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The geometry of nodal sets and outlier detection

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arXiv 2017年

作者： Cheng, Xiuyuan Mishne, Gal Steinerberger, Stefan Applied Mathematics Program Yale University New HavenCT06510 United States Applied Mathematics Program Yale University New HavenCT06510 United States Department of Mathematics Yale University New HavenCT06510 United States

Let (M, g) be a compact manifold and let −∆φk = λkφk be the sequence of Laplacian eigenfunctions. We present a curious new phenomenon which, so far, we only managed to understand in a few highly specialized cases: the family of functions fN : M → R≥01 |φk(x)| fN(x) = kX≤N √λk kφkkL∞(M)seems strangely suited for the detection of anomalous points on the manifold. It may be heuristically interpreted as the sum over distances to the nearest nodal line and potentially hints at a new phenomenon in spectral geometry. We give rigorous statements on the unit square [0, 1]2 (where minima localize in Q2) and on Paley graphs (where fN recovers the geometry of quadratic residues of the underlying finite field Fp). Numerical examples show that the phenomenon seems to arise on fairly generic manifolds.2010 mathematics Subject Classification. 35J05, 35P30, 58J50 (primary) and 05C25, 05C50 (secondary). Copyright © 2017, The Authors. All rights reserved.

关键词： Geometry

Roadmap on Nonlocality in Photonic Materials and Metamaterials

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arXiv 2025年

作者： Monticone, Francesco Mortensen, N. Asger Fernández-Domínguez, Antonio I. Luo, Yu Tserkezis, Christos Khurgin, Jacob B. Shahbazyan, Tigran V. Chaves, André J. Peres, Nuno M.R. Wegner, Gino Busch, Kurt Hu, Huatian della Sala, Fabio Zhang, Pu Ciracì, Cristian Aizpurua, Javier Babaze, Antton Borisov, Andrei G. Chen, Xue-Wen Christensen, Thomas Yan, Wei Yang, Yi Hohenester, Ulrich Huber, Lorenz Wubs, Martijn de Liberato, Simone Gonçalves, P.A.D. de Abajo, F. Javier García Hess, Ortwin Tarasenko, Illya Cox, Joel D. Jelver, Line Dias, Eduardo J.C. Sánchez, Miguel Sánchez Margetis, Dionisios Gómez-Santos, Guillermo Stauber, Tobias Tretyakov, Sergei Simovski, Constantin Pakniyat, Samaneh Gómez-Díaz, J. Sebastián Bondarev, Igor V. Biehs, Svend-Age Boltasseva, Alexandra Shalaev, Vladimir M. Krasavin, Alexey V. Zayats, Anatoly V. Alù, Andrea Song, Jung-Hwan Brongersma, Mark L. Levy, Uriel Long, Olivia Y. Guo, Cheng Fan, Shanhui Bozhevolnyi, Sergey I. Overvig, Adam Prudêncio, Filipa R. Silveirinha, Mário G. Gangaraj, S. Ali Hassani Argyropoulos, Christos Huidobro, Paloma A. Galiffi, Emanuele Yang, Fan Pendry, John B. Miller, David A.B. School of Electrical and Computer Engineering Cornell University IthacaNY14853 United States POLIMA Center for Polariton-driven Light–Matter Interactions University of Southern Denmark Campusvej 55 Odense MDK-5230 Denmark D-IAS—Danish Institute for Advanced Study University of Southern Denmark Campusvej 55 Odense MDK-5230 Denmark Departamento de Física Teórica de la Materia Condensada Universidad Autónoma de Madrid MadridE-28049 Spain Universidad Autónoma de Madrid MadridE-28049 Spain School of Electrical and Electronic Engineering Nanyang Technological University Singapore639798 Singapore Key Laboratory of Radar Imaging and Microwave Photonics Ministry of Education College of Electronic and Information Engineering Nanjing University of Aeronautics and Astronautics Nanjing211106 China Department of Electrical and Computer Engineering Johns Hopkins University BaltimoreMD21218 United States Department of Physics Jackson State University JacksonMS39217 United States Department of Physics Aeronautics Institute of Technology SP So Jos dos Campos12228-900 Brazil Departamento de Física Universidade do Minho BragaP-4710-057 Portugal Av Mestre José Veiga Braga4715-330 Portugal Max-Born-Institut Berlin12489 Germany Humboldt-Universität zu Berlin Institut für Physik AG Theoretische Optik and Photonik Berlin12489 Germany Center for Biomolecular Nanotechnologies Istituto Italiano di Tecnologia Via Barsanti 14 Arnesano73010 Italy Via Monteroni Campus Unisalento Lecce73100 Italy School of physics Huazhong University of Science and Technology Luoyu Road 1037 Wuhan430074 China Donostia International Physics Center DIPC Donostia-San Sebastián20018 Spain Ikerbasque Basque Foundation for Science Bilbao48009 Spain Department of Electricity and Electronics University of the Basque Country Leioa48940 Spain Department of Applied Physics University of the Basque Country Donostia20018 Spain Université Paris-Saclay CNRS Institut des Scienc

Photonic technologies continue to drive the quest for new optical materials with unprecedented responses. A major frontier in this field is the exploration of nonlocal (spatially dispersive) materials, going beyond the local, wavevector-independent assumption traditionally made in optical material modeling. On one end, the growing interest in plasmonic, polaritonic and quantum materials has revealed naturally occurring nonlocalities, emphasizing the need for more accurate models to predict and design their optical responses. This has major implications also for topological, nonreciprocal, and time-varying systems based on these material platforms. Beyond natural materials, artificially structured materials—metamaterials and metasurfaces–can provide even stronger and engineered nonlocal effects, emerging from long-range interactions or multipolar effects. This is a rapidly expanding area in the field of photonic metamaterials, with open frontiers yet to be explored. In the case of metasurfaces, in particular, nonlocality engineering has become a powerful tool for designing strongly wavevector-dependent responses, enabling enhanced wavefront control, spatial compression, multifunctional devices, and wave-based computing. Furthermore, nonlocality and related concepts play a critical role in defining the ultimate limits of what is possible in optics, photonics, and wave physics. This Roadmap aims to survey the most exciting developments in nonlocal photonic materials, highlight new opportunities and open challenges, and chart new pathways that will drive this emerging field forward–toward new scientific discoveries and technological advancements. Copyright © 2025, The Authors. All rights reserved.

关键词： Nanocrystals

Geometry-based data generation

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arXiv 2018年

作者： Lindenbaum, Ofir Stanley, Jay S. Wolf, Guy Krishnaswamy, Smita Department of Genetics Yale University New HavenCT United States Applied Mathematics Program Yale University New HavenCT United States Computational Biology & Bioinformatics Program Yale University New HavenCT United States Department of Computer Science Yale University New HavenCT United States

We propose a new type of generative model of high-dimensional data that learns a manifold geometry of the data, rather than density, and can generate points evenly along this manifold. This is in contrast to existing generative models that represent data density, and are strongly affected by noise and other artifacts of data collection. We demonstrate how this approach corrects sampling biases and artifacts, and improves several downstream data analysis tasks such as clustering and classification. Finally, we demonstrate that this approach is especially useful in biology where, despite the advent of single-cell technologies, rare subpopulations and gene-interaction relationships are affected by biased sampling. We show that SUGAR can generate hypothetical populations and reveal intrinsic patterns and mutual-information relationships between genes on a single cell dataset of hematopoiesis. Copyright © 2018, The Authors. All rights reserved.

关键词： Clustering algorithms

Cosmological tests with the joint lightcurve analysis

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arXiv 2018年

作者： Melia, F. Wei, J.-J. Maier, R.S. Wu, X.-F. Depts. of Physics and Astronomy Applied Math Program University of Arizona AZ85721 United States Purple Mountain Observatory Chinese Academy of Sciences Nanjing210008 China Depts. of Mathematics and Physics Statistics Program University of Arizona AZ85721 United States Purple Mountain Observatory Nanjing210008 China

We examine whether a comparison between wCDM and Rh = ct using merged Type Ia SN catalogs produces results consistent with those based on a single homogeneous sample. Using the Betoule et al. [1] joint lightcurve analysis (JLA) of a combined sample of 613 events from SNLS and SDSS-II, we estimate the parameters of the two models and compare them. We find that the improved statistics can alter the model selection in some cases, but not others. In addition, based on the model fits, we find that there appears to be a lingering systematic offset of ∼ 0.04-0.08 mag between the SNLS and SDSS-II sources, in spite of the cross-calibration in the JLA. Treating wCDM, ΛCDM and Rh = ct as separate models, we find in an unbiased pairwise statistical comparison that the Bayes Information Criterion (BIC) favors the Rh = ct Universe with a likelihood of 82.8% versus 17.2% for wCDM, but the ratio of likelihoods is reversed (16.2% versus 83.8%) when wde = −1 (i.e., ΛCDM) and strongly reversed (1.0% versus 99.0%) if in addition k = 0 (i.e., flat ΛCDM). We point out, however, that the value of k is a measure of the net energy (kinetic plus gravitational) in the Universe and is not constrained theoretically, though some models of inflation would drive k → 0 due to an expansion-enforced dilution. Since we here consider only the basic ΛCDM model, the value of k needs to be measured and, therefore, the pre-assumption of flatness introduces a significant bias into the BIC. Copyright © 2018, The Authors. All rights reserved.

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EU-Level Protection of Designs-The Registered Community Design

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IEEE CONSUMER ELECTRONICS MAGAZINE 2016年第1期5卷 129-+页

作者： Heneghan, Kieran Program in Applied and Computational Mathematics Princeton University KLS OPTIM Villebon sur Yvette France

The design of most consumer products can play a key role in their commercial success, and, therefore, the ability to protect new and innovative designs is an important part of a modern competitive marketplace. Since 1 April 2003, it has been possible to obtain European Union (EU)-wide design protection by means of a single registration made under the Community Design regulation (Council Regulation 6/2002 of 12 December 2001), usually referred to as a Registered Community Design (RCD). Before the advent of the RCD, a designer seeking protection across the EU was restricted to seeking registration at a national level. Now, with 28 member states in the EU, the ability to register a design with just a single filing is a major step forward in terms of protection of intellectual property rights in the EU.

关键词： Registers Intellectual property Consumer electronics Visualization Packaging Computers Software

Investigating the Nature of the Luminous Ambiguous Nuclear Transient ASASSN-17jz

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arXiv 2021年

作者： Holoien, Thomas W.-S. Neustadt, Jack M.M. Vallely, Patrick J. Auchettl, Katie Hinkle, Jason T. Romero-Cañizales, Cristina Shappee, Benjamin J. Kochanek, Christopher S. Stanek, K.Z. Chen, Ping Dong, Subo Prieto, Jose L. Thompson, Todd A. Brink, Thomas G. Filippenko, Alexei V. Zheng, Weikang Bersier, David Bose, Subhash Burgasser, Adam J. Channa, Sanyum De Jaeger, Thomas Hestenes, Julia Im, Myungshin Jeffers, Benjamin Jun, Hyunsung D. Lansbury, George Post, Richard S. Ross, Timothy W. Stern, Daniel Tang, Kevin Tucker, Michael A. Valenti, Stefano Yunus, Sameen Zhang, Keto D. The Observatories of the Carnegie Institution for Science 813 Santa Barbara St. PasadenaCA91101 United States Department of Astronomy The Ohio State University 140 West 18th Avenue ColumbusOH43210 United States School of Physics The University of Melbourne ParkvilleVIC3010 Australia Department of Astronomy and Astrophysics University of California Santa CruzCA95064 United States Institute for Astronomy University of Hawai'i 2680 Woodlawn Drive HonoluluHI96822 United States Institute of Astronomy and Astrophysics Academia Sinica 11F of Astronomy-Mathematics Building AS/NTU No. 1 Roosevelt Rd Taipei10617 Taiwan The Ohio State University 191 W. Woodruff Ave. ColumbusOH43210 United States Kavli Institute for Astronomy and Astrophysics Peking University Yi He Yuan Road 5 Hai Dian District Beijing100871 China Núcleo de Astronomía de la Facultad de Ingeniería y Ciencias Universidad Diego Portales Av. Ejército 441 Santiago Chile Millennium Institute of Astrophysics Santiago Chile Department of Astronomy University of California BerkeleyCA94720-3411 United States Miller Institute for Basic Research in Science University of California BerkeleyCA94720 United States Astrophysics Research Institute Liverpool John Moores University 146 Brownlow Hill LiverpoolL3 5RF United Kingdom Center for Astrophysics and Space Sciences University of California San Diego La Jolla CA92093 United States Department of Applied Physics and Applied Mathematics Columbia University New YorkNY10027-6623 United States SNU Astronomy Research Center Astronomy Program Dept. of Physics and Astronomy Seoul National University Seoul08826 Korea Republic of School of Physics Korea Institute for Advanced Study 85 Hoegiro Dongdaemun-gu Seoul02455 Korea Republic of European Southern Observatory Karl-Schwarzschild-Strasse 2 GarchingD-85748 Germany Post Observatory LexingtonMA02421 United States Jet Propulsion Laboratory California Institute of Technology 4800 Oa

We present observations of the extremely luminous but ambiguous nuclear transient (ANT) ASASSN-17jz, spanning roughly 1200 days of the object's evolution. ASASSN-17jz was discovered by the All-Sky Automated Survey for Supernovae (ASAS-SN) in the galaxy SDSS J171955.84+414049.4 on UT 2017 July 27 at a redshift of z = 0:1641. The transient peaked at an absolute B-band magnitude of MB,peak=-22:81, corresponding to a bolometric luminosity of Lbol,peak= 8.3×1044erg s-1, and exhibited late-time ultraviolet emission that was still ongoing in our latest observations. Integrating the full light curve gives a total emitted energy of Etot= (1.36±0.08)×1052erg, with (0.80±0.02)×1052erg of this emitted within 200 days of peak light. This late-time ultraviolet emission is accompanied by increasing X-ray emission that becomes softer as it brightens. ASASSN-17jz exhibited a large number of spectral emission lines most commonly seen in active galactic nuclei (AGNs) with little evidence of evolution. It also showed transient Balmer features which became fainter and broader over time, and are still being detected > 1000 days after peak brightness. We consider various physical scenarios for the origin of the transient, including supernovae (SNe), tidal disruption events (TDEs), AGN outbursts, and ANTs. We find that the most likely explanation is that ASASSN-17jz was an SN IIn occurring in or near the disk of an existing AGN, and that the late-time emission is caused by the AGN transitioning to a more active state. Copyright © 2021, The Authors. All rights reserved.

关键词： Supernovae

MANIFOLD LEARNING WITH BI-STOCHASTIC KERNELS

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arXiv 2017年

作者： Marshall, Nicholas F. Coifman, Ronald R. Program in Applied Mathematics Yale University New HavenCT06511 United States

In this paper we answer the following question: what is the infinitesimal generator of the diffusion process defined by a kernel that is normalized such that it is bi-stochastic with respect to a specified measure? More precisely, under the assumption that data is sampled from a Riemannian manifold we determine how the resulting infinitesimal generator depends on the potentially nonuniform distribution of the sample points, and the specified measure for the bi-stochastic normalization. In a special case, we demonstrate a connection to the heat kernel. We consider both the case where only a single data set is given, and the case where a data set and a reference set are given. The spectral theory of the constructed operators is studied, and Nyström extension formulas for the gradients of the eigenfunctions are computed. Applications to discrete point sets and manifold learning are discussed. Copyright © 2017, The Authors. All rights reserved.

关键词： Stochastic systems

Novel Polarimetric Analysis of Near Horizon Flaring Episodes in M87* in Millimeter Wavelength

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arXiv 2025年

作者： Emami, Razieh Liska, Matthew Chatterjee, Koushik Bower, Geoffrey C. Benbow, Wystan Finkbeiner, Douglas Wielgus, Maciek Hernquist, Lars Smith, Randall Tremblay, Grant Ricarte, Angelo Steiner, James F. Broderick, Avery E. Saurabh Davelaar, Jordy Grindlay, Josh Vogelsberger, Mark Chan, Chi-Kwan Center for Astrophysics Harvard & Smithsonian 60 Garden Street CambridgeMA02138 United States Center for Relativistic Astrophysics Georgia Institute of Technology Howey Physics Bldg 837 State Street NW AtlantaGA30332 United States Department of Physics University of Maryland College ParkMD United States Academia Sinica Institute of Astronomy and Astrophysics 645 N. A’ohoku Place HiloHI96720 United States Instituto de Astrofísica de Andalucía-CSIC Glorieta de la Astronomía s/n GranadaE-18008 Spain Black Hole Initiative Harvard University 20 Garden Street CambridgeMA02138 United States Perimeter Institute for Theoretical Physics 31 Caroline Street North WaterlooONN2L 2Y5 Canada Department of Physics and Astronomy University of Waterloo 200 University Avenue West WaterlooONN2L 3G1 Canada Max-Planck-Institut für Radioastronomie Auf dem Hügel 69 BonnD-53121 Germany Department of Astrophysical Sciences Peyton Hall Princeton University PrincetonNJ08544 United States NASA Hubble Fellowship Program Department of Physics Kavli Institute for Astrophysics and Space Research Massachusetts Institute of Technology CambridgeMA02139 United States Department of Astronomy and Steward Observatory University of Arizona 933 North Cherry Avenue TucsonAZ85721 United States Data Science Institute University of Arizona 1230 N. Cherry Ave. TucsonAZ85721 United States Program in Applied Mathematics University of Arizona 617 N. Santa Rita TucsonAZ85721 United States

Recent multi-wavelength observations of M87* (Algaba et al. 2024) revealed a high-energy γ-ray flare without a corresponding millimeter counterpart. We present a theoretical polarimetric study to evaluate the presence and nature of a potential millimeter flare in M87*, using a suite of general relativistic magnetohydrodynamical simulations with varying black hole (BH) spins and magnetic field configurations. We find that the emergence of a millimeter flare is strongly influenced by both spin and magnetic structure, with limited sensitivity to the electron distribution (thermal vs. non-thermal). We model the intensity light curve with a damped random walk (DRW) and compare the characteristic timescale (τ) with recent SMA observations, finding that the simulated τ exceeds observed values by over an order of magnitude. In a flaring case with BH spin a=+0.5, we identify a distinct millimeter flare followed by an order-of-magnitude flux drop. All Stokes parameters show variability near the flare, including a sign reversal in the electric vector position angle. While most βm modes remain stable, the EB-correlation phase is highly sensitive to both the flare peak and decay. We examine polarimetric signatures in photon sub-rings, focusing on modes ns=0 and ns=1. The ns=0 signal closely matches the full image, while ns=1 reveals distinct behaviors, highlighting the potential of space VLBI to isolate sub-ring features. Finally, we analyze the magnetic and velocity field evolution during the flare, finding that magnetic reconnection weakens during the flux decay, and the clockwise velocity flow transitions into an outflow-dominated regime. These results suggest that transient radio variability near flares encodes key information about BH spin and magnetic field structure, offering a novel probe into the physics of active galactic nuclei. © 2025, CC BY.

关键词： Magnetohydrodynamics

Global weak solutions for generalized SQG in bounded domains

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arXiv 2017年

作者： Nguyen, Huy Quang PROGRAM IN APPLIED AND COMPUTATIONAL MATHEMATICS PRINCETON UNIVERSITY PRINCETONNJ08544 United States

We prove the existence of global L2 weak solutions for a family of generalized inviscid surface-quasi geostrophic (SQG) equations in bounded domains of the plane. In these equations, the active scalar is transported by a velocity field which is determined by the scalar through a more singular nonlocal operator compared to the SQG equation. The result is obtained by establishing appropriate commutator representations for the weak formulation together with good bounds for them in bounded *** Classification: 35Q35, 35Q86.35Q35, 35Q86 Copyright © 2017, The Authors. All rights reserved.

关键词： Velocity