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arXiv 2019年

作者： Kim, J. Torquato, S. Department of Physics Princeton University PrincetonNJ08544 United States Department of Chemistry Princeton University PrincetonNJ08544 United States Princeton Institute for the Science and Technology of Materials Princeton University PrincetonNJ08544 United States Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

Disordered hyperuniform dispersions are exotic amorphous two-phase materials characterized by an anomalous suppression of long-wavelength volume-fraction fluctuations, which endows them with novel physical properties. While such unusual materials have received considerable attention, a stumbling block has been an inability to create large samples that are truly hyperuniform due to current computational and experimental limitations. To overcome such barriers, we introduce a new and simple construction procedure that guarantees perfect hyperuniformity for very large sample sizes. This methodology involves tessellating space into cells and then inserting a particle into each cell such that the local-cell particle packing fractions are identical to the global packing fraction. We analytically prove that such dispersions are perfectly hyperuniform in the infinite-sample-size limit. Our methodology enables a remarkable mapping that converts a very large nonhyperuniform disordered dispersion into a perfectly hyperuniform one, which we numerically demonstrate in two and three dimensions. A similar analysis also establishes the hyperuniformity of the famous Hashin-Shtrikman multiscale dispersions, which possess optimal transport and elastic properties. Our hyperuniform designs can be readily fabricated using modern photolithographic and 3D printing technologies. The exploration of the enormous class of hyperuniform dispersions that can be designed and tuned by our tessellation-based methodology paves the way for accelerating the discovery of novel hyperuniform materials. Copyright © 2019, The Authors. All rights reserved.

关键词： 3D printers

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An analysis of reconstruction noise from undersampled 4D flow MRI

arXiv

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arXiv 2022年

作者： Partin, Lauren Schiavazzi, Daniele E. Long, Carlos A. Sing Department Of Applied And Computational Mathematics And Statistics University Of Notre Dame Notre DameIN United States Institute For Mathematical And Computational Engineering Institute For Biological And Medical Engineering Pontificia Universidad Católica De Chile Santiago Chile ANID - Millennium Science Initiative Program Millennium Nucleus Center For The Discovery Of Structures In Complex Data Chile ANID - Millennium Science Initiative Program Millennium Nucleus Center For Cardiovascular Magnetic Resonance Chile

Novel Magnetic Resonance (MR) imaging modalities can quantify hemodynamics but require long acquisition times, precluding its widespread use for early diagnosis of cardiovascular disease. To reduce the acquisition times, reconstruction methods from undersampled measurements are routinely used, that leverage representations designed to increase image compressibility. Reconstructed anatomical and hemodynamic images may present visual artifacts. Although some of these artifact are essentially reconstruction errors, and thus a consequence of undersampling, others may be due to measurement noise or the random choice of the sampled frequencies. Said otherwise, a reconstructed image becomes a random variable, and both its bias and its covariance can lead to visual artifacts;the latter leads to spatial correlations that may be misconstrued for visual information. Although the nature of the former has been studied in the literature, the latter has not received as much attention. In this study, we investigate the theoretical properties of the random perturbations arising from the reconstruction process, and perform a number of numerical experiments on simulated and MR aortic flow. Our results show that the correlation length remains limited to two to three pixels when a Gaussian undersampling pattern is combined with recovery algorithms based on 1-norm minimization. However, the correlation length may increase significantly for other undersampling patterns, higher undersampling factors (i.e., 8× or 16× compression), and different reconstruction methods. © 2022, CC BY-NC-ND.

关键词： Magnetic resonance

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Bilateral discrete and continuous orthogonality relations in the q−1-symmetric Askey scheme

arXiv

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arXiv 2024年

作者： Cohl, Howard S. Volkmer, Hans Applied and Computational Mathematics Division National Institute of Standards and Technology GaithersburgMD20899-8910 United States Department of Mathematical Sciences University of Wisconsin–Milwaukee PO Box 413 MilwaukeeWI53201 United States

In the q−1-symmetric Askey scheme, namely the q−1-Askey–Wilson, continuous dual q−1-Hahn, q−1-Al-Salam–Chihara, continuous big q−1-Hermite and continuous q−1Hermite polynomials, we compute bilateral discrete and continuous orthogonality relations. We also derive a q-beta integral which comes from the continuous orthogonality relation for the q−1-Askey–Wilson polynomials. In the q → 1− limit, this q-beta integral corresponds to a beta integral of Ramanujan-type which we present and provide two proofs *** Codes 33D45, 05A15, 42C05, 05E05, 33D15 © 2024, CC0.

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Nonlocal effective electromagnetic wave characteristics of composite media: Beyond the quasistatic regime

arXiv

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arXiv 2020年

作者： Torquato, Salvatore Kim, Jaeuk Department of Physics Princeton University PrincetonNJ08544 United States Department of Chemistry Princeton University PrincetonNJ08544 United States Princeton Institute Science and Technology of Materials Princeton University PrincetonNJ08544 United States Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

We derive exact nonlocal homogenized constitutive relations for the effective electromagnetic wave properties of disordered two-phase composites and metamaterials from first principles. This exact formalism enables us to extend the long-wavelength limitations of conventional homogenization estimates of the effective dynamic dielectric constant tensor Εe(kI, ω) for arbitrary microstructures so that it can capture spatial dispersion well beyond the quasistatic regime (where ω and kI are the frequency and wave vector of the incident radiation). We accomplish this task by deriving nonlocal strong-contrast expansions that exactly account for complete microstructural information (infinite set of n-point correlation functions) and hence multiple scattering to all orders for the range of wave numbers for which our extended homogenization theory applies, i.e., 0 ≤ |kI|. 1 (where is a characteristic heterogeneity length scale). Because of the fast-convergence properties of such expansions, their lower-order truncations yield accurate closed-form approximate formulas for Εe(kI, ω) that apply for a wide class of microstructures. These nonlocal formulas are resummed representations of the strong-contrast expansions that still accurately capture multiple scattering to all orders via the microstructural information embodied in the spectral density, which is easy to compute for any composite. The accuracy of these microstructure-dependent approximations is validated by comparison to full-waveform simulation computations for both 2D and 3D ordered and disordered models of composite media. Thus, our closed-form formulas enable one to predict accurately and efficiently the effective wave characteristics well beyond the quasistatic regime for a wide class of composite microstructures without having to perform full-blown simulations. We find that disordered hyperuniform media are generally less lossy than their nonhyperuniform counterparts. We also show that certain disordered hyperunif

关键词： Microstructure

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Impact of time-correlated noise on zero-noise extrapolation

引用

Physical Review A 2022年第5期106卷 052406-052406页

作者： Kevin Schultz Ryan LaRose Andrea Mari Gregory Quiroz Nathan Shammah B. David Clader William J. Zeng Johns Hopkins University Applied Physics Laboratory Laurel Maryland 20723 USA Department of Computational Mathematics Science and Engineering Michigan State University East Lansing Michigan 48823 USA Unitary Fund Walnut California 91789 USA Goldman Sachs & Co New York New York 10282 USA

Zero-noise extrapolation is a quantum error mitigation technique that has typically been studied under the ideal approximation that the noise acting on a quantum device is not time correlated. In this paper, we investigate the feasibility and performance of zero-noise extrapolation in the presence of time-correlated noise. We show that, in contrast to white noise, time-correlated noise is harder to mitigate via zero-noise extrapolation because it is difficult to scale the noise level without also modifying its spectral distribution. This limitation is particularly strong if “local” gate-level methods are applied for noise scaling. However, we find that “global” noise-scaling methods, e.g., global unitary folding, can be sufficiently reliable even in the presence of time-correlated noise.

关键词： Quantum algorithms Quantum benchmarking Quantum control

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Random scalar fields and hyperuniformity

arXiv

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arXiv 2017年

作者： Ma, Zheng Torquato, Salvatore Department of Physics Princeton University PrincetonNJ08544 United States Department of Chemistry Princeton University PrincetonNJ08544 United States Princeton Institute for the Science and Technology of Materials Princeton University PrincetonNJ08544 United States Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

Disordered many-particle hyperuniform systems are exotic amorphous states of matter that lie between crystals and liquids. Hyperuniform systems have attracted recent attention because they are endowed with novel transport and optical properties. Recently, the hyperuniformity concept has been generalized to characterize two-phase media, scalar fields and random vector fields. In this paper, we devise methods to explicitly construct hyperuniform scalar fields. Specifically, we analyze spatial patterns generated from Gaussian random fields, which have been used to model the microwave background radiation and heterogeneous materials, the Cahn-Hilliard equation for spinodal decomposition, and Swift-Hohenberg equations that have been used to model emergent pattern formation, including Rayleigh-Bénard convection. We show that the Gaussian random scalar fields can be constructed to be hyperuniform. We also numerically study the time evolution of spinodal decomposition patterns and demonstrate that they are hyperuniform in the scaling regime. Moreover, we find that labyrinth-like patterns generated by the Swift-Hohenberg equation are effectively hyperuniform. We show that thresholding (level-cutting) a hyperuniform Gaussian random field to produce a two-phase random medium tends to destroy the hyperuniformity of the progenitor scalar field. We then propose guidelines to achieve effectively hyperuniform two-phase media derived from thresholded non-Gaussian fields. Our investigation paves the way for new research directions to characterize the large-structure spatial patterns that arise in physics, chemistry, biology and ecology. Moreover, our theoretical results are expected to guide experimentalists to synthesize new classes of hyperuniform materials with novel physical properties via coarsening processes and using state-of-the-art techniques, such as stereolithography and 3D printing. Copyright © 2017, The Authors. All rights reserved.

关键词： Gaussian distribution

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Characterization of void space, large-scale structure, and transport properties of maximally random jammed packings of superballs

arXiv

引用

arXiv 2021年

作者： Maher, Charles Emmett Stillinger, Frank H. Torquato, Salvatore Department of Chemistry Princeton University PrincetonNJ08544 United States Department of Physics Princeton University PrincetonNJ08544 United States Princeton Institute for the Science and Technology of Materials Princeton University PrincetonNJ08544 United States Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

Dense, disordered packings of particles are useful models of low-temperature amorphous phases of matter, biological systems, granular media, and colloidal systems. The study of dense packings of nonspherical particles enables one to ascertain how rotational degrees of freedom affect packing behavior. Here, we study superballs, a large family of deformations of the sphere, defined in three dimensions by |x1|2p + |x2|2p + |x3|2p ≤ 1, where p ∈ (0, ∞) is a deformation parameter indicating to what extent the shape deviates from a sphere. As p increases from the sphere point (p = 1), the superball tends to a cuboidal shape and approaches a cube in the p → ∞ limit. As p → 0.5, it approaches an octahedron, becomes a concave body with octahedral symmetry for p ¯) vary nonanalytically as p diverges from unity. Here, we use an event-driven molecular dynamics algorithm to produce MRJ superball packings with 0.85 ≤ p ≤ 1.50. To supplement the previous work on such packings, we characterize their large-scale structure by examining the behaviors of their structure factors S(Q) and spectral densities χV (Q), as the wave number Q tends to zero, and find that these packings are effectively hyperuniform for all values of p examined. We show that the mean width w¯ is a useful length scale to make distances dimensionless in order to compare systematically superballs of different shape. Moreover, we compute the complementary cumulative pore-size distribution F (δ) and find that the pore sizes tend to decrease as |1 − p| increases. From F (δ), we estimate how the fluid permeability, mean survival time, and principal diffusion relaxation time vary as a function of p. Additionally, we compute the diffusion "spreadability" S(t) [Torquato, Phys. Rev. E, 104, 054102, (2021)] of these packings and find that the long-time power-law scaling indicates these packings are hyperuniform with a small-Q power law scaling of the spectral density χV (Q) ∼ Qα with an exponent α that ranges from 0.64 at th

关键词： Spheres

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Convergence of Optimal Values Past Critical Threshold

Convergence of Optimal Values Past Critical Threshold

引用

Annual Allerton Conference on Communication, Control, and Computing

作者： Steven Jaslar Sekhar Tatikonda Program in Applied Mathematics Yale University Department of Electrical Engineering Yale University

ISBN: (纸本)9781424429257

In this paper we examine maximum weight independent sets (MWIS) on branching processes and Erdos-Renyi graphs. Gamarnik, Nowicki, and Swirscsz [1], showed that, if a particular fixed point equation had a unique solution, then the MWIS value converged. This followed by showing that there was no long range dependencies in the MWIS solution. Here we examine what happens when there are multiple solutions to the fixed point equation, and hence there are long range dependencies. Specifically, we show that, under surprisingly weak conditions - which include the possibility of long range dependencies - that the MWIS value converges in probability to a random variable. We give a new method for calculating the MWIS value for Erdos-Renyi graphs. In the case of vertex weights independently drawn from the standard exponential distribution and with multiple solutions to the fixed point equation, we use our method to show improved upper and lower bounds of the MWIS value. Finally, we show how our proof techniques are general enough to be applied to other optimization problems with local constraints such as Maximum Weight Matching and Maximum Weight Color.

关键词： Convergence Equations mathematics Random variables Exponential distribution Constraint optimization Upper bound Tree graphs

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Smoothing the Conditional Value-at-Risk based Pickands Estimators

arXiv

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arXiv 2024年

作者： Li, Yizhou Polak, Pawel Center of Excellence in Wireless and Information Technology Stony Brook University United States Department of Applied Mathematics and Statistics Stony Brook University United States Institute for Advanced Computational Science Stony Brook University United States

We incorporate the conditional value-at-risk (CVaR) quantity into a generalized class of Pickands estimators. By introducing CVaR, the newly developed estimators not only retain the desirable properties of consistency, location, and scale invariance inherent to Pickands estimators, but also achieve a reduction in mean squared error (MSE). To address the issue of sensitivity to the choice of the number of top order statistics used for the estimation, and ensure robust estimation, which are crucial in practice, we first propose a beta measure, which is a modified beta density function, to smooth the estimator. Then, we develop an algorithm to approximate the asymptotic mean squared error (AMSE) and determine the optimal beta measure that minimizes AMSE. A simulation study involving a wide range of distributions shows that our estimators have good and highly stable finite-sample performance and compare favorably with the other estimators. Copyright © 2024, The Authors. All rights reserved.

关键词： Mean square error

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Visualization of processes in self-learning systems

Visualization of processes in self-learning systems

引用

Annual Conference on Privacy, Security and Trust, PST

作者： Gabriele Peters Kerstin Bunte Marc Strickert Michael Biehl Thomas Villmann Human-Computer Interaction FernUniversität in Hagen Germany CITEC-Cognitive Interaction Technology Center of Excellence Bielefeld University Germany Department of Computer Science and Mathematics University of Marburg Germany Johann Bernoulli Inst. for Mathematics & Computer Science University of Groningen The Netherlands Computational Intelligence Group University of Applied Sciences Mittweida Germany

One aspect of self-organizing systems is their desired ability to be self-learning, i.e., to be able to adapt dynamically to conditions in their environment. This quality is awkward especially if it comes to applications in security or safety-sensitive areas. Here a step towards more trustful systems could be taken by providing transparency of the processes of a system. An important means of giving feedback to an operator is the visualization of the internal processes of a system. In this position paper we address the problem of visualizing dynamic processes especially in self-learning systems. We take an existing self-learning system from the field of computer vision as an example from which we derive questions of general interest such as possible options to visualize the flow of information in a dynamic learning system or the visualization of symbolic data. As a side effect the visualization of learning processes may provide a better understanding of underlying principles of learning in general, i.e, also in biological systems. That may also facilitate improved designs of future self-learning systems.

关键词： Visualization Data visualization Learning systems Shape Electronic mail Machine learning Security

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