Search

Cutting mechanics of soft solids is gaining rapid attention thanks to its promising benefits in material characterization and other applications. However, a full understanding of the physical phenomena is still missing, and several questions remain outstanding. E.g.: How can we directly and reliably measure toughness from cutting experiments? What is the role of blade sharpness? In this paper, we explore the simple problem of wire cutting, where blade sharpness is only defined by the wire radius. Through finite element analysis, we obtain a simple scaling relation between the wire radius and the steady-state cutting force per unit sample thickness. The cutting force is independent of the wire radius if the latter is below a transition length, while larger radii produce a linear force-radius correlation. The minimum cutting force, for small radii, is given by cleavage toughness, i.e., the surface energy required to break covalent bonds in the crack plane. The force-radius slope is instead given by the wear shear strength in the material. Via cutting experiments on polyacrylamide gels, we find that the magnitude of shear strength is close to the work of fracture of the material, i.e., the critical strain energy density required to break a pristine sample in uniaxial tension. The work of fracture characterizes the toughening contribution from the fracture process zone (FPZ), which adds to cleavage toughness. Our study provides two important messages, that answer the above questions: Toughness can be estimated from wire-cutting experiments from the intercept of the force-radius linear correlation, as previously explored. However, as we discovered, this only estimates cleavage toughness. Additionally, the force-radius slope is correlated with the work of fracture, giving an estimation of the dissipative contributions from the FPZ.

We present a simple universal algorithm for high-dimensional integration which has the optimal error rate (independent of the dimension) in all weighted Korobov classes both in the randomized and the deterministic setting. Our theoretical findings are complemented by numerical tests., Comment: 18 pages. MATLAB code for numerical tests is attached

In this paper, we present some new (in-)tractability results related to the integration problem in subspaces of the Wiener algebra over the $d$-dimensional unit cube. We show that intractability holds for multivariate integration in the standard Wiener algebra in the deterministic setting, in contrast to polynomial tractability in an unweighted subspace of the Wiener algebra recently shown by Goda (2023). Moreover, we prove that multivariate integration in the subspace of the Wiener algebra introduced by Goda is strongly polynomially tractable if we switch to the randomized setting. We also identify subspaces in which multivariate integration in the deterministic setting are (strongly) polynomially tractable and we compare these results with the bound which can be obtained via Hoeffding's inequality., Comment: 16 pages

We propose a randomized lattice algorithm for approximating multivariate periodic functions over the $d$-dimensional unit cube from the weighted Korobov space with mixed smoothness $\alpha > 1/2$ and product weights $\gamma_1,\gamma_2,\ldots\in [0,1]$. Building upon the deterministic lattice algorithm by Kuo, Sloan, and Wo\'{z}niakowski (2006), we incorporate a randomized quadrature rule by Dick, Goda, and Suzuki (2022) to accelerate the convergence rate. This randomization involves drawing the number of points for function evaluations randomly, and selecting a good generating vector for rank-1 lattice points using the randomized component-by-component algorithm. We prove that our randomized algorithm achieves a worst-case root mean squared $L_2$-approximation error of order $M^{-\alpha/2 - 1/8 + \varepsilon}$ for an arbitrarily small $\varepsilon > 0$, where $M$ denotes the maximum number of function evaluations, and that the error bound is independent of the dimension $d$ if the weights satisfy $\sum_{j=1}^\infty \gamma_j^{1/\alpha} < \infty$. Our upper bound converges faster than a lower bound on the worst-case $L_2$-approximation error for deterministic rank-1 lattice-based approximation proved by Byrenheid, K\"{a}mmerer, Ullrich, and Volkmer (2017). We also show a lower error bound of order $M^{-\alpha/2-1/2}$ for our randomized algorithm, leaving a slight gap between the upper and lower bounds open for future research., Comment: 22 pages

Matching users with mutual preferences is a critical aspect of services driven by reciprocal recommendations, such as job search. To produce recommendations in such scenarios, one can predict match probabilities and construct rankings based on these predictions. However, this direct match prediction approach often underperforms due to the extreme sparsity of match labels. Therefore, most existing methods predict preferences separately for each direction (e.g., job seeker to employer and employer to job seeker) and then aggregate the predictions to generate overall matching scores and produce recommendations. However, this typical approach often leads to practical issues, such as biased error propagation between the two models. This paper introduces and demonstrates a novel and practical solution to improve reciprocal recommendations in production by leveraging pseudo-match scores. Specifically, our approach generates dense and more directly relevant pseudo-match scores by combining the true match labels, which are accurate but sparse, with relatively inaccurate but dense match predictions. We then train a meta-model to output the final match predictions by minimizing the prediction loss against the pseudo-match scores. Our method can be seen as a best-of-both (BoB) approach, as it combines the high-level ideas of both direct match prediction and the two separate models approach. It also allows for user-specific weights to construct personalized pseudo-match scores, achieving even better matching performance through appropriate tuning of the weights. Offline experiments on real-world job search data demonstrate the superior performance of our BoB method, particularly with personalized pseudo-match scores, compared to existing approaches in terms of finding potential matches.

Cutting mechanics in soft solids present a complex mechanical challenge due to the intricate behavior of soft ductile materials as they undergo crack nucleation and propagation. Recent research has explored the relationship between the cutting force needed to continuously cut a soft material and the radius of the wire (blade). A typical simplifying assumption is that of material incompressibility, albeit no material in nature is really incompressible. In this study, we relax this assumption and examine how material (in)compressibility influences the correlation between cutting forces and material properties like toughness and modulus. The ratio {\mu}/\k{appa}, where {\mu} and \k{appa} are the shear and bulk moduli, indicates the material's degree of compressibility, where incompressible materials have {\mu}/\k{appa}=0, and larger {\mu}/\k{appa} provide higher volumetric compressibility. Following previous observations, we obtain two cutting regimes: (i) high toughness or small wire radius, and (ii) low toughness or large wire radius. Regime (i) is dominated by frictional dissipation, while regime (ii) is dominated by adhesive debonding and/or the wear resistance of the material. These regimes are controlled by the ratio between the wire radius and the elasto-cohesive length of the material: the critical crack opening displacement at fast fracture under uniaxial tension. In the large radius, regime (ii), our theoretical findings reveal that incompressible materials require larger forces. Notably, however, the elasto-cohesive length of the material, defining the transition wire radius between regimes (i) and (ii), is larger for compressible materials, which are therefore more likely to be cut in regime (i), and thus requiring larger cutting forces.

Phonocardiography has recently gained popularity in low-cost and remote monitoring, including passive fetal heart monitoring. Development for methods which analyse phonocardiographical data try to capitalize on this opportunity, and in recent years a multitude of such algorithms and models have been published. Although there is little to no standardization in these published algorithms and multiple parts of these models have to be reimplemented on a case-by-case basis. Datasets containing heart sound recordings also lack standardization in both data storage and labeling, especially in fetal phonocardiography. We are presenting a toolbox that can serve as a basis for a future standard framework for heart sound analysis. This toolbox contains some of the most widely used processing steps, and with these, complex analysis processes can be created. These functions can be individually tested. Due to the interdependence of the steps, we validated the current segmentation stage using a manually labeled fetal phonocardiogram dataset comprising 50 one-minute abdominal PCG recordings, which include 6,758 S1 and 6,729 S2 labels. Our results were compared to other common and available segmentation methods, peak detection with the Neurokit2 library, and the Hidden Semi-Markov Model by Springer et al. With a 30 ms tolerance our best model achieved a 97.1% F1 score and 10.8 +/- 7.9 ms mean absolute error for S1 detection. This detection accuracy outperformed all tested methods. With this a more accurate S2 detection method can be created as a multi-step process. After an accurate segmentation the extracted features should be representative of the selected segments, which allows for more accurate statistics or classification models. The toolbox contains functions for both feature extraction and statistics creation which are compatible with the previous steps., Comment: 25 pages, 11 figures, submitted to Open Source and Validated Computational Tools for Physiological Time Series Analysis, for associated program documentation, see https://pypcg-toolbox.readthedocs.io/en/latest/

We study the multivariate integration problem for periodic functions from the weighted Korobov space in the randomized setting. We introduce a new randomized rank-1 lattice rule with a randomly chosen number of points, which avoids the need for component-by-component construction in the search for good generating vectors while still achieving nearly the optimal rate of the randomized error. Our idea is to exploit the fact that at least half of the possible generating vectors yield nearly the optimal rate of the worst-case error in the deterministic setting. By randomly choosing generating vectors $r$ times and comparing their corresponding worst-case errors, one can find one generating vector with a desired worst-case error bound with a very high probability, and the (small) failure probability can be controlled by increasing $r$ logarithmically as a function of the number of points. Numerical experiments are conducted to support our theoretical findings., Comment: revision, 22 pages, 3 figures

Numerical integration over the real line for analytic functions is studied. Our main focus is on the sharpness of the error bounds. We first derive two general lower estimates for the worst-case integration error, and then apply these to establish lower bounds for various quadrature rules. These bounds turn out to be either novel or improve upon existing results, leading to lower bounds that closely match upper bounds for various formulas. Specifically, for the suitably truncated trapezoidal rule, we improve upon general lower bounds on the worst-case error obtained by Sugihara [\textit{Numer. Math.}, 75 (1997), pp.~379--395] and provide exceptionally sharp lower bounds apart from a polynomial factor, and in particular show that the worst-case error for the trapezoidal rule by Sugihara is not improvable by more than a polynomial factor. Additionally, our research reveals a discrepancy between the error decay of the trapezoidal rule and Sugihara's lower bound for general numerical integration rules, introducing a new open problem. Moreover, Gauss--Hermite quadrature is proven sub-optimal under the decay conditions on integrands we consider, a result not deducible from upper-bound arguments alone. Furthermore, to establish the near-optimality of the suitably scaled Gauss--Legendre and Clenshaw--Curtis quadratures, we generalize a recent result of Trefethen [\textit{SIAM Rev.}, 64 (2022), pp.~132--150] for the upper error bounds in terms of the decay conditions., Comment: 23 pages, no figure

Are common quasi-Monte Carlo sequences quasi-uniform? While this question remains widely open, in this short note, we prove that the two-dimensional Sobol' sequence is not quasi-uniform. This result partially answers an unsolved problem of Sobol' and Shukhman (2007) in a negative manner., Comment: 5 pages