In this paper we derive quantitative boundary Hölder estimates, with explicit constants, for the inhomogeneous Poisson problem in a bounded open set $D\subset \mathbb{R}^d$. Our approach has two main steps: firstly, we consider an arbitrary $D$ as above and prove that the boundary $\alpha$-Hölder regularity of the solution the Poisson equation is controlled, with explicit constants, by the Hölder seminorm of the boundary data, the $L^ \gamma$-norm of the forcing term with $\gamma>d/2$, and the $\alpha/2$-moment of the exit time from $D$ of the Brownian motion. Secondly, we derive explicit estimates for the $\alpha/2$-moment of the exit time in terms of the distance to the boundary, the regularity of the domain $D$, and $\alpha$. Using this approach, we derive explicit estimates for the same problem in domains satisfying exterior ball conditions, respectively exterior cone/wedge conditions, in terms of simple geometric features. As a consequence we also obtain explicit constants for pointwise estimates for the Green function and for the gradient of the solution. The obtained estimates can be employed to bypass the curse of high dimensions when aiming to approximate the solution of the Poisson problem using neural networks, obtaining polynomial scaling with dimension, which in some cases can be shown to be optimal.

We present the first fixed-length elementary closed-form expressions for the prime-counting function, $\pi(n)$, and the $n$-th prime number, $p(n)$. These expressions are arithmetic terms, requiring only a finite and fixed number of elementary arithmetic operations from the set: addition, subtraction, multiplication, integer division, and exponentiation. Mazzanti proved that every Kalmar function can be represented as an arithmetic term. We develop an arithmetic term representing the prime omega function, $\omega(n)$, which counts the number of distinct prime divisors of a positive integer $n$. From this term, we find immediately an arithmetic term for the prime-counting function, $\pi(n)$. Combining these results with a new arithmetic term for binomial coefficients and novel prime-related exponential Diophantine equations, we manage to develop an arithmetic term for the $n$-th prime number, $p(n)$, thereby providing a constructive solution to the fundamental question: Is there an order to the primes?

Solving fish segmentation in underwater videos, a real-world problem of great practical value in marine and aquaculture industry, is a challenging task due to the difficulty of the filming environment, poor visibility and limited existing annotated underwater fish data. In order to overcome these obstacles, we introduce a novel two stage unsupervised segmentation approach that requires no human annotations and combines artificially created and real images. Our method generates challenging synthetic training data, by placing virtual fish in real-world underwater habitats, after performing fish transformations such as Thin Plate Spline shape warping and color Histogram Matching, which realistically integrate synthetic fish into the backgrounds, making the generated images increasingly closer to the real world data with every stage of our approach. While we validate our unsupervised method on the popular DeepFish dataset, obtaining a performance close to a fully-supervised SoTA model, we further show its effectiveness on the specific case of salmon segmentation in underwater videos, for which we introduce DeepSalmon, the largest dataset of its kind in the literature (30 GB). Moreover, on both datasets we prove the capability of our approach to boost the performance of the fully-supervised SoTA model.

Frobenius monoidal functors preserve duals. We show that conversely, (co)monoidal functors between autonomous categories which preserve duals are Frobenius monoidal. We apply this result to linearly distributive functors between autonomous categories.

This note proves a result on the existence of barycenters in a class of uniformly convex geodesic spaces.

In this paper, we prove that in the overparametrized regime, deep neural network provide universal approximations and can interpolate any data set, as long as the activation function is locally in $L^1(\RR)$ and not an affine function. Additionally, if the activation function is smooth and such an interpolation networks exists, then the set of parameters which interpolate forms a manifold. Furthermore, we give a characterization of the Hessian of the loss function evaluated at the interpolation points. In the last section, we provide a practical probabilistic method of finding such a point under general conditions on the activation function.

In these notes we propose a new, simpler proof system for first-order matching logic with application and definedness. The new proof system is inspired by Tarski's axiomatization for first order-logic with equality (simplified by Kalish and Montague), that does not involve the notions of a free variable and free substitution. We give also a proof system for first-order matching logic with application, obtained by adapting to matching logic Gödel's proof system for first-order intuitionistic logic.

Finding a low-weight multiple (LWPM) of a given polynomial is very useful in the cryptanalysis of stream ciphers and arithmetic in finite fields. There is no known deterministic polynomial time complexity algorithm for solving this problem, and the most efficient algorithms are based on a time/memory trade-off. The widespread perception is that this problem is difficult. In this paper, we establish a relationship between the LWPM problem and the MAX-SAT problem of determining an assignment that maximizes the number of valid clauses of a system of affine Boolean clauses. This relationship shows that any algorithm that can compute the optimum of a MAX-SAT instance can also compute the optimum of an equivalent LWPM instance. It also confirms the perception that the LWPM problem is difficult.

We introduce Generalized Test-Time Augmentation (GTTA), a highly effective method for improving the performance of a trained model, which unlike other existing Test-Time Augmentation approaches from the literature is general enough to be used off-the-shelf for many vision and non-vision tasks, such as classification, regression, image segmentation and object detection. By applying a new general data transformation, that randomly perturbs multiple times the PCA subspace projection of a test input, GTTA forms robust ensembles at test time in which, due to sound statistical properties, the structural and systematic noises in the initial input data is filtered out and final estimator errors are reduced. Different from other existing methods, we also propose a final self-supervised learning stage in which the ensemble output, acting as an unsupervised teacher, is used to train the initial single student model, thus reducing significantly the test time computational cost, at no loss in accuracy. Our tests and comparisons to strong TTA approaches and SoTA models on various vision and non-vision well-known datasets and tasks, such as image classification and segmentation, speech recognition and house price prediction, validate the generality of the proposed GTTA. Furthermore, we also prove its effectiveness on the more specific real-world task of salmon segmentation and detection in low-visibility underwater videos, for which we introduce DeepSalmon, the largest dataset of its kind in the literature.

Quantifying the gap between synthetic and real-world imagery is essential for improving both transformer-based models - that rely on large volumes of data - and datasets, especially in underexplored domains like aerial scene understanding where the potential impact is significant. This paper introduces a novel methodology for scene complexity assessment using Multi-Model Consensus Metric (MMCM) and depth-based structural metrics, enabling a robust evaluation of perceptual and structural disparities between domains. Our experimental analysis, utilizing real-world (Dronescapes) and synthetic (Skyscenes) datasets, demonstrates that real-world scenes generally exhibit higher consensus among state-of-the-art vision transformers, while synthetic scenes show greater variability and challenge model adaptability. The results underline the inherent complexities and domain gaps, emphasizing the need for enhanced simulation fidelity and model generalization. This work provides critical insights into the interplay between domain characteristics and model performance, offering a pathway for improved domain adaptation strategies in aerial scene understanding.

We adapt a continuous logic axiomatization of tracial von Neumann algebras due to Farah, Hart and Sherman in order to prove a metatheorem for this class of structures in the style of proof mining, a research program that aims to obtain the hidden computational content of ordinary mathematical proofs using tools from proof theory.

The main non-associative algebras are Lie algebras and Jordan algebras. There are several ways to unify these non-associative algebras and associative algebras.

In this paper we deal with the problem of characterizing those generalized Mehler semigroups that do correspond to càdlàg Markov processes, which is highly non-trivial and has remained open for more than a decade. Our approach is to reconsider the {\it càdlàg problem} for generalized Mehler semigroups as a particular case of the much broader problem of constructing Hunt processes from a given Markov semigroup. Following this strategy, a consistent part of this work is devoted to prove that starting from a Markov semigroup on a general (possibly non-metrizable) state space, the existence of a suitable Lyapunov function with relatively compact sub/sup-sets in conjunction with a local Feller-type regularity of the resolvent are sufficient to ensure the existence of an associated càdlàg Markov process; if the topology is locally generated by potentials, then the process is in fact Hunt. Other results of fine potential theoretic nature are also pointed out, an important one being the fact that the Hunt property of a process is stable under the change of the topology, as long as it is locally generated by potentials. Then, we derive sufficient conditions for a large class of generalized Mehler semigroups in order to posses an associated Hunt process with values in the original space. To this end, we first construct explicit Lyapunov functions whose sub-level sets are relatively compact with respect to the (non-metrizable) weak topology, and then we use the above mentioned stability to deduce the Hunt property with respect to the stronger norm topology. We test these conditions on a stochastic heat equation on $L^2(D)$ whose drift is the Dirichlet Laplacian on a bounded domain $D \subset \mathbb{R}^d$, driven by a (non-diagonal) Lévy noise whose characteristic exponent is not necessarily Sazonov continuous.

We consider periodic (pseudo)differential {elliptic operators of Schrödinger type} perturbed by weak magnetic fields not vanishing at infinity, and extend our previous analysis in \cite{CIP,CHP-2,CHP-4} to the case {of a semimetal having a finite family of Bloch eigenvalues whose range may overlap with the other Bloch bands but remains isolated at each fixed quasi-momentum.} We do not make any assumption of triviality for the associated Bloch bundle. In this setting, we formulate a general form of the Peierls-Onsager substitution {via strongly localized tight-frames and magnetic matrices. We also} prove the existence of an approximate time evolution for initial states supported inside the range of the isolated Bloch family, with a precise error control.

In this paper we use proof mining methods to compute rates of ($T$-)asymptotic regularity of the generalized Krasnoselskii-Mann-type iteration associated to a nonexpansive mapping $T:X\to X$ in a uniformly convex normed space $X$. For special choices of the parameter sequences, we obtain quadratic rates.

We introduce and study a three-folded linear operator depending on three parameters that has associated a triangular number tilling of the plane. As a result the set of all triples of integers is decomposed in classes of equivalence organized in four towers of two-dimensional triangulations. We provide the full characterization of the represented integers belonging to each network as families of certain quadratic forms. We note that one of the towers is generated by a germ that produces a covering of the plane with {Löschian} numbers.

In 1983, Z\u{a}linescu showed that the squared norm of a uniformly convex normed space is uniformly convex on bounded subsets. We extend this result to the metric setting of uniformly convex hyperbolic spaces. We derive applications to the convergence of shadow sequences and to proximal minimization.

Generating novel views from recorded videos is crucial for enabling autonomous UAV navigation. Recent advancements in neural rendering have facilitated the rapid development of methods capable of rendering new trajectories. However, these methods often fail to generalize well to regions far from the training data without an optimized flight path, leading to suboptimal reconstructions. We propose a self-supervised cyclic neural-analytic pipeline that combines high-quality neural rendering outputs with precise geometric insights from analytical methods. Our solution improves RGB and mesh reconstructions for novel view synthesis, especially in undersampled areas and regions that are completely different from the training dataset. We use an effective transformer-based architecture for image reconstruction to refine and adapt the synthesis process, enabling effective handling of novel, unseen poses without relying on extensive labeled datasets. Our findings demonstrate substantial improvements in rendering views of novel and also 3D reconstruction, which to the best of our knowledge is a first, setting a new standard for autonomous navigation in complex outdoor environments.

In this paper, we investigate the following nonlinear Schr\"odinger equation with Neumann boundary conditions: \begin{equation*} \begin{cases} -\Delta u+ \lambda u= f(u) & {\rm in} \,~ \Omega,\\ \displaystyle\frac{\partial u}{\partial \nu}=0 \, &{\rm on}\,~\partial \Omega \end{cases} \end{equation*} coupled with a constraint condition: \begin{equation*} \int_{\Omega}|u|^2 dx=c, \end{equation*} where $\Omega\subset \mathbb{R}^N(N\ge3)$ denotes a smooth bounded domain, $\nu$ represents the unit outer normal vector to $\partial \Omega$,$c$is a positive constant, and$\lambda$ acts as a Lagrange multiplier. When the nonlinearity $f$ exhibits a general mass supercritical growth at infinity, we establish the existence of normalized solutions, which are not necessarily positive solutions and can be characterized as mountain pass type critical points of the associated constraint functional. Our approach provides a uniform treatment of various nonlinearities, including cases such as $f(u)=|u|^{p-2}u$, $|u|^{q-2}u+ |u|^{p-2}u$, and $-|u|^{q-2}u+|u|^{p-2}u$, where $2

This article presents a construction of the concept of stochastic integration in Riemannian manifolds from a purely functional-analytic point of view. We show that there are infinitely many such integrals, and that any two of them are related by a simple formula. We also find that the Stratonovich and Itô integrals known to probability theorists are two instances of the general concept constructed herein.