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It can be shown that Stable Diffusion has a permutation-invariance property with respect to the rows of Contrastive Language-Image Pretraining (CLIP) embedding matrices. This inspired the novel observation that these embeddings can…
This paper is concerned with various aspects of the Slepian process $(B_{t+1} - B_t, t \ge 0)$ derived from a one-dimensional Brownian motion $(B_t, t \ge 0 )$. In particular, we offer an analysis of the local structure of the Slepian zero…
The paper is devoted to the construction of an optimal interpolation formula in $K_2(P_2)$ Hilbert space. Here the interpolation formula consists of a linear combination $\sum_{\beta=0}^NC_{\beta}(z)\varphi(x_\beta)$ of given values of a…
In this work, we introduce the Kaiser-Bessel interpolation basis for the particle-mesh interpolation in the fast Ewald method. A reliable a priori error estimate is developed to measure the accuracy of the force computation in correlated…
This paper presents a general high-order kernel regularization technique applicable to all four integral operators of Calder\'on calculus associated with linear elliptic PDEs in two and three spatial dimensions. Like previous density…
The generalization ability of kernel interpolation in large dimensions (i.e., $n \asymp d^{\gamma}$ for some $\gamma>0$) might be one of the most interesting problems in the recent renaissance of kernel regression, since it may help us…
Multiresolution analyses based upon interpolets, interpolating scaling functions introduced by Deslauriers and Dubuc, are particularly well-suited to physical applications because they allow exact recovery of the multiresolution…
In this paper we consider a punctured Riemann surface endowed with a Hermitian metric which equals the Poincar\'e metric near the punctures and a holomorphic line bundle which polarizes the metric. We show that the Bergman kernel can be…
We present a novel derivative-free interpolation based optimization algorithm. A trust-region method is used where a surrogate model is realized via an interpolation framework. The framework for interpolation is provided by Universal…
We study in this paper the well-posedness and stability of three structures with interfacial slip and two infinite memories effective on the transverse displacement and the rotation angle. We consider a large class of kernels and prove that…
We provide a new construction of Brownian disks in terms of forests of continuous random trees equipped with nonnegative labels corresponding to distances from a distinguished point uniformly distributed on the boundary of the disk. This…
This short paper announces the main results of \cite{SBB2026}, where the Schr\"odinger--Bass Bridge (SBB) problem is introduced and studied in full generality. Here we provide a direct PDE derivation of the SBB system in dimension one,…
We prove that the spatial coagulation equation with bounded coagulation rate is well-posed for all times in a given class of kernels if the convection term of the underlying particle dynamics has divergence bounded below by a positive…
The polynomial kernels are widely used in machine learning and they are one of the default choices to develop kernel-based classification and regression models. However, they are rarely used and considered in numerical analysis due to their…
We prove the existence of solutions to a non-linear, non-local, degenerate equation which was previously derived as the formal hydrodynamic limit of an active Brownian particle system, where the particles are endowed with a position and an…
We develop a theoretical framework for generalization in the interpolating regime of statistical learning. The central question is why highly overparameterized estimators can attain zero empirical risk while still achieving nontrivial…
In this note we present a new proof of the Carleson Embedding Theorem on the unit disc and unit ball. The only technical tool used in the proof of this fact is Green's formula. The starting point is that every Carleson measure gives rise to…
In this paper we develop a novel framework for numerically solving scalar conservation laws in one space dimension. Utilizing the method of characteristics in conjunction with the equal area principle we develop an approach where the weak…
The fast Ewald methods are widely used to compute the point-charge electrostatic interactions in molecular simulations. The key step that introduces errors in the computation is the particle-mesh interpolation. In this work, the optimal…
Current Lagrangian (particle-tracking) algorithms used to simulate diffusion-reaction equations must employ a certain number of particles to properly emulate the system dynamics---particularly for imperfectly-mixed systems. The number of…