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In this paper, we show Hardy-Rellich identities for polyharmonic operators $\Delta^m$ and radial Laplacian $\Delta_r^m$ in $\mathbb{R}^n$ with Hardy-H\'enon weight $|x|^\alpha$ for all $m, n\in \mathbb{N}, \alpha\in \mathbb{R}$. Moreover, the iterative method is applied to give Hardy-Rellich equalities with general weights on Riemannian manifolds. These identities provide naturally an alternative approach to obtain and improve Hardy-Rellich type inequalities. As example of application, we extend several Rellich inequalities of Tertikas-Zographopoulos (Adv. Math. 2007) to the weighted case; using equality with weights involving logarithmic, we show another new weighted Rellich estimate between integrals of $\Delta u$ and $|\nabla u|$; we establish also a Rellich identity involving the Laplace-Beltrami operator $\Delta_\mathbb{H}$ and the radial Laplacian $\Delta_{\rho, \mathbb{H}}$ of the hyperbolic space $\mathbb{H}^n$, which yields in particular brand-new Rellich inequalities for $\|\Delta_\mathbb{H} u\|$ in $\mathbb{H}^3$ and $\mathbb{H}^4$.

In this paper, we are devoted to studying the positive weak, punctured or distributional solutions to the biharmonic Lane-Emden equation \begin{equation*} \Delta^{2} u=u^{p} \quad \quad \text{in} \ \mathbb{R}^{N}\setminus Z, \end{equation*} where $N\geq5$, $1

We present a novel approach to predicting the pressure and flow rate of flexible electrohydrodynamic pumps using the Kolmogorov-Arnold Network. Inspired by the Kolmogorov-Arnold representation theorem, KAN replaces fixed activation functions with learnable spline-based activation functions, enabling it to approximate complex nonlinear functions more effectively than traditional models like Multi-Layer Perceptron and Random Forest. We evaluated KAN on a dataset of flexible EHD pump parameters and compared its performance against RF, and MLP models. KAN achieved superior predictive accuracy, with Mean Squared Errors of 12.186 and 0.001 for pressure and flow rate predictions, respectively. The symbolic formulas extracted from KAN provided insights into the nonlinear relationships between input parameters and pump performance. These findings demonstrate that KAN offers exceptional accuracy and interpretability, making it a promising alternative for predictive modeling in electrohydrodynamic pumping.

Web image datasets curated online inherently contain ambiguous in-distribution (ID) instances and out-of-distribution (OOD) instances, which we collectively call non-conforming (NC) instances. In many recent approaches for mitigating the negative effects of NC instances, the core implicit assumption is that the NC instances can be found via entropy maximization. For "entropy" to be well-defined, we are interpreting the output prediction vector of an instance as the parameter vector of a multinomial random variable, with respect to some trained model with a softmax output layer. Hence, entropy maximization is based on the idealized assumption that NC instances have predictions that are "almost" uniformly distributed. However, in real-world web image datasets, there are numerous NC instances whose predictions are far from being uniformly distributed. To tackle the limitation of entropy maximization, we propose $(\alpha, \beta)$-generalized KL divergence, $\mathcal{D}_{\text{KL}}^{\alpha, \beta}(p\|q)$, which can be used to identify significantly more NC instances. Theoretical properties of $\mathcal{D}_{\text{KL}}^{\alpha, \beta}(p\|q)$ are proven, and we also show empirically that a simple use of $\mathcal{D}_{\text{KL}}^{\alpha, \beta}(p\|q)$ outperforms all baselines on the NC instance identification task. Building upon $(\alpha,\beta)$-generalized KL divergence, we also introduce a new iterative training framework, GenKL, that identifies and relabels NC instances. When evaluated on three web image datasets, Clothing1M, Food101/Food101N, and mini WebVision 1.0, we achieved new state-of-the-art classification accuracies: $81.34\%$, $85.73\%$ and $78.99\%$/$92.54\%$ (top-1/top-5), respectively., Comment: Published (with open access) at International Journal of Computer Vision (IJCV, 2023). 25 pages, 8 figures. Code is available at: https://github.com/codetopaper/GenKL

In this paper, we establish discrete Hardy-Rellich inequalities on $\mathbb{N}$ with $\Delta^\frac{\ell}{2}$ and optimal constants, for any $\ell \geq 1$. As far as we are aware, these sharp inequalities are new for $\ell \geq 3$. Our approach is to use weighted equalities to get some sharp Hardy inequalities using shifting weights, then to settle the higher order cases by iteration. We provide also a new Hardy-Leray type inequality on $\mathbb{N}$ with the same constant as the continuous setting. Furthermore, the main ideas work also for general graphs or the $\ell^p$ setting.

We present a near-real-time global gridded daily CO$_2$ emissions dataset (GRACED) throughout 2021. GRACED provides gridded CO$_2$ emissions at a 0.1degree*0.1degree spatial resolution and 1-day temporal resolution from cement production and fossil fuel combustion over seven sectors, including industry, power, residential consumption, ground transportation, international aviation, domestic aviation, and international shipping. GRACED is prepared from a near-real-time daily national CO$_2$ emissions estimates (Carbon Monitor), multi-source spatial activity data emissions and satellite NO$_2$ data for time variations of those spatial activity data. GRACED provides the most timely overview of emissions distribution changes, which enables more accurate and timely identification of when and where fossil CO$_2$ emissions have rebounded and decreased. Uncertainty analysis of GRACED gives a grid-level two-sigma uncertainty of value of 19.9% in 2021, indicating the reliability of GRACED was not sacrificed for the sake of higher spatiotemporal resolution that GRACED provides. Continuing to update GRACED in a timely manner could help policymakers monitor energy and climate policies' effectiveness and make adjustments quickly.

In this paper, we are devoted to studying the positive solutions of the following higher order Hardy-H\'enon equation $$ (-\Delta)^{m}u=|x|^{\alpha}u^{p} \quad\mbox{in}~ B_{1}\setminus\{0\}\subset\mathbb{R}^{n} $$ with an isolated singularity at the origin, where $\alpha>-2m$, $m\geq1$ is an integer and $n>2m$. For $10$ is an exact constant.

In this paper, we consider the first order Hardy inequalities using simple equalities. This basic setting not only permits to derive quickly many well-known Hardy inequalities with optimal constants, but also supplies improved or new estimates in miscellaneous situations, such as multipolar potential, the exponential weight, hyperbolic space, Heisenberg group, the edge Laplacian, or the Grushin type operator., Comment: To appear in Commun.Math.Reasearch

A high contrast imaging technique based on an optical vortex coronagraph (OVC) is used to measure the spatial phase profile induced by an air plasma generated by a femtosecond laser pulse. The sensitivity of the OVC method significantly surpassed both in-line holographic and direct imaging methods based on air plasma fluorescence. The estimated phase sensitivity of 0.046 waves provides opportunities for OVC applications in areas such as bioimaging, material characterization, as well as plasma diagnostics.