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1. Recovery of a time-dependent potential in hyperbolic equations on conformally transversally anisotropic manifolds

   https://doi.org/10.4171/jst/547  

   Liu, Boya; Saksala, Teemu; Yan, Lili (March 2025, Journal of Spectral Theory)

   We study an inverse problem of determining a time-dependent potential appearing in the wave equation on conformally transversally anisotropic manifolds of dimension three or higher. These are compact Riemannian manifolds with boundary that are conformally embedded in a product of the real line and a transversal manifold. Under the assumption of the attenuated geodesic ray transform being injective on the transversal manifold, we prove the unique determination of time-dependent potentials from the knowledge of a certain partial Cauchy data set. 

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2. Partial Data Inverse Problem for Hyperbolic Equation with Time-Dependent Damping Coefficient and Potential

   https://doi.org/10.1137/23M1588676  

   Liu, Boya; Saksala, Teemu; Yan, Lili (August 2024, SIAM Journal on Mathematical Analysis)

   We study an inverse problem of determining a time-dependent damping coefficient and potential appearing in the wave equation in a compact Riemannian manifold of dimension three or higher. More specifically, we are concerned with the case of conformally transversally anisotropic manifolds, or in other words, compact Riemannian manifolds with boundary conformally embedded in a product of the Euclidean line and a transversal manifold. With an additional assumption of the attenuated geodesic ray transform being injective on the transversal manifold, we prove that the knowledge of a certain partial Cauchy data set determines the time-dependent damping coefficient and potential uniquely. 

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3. Universal deformations and inhomogeneities in isotropic Cauchy elasticity

   https://doi.org/10.1098/rspa.2024.0229  

   Yavari, Arash (July 2024, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   For a given class of materials, universal deformations are those deformations that can be maintained in the absence of body forces and by applying solely boundary tractions. For inhomogeneous bodies, in addition to the universality constraints that determine the universal deformations, there are extra constraints on the form of the material inhomogeneities—universal inhomogeneity constraints. Those inhomogeneities compatible with the universal inhomogeneity constraints are called universal inhomogeneities. In a Cauchy elastic solid, stress at a given point and at an instance of time is a function of strain at that point and that exact moment in time, without any dependence on prior history. A Cauchy elastic solid does not necessarily have an energy function, i.e. Cauchy elastic solids are, in general, non-hyperelastic (or non-Green elastic). In this paper, we characterize universal deformations in both compressible and incompressible inhomogeneous isotropic Cauchy elasticity. As Cauchy elasticity includes hyperelasticity, one expects the universal deformations of Cauchy elasticity to be a subset of those of hyperelasticity both in compressible and incompressible cases. It is also expected that the universal inhomogeneity constraints to be more stringent than those of hyperelasticity, and hence, the set of universal inhomogeneities to be smaller than that of hyperelasticity. We prove the somewhat unexpected result that the sets of universal deformations of isotropic Cauchy elasticity and isotropic hyperelasticity are identical, in both the compressible and incompressible cases. We also prove that their corresponding universal inhomogeneities are identical as well. 

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4. The Cauchy Problem for Boltzmann Bi-linear Systems: The Mixing of Monatomic and Polyatomic Gases

   https://doi.org/10.1007/s10955-023-03221-4  

   Alonso, Ricardo_J; Čolić, Milana; Gamba, Irene_M (January 2024, Journal of Statistical Physics)

   Abstract From a unified vision of vector valued solutions in weighted Banach spaces, this paper establishes the existence and uniqueness for space homogeneous Boltzmann bi-linear systems with conservative collisional forms arising in complex gas dynamical structures. This broader vision is directly applied to dilute multi-component gas mixtures composed of both monatomic and polyatomic gases. Such models can be viewed as extensions of scalar Boltzmann binary elastic flows, as much as monatomic gas mixtures with disparate masses and single polyatomic gases, providing a unified approach for vector valued solutions in weighted Banach spaces. Novel aspects of this work include developing the extension of a general ODE theory in vector valued weighted Banach spaces, precise lower bounds for the collision frequency in terms of the weighted Banach norm, energy identities, angular or compact manifold averaging lemmas which provide coerciveness resulting into global in time stability, a new combinatorics estimate forp-binomial forms producing sharper estimates for thek-moments of bi-linear collisional forms. These techniques enable the Cauchy problem improvement that resolves the model with initial data corresponding to strictly positive and bounded initial vector valued mass and total energy, in addition to only a$$2^+$$ $2^{+}$moment determined by the hard potential rates discrepancy, a result comparable in generality to the classical Cauchy theory of the scalar homogeneous Boltzmann equation. 

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5. A unified combination framework for dependent tests with applications to microbiome association studies

   https://doi.org/10.1093/biomtc/ujaf001  

   Yu, Xiufan; Zhang, Linjun; Srinivasan, Arun; Xie, Min-ge; Xue, Lingzhou (January 2025, Biometrics)

   ABSTRACT We introduce a novel meta-analysis framework to combine dependent tests under a general setting, and utilize it to synthesize various microbiome association tests that are calculated from the same dataset. Our development builds upon the classical meta-analysis methods of aggregating P-values and also a more recent general method of combining confidence distributions, but makes generalizations to handle dependent tests. The proposed framework ensures rigorous statistical guarantees, and we provide a comprehensive study and compare it with various existing dependent combination methods. Notably, we demonstrate that the widely used Cauchy combination method for dependent tests, referred to as the vanilla Cauchy combination in this article, can be viewed as a special case within our framework. Moreover, the proposed framework provides a way to address the problem when the distributional assumptions underlying the vanilla Cauchy combination are violated. Our numerical results demonstrate that ignoring the dependence among the to-be-combined components may lead to a severe size distortion phenomenon. Compared to the existing P-value combination methods, including the vanilla Cauchy combination method and other methods, the proposed combination framework is flexible and can be adapted to handle the dependence accurately and utilizes the information efficiently to construct tests with accurate size and enhanced power. The development is applied to the microbiome association studies, where we aggregate information from multiple existing tests using the same dataset. The combined tests harness the strengths of each individual test across a wide range of alternative spaces, enabling more efficient and meaningful discoveries of vital microbiome associations. 

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