target InfoNCE loss is conceptually simple, efficient to compute, and can be motivated through the framework of noise contrastive estimation. Using a toy example, we demonstrate shortcomings of the categorical distribution assumption of cross-entropy, and discuss implications of sampling from soft distributions. We observe that soft target InfoNCE performs on par with strong soft target cross-entropy baselines and outperforms hard target NLL and InfoNCE losses on popular benchmarks, including ImageNet. Finally, we provide a simple implementation of our loss, geared towards supervised classification and fully compatible with deep classification models trained with cross-entropy.

This report evaluates the efficiency of Graph Edit Distance (GED) computation for graph similarity search, comparing Cascading Metric Trees (CMT) with brute-force verification. Despite the anticipated advantages of CMT, our findings indicate it does not consistently outperform brute-force methods in speed. The study, based on graph data from PubChem, suggests that the computational complexity of GED-based GSS remains a challenge.

Consider a compact Riemannian manifold in dimension $n$ with strictly convex boundary. We show the local invertibility near a boundary point of the transverse ray transform of $2$ tensors for $n\geq 3$ and the mixed ray transform of $2+2$ tensors for $n=3$. When the manifold admits a strictly convex function, this local invertibility result leads to global invertibility.

We study inverse boundary problems for third-order nonlinear tensorial perturbations of biharmonic operators on a bounded domain in $\mathbb{R}^n$, where $n\geq 3$. By imposing appropriate assumptions on the nonlinearity, we demonstrate that the Dirichlet-to-Neumann map, known on the boundary of the domain, uniquely determines the genuinely nonlinear tensorial third-order perturbations of the biharmonic operator. The proof relies on the inversion of certain generalized momentum ray transforms on symmetric tensor fields. Notably, the corresponding inverse boundary problem for linear tensorial third-order perturbations of the biharmonic operator remains an open question.

Artificial Intelligence methods are powerful tools for biological image analysis and processing. High-quality annotated images are key to training and developing new methods, but access to such data is often hindered by the lack of standards for sharing datasets. We brought together community experts in a workshop to develop guidelines to improve the reuse of bioimages and annotations for AI applications. These include standards on data formats, metadata, data presentation and sharing, and incentives to generate new datasets. We are positive that the MIFA (Metadata, Incentives, Formats, and Accessibility) recommendations will accelerate the development of AI tools for bioimage analysis by facilitating access to high quality training data., Comment: 16 pages, 3 figures

We conducted direct numerical simulations of turbulent open channel flow (OCF) and closed channel flow (CCF) of friction Reynolds numbers up to $\mathrm{Re}_\tau \approx 900$ in large computational domains up to $L_x\times L_z=12\pi h \times 4\pi h$ to analyse the Reynolds number scaling of turbulence intensities. Unlike CCF, our data suggests that the streamwise turbulence intensity in OCF scales with the bulk velocity for $\mathrm{Re}_\tau \gtrsim 400$. The additional streamwise kinetic energy in OCF with respect to CCF is provided by larger and more intense very-large-scale motions in the former type of flow. Therefore, compared to CCF, larger computational domains of $L_x\times L_z=12\pi h\times 4\pi h$ are required to faithfully capture very-large-scale motions in OCF -- and observe the reported scaling. OCF and CCF turbulence statistics data sets are available at https://doi.org/10.4121/88678f02-2a34-4452-8534-6361fc34d06b ., Comment: Submitted to Progress in Turbulence X: Proceedings of the iTi Conference on Turbulence 2023. Data: https://doi.org/10.4121/88678f02-2a34-4452-8534-6361fc34d06b

We propose a general method for optimally approximating an arbitrary matrix $\mathbf{M}$ by a structured matrix $\mathbf{T}$ (circulant, Toeplitz/Hankel, etc.) and examine its use for estimating the spectra of genomic linkage disequilibrium matrices. This application is prototypical of a variety of genomic and proteomic problems that demand robustness to incomplete biosequence information. We perform a simulation study and corroborative test of our method using real genomic data from the Mouse Genome Database. The results confirm the predicted utility of the method and provide strong evidence of its potential value to a wide range of bioinformatics applications. Our optimal general matrix approximation method is expected to be of independent interest to an even broader range of applications in applied mathematics and engineering.

We critique the formulation of Arrow's no-dictator condition to show that it does not correspond to the accepted informal/intuitive interpretation. This has implications for the theorem's scope of applicability.

In this note we observe that automated theorem provers (ATPs) that recursively enumerate theorems in a particular way will fail to identify some valid theorems for a reason that is analogous to how G\"odel proved the existence of what are now referred to as G\"odel statements. This observation has no significant practical or theoretical implications, but it may be of pedagogical value for honing the intuition of students about recursive enumeration in the context of ATP., Comment: Converted to a pedagogical focus

In this paper we report on new results relating to a conjecture regarding properties of $n\times n$, $n\leq 6$, positive definite matrices. The conjecture has been proven for $n\leq 4$ using computer-assisted sum of squares (SoS) methods for proving polynomial nonnegativity. Based on these proven cases, we report on the recent identification of a new family of matrices with the property that their diagonals majorize their spectrum. We then present new results showing that this family can extended via Kronecker composition to $n>6$ while retaining the special majorization property. We conclude with general considerations on the future of computer-assisted and AI-based proofs.

This is a continuation of our study [Uhlmann-Zhai, JMPA, 2021] on an inverse boundary value problem for a nonlinear elastic wave equation. We prove that all the linear and nonlinear coefficients can be recovered from the displacement-to-traction map, including the density, under some natural geometric conditions on the wavespeeds.

In this article, we investigate the Calder\'on problem for nonlocal parabolic equations, where we are interested to recover the leading coefficient of nonlocal parabolic operators. The main contribution is that we can relate both (anisotropic) variable coefficients local and nonlocal Calder\'on problem for parabolic equations. More concretely, we show that the (partial) Dirichlet-to-Neumann map for the nonlocal parabolic equation determines the (full) Dirichlet-to-Neumann map for the local parabolic equation. This article extends our earlier results [LLU22] by using completely different methods. Moreover, the results hold for any spatial dimension $n\geq 2$., Comment: 46 pages. All comments are welcome

In this paper, we review and compare several velocity-level and acceleration-level Pseudo-Inverse-based Path Planning (PPP) and Pseudo-Inverse-based Repetitive Motion Planning (PRMP) schemes based on the kinematic model of robotic manipulators. We show that without unit consistency in the pseudo-inverse computation, path planning of incommensurate robotic manipulators will fail. Also, we investigated the robustness and noise tolerance of six PPP and PRMP schemes in the literature against various noise types (i.e. zero, constant, time-varying and random noises). We compared the simulated results using two redundant robotic manipulators: a 3DoF (2RP), and a 7DoF (2RP4R). These experimental results demonstrate that the improper Generalized Inverse (GI) with arbitrary selection of unit and/or in the presence of noise can lead to unexpected behavior of the robot, while producing wrong instantaneous outputs in the task space, which results in distortions and/or failures in the execution of the planned path. Finally, we propose and demonstrate the efficacy of the Mixed Inverse (MX) as the proper GI to achieve unit-consistency in path planning.

Numerical methods for Inverse Kinematics (IK) employ iterative, linear approximations of the IK until the end-effector is brought from its initial pose to the desired final pose. These methods require the computation of the Jacobian of the Forward Kinematics (FK) and its inverse in the linear approximation of the IK. Despite all the successful implementations reported in the literature, Jacobian-based IK methods can still fail to preserve certain useful properties if an improper matrix inverse, e.g. Moore-Penrose (MP), is employed for incommensurate robotic systems. In this paper, we propose a systematic, robust and accurate numerical solution for the IK problem using the Mixed (MX) Generalized Inverse (GI) applied to any type of Jacobians (e.g., analytical, numerical or geometric) derived for any commensurate and incommensurate robot. This approach is robust to whether the system is under-determined (less than 6 DoF) or over-determined (more than 6 DoF). We investigate six robotics manipulators with various Degrees of Freedom (DoF) to demonstrate that commonly used GI's fail to guarantee the same system behaviors when the units are varied for incommensurate robotics manipulators. In addition, we evaluate the proposed methodology as a global IK solver and compare against well-known IK methods for redundant manipulators. Based on the experimental results, we conclude that the right choice of GI is crucial in preserving certain properties of the system (i.e. unit-consistency).

In this paper, we propose a new constraint, called shift-consistency, for solving matrix/tensor completion problems in the context of recommender systems. Our method provably guarantees several key mathematical properties: (1) satisfies a recently established admissibility criterion for recommender systems; (2) satisfies a definition of fairness that eliminates a specific class of potential opportunities for users to maliciously influence system recommendations; and (3) offers robustness by exploiting provable uniqueness of missing-value imputation. We provide a rigorous mathematical description of the method, including its generalization from matrix to tensor form to permit representation and exploitation of complex structural relationships among sets of user and product attributes. We argue that our analysis suggests a structured means for defining latent-space projections that can permit provable performance properties to be established for machine learning methods., Comment: arXiv admin note: text overlap with arXiv:2204.01815

In this paper we discuss pre- and post-processing methods to induce desired consistency and/or invariance properties in blackbox systems, e.g., AI-based. We demonstrate our approach in the context of blackbox SVD-based matrix-completion methods commonly used in recommender system (RS) applications. We provide empirical results showing that enforcement of unit-consistency and shift-consistency, which have provable RS-relevant properties relating to robustness and fairness, also lead to improved performance according to generic RMSE and MAE performance metrics, irrespective of the initial chosen hyperparameter.

A computational framework is presented to numerically simulate the effects of antihypertensive drugs, in particular calcium channel blockers, on the mechanical response of arterial walls. A stretch-dependent smooth muscle model by Uhlmann and Balzani is modified to describe the interaction of pharmacological drugs and the inhibition of smooth muscle activation. The coupled deformation-diffusion problem is then solved using the finite element software FEDDLib and overlapping Schwarz preconditioners from the Trilinos package FROSch. These preconditioners include highly scalable parallel GDSW (generalized Dryja-Smith-Widlund) and RDSW (reduced GDSW) preconditioners. Simulation results show the expected increase in the lumen diameter of an idealized artery due to the drug-induced reduction of smooth muscle contraction, as well as a decrease in the rate of arterial contraction in the presence of calcium channel blockers. Strong and weak parallel scalability of the resulting computational implementation are also analyzed.

We present a simple modification of the direct-forcing immersed boundary method (IBM) proposed by Uhlmann [J. Comput. Phys, 2005] in order to enable it to be applied to particulate flows with solid-to-fluid density ratios around unity. The main difference with respect to the original formulation lies in the particle velocity update which is performed directly after the preliminary velocity field has been computed in the absence of any IBM volume forcing term. In addition, we apply the forcing term to the entire space occupied by the immersed solid object (instead of to the vicinity of its interface only). The present approach requires the evaluation of integrals of the velocity field over the volume occupied by the solid particle, which are evaluated efficiently as sums over the respective quantities available at particle-attached force points. The resulting method can be used seamlessly for density ratios down to $\rho_p/\rho_f>0.5$. The new formulation has been validated using three configurations: (i) lateral migration of a neutrally buoyant circular particle in two-dimensional Couette flow; (ii) the release from rest of a neutrally buoyant sphere in a free stream; (iii) the release of a particle in a free stream after an initial phase in which it is translationally fixed with an imposed angular velocity. In all three test cases the present IBM formulation yields a very good agreement with the available reference data. Thus, the proposed approach is a cost-efficient and accurate modification of the original method which allows for the simulation of fluid systems involving density-matched solid particles., Comment: accepted for publication in Computers & Fluids

We relate the (anisotropic) variable coefficient local and nonlocal Calder\'on problems by means of the Caffarelli-Silvestre extension. In particular, we prove that (partial) Dirichlet-to-Neumann data for the fractional Calder\'on problem in three and higher dimensions determine the (full) Dirichlet-to-Neumann data for the local Calder\'on problem. As a consequence, any (variable coefficient) uniqueness result for the local problem also implies a uniqueness result for the nonlocal problem. Moreover, our approach is constructive and associated Tikhonov regularization schemes can be used to recover the data. Finally, we highlight obstructions for reversing this procedure, which essentially consist of two one-dimensional averaging processes., Comment: 34 pages, comments welcome, corrected some typos

In this paper we derive and analyze an algorithm for inverting quaternion matrices. The algorithm is an analogue of the Frobenius algorithm for the complex matrix inversion. On the theory side, we prove that our algorithm is more efficient that other existing methods. Moreover, our algorithm is optimal in the sense of the least number of complex inversions. On the practice side, our algorithm outperforms existing algorithms on randomly generated matrices. We argue that this algorithm can be used to improve the practical utility of recursive Strassen-type algorithms by providing the fastest possible base case for the recursive decomposition process when applied to quaternion matrices.

We have performed particle-resolved direct numerical simulations of many heavy non-spherical particles settling under gravity in the dilute regime. The particles are oblate spheroids of aspect ratio 1.5 and density ratio 1.5. Two Galileo numbers are considered, namely 111 and 152, for which a single oblate spheroid follows a steady vertical and a steady oblique path, respectively. In both cases, a strongly inhomogeneous spatial distribution of the disperse phase in the form of columnar clusters is observed, with a significantly enhanced average settling velocity as a consequence. Thus, in contrast to previous results for spheres, the qualitative difference in the single particle regime does not result in a qualitatively different behavior of the many-particle cases. In addition, we have carried out an analysis of pairwise interactions of particles in the well-known drafting-kissing-tumbling setup, for oblate spheroids of aspect ratio 1.5 and for spheres. We have varied systematically the relative initial position between the particle pair and we have considered free-to-rotate particles and rotationally-locked ones. We have found that the region of attraction for both particle shapes, with and without rotation, is very similar. However, significant differences occur during the drafting and tumbling phases. In particular, free-to-rotate spheres present longer drafting phases and separate quickly after the collision. Spheroids remain close to each other for longer times after the collision, and free-to-rotate ones experience two or more collision events. Therefore, we have observed a shape-induced increase in the interaction which might explain the increased tendency to cluster of the many-particle cases., Comment: 40 pages, 24 figures

We prove that the Dirichlet-to-Neumann map of the linear wave equation determines the topological, differentiable and conformal structure of the underlying Lorentzian manifold, under mild technical assumptions. With more stringent geometric assumptions, the full Lorentzian structure of the manifold can be recovered as well. The key idea of the proof is to show that the singular support of the Schwartz kernel of the Dirichlet-to-Neumann map of a manifold completely determines the so-called boundary light observation set of the manifold together with its natural causal structure., Comment: There is an error in the proof