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1. Free, publicly-accessible full text available July 2, 2025
2. In this paper we derive the best constant for the following$L^{\infty }$-type Gagliardo-Nirenberg interpolation inequality$\begin{equation*} \|u\|_{L^{\infty }}\leq C_{q,\infty ,p} \|u\|^{1-\theta }_{L^{q+1}}\|\nabla u\|^{\theta }_{L^p},\quad \theta =\frac {pd}{dp+(p-d)(q+1)}, \end{equation*}$where parameters$q$and$p$satisfy the conditions$p>d\geq 1$,$q\geq 0$. The best constant$C_{q,\infty ,p}$is given by$\begin{equation*} C_{q,\infty ,p}=\theta ^{-\frac {\theta }{p}}(1-\theta )^{\frac {\theta }{p}}M_c^{-\frac {\theta }{d}},\quad M_câ\int _{\mathbb {R}^d}u_{c,\infty }^{q+1} dx, \end{equation*}$where$u_{c,\infty }$is the unique radial non-increasing solution to a generalized Lane-Emden equation. The case of equality holds when$u=Au_{c,\infty }(\lambda (x-x_0))$for any real numbers$A$,$\lambda >0$and$x_{0}\in \mathbb {R}^d$. In fact, the generalized Lane-Emden equation in$\mathbb {R}^d$contains a delta function as a source and it is a Thomas-Fermi type equation. For$q=0$or$d=1$,$u_{c,\infty }$have closed form solutions expressed in terms of the incomplete Beta functions. Moreover, we show that$u_{c,m}\to u_{c,\infty }$and$C_{q,m,p}\to C_{q,\infty ,p}$as$m\to +\infty$for$d=1$, where$u_{c,m}$and$C_{q,m,p}$are the function achieving equality and the best constant of$L^m$-type Gagliardo-Nirenberg interpolation inequality, respectively.

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Free, publicly-accessible full text available June 1, 2025
3. Free, publicly-accessible full text available May 1, 2025
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5. Free, publicly-accessible full text available December 31, 2024
6. Free, publicly-accessible full text available January 1, 2025
7. Abstract Background

Delta radiomics is a highâthroughput computational technique used to describe quantitative changes in serial, timeâseries imaging by considering the relative change in radiomic features of images extracted at two distinct time points. Recent work has demonstrated a lack of prognostic signal of radiomic features extracted using this technique. We hypothesize that this lack of signal is due to the fundamental assumptions made when extracting features via delta radiomics, and that other methods should be investigated.

Purpose

The purpose of this work was to show a proofâofâconcept of a new radiomics paradigm for sparse, timeâseries imaging data, where features are extracted from a spatialâtemporal manifold modeling the time evolution between images, and to assess the prognostic value on patients with oropharyngeal cancer (OPC).

Methods

To accomplish this, we developed an algorithm to mathematically describe the relationship between two images acquired at time and . These images serve as boundary conditions of a partial differential equation describing the transition from one image to the other. To solve this equation, we propagate the position and momentum of each voxel according to FokkerâPlanck dynamics (i.e., a technique common in statistical mechanics). This transformation is driven by an underlying potential force uniquely determined by the equilibrium image. The solution generates a spatialâtemporal manifold (3 spatial dimensions + time) from which we define dynamic radiomic features. First, our approach was numerically verified by stochastically sampling dynamic Gaussian processes of monotonically decreasing noise. The transformation from high to low noise was compared between our FokkerâPlanck estimation and simulated groundâtruth. To demonstrate feasibility and clinical impact, we applied our approach to18FâFDGâPET images to estimate early metabolic response of patients (nÂ =Â 57) undergoing definitive (chemo)radiation for OPC. Images were acquired preâtreatment and 2âweeks intraâtreatment (after 20Â Gy). Dynamic radiomic features capturing changes in texture and morphology were then extracted. Patients were partitioned into two groups based on similar dynamic radiomic feature expression via kâmeans clustering and compared by KaplanâMeier analyses with logârank tests (pÂ <Â 0.05). These results were compared to conventional delta radiomics to test the added value of our approach.

Results

Numerical results confirmed our technique can recover image noise characteristics given sparse input data as boundary conditions. Our technique was able to model tumor shrinkage and metabolic response. While no delta radiomics features proved prognostic, KaplanâMeier analyses identified nine significant dynamic radiomic features. The most significant feature was GrayâLevelâSizeâZoneâMatrix grayâlevel variance (pÂ =Â 0.011), which demonstrated prognostic improvement over its corresponding delta radiomic feature (pÂ =Â 0.722).

Conclusions

We developed, verified, and demonstrated the prognostic value of a novel, physicsâbased radiomics approach over conventional delta radiomics via data assimilation of quantitative imaging and differential equations.

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Free, publicly-accessible full text available May 1, 2025
8. Free, publicly-accessible full text available January 1, 2025
9. We propose a high-order stochasticâstatistical moment closure model for efficient ensemble prediction of leading-order statistical moments and probability density functions in multiscale complex turbulent systems. The statistical moment equations are closed by a precise calibration of the high-order feedbacks using ensemble solutions of the consistent stochastic equations, suitable for modeling complex phenomena including non-Gaussian statistics and extreme events. To address challenges associated with closely coupled spatiotemporal scales in turbulent states and expensive large ensemble simulation for high-dimensional systems, we introduce efficient computational strategies using the random batch method (RBM). This approach significantly reduces the required ensemble size while accurately capturing essential high-order structures. Only a small batch of small-scale fluctuation modes is used for each time update of the samples, and exact convergence to the full model statistics is ensured through frequent resampling of the batches during time evolution. Furthermore, we develop a reduced-order model to handle systems with really high dimensions by linking the large number of small-scale fluctuation modes to ensemble samples of dominant leading modes. The effectiveness of the proposed models is validated by numerical experiments on the one-layer and two-layer Lorenz â96 systems, which exhibit representative chaotic features and various statistical regimes. The full and reduced-order RBM models demonstrate uniformly high skill in capturing the time evolution of crucial leading-order statistics, non-Gaussian probability distributions, while achieving significantly lower computational cost compared to direct Monte-Carlo approaches. The models provide effective tools for a wide range of real-world applications in prediction, uncertainty quantification, and data assimilation.

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Free, publicly-accessible full text available October 1, 2024