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Free, publiclyaccessible full text available July 2, 2025

In this paper we derive the best constant for the following
$L^{\infty }$ type GagliardoNirenberg interpolation inequality$\begin{equation*} \u\_{L^{\infty }}\leq C_{q,\infty ,p} \u\^{1\theta }_{L^{q+1}}\\nabla u\^{\theta }_{L^p},\quad \theta =\frac {pd}{dp+(pd)(q+1)}, \end{equation*}$ 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 nonincreasing solution to a generalized LaneEmden equation. The case of equality holds when$u=Au_{c,\infty }(\lambda (xx_0))$ for any real numbers$A$ ,$\lambda >0$ and$x_{0}\in \mathbb {R}^d$ . In fact, the generalized LaneEmden equation in$\mathbb {R}^d$ contains a delta function as a source and it is a ThomasFermi 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 GagliardoNirenberg interpolation inequality, respectively.Free, publiclyaccessible full text available June 1, 2025 
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Free, publiclyaccessible full text available January 1, 2025

Abstract Background Delta radiomics is a highâthroughput computational technique used to describe quantitative changes in serial, timeâseries 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 proofâofâconcept of a new radiomics paradigm for sparse, timeâseries imaging data, where features are extracted from a spatialâtemporal 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 FokkerâPlanck 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 spatialâtemporal 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 FokkerâPlanck estimation and simulated groundâtruth. To demonstrate feasibility and clinical impact, we applied our approach to^{18}FâFDGâPET images to estimate early metabolic response of patients (
n Â =Â 57) undergoing definitive (chemo)radiation for OPC. Images were acquired preâtreatment and 2âweeks intraâtreatment (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 kâmeans clustering and compared by KaplanâMeier analyses with logârank tests (p Â <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, KaplanâMeier analyses identified nine significant dynamic radiomic features. The most significant feature was GrayâLevelâSizeâZoneâMatrix grayâlevel variance (
p Â =Â 0.011), which demonstrated prognostic improvement over its corresponding delta radiomic feature (p Â =Â 0.722).Conclusions We developed, verified, and demonstrated the prognostic value of a novel, physicsâbased radiomics approach over conventional delta radiomics via data assimilation of quantitative imaging and differential equations.
Free, publiclyaccessible full text available May 1, 2025 
Free, publiclyaccessible full text available January 1, 2025

We propose a highorder stochasticâstatistical moment closure model for efficient ensemble prediction of leadingorder statistical moments and probability density functions in multiscale complex turbulent systems. The statistical moment equations are closed by a precise calibration of the highorder feedbacks using ensemble solutions of the consistent stochastic equations, suitable for modeling complex phenomena including nonGaussian statistics and extreme events. To address challenges associated with closely coupled spatiotemporal scales in turbulent states and expensive large ensemble simulation for highdimensional systems, we introduce efficient computational strategies using the random batch method (RBM). This approach significantly reduces the required ensemble size while accurately capturing essential highorder structures. Only a small batch of smallscale 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 reducedorder model to handle systems with really high dimensions by linking the large number of smallscale fluctuation modes to ensemble samples of dominant leading modes. The effectiveness of the proposed models is validated by numerical experiments on the onelayer and twolayer Lorenz â96 systems, which exhibit representative chaotic features and various statistical regimes. The full and reducedorder RBM models demonstrate uniformly high skill in capturing the time evolution of crucial leadingorder statistics, nonGaussian probability distributions, while achieving significantly lower computational cost compared to direct MonteCarlo approaches. The models provide effective tools for a wide range of realworld applications in prediction, uncertainty quantification, and data assimilation.
Free, publiclyaccessible full text available October 1, 2024