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Scaled Brownian motion with random anomalous diffusion exponent

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arXiv 2024年

作者： Woszczek, Hubert Chechkin, Aleksei Wylomańska, Agnieszka Faculty of Pure and Applied Mathematics Hugo Steinhaus Cenetr Wroclaw University of Science and Technology Wyspiańskiego 27 Wroclaw50-370 Poland Institute for Physics & Astronomy University of Potsdam Potsdam-Golm14476 Germany Akhiezer Institute for Theoretical Physics National Science Center Kharkov Institute of Physics and Technology’ Akademicheskaya st.1 Kharkov61108 Ukraine

The scaled Brownian motion (SBM) is regarded as one of the paradigmatic random processes, featuring the anomalous diffusion property characterized by the diffusion exponent. It is a Gaussian, self-similar process with independent increments, which has found applications across various fields, from turbulence and stochastic hydrology to biophysics. In our paper, inspired by recent single particle tracking biological experiments, we introduce a process termed the scaled Brownian motion with random exponent (SBMRE), which preserves SBM characteristics at the level of individual trajectories, albeit with randomly varying anomalous diffusion exponents across the trajectories. We discuss the main probabilistic properties of SBMRE, including its probability density function (pdf), and the q-th absolute moment. Additionally, we present the expected value of the time-averaged mean squared displacement (TAMSD) and the ergodicity breaking parameter. Furthermore, we analyze the pdf of the first hitting time in a semi-infinite domain, the martingale property of SBMRE, and its stochastic exponential. As special cases, we consider two distributions of the anomalous diffusion exponent, namely the two-point and beta distributions, and discuss the asymptotics of the presented characteristics in such cases. Theoretical results for SBMRE are validated through numerical simulations and compared with the corresponding characteristics for SBM. © 2024, CC BY.

关键词： Brownian movement

BUBBLING PHENOMENON FOR SEMILINEAR NEUMANN ELLIPTIC EQUATIONS OF CRITICAL EXPONENTIAL GROWTH

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arXiv 2023年

作者： Chen, Lu Lu, Guozhen Zhang, Caifeng School of Mathematics and Statistics Beijing Institute of Technology Beijing100081 China Department of Mathematics University of Connecticut StorrsCT06269 United States Department of Applied Mathematics School of Mathematics and Physics University of Science and Technology of Beijing Beijing100083 China

In the past few decades, much attention has been paid to the bubbling problem for semilinear Neumann elliptic equation with the critical and subcritical polynomial nonlinearity, much less is known if the polynomial nonlinearity is replaced by the exponential nonlinearity. In this paper, we consider the following semilinear Neumann elliptic problem with the Trudinger-Moser exponential growth: (− ∂u ∂ν d d ∆ = ud 0 + ud = ud(eu2 d − 1) on in ∂Ω Ω , , where d > 0 is a parameter, Ω is a smooth bounded domain in R2, ν is the unit outer normal to ∂Ω. We first prove the existence of a ground state solution to the above equation. If d is sufficiently small, we prove that any ground state solution ud has at most one maximum point which is located on the boundary of Ω and characterize the shape of ground state solution ud around the condensation point Pd. The key point of the proof lies in proving that the maximum point Pd is close to the boundary at the speed of √d when d → 0 and ud under suitable scaling transform converges strongly to the ground state solution of the limit equation ∆w + w = w(ew2 − 1). Our proof is based on the energy threshold of cutoff function, the concentration compactness principle for the Trudinger-Moser inequality, regularity theory for elliptic equation and an accurate analysis for the energy of the ground state solution ud as d → 0. Furthermore, by assuming that Ω is a unit disk, we remove the smallness assumption on d and show the maximum point of ground state solution ud must lie on the boundary of Ω for any d > 0. Copyright © 2023, The Authors. All rights reserved.

关键词： Bubble formation

Improving Explainability of Softmax Classifiers Using a Prototype-Based Joint Embedding Method

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arXiv 2024年

作者： Sit, Hilarie Keith, Brendan Bergen, Karianne Department of Earth Environmental and Planetary Sciences Brown University ProvidenceRI United States Data Science Institute Brown University ProvidenceRI United States Division of Applied Mathematics Brown University ProvidenceRI United States

We propose a prototype-based approach for improving explainability of softmax classifiers that provides an understandable prediction confidence, generated through stochastic sampling of prototypes, and demonstrates potential for out of distribution detection (OOD). By modifying the model architecture and training to make predictions using similarities to any set of class examples from the training dataset, we acquire the ability to sample for prototypical examples that contributed to the prediction, which provide an instance-based explanation for the model’s decision. Furthermore, by learning relationships between images from the training dataset through relative distances within the model’s latent space, we obtain a metric for uncertainty that is better able to detect out of distribution data than softmax confidence. © 2024, CC BY.

关键词： Forecasting

Developing Highly Porous Glass Microspheres via a Single-Stage Flame Spheroidisation Process 9

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Developing Highly Porous Glass Microspheres via a Single-Sta...

9th Asian physics Symposium 2021, APS 2021

作者： Nuzulia, N.A. Islam, T. Saputra, A. Sudiro, T. Timuda, G.E. Mart, T. Sari, Y.W. Ahmed, I. Department of Physics Faculty of Mathematics and Natural Sciences Universitas Indonesia Indonesia Department of Physics Faculty of Mathematics and Natural Sciences IPB University Indonesia Advanced Materials Research Group Faculty of Engineering University of Nottingham United Kingdom Department of Applied Chemistry and Chemical Engineering Faculty of Engineering Noakhali Science and Technology University Noakhali3814 Bangladesh Research Center for Physics Indonesian Institute of Sciences Indonesia

Glass microspheres are gaining attention in bone tissue engineering due to their ability to convert into hydroxyapatite-like materials, resembling the inorganic mineral of natural bone. The morphology of glass microspheres as starting material has been considered to influence the conversion rate and the resulting product where porous microspheres could promote faster conversion to hydroxyapatite than solid microspheres. This paper reports on manufacturing glass microspheres (solid and porous) using a flame spheroidisation process. The effect of various gas ratios of acetylene and oxygen on the morphological changes of glass microspheres was investigated. Irregular shaped glass particles with starting particle size ranges of 63 - 125 μm were used as feed and delivered to a hot flame to produce solid microspheres. To manufacture porous glass microspheres via a single-stage flame spheroidisation process, calcium carbonate was utilised as a porogen and mixed with the glass particles. Solid and porous glass microspheres were successfully produced, exploring various gas ratios of 3:3, 4:7, 5:7 and 6:7 with a mean particle size range between 73 - 105 μm. Moreover, the average pore size of 6 μm and 9 μm was obtained using 5:7 and 6:7 gas ratios, respectively. This single-stage flame spheroidisation process is a promising method for producing both solid and porous bioactive glass microspheres. © Published under licence by IOP Publishing Ltd.

关键词： Hydroxyapatite

Tuning Frequency Bias of State Space Models

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arXiv 2024年

作者： Yu, Annan Lyu, Dongwei Lim, Soon Hoe Mahoney, Michael W. Benjamin Erichson, N. Center for Applied Mathematics Cornell University United States Data Science Institute University of Chicago United States Department of Mathematics KTH Royal Institute of Technology Sweden Nordita KTH Royal Institute of Technology Stockholm University Sweden Lawrence Berkeley National Laboratory United States International Computer Science Institute Department of Statistics University of California at Berkeley United States

State space models (SSMs) leverage linear, time-invariant (LTI) systems to effectively learn sequences with long-range dependencies. By analyzing the transfer functions of LTI systems, we find that SSMs exhibit an implicit bias toward capturing low-frequency components more effectively than high-frequency ones. This behavior aligns with the broader notion of frequency bias in deep learning model training. We show that the initialization of an SSM assigns it an innate frequency bias and that training the model in a conventional way does not alter this bias. Based on our theory, we propose two mechanisms to tune frequency bias: either by scaling the initialization to tune the inborn frequency bias;or by applying a Sobolev-norm-based filter to adjust the sensitivity of the gradients to high-frequency inputs, which allows us to change the frequency bias via training. Using an image-denoising task, we empirically show that we can strengthen, weaken, or even reverse the frequency bias using both mechanisms. By tuning the frequency bias, we can also improve SSMs' performance on learning long-range sequences, averaging an 88.26% accuracy on the Long-Range Arena (LRA) benchmark tasks. © 2024, CC BY.

关键词： State space methods

Superconducting Two-Dimensional Penta Materials

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SSRN

SSRN 2023年

作者： Liu, Hao-Dong Si, Jianguo Jiao, Na Han, Yu-Lin Zheng, Mengmeng Lu, Hong-Yan Wang, Baotian Zhang, Ping Beijing100049 China Spallation Neutron Source Science Center Dongguan523803 China School of Physics and Physical Engineering Qufu Normal University Qufu273165 China Institute of Applied Physics and Computational Mathematics Beijing1000 China Collaborative Innovation Center of Extreme Optics Shanxi University Shanxi Taiyuan030006 China Institute of Applied Physics and Computational Mathematics Beijing100088 China

Successful experimental synthesis and theoretical prediction of two-dimensional (2D) penta materials with various properties have enriched the family of 2D materials. In this work, based on first-principles calculations, we systematically study the electronic structures, phonon dispersions, and electron-phonon coupling (EPC) of four metallic penta materials `XY2' (NC2, PC2, BN2, and BP2). Three of them display weak phonon-mediated superconductivity with different low superconducting transition temperatures (Tc). The highest Tc among them is 12 K for BP2. Through applying biaxial tensile strain, the λ and Tc of PC2 (BN2) can be enhanced up to 1.64 (2.68) and 49.77 (52.29) K, respectively. By solving the anisotropic Eliashberg equations, single-gap superconducting nature is predicted for PC2 and BN2. The high sensitive of superconductivity in these systems clearly demonstrate that they are potential candidates for applications in using as superconducting nanodevices. Our work will stimulate further searching of high-Tc superconductivity in low-dimensional materials. © 2023, The Authors. All rights reserved.

关键词： Electron-phonon interactions

Towards Attenuation Imaging with Computed Ultrasound Tomography in Echo Mode (CUTE)

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TechRxiv

TechRxiv 2022年

作者： Martiartu, Naiara Korta Yolgunlu, Parisa Salemi Stähli, Patrick Frenz, Martin Jaeger, Michael Institute of Applied Physics University of Bern Bern Switzerland Insel Data Science Center Bern University Hospital Bern Switzerland

This work presents a novel attenuation imaging technique for pulse-echo ultrasound systems. In contrast to state-of-the-art techniques, we formulate the reconstruction in two dimensions relying on tissue insonifications with different steering angles. By beamforming backscattered echoes recorded by the transducer, we measure at each location the changes in the amplitudes of detected echoes with different transmissions and relate them to local tissue attenuation. This relationship assumes ultrasound waves propagate in straight paths;thus, we linearize the forward problem to provide suitable time-to-solutions for clinical practice. The presented technique is the natural extension of computed tomography in echo mode (CUTE), initially developed for tissue speed-of-sound quantification. The performance of our method is demonstrated in numerical examples with data computed using the k-Wave numerical solver for wave-propagation simulations. These examples consider tissue-mimicking media with varying heterogeneity in attenuation and echogenicity. The results show that our method can provide images with promising spatial and contrast resolution, as well as attenuation estimates independent of tissue echogenicity. This work represents a necessary first step towards multi-modal CUTE imaging of speed of sound and attenuation in tissue. © 2022, CC BY.

关键词： Tissue

A Two-Phase Image Dejittering Approach on the Perspective of Mixed Noise Removal

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SSRN

SSRN 2023年

作者： Zhang, Yingxin Zhang, Wenxing Yin, Junping School of Mathematical Sciences University of Electronic Science and Technology of China Chengdu611731 China Institute of Applied Physics and Computational Mathematics Beijing100094 China Shanghai Zhangjiang Academy of Mathematics Shanghai201203 China

The jittery image, which is visually abnormal in jags of edge and loss of coherence, arises commonly from the area of wireless video transmission and multimedia data compression. The problem of image dejittering is challenging to resolve because of the ubiquitous noise and blur in jittery data. In this paper, we are devoted to the pixel-jitter (possibly blurry) image recovery on the perspective of mixed noise removal. By viewing a jittery image as the contamination of an ideal image with spatially-varying Gaussian+impulse noise, we proposed a two-phase (including filtering and diffusing phases) approach for recovering pixel-jitter images. At the filtering phase, outliers posed by jitters in the vicinity of edges are inhibited and inspected by median filters. At the diffusing phase, structure tensor based anisotropic diffusion is exploited to reduce the spatially-varying perturbations in piecewise smooth image regions. Upon the spectral decomposition of structure tensor and algorithmic advances in separable convex optimization, the anisotropic diffusion model can be efficiently solved with all subproblems admitting closed-form solutions. Numerical simulations on recovering jittery data demonstrate the compelling performance of the two-phase approach. © 2023, The Authors. All rights reserved.

关键词： Pixels

Novelty of Different Distance Approach for Multi-Criteria Decision-Making Challenges Using q-Rung Vague Sets

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Computer Modeling in Engineering & sciences 2024年第6期139卷 3353-3385页

作者： Murugan Palanikumar Nasreen Kausar Dragan Pamucar Seifedine Kadry Chomyong Kim Yunyoung Nam Saveetha School of Engineering Saveetha Institute of Medical and Technical SciencesChennai602105India Department of Mathematics Faculty of Arts and ScienceYildiz Technical UniversityEsenlerIstanbul34220Turkey Department of Operations Research and Statistics Faculty of Organizational SciencesUniversity of BelgradeBelgrade11000Serbia College of Engineering Yuan Ze UniversityTaoyuan City320315TaiwanChina Department of Applied Data Science Noroff University CollegeKristiansand4612Norway Artificial Intelligence Research Center(AIRC) Ajman UniversityAjman346United Arab Emirates Department of Electrical and Computer Engineering Lebanese American UniversityByblos1102-2801Lebanon ICT Convergence Research Center Soonchunhyang UniversityAsan31538South Korea Department of Computer Science and Engineering Soonchunhyang UniversityAsan31538South Korea

In this article,multiple attribute decision-making problems are solved using the vague normal set(VNS).It is possible to generalize the vague set(VS)and q-rung fuzzy set(FS)into the q-rung vague set(VS).A log q-rung normal vague weighted averaging(log q-rung NVWA),a log q-rung normal vague weighted geometric(log q-rung NVWG),a log generalized q-rung normal vague weighted averaging(log Gq-rung NVWA),and a log generalized q-rungnormal vagueweightedgeometric(logGq-rungNVWG)operator are discussed in this *** is provided of the scoring function,accuracy function and operational laws of the log q-rung *** algorithms underlying these functions are also described.A numerical example is provided to extend the Euclidean distance and the Humming ***,idempotency,boundedness,commutativity,and monotonicity of the log q-rung VS are examined as they facilitate recognizing the optimal alternative more quickly and help clarify *** chose five anemia patients with four types of symptoms including seizures,emotional shock or hysteria,brain cause,and high fever,who had either retrograde amnesia,anterograde amnesia,transient global amnesia,post-traumatic amnesia,or infantile *** numbers q are used to express the results of the *** demonstrate the effectiveness and accuracy of the models we are investigating,we compare several existing models with those that have been developed.

关键词： Vague set aggregating operators euclidean distance hamming distance decision making