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arXiv 2025年

作者： Whitman, Sheila E. Latypov, Marat I. Graduate Interdisciplinary Program in Applied Mathematics University of Arizona TucsonAZ85721 United States Department of Materials Science and Engineering University of Arizona TucsonAZ85721 United States

Machine learning of microstructure–property relationships from data is an emerging approach in computational materials science. Most existing machine learning efforts focus on the development of task-specific models for each microstructure–property relationship. We propose utilizing pre-trained foundational vision transformers for the extraction of task-agnostic microstructure features and subsequent light-weight machine learning of a microstructure-dependent property. We demonstrate our approach with pre-trained state-of-the-art vision transformers (CLIP, DINOV2, SAM) in two case studies on machine-learning: (i) elastic modulus of two-phase microstructures based on simulations data;and (ii) Vicker’s hardness of Ni-base and Co-base superalloys based on experimental data published in literature. Our results show the potential of foundational vision transformers for robust microstructure representation and efficient machine learning of microstructure–property relationships without the need for expensive task-specific training or fine-tuning of bespoke deep learning models. © 2025, CC BY.

关键词： Elastic moduli

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Fairness Through Matching

arXiv

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arXiv 2025年

作者： Kim, Kunwoong Kong, Insung Lee, Jongjin Chae, Minwoo Park, Sangchul Kim, Yongdai Department of Statistics Seoul National University Korea Republic of Department of Applied Mathematics University of Twente Netherlands Samsung Research Korea Republic of School of Law Seoul National University Korea Republic of Department of Statistics Interdisciplinary Program in Artificial Intelligence Seoul National University Korea Republic of Seoul National University Korea Republic of

Group fairness requires that different protected groups, characterized by a given sensitive attribute, receive equal outcomes overall. Typically, the level of group fairness is measured by the statistical gap between predictions from different protected groups. In this study, we reveal an implicit property of existing group fairness measures, which provides an insight into how the group-fair models behave. Then, we develop a new group-fair constraint based on this implicit property to learn group-fair models. To do so, we first introduce a notable theoretical observation: every group-fair model has an implicitly corresponding transport map between the input spaces of each protected group. Based on this observation, we introduce a new group fairness measure termed Matched Demographic Parity (MDP), which quantifies the averaged gap between predictions of two individuals (from different protected groups) matched by a given transport map. Then, we prove that any transport map can be used in MDP to learn group-fair models, and develop a novel algorithm called Fairness Through Matching (FTM)1, which learns a group-fair model using MDP constraint with an user-specified transport map. We specifically propose two favorable types of transport maps for MDP, based on the optimal transport theory, and discuss their advantages. Experiments reveal that FTM successfully trains group-fair models with certain desirable properties by choosing the transport map accordingly. © 2025, CC BY.

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A SIZE-STRUCTURED MODEL FOR CANNIBALISM

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THEORETICAL POPULATION BIOLOGY 1992年第3期42卷 347-361页

作者： CUSHING, JM Department of Mathematics Interdisciplinary Program in Applied Mathematics Building 89 University of Arizona Tucson Arizona 85721 USA

A size-structured model for the dynamics of a cannibalistic population is derived under the assumption that cannibals (successfully) attack only smaller bodied victims, as is generally the case in the biological world. In addition to the resulting size-dependent death rate, the model incorporates the positive feedback mechanism resulting from the added resource energy obtained by the cannibal from the consumption of the victim. From the nonlinear partial integro-differential equation model, it is shown how to obtain a complete analysis of the global dynamics of the total population biomass. This analysis yields many dynamical features that have been attributed to cannibalism in the literature, including density self-regulation, a “life-boat strategy” phenomenon by which a population avoids extinction by practicing cannibalism under circumstances when it would otherwise go extinct, and multiple stable positive equilibrium states and hysteresis.

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A DISCRETE MODEL FOR COMPETING STAGE-STRUCTURED SPECIES

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THEORETICAL POPULATION BIOLOGY 1992年第3期41卷 372-387页

作者： CUSHING, JM Department of Mathematics Interdisciplinary Program in Applied Mathematics Building 89 University of Arizona Tucson Arizona 85721 USA

This paper deals with the problem of relating physiological properties of individual organisms to the dynamics at the total population level. A general nonlinear matrix difference equation is described which accounts for the dynamics of stage-structured populations under the assumption that individuals in the populations can be placed into well defined descriptive stages. Density feedback is modeled through an assumption that (stage-specific) fertilities and transitions are proportional to a resource uptake functional which is dependent upon a total weighted population size. It is shown how, if stage-specific differences in mortality are insignificant compared to stage-specific differences in fertility and inter-stage transitions, a nonlinear version of the strong ergodic theorem of demography mathematically separates the population level dynamics from the dynamics of the stage distribution vector, which is shown to stabilize independently of the population level dynamics. The nonlinear dynamics at the population level are governed by a key parameter π that encapsulates the stage-specific parameters and thereby affords a means by which population level dynamics can be linked to properties of individual organisms. The method is applied to a community of stagestructured populations competing for a common limiting resource, and it is seen how the parameter π determines the competitively superior species. An example of size structured competitors illustrates how the method can relate the competitive success of a species to such size-specific properties as resource conversion efficiencies and allocation fractions for individual growth and reproduction, largest adult body size, and size at birth and maturation.

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A SIMPLE-MODEL OF CANNIBALISM

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MATHEMATICAL BIOSCIENCES 1991年第1期107卷 47-71页

作者： CUSHING, JM Department of Mathematics Interdisciplinary Program in Applied Mathematics University of Arizona Tucson Arizona 85721 U.S.A.

A simple nonlinear discrete model is derived for the dynamics of a two-age class population consisting of juveniles and adults that includes cannibalism of juveniles by adults. The model is investigated analytically and numerically. It is shown how even this very simple model, by incorporating the negative and positive feedbacks due to cannibalism, can account for several important phenomena concerning the dynamics of cannibalistic populations that have been discussed and studied in the literature. These include the possibilities that the practice of cannibalism can (1) in certain circumstances be a form of self-regulation that promotes stable equilibration, while in other Circumstances it can lead to population oscillations;(2) result in a viable population in circumstances when its absence would otherwise result in extinction;and (3) be the source of multiple stable equilibria and hysteresis effects.

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THE DYNAMICS OF HIERARCHICAL AGE-STRUCTURED POPULATIONS

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JOURNAL OF MATHEMATICAL BIOLOGY 1994年第7期32卷 705-729页

作者： CUSHING, JM 1. Department of Mathematics Interdisciplinary Program on Applied Mathematics Building 89 University of Arizona 85721 Tucson AZ USA

An age-structured population is considered in which the birth and death rates of an individual of age a is a function of the density of individuals older and/or younger than a. An existence/uniqueness theorem is proved for the McKendrick equation that governs the dynamics of the age distribution function. This proof shows how a decoupled ordinary differential equation for the total population size can be derived. This result makes a study of the population's asymptotic dynamics (indeed, often its global asymptotic dynamics) mathematically tractable. Several applications to models for intra-specific competition and predation are given.

关键词： AGE-STRUCTURED POPULATION DYNAMICS MCKENDRICK EQUATIONS HIERARCHICAL MODELS EXISTENCE/UNIQUENESS ASYMPTOTIC DYNAMICS GLOBAL STABILITY INTRA-SPECIFIC COMPETITION CANNIBALISM

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Truncated Gaussian RBF Differences are Always Inferior to Finite Differences of the Same Stencil Width

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Communications in Computational Physics 2009年第1期5卷 42-60页

作者： John P.Boyd Lei Wang Department of Atmospheric Oceanic and Space ScienceUniversity of MichiganAnn Arbor MI 48109USA Department of Mathematics and Program in Applied and Interdisciplinary Mathematics University of MichiganAnn Arbor MI 48109USA

Radial basis functions(RBFs)can be used to approximate derivatives and solve differential equations in several ***,we compare one important scheme to ordinary finite differences by a mixture of numerical experiments and theoretical Fourier analysis,that is,by deriving and discussing analytical formulas for the error in differentiating exp(ikx)for arbitrary k.‘Truncated RBF differences”are derived from the same strategy as Fourier and Chebyshev pseudospectral methods:Differentiation of the Fourier,Chebyshev or RBF interpolant generates a differentiation matrix that maps the grid point values or samples of a function u(x)into the values of its derivative on the *** Fourier and Chebyshev interpolants,the action of the differentiation matrix can be computed indirectly but efficiently by the Fast Fourier Transform(FFT).For RBF functions,alas,the FFT is inapplicable and direct use of the dense differentiation matrix on a grid of N points is prohibitively expensive(O(N2))unless N is ***,for Gaussian RBFs,which are exponentially localized,there is another option,which is to truncate the dense matrix to a banded matrix,yielding“truncated RBF differences”.The resulting formulas are identical in form to finite differences except for the difference *** a grid of spacing h with the RBF asφ(x)=exp(−α^(2)(x/h)^(2)),d f dx(0)≈∑^(∞)_(m)=1 wm{f(mh)−f(−mh)},where without approximation wm=(−1)m+12α^(2)/sinh(mα^(2)).We derive explicit formula for the differentiation of the linear function,f(X)≡X,and the errors *** show that Gaussian radial basis functions(GARBF),when truncated to give differentiation formulas of stencil width(2M+1),are significantly less accurate than(2M)-th order finite differences of the same stencil *** error of the infinite series(M=∞)decreases exponentially asα→***,truncated GARBF series have a second error(truncation error)that grows exponentially asα→*** forα∼O(1)where the sum of these two errors is minimized,it is

关键词： Pseudospectral radial basis function high order finite difference nonstandard finite differences spectral differences

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Periodic cycles of nonlinear discrete renewal equations

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Journal of Difference Equations and Applications 1996年第2期2卷 117-137页

作者： J. M. Cushing - [a] [a] Department of Mathematics Interdisciplinary Program on Applied Mathematics Tucson AZ

A general class of discrete, nonlinear renewal equations containing a real parameter is studied. Bifurcation theory methods are used to prove the existence of nontrivial periodic solutions and asymptotically periodic solutions. Fundamental to the approach is the “limit equation” whose periodic solutions are shown to be asymptotic limits of solutions of the renewal equation. An application is made to a model of age-structured population dynamics in which the bifurcation of nontrivial equilibria and 2-cycles is shown to occur with increasing inherent net reproductive value.

关键词： 39A10 difference equations renewal equation asymptotic dynamics equilibrium periodic solutions bifurcation

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An Evolutionary Beverton-Holt Model

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19th International Conference on Difference Equations and Applications, ICDEA 2013

作者： Cushing, J.M. Department of Mathematics and The Interdisciplinary Program in Applied Mathematics University of Arizona 617 N. Santa Rita Tucson AZ 85721 United States

ISBN: (纸本)9783662441398

The classic Beverton-Holt (discrete logistic) difference equation, which arises in population dynamics, has a globally asymptotically stable equilibrium (for positive initial conditions) if its coefficients are constants. If the coefficients change in time, then the equation becomes nonautonomous and the asymptotic dynamics might not be as simple. One reason the coefficients can change in time is their evolution by natural selection. If the model coefficients are functions of a heritable phenotypic trait subject to natural selection then, by standard methods for modeling evolution, the model becomes a planar system of coupled difference equations, consisting of a Beverton-Holt type equation for the population dynamics and a difference equation for the dynamics of the mean phenotypic trait. We consider a case when the trait equation uncouples from the population dynamic equation and obtain criteria under which the evolutionary system has globally asymptotically stable equilibria or periodic solutions. © Springer-Verlag Berlin Heidelberg 2014.

关键词： Population dynamics

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Preface to the special issue dedicated to Saber Elaydi on the occasion of his 80th birthday

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Journal of Difference Equations and Applications 2024年第10期30卷 1397-1404页

作者： Cushing, Jim Michael Sasu, Adina Luminiţa Department of Mathematics Interdisciplinary Program in Applied Mathematics University of Arizona Tucson AZ United States Department of Mathematics West University of Timişoara Timişoara Romania Academy of Romanian Scientists Bucharest Romania

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