Tootfinder

I was recently asked where I would see useful applications of CCS. My answer was: Cement🧱🪨 and Biomethane🐄🌱🏭
Most won't be surprised that I said Cement. Yet, Biomethane is not what people usually have in mind. But bear with me, it makes a lot of sense.
CCS is a controversial topic to begin with. That has a lot to do with the fact that many see it - often rightfully so - as a delay strategy by the fossil fuel🛢️ industry.

Will AI in the classroom mean students asking ChatGPT for help with algebra homework, or strapping on a headband that streams their brainwaves to teachers and parents? In China, teachers are already policing attention with the latter
https://www.linked…

Publish to Perish: Prompt Injection Attacks on LLM-Assisted Peer Review
Matteo Gioele Collu, Umberto Salviati, Roberto Confalonieri, Mauro Conti, Giovanni Apruzzese
https://arxiv.org/abs/2508.20863

Publish to Perish: Prompt Injection Attacks on LLM-Assisted Peer Review
Large Language Models (LLMs) are increasingly being integrated into the scientific peer-review process, raising new questions about their reliability and resilience to manipulation. In this work, we investigate the potential for hidden prompt injection attacks, where authors embed adversarial text within a paper's PDF to influence the LLM-generated review. We begin by formalising three distinct threat models that envision attackers with different motivations -- not all of which implying malicio…

I left a comment last week on Smashing’s “Design Patterns For AI Interfaces” (which is still in moderation):
https://www.smashingmagazine.com/2025/07/design-patterns-ai-interfaces/
Essentially, if your UI doesn’t also try to prevent harm then it might …

Design Patterns For AI Interfaces — Smashing Magazine
Designing a new AI feature? Where do you even begin? Here’s a simple, practical overview with useful design patterns for better AI experiences.

A Crime/S.I.R. optimal control problem
Mariana \'Alvarez, Alexander Alegr\'ia, Andr\'es Rivera, Sebasti\'an Pedersen
https://arxiv.org/abs/2509.21406 https://

A Crime/S.I.R. optimal control problem
This paper presents and discusses a mathematical model inspired by control theory to derive optimal public policies for minimizing costs associated with the reduction and control of criminal activity in a population. Specifically, we analyze the optimal control problem \begin{equation*} \min G(u_1, u_2, u_3) = \int_{0}^{t_{\text{F}}} \left( I(t) - R(t) + \frac{B_1}{2} u_1^2(t) + \frac{B_2}{2} u_2^2(t) + \frac{B_3}{2} u_3^2(t) \right) \, dt. \end{equation*} where $I=I(t)$ and $R=R(t)$ satisfies …

'Negotiation can begin where the front line currently stands' — Zelensky says ahead of meeting with Trump: https://benborges.xyz/2025/08/17/negotiation-can-begin-where-the.html

Multiplicative trace and spectrum preservers on stochastic matrices
Ming-Cheng Tsai, Huajun Huang
https://arxiv.org/abs/2509.22743 https://arxiv.org/pdf/25…

Multiplicative trace and spectrum preservers on stochastic matrices
We characterize maps $ϕ_i: \mathcal{S} \to \mathcal{S}$, $i=1, \ldots, m$ and $m\ge 1$, that have the multiplicative spectrum or trace preserving property: \begin{eqnarray*} \textrm{spec} (ϕ_1(A_1)\cdots ϕ_m(A_m)) &=& \textrm{spec} (A_1\cdots A_m),\quad\text{or}\quad \textrm{tr} (ϕ_1(A_1)\cdots ϕ_m(A_m)) &=& \textrm{tr} (A_1\cdots A_m), \end{eqnarray*} where $\mathcal{S}$ is the set of $n\times n$ doubly stochastic, row stochastic, or column stochastic matrices, or the space span…

Robust Data-Driven Quasiconcave Optimization
Jian Wu, William B. Haskell, Wenjie Huang, Huifu Xu
https://arxiv.org/abs/2508.19787 https://arxiv.org/pdf/250…

Robust Data-Driven Quasiconcave Optimization
We investigate a data-driven quasiconcave maximization problem where information about the objective function is limited to a finite sample of data points. We begin by defining an ambiguity set for admissible objective functions based on available partial information about the objective. This ambiguity set consists of those quasiconcave functions that majorize a given data sample, and that satisfy additional functional properties (monotonicity, Lipschitz continuity, and permutation invariance).…

need to figure out where my roommate put the legs for my bedframe so I can maybe begin to give my room some semblance of logic

Gosh where to begin?... - Dr. John Kuhn - Abilene ISD Superintendent | Facebook https://www.facebook.com/permalink.php?story_fbid=pfbid02QrSwNt1KoQvvDyw3itEaeF7V6SfAqmuwKXR8t3BfQbcrX1Mcq6MYsrttsyPZK1KU…

Can You Hear Us?: Five Voices from Gaza – Hot Press Magazine, Ireland
https://www.hotpress.com/opinion/can-you-hear-us-five-voices-from-gaza-23113207
The five voices are those of my partner at Gaza Verified, @…

Can You Hear Us?: Five Voices from Gaza
It is hard for us, here in the relative comfort and security of Ireland, to even begin to imagine what is has been – and what it is – like in Gaza, where...

Hook immanantal equalities for linear combination matrices of (di)graphs and their applications
Xiangshuai Dong, Tingzeng Wu, HongJian Lai
https://arxiv.org/abs/2508.17743 https…

Hook immanantal equalities for linear combination matrices of (di)graphs and their applications
Let $χ_λ$ be an irreducible character of the symmetric group $S_n$. For an $n \times n$ matrix $M = (m_{ij})$, define the immanant of $M$ corresponding to $χ_λ$ by \begin{eqnarray*} d_λ(M) = \sum_{σ\in S_n} χ_λ(σ) \prod_{i=1}^n m_{iσ(i)}. \end{eqnarray*} For $λ= (k, 1^{n-k})$, the immanant $d_{(k, 1^{n-k})}(M)$ is called the hook immanant and denoted by $d_k(M)$. The hook immanant polynomial of matrix $M$ is defined as $d_{k}(xI_n - M)$, where $I_n$ is the $n \times n$ identity matri…

Toric surface codes and the periodicity of polytopes
Amelia Gibbs, Eliza Hogan, Kelly Jabbusch, Jenna Plute, Nicholas Toloczko
https://arxiv.org/abs/2509.21178 https://

Toric surface codes and the periodicity of polytopes
Toric codes are error-correcting codes that are derived from toric varieties, which hold a unique correspondence to integral convex polytopes. In this paper, we focus on integral convex polytopes $P \subseteq \mathbb{R}^2$ and the toric codes they define. We begin by studying period-1 polytopes -- polytopes satisfying the property $L(tP)$ = $tL(P)$ for all $t \in \mathbb{Z}^+$, where $tP$ is the $t$-dilate of $P$, and we prove an explicit formula for the minimum distance of toric codes associat…

Where to watch Buccaneers vs. Falcons: TV channel, kickoff time, odds, prediction

https://www.cbssports.com/nfl/news/where-to-watch-buccaneers-vs-falcons…

“Additionally, Cursor’s Amazon Web Services bills more than doubled from $6.2 million in May 2025 to $12.6 million in June 2025, exacerbating a cash crunch that began when Anthropic introduced Priority Service Tiers, an aggressive rent-seeking measure that begun what I call the Subprime AI Crisis, where model providers begin jacking up the prices on their previously subsidized rates.”
#SubprimeAI
https://www.wheresyoured.at/costs/?ref=ed-zitrons-wheres-your-ed-at-newsletter

Global classical solutions to a two-dimensional chemotaxis-fluid system involving signal-dependent degenerate diffusion
Yansheng Ma, Peter Y. H. Pang, Yifu Wang
https://arxiv.org/abs/2509.17073

Global classical solutions to a two-dimensional chemotaxis-fluid system involving signal-dependent degenerate diffusion
This paper is concerned with the two-dimensional chemotaxis-fluid model \begin{equation*} \begin{cases} n_t+u\cdot\nabla n=Δ(nϕ(v))+μn(1-n),\\ v_t+u\cdot\nabla v=Δv-nv,\\ u_t+ κ(u\cdot\nabla) u=Δu+n\nablaΦ-\nabla P, \quad\nabla\cdot u=0, \end{cases} \end{equation*} accounting for signal-dependent motilities of microbial populations interacting with an incompressible liquid through transport and buoyancy, where the suitably smooth function $ϕ$ satisfies $ϕ>0$ on $(0,\infty)$ with $�…

Table2LaTeX-RL: High-Fidelity LaTeX Code Generation from Table Images via Reinforced Multimodal Language Models
Jun Ling, Yao Qi, Tao Huang, Shibo Zhou, Yanqin Huang, Jiang Yang, Ziqi Song, Ying Zhou, Yang Yang, Heng Tao Shen, Peng Wang
https://arxiv.org/abs/2509.17589

Table2LaTeX-RL: High-Fidelity LaTeX Code Generation from Table Images via Reinforced Multimodal Language Models
In this work, we address the task of table image to LaTeX code generation, with the goal of automating the reconstruction of high-quality, publication-ready tables from visual inputs. A central challenge of this task lies in accurately handling complex tables -- those with large sizes, deeply nested structures, and semantically rich or irregular cell content -- where existing methods often fail. We begin with a comprehensive analysis, identifying key challenges and highlighting the limitations …

The Recap: Looking at the (many) game-changing plays https://www.giants.com/news/giants-vs-cowboys-highlights-week-2-nfl-2025-russell-wilson-malik-nabers

A Malmquist-Yosida type theorem for Schwarzian differential equations
Xiong-Feng Liu
https://arxiv.org/abs/2509.18649 https://arxiv.org/pdf/2509.18649

StoryBox: Collaborative Multi-Agent Simulation for Hybrid Bottom-Up Long-Form Story Generation Using Large Language Models
Zehao Chen, Rong Pan, Haoran Li
https://arxiv.org/abs/2510.11618

StoryBox: Collaborative Multi-Agent Simulation for Hybrid Bottom-Up Long-Form Story Generation Using Large Language Models
Human writers often begin their stories with an overarching mental scene, where they envision the interactions between characters and their environment. Inspired by this creative process, we propose a novel approach to long-form story generation, termed hybrid bottom-up long-form story generation, using multi-agent simulations. In our method, agents interact within a dynamic sandbox environment, where their behaviors and interactions with one another and the environment generate emergent events…

AFC West Preseason Position Rankings: Where Does Geno Smith Land? https://raiderramble.com/2025/08/07/afc-west-preseason-position-rankings-where-does-geno-smith-land/

Multiple nodal solutions to a scalar field equation with double-power nonlinearity and zero mass at infinity
M\'onica Clapp, Carlos Culebro
https://arxiv.org/abs/2508.15167 …

Multiple nodal solutions to a scalar field equation with double-power nonlinearity and zero mass at infinity
We consider the nonlinear elliptic equation \begin{equation*} -Δu + V(x)u = f(u), \qquad u\in D^{1,2}_0(Ω), \end{equation*} in an exterior domain $Ω$ of $\mathbb{R}^N$, where $V$ is a scalar potential that decays to zero at infinity and the nonlinearity $f$ is subcritical at infinity and supercritical near the origin. Under weak symmetry assumptions, we provide conditions that guarantee that this problem has a prescribed number of sign-changing solutions. In particular, we show that in dimen…

Rankin-Selberg integrals for $\mathrm{GSpin}$ groups with application to the global Gan-Gross-Prasad conjecture
Pan Yan
https://arxiv.org/abs/2508.09066 https://

Rankin-Selberg integrals for $\mathrm{GSpin}$ groups with application to the global Gan-Gross-Prasad conjecture
We construct Rankin-Selberg integrals using Bessel models for a product of tensor product partial $L$-functions \begin{equation*} L^S(s,π\timesτ_1) L^S(s,π\timesτ_2)\cdots L^S(s,π\timesτ_r) \end{equation*} where $π$ is an irreducible cuspidal automorphic representation of a quasi-split $\mathrm{GSpin}$ group, and $τ_1, \cdots, τ_r$ are irreducible unitary cuspidal automorphic representations of $\mathrm{GL}_{k_1}, \cdots, \mathrm{GL}_{k_r}$ respectively. As an application, we prove one…

Higher-order Chern-Simons extensions to QED in $2 1$ dimensions
Ricardo Avila, Albert Yu. Petrov, Carlos M. Reyes, C\'esar Riquelme, Angel Sanchez
https://arxiv.org/abs/2508.21266

Higher-order Chern-Simons extensions to QED in $2+1$ dimensions
In this work, we investigate radiative corrections in the higher-order extension of the Maxwell-Chern-Simons model coupled to standard spinor matter in $2+1$ dimensions. We begin analyzing the higher-order gauge sector, where we find the modes and the polarizations vectors associated to a massive photon and ghost field. The higher-order gauge model is canonically quantized and as expected the resulting algebra of creation and annihilation operators corresponds to an indefinite metric in Hilbert…

Periodic solutions in a tumor-immune competition system with time-delay and chemotherapy effects
Pablo Amster, Andr\'es Rivera, Jhon A. Arredondo
https://arxiv.org/abs/2510.11135

Periodic solutions in a tumor-immune competition system with time-delay and chemotherapy effects
The main purpose of this paper is to analyze the dynamics of the system of time-delay differential equations (DDEs) \begin{equation*} \begin{split} \dot{T}(t)&=T(t) f(t,T(t))-γE(t)T(t),\\ \dot{E}(t)&=σ+ \frac{pE(t)T(t-τ_1)}{g+a T(t-τ_1)}-\frac{mE(t)T(t-τ_2)}{g+a T(t-τ_2)}-ηE(t), \end{split} \end{equation*} where $T=T(t)$ and $E=E(t)$ represent the concentrations of tumor and effector cells at the time $t$. The coefficients $σ$, $μ$, $γ$, and $η$ are all positive, and $f(t, T)$ repres…

Subdivision Schemes in Metric Spaces
Nira Dyn, Nir Sharon
https://arxiv.org/abs/2509.08070 https://arxiv.org/pdf/2509.08070

Subdivision Schemes in Metric Spaces
We develop a unified framework for nonlinear subdivision schemes on complete metric spaces (CMS). We begin with CMS preliminaries and formalize refinement in CMS, retaining key structural properties, such as locality. We prove a convergence theorem under contractivity and demonstrate its applicability. To address schemes where contractivity is unknown, we introduce two notions of proximity. Our proximity methods relate a nonlinear scheme to another nonlinear scheme with known contractivity, rat…

The Gamma Expansion of the Level Two Large Deviation Rate Functional for Reversible Diffusion Processes
Claudio Landim, Jungkyoung Lee, Mauro Mariani
https://arxiv.org/abs/2509.13222

The Gamma Expansion of the Level Two Large Deviation Rate Functional for Reversible Diffusion Processes
Fix a smooth Morse function $U\colon \mathbb{R}^{d}\to\mathbb{R}$ with finitely many critical points, and consider the solution of the stochastic differential equation \begin{equation*} d\bm{x}_ε(t)=-\nabla U(\bm{x}_ε(t))\,dt \,+\,\sqrt{2ε}\, d\bm{w}_{t}\,, \end{equation*} where $(\bm{w}_{t})_{t\ge0}$ represents a $d$-dimensional Brownian motion, and $ε>0$ a small parameter. Denote by $\mathcal{P}(\mathbb{R}^{d})$ the space of probability measures on $\bb R^d$, and by $\mathcal{I}_ε \colon…

SHANKS: Simultaneous Hearing and Thinking for Spoken Language Models
Cheng-Han Chiang, Xiaofei Wang, Linjie Li, Chung-Ching Lin, Kevin Lin, Shujie Liu, Zhendong Wang, Zhengyuan Yang, Hung-yi Lee, Lijuan Wang
https://arxiv.org/abs/2510.06917

SHANKS: Simultaneous Hearing and Thinking for Spoken Language Models
Current large language models (LLMs) and spoken language models (SLMs) begin thinking and taking actions only after the user has finished their turn. This prevents the model from interacting during the user's turn and can lead to high response latency while it waits to think. Consequently, thinking after receiving the full input is not suitable for speech-to-speech interaction, where real-time, low-latency exchange is important. We address this by noting that humans naturally "think while liste…

The H\'enon equation in Orlicz-Sobolev spaces
Pablo Ochoa, Ariel Salort
https://arxiv.org/abs/2509.17923 https://arxiv.org/pdf/2509.17923

The Hénon equation in Orlicz-Sobolev spaces
In this paper, we consider the Hénon problem in the setting of Orlicz-Sobolev spaces: \begin{equation*} \begin{cases} -Δ_g u= |x|^αh( u) \quad \text{in }B\\ u>0 \quad \text{in }B\\ u= 0 \quad \text{on }\partial B\\ \end{cases} \end{equation*}where $B$ is the unit ball in $\mathbb{R}^n$, $g=G'$, $h=H'$ are N-functions and the operator $-Δ_g$ is the $g$-Laplacian. We show that the symmetric term $|x|^α$, for $α>0$, allows to have radial solutions even for supercritical $…

Approximate Proportionality in Online Fair Division
Davin Choo, Winston Fu, Derek Khu, Tzeh Yuan Neoh, Tze-Yang Poon, Nicholas Teh
https://arxiv.org/abs/2508.03253 https://

Approximate Proportionality in Online Fair Division
We study the online fair division problem, where indivisible goods arrive sequentially and must be allocated immediately and irrevocably to agents. Prior work has established strong impossibility results for approximating classic fairness notions, such as envy-freeness and maximin share fairness, in this setting. In contrast, we focus on proportionality up to one good (PROP1), a natural relaxation of proportionality whose approximability remains unresolved. We begin by showing that three natura…

New upper bound for lattice covering by spheres
Jun Gao, Xizhi Liu, Oleg Pikhurko, Shumin Sun
https://arxiv.org/abs/2508.06446 https://arxiv.org/pdf/2508.0…

New upper bound for lattice covering by spheres
We show that there exists a lattice covering of $\mathbb{R}^n$ by Eucledian spheres of equal radius with density $O\big(n \ln^β n \big)$ as $n\to\infty$, where \begin{align*} β:= \frac{1}{2} \log_2 \left(\frac{8 π\mathrm{e}}{3\sqrt 3}\right)=1.85837...\,. \end{align*} This improves upon the previously best known upper bound by Rogers from 1959 of $O\big(n \ln^α n \big)$, where $α:= \frac{1}{2} \log_{2}(2π\mathrm{e})=2.0471...\,.$

On the verification of a Nicolas inequality
Orlando Galdames-Bravo
https://arxiv.org/abs/2509.11182 https://arxiv.org/pdf/2509.11182

On the verification of a Nicolas inequality
Nicolas inequality we deal can be written as \begin{equation}\label{Nicineq} e^γ\log\log N_x < \dfrac{N_x}{φ(N_x)}\,, \end{equation} where $x\ge 2$, $N_x$ denotes the product of the primes less or equal than $x$, $γ$ is the Euler constant and $φ$ is the Euler totient function. We show that there is a large $x_0>0$ such that this inequality fails infinitely often for integers $x\ge x_0$. To this aim we analyze the sign of the Big-o function in the Mertens estimate for the sum of reciprocals …

The Uranus Flagship: Investigating New Paradigms for Outer Solar System Exploration Workshop Summary Report
Amy Simon, Louise Prockter, Ian Cohen, Kathleen Mandt, Lynnae Quick
https://arxiv.org/abs/2508.21074

The Uranus Flagship: Investigating New Paradigms for Outer Solar System Exploration Workshop Summary Report
This white paper is a summary of the Uranus Flagship Workshop that took place 21 to 23 May 2024 at NASA's Goddard Space Flight Center. Co-led by Goddard and Johns Hopkins Applied Physics Lab conveners, we had a broad, international, Science Organizing Committee, and a largely early career Local Organizing Committee from APL and GSFC. From prior workshops, it was apparent that the community was wildly enthusiastic about starting a mission, but lacked focus on what was possible or where to begin.…

On a zero mass Schr\"odinger-Bopp-Podolsky system: ground states, nonexistence results and asymptotic behaviour
Alessio Pomponio, Lianfeng Yang
https://arxiv.org/abs/2509.17479

On a zero mass Schrödinger-Bopp-Podolsky system: ground states, nonexistence results and asymptotic behaviour
In this paper, we consider the following zero mass Schrödinger-Bopp-Podolsky system \[ \begin{cases} -Δu +q^2ϕu=|u|^{p-2}u, -Δϕ+a^2Δ^2ϕ=4πu^2, \end{cases} \text{ in } \mathbb{R}^3, \] where $a>0$ and $q\ne 0$. We complete the study initiated in [2], which relied on a perturbation argument to establish the existence of weak solutions. Here, in contrast, our approach, based on the Mountain Pass Theorem and the splitting lemma, directly yields a ground state solut…

A Probabilistic Framework for the Erdos-Kac Theorem
Mantha Sai Gopal
https://arxiv.org/abs/2509.04102 https://arxiv.org/pdf/2509.04102

A Probabilistic Framework for the Erdos-Kac Theorem
The Erdos-Kac theorem, a foundational result in probabilistic number theory, states that the number of prime factors of an integer follows a Gaussian distribution. In this paper we develop and analyze probabilistic models for "random integers" to study the mechanisms underlying this theorem. We begin with a simple model, where each prime p is chosen as a divisor with probability 1/p in a sequence of independent trials. A preliminary analysis shows that this construction almost surely yields an …

Least zero of pairs of additive cubic equations
Yixiu Xiao, Hongze Li
https://arxiv.org/abs/2510.10001 https://arxiv.org/pdf/2510.10001

Least zero of pairs of additive cubic equations
An effective upper bound is established for the least non-trivial integer solution to the system of cubic forms \[ \begin{cases} F = c_{1}x_1^3 + c_{2}x_2^3 + \cdots + c_{n}x_n^3 = 0, \\ G = d_{1}x_1^3 + d_{2}x_2^3 + \cdots + d_{n}x_n^3 = 0, \end{cases} \] under the "$M$-good" condition for $n \ge 16$, where $c_{1}, \dots, c_{n}$ and $d_{1}, \dots, d_{n}$ are integers. Additionally, a range is derived for the probability that randomly selected simultaneous equations satisfy the $M$-good conditi…

Normalized solution for Kirchhoff equation with upper critical exponent and mixed Choquard type nonlinearities
Jinyuan Shang, Wenting Zhao, Xianjiu Huang
https://arxiv.org/abs/2509.14681

Normalized solution for Kirchhoff equation with upper critical exponent and mixed Choquard type nonlinearities
In this paper, we consider the existence of normalized solution to the following Kirchhoff equation with mixed Choquard type nonlinearities: \begin{equation*} \begin{cases} -\left(a + b \int_{\mathbb{R}^3} |\nabla u|^2 \, dx\right) Δu - λu = μ|u|^{q-2} u + (I_α* |u|^{α+ 3}) |u|^{α+1} u, \quad x \in \mathbb{R}^3, \\ \int_{\mathbb{R}^3} u^2 \, dx = ρ^2, \end{cases} \end{equation*} where $a,b,ρ>0$, $α\in \left(0, 3\right)$, $\frac{14}{3} < q < 6$ an…

Normalized solutions to Kirchhoff equation with the Sobolev critical exponent in high dimensional spaces
Ruikang Lu, Qilin Xie, Jianshe Yu
https://arxiv.org/abs/2509.13663 https…

Normalized solutions to Kirchhoff equation with the Sobolev critical exponent in high dimensional spaces
The following well-known Kirchhoff equation with the Sobolev critical exponent has been extensively studied, \begin{equation*} -\Big(a+b\int_{\mathbb R^N} | \nabla u|^2dx\Big) Δu+λu=μ|u|^{q-2}u+|u|^{2^*-2}u \ \ {\rm in}\ \ \mathbb{R}^N, \ \ N\geq4, \end{equation*} having prescribed mass $\int_{\mathbb R^N}|u|^2dx=c$, where $a$, $c$ are two positive constants, $b,μ$ are two parameters, $λ$ appears as a real Lagrange multiplier and $2

Multiple solutions to a class of $p$-Laplacian Schr\"{o}dinger equations
Lin Zhang
https://arxiv.org/abs/2509.05556 https://arxiv.org/pdf/2509.05556…

Multiple solutions to a class of $p$-Laplacian Schrödinger equations
In this paper, we will prove the existence of infinitely many solutions to the following equation by utilizing the variational perturbation method \begin{equation*} -div(A(x,u)|\nabla u|^{p-2}\nabla u)+\frac{1}{p}A_{t}(x,u)|\nabla u|^{p}+V(x)|u|^{p-2}u=g(x, u), ~~ x\in\mathbb{R}^{N}, \end{equation*} where $2

Nonlocal Harnack inequalities for nonlocal double phase equations I ; with positive bounded modulating coefficient with no H\"older condition
Yong-Cheol Kim
https://arxiv.org/abs/2509.07433

Nonlocal Harnack inequalities for nonlocal double phase equations I ; with positive bounded modulating coefficient with no Hölder condition
In this paper, by applying the De Giorgi-Nash-Moser theory we prove nonlocal Harnack inequalities for (locally nonnegative in $Ω$) weak solutions to nolocal double phase equations \begin{equation*}\begin{cases}\cL u =0 & \text{ in $Ω$,} \\ u=g & \text{ in $\BR^n\sΩ$ } \end{cases}\end{equation*} where $Ω\subset\BR^n$ ($n\ge 2$) is a bounded domain with Lipschitz boundary, $\cL$ is the nonlocal double phase operator $\cL$ given by \begin{equation*}\begin{split}\cL u(x)=&\pv\int_{\BR^n}|u…

Global solvability for doubly degenerate nutrient taxis system with a wide range of bacterial responses in physical dimension
Bao-Ngoc Tran, Juan Yang
https://arxiv.org/abs/2508.03268

Global solvability for doubly degenerate nutrient taxis system with a wide range of bacterial responses in physical dimension
Motivated by the study of bacteria's response to environmental conditions, we consider the doubly degenerate nutrient taxis system \begin{align*} \begin{cases} u_t=\nabla\cdot(uv\nabla u)-χ\nabla\cdot(u^αv\nabla v)+\ell uv,\\ v_t=Δv-uv, \end{cases} \end{align*} subjected to no-flux boundary conditions and smooth initial data, where $α\in\mathbb{R}$ is the bacterial response parameter. Global solvability of weak solutions to this taxis system is highly challenging due to not only the doubly …

H\"older regularity of weak solutions to nonlocal doubly degenerate parabolic equations
Qifan Li
https://arxiv.org/abs/2509.05914 https://arxiv.org/pd…

Hölder regularity of weak solutions to nonlocal doubly degenerate parabolic equations
We study local regularity for nonlocal doubly degenerate parabolic equations. The model equation is \begin{equation*}\begin{split} \partial_t(|u|^{q-1}u)+\mathrm{P}.\mathrm{V}.\int_{\mathbb{R}^n}\frac{|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))}{|x-y|^{n+sp}}\,\mathrm{d}y=0, \end{split} \end{equation*} where $02$ and $0

Nondegeneracy of bubble solutions to the Choquard equation in two dimension
Jinkai Gao, Xinfu Li, Shiwang Ma
https://arxiv.org/abs/2508.02286 https://arxiv…

Nondegeneracy of bubble solutions to the Choquard equation in two dimension
In this paper, we study the following Choquard equation with exponential nonlinearity \begin{equation*} -Δu=\left(\int_{\R^{2}}\frac{e^{u(y)}}{|x-y|^α}dy\right)e^{u(x)},\quad \text{~in~}\R^{2}, \end{equation*} where $α\in (0,2)$. Although the classification of solutions to this equation has been established recently, the nondegeneracy of its solutions remains open. Here, we prove the nondegeneracy by combining the integral representation of solutions with the spherical harmonic d…

Asymptotic behavior of the least energy solutions to the Choquard equation in dimension two
Jinkai Gao, Xinfu Li, Shiwang Ma
https://arxiv.org/abs/2508.02139 https://

Asymptotic behavior of the least energy solutions to the Choquard equation in dimension two
In this paper, we are interested in the following planar Choquard equation \begin{equation*} \begin{cases} -Δu=\displaystyle\left(\int\limits_Ω\frac{u^{p+1}(y)}{|x-y|^α}dy\right)u^{p},\quad u>0,\ \ &\mbox{in}\ Ω, \quad \ \ u=0, \ \ &\mbox{on}\ \partial Ω, \end{cases} \end{equation*} where $Ω$ is a smooth bounded domain in $\mathbb{R}^2$, $α\in (0,2)$ and $p>1$ is a positive parameter. Unlike the higher-dimensional case, we prove that the least energy solutions $u_{p}$ n…

Gradient regularity for widely degenerate parabolic equations
Michael Strunk
https://arxiv.org/abs/2510.07999 https://arxiv.org/pdf/2510.07999

Gradient regularity for widely degenerate parabolic equations
In this paper, we are interested in the regularity of weak solutions $u\colonΩ_T\to\mathbb{R}$ to parabolic equations of the type \begin{equation*} \partial_t u - \mathrm{div} \nabla \mathcal{F}(x,t,Du) = f\qquad\mbox{in $Ω_T$}, \end{equation*} where $\mathcal{F}$ is only elliptic for values of $Du$ outside a bounded and convex set $E\subset \mathbb{R}^n$ with the property that $0\in \mathrm{Int}{E}$. Here, $Ω_T :=Ω\times(0,T)\subset\mathbb{R}^{n+1}$ denotes a space-time cylinder taken ov…

Fujita-type results for parabolic equations with Hartree-type nonlinearities
Ahmad Z. Fino, Berikbol T. Torebek
https://arxiv.org/abs/2510.11648 https://ar…

Fujita-type results for parabolic equations with Hartree-type nonlinearities
This paper investigates the critical behavior of global solutions to a parabolic equation with a Hartree-type nonlinearity of the form $$\left\{\begin{array}{ll} u_{t}+(-Δ)^{\fracβ{2}} u= (\mathcal{K}\ast |u|^{p})|u|^{q},&\qquad x\in \mathbb{R}^n,\,\,\,t>0, u(x,0)=u_{0}(x),& \qquad x\in \mathbb{R}^n,\end{array} \right.$$ where $β\in(0,2]$, $n\geq1$, $p>1$, $q\geq 1$, $(-Δ)^{\fracβ{2}},\,β\in(0,2)$ denotes the fractional Laplacian, the symbol $\ast$ denotes the conv…

Regularizing effect of the natural growth term in quasilinear problems with sign-changing nonlinearities
Jos\'e Carmona Tapia, Paolo Malanchini, Antonio J. Mart\'inez Aparicio, Pedro J. Mart\'inez-Aparicio
https://arxiv.org/abs/2509.01355

Regularizing effect of the natural growth term in quasilinear problems with sign-changing nonlinearities
We investigate the existence and nonexistence of solutions to the Dirichlet problem \begin{equation*} \tag{$P$} \label{pba} \left\{ \begin{alignedat}{2} -Δ_p u + g(u) |\nabla u|^p &= λf(u) \quad &&\mbox{in} \;\; Ω, \\ u &= 0 \quad &&\mbox{on} \;\; \partialΩ, \end{alignedat} \right. \end{equation*} where $Ω\subset \mathbb{R}^N$ is a smooth bounded domain, $p\in (1,\infty)$, $λ>0$ and $g\in C(\mathbb{R})$. Our main assumption is that $:f \mathbb{R}\to \mathbb{R}$ is a continuous f…

Chemotaxis Models with Nonlinear/Porous Medium Diffusion, Consumption, and Logistic source on $\mathbb{R}^N$: I. Global Solvability and Boundedness
Zulaihat Hassan, Wenxian Shen, Yuming Paul Zhang
https://arxiv.org/abs/2509.05494

Chemotaxis Models with Nonlinear/Porous Medium Diffusion, Consumption, and Logistic source on $\mathbb{R}^N$: I. Global Solvability and Boundedness
This series of papers is concerned with the global solvability, boundedness, regularity, and uniqueness of weak solutions to the following parabolic-parabolic chemotaxis system with a logistic source and chemical consumption: \begin{equation*} \begin{cases} u_t = m\nabla\cdot \left((\eps+u)^{m-1}\nabla u\right) - χ\nabla \cdot (u \nabla v) + u(a - b u), & \text{ in } (0,\infty)\times\mathbb{R}^N, \\ v_t = Δv - uv, & \text{ in } (0,\infty)\times\mathbb{R}^N, \end{cases} \end{equation*} where $…

Asymptotic behaviour and existence of positive solutions for mixed local nonlocal elliptic equations with Hardy potential
Shammi Malhotra, Sarika Goyal, K. Sreenadh
https://arxiv.org/abs/2510.04763

Asymptotic behaviour and existence of positive solutions for mixed local nonlocal elliptic equations with Hardy potential
We investigate the existence and multiplicity of positive solutions to the following problem driven by the superposition of the Laplacian and the fractional Laplacian with Hardy potential \begin{equation*} \left\{ \begin{aligned} -Δu + (-Δ)^s u - μ\frac{u}{|x|^2} &= λ|u|^{p-2} u + |u|^{2^*-2} u \quad \text{in } Ω\subset \mathbb{R}^N, u &= 0 \quad \text{in } \mathbb{R}^N \setminus Ω, \end{aligned} \right. \end{equation*} where $ Ω\subset \mathbb{R}^N $ is…

Existence and nonexistence of solutions for singular quadratic quasilinear equations
David Arcoya, Jos\'e Carmona, Tommaso Leonori, Pedro J. Mart\'inez-Aparicio, Luigi Orsina, Francesco Petitta
https://arxiv.org/abs/2508.06375

Existence and nonexistence of solutions for singular quadratic quasilinear equations
We study both existence and nonexistence of nonnegative solutions for nonlinear elliptic problems with singular lower order terms that have natural growth with respect to the gradient, whose model is $$ \begin{cases} -Δu + \frac{|\nabla u|^2}{u^γ} = f & \mbox{in } Ω,\newline \hfill u=0 \hfill & \mbox{on } \partial Ω, \end{cases} $$ where $Ω$ is an open bounded subset of $\mathbb{R}^N $, $γ> 0$ and $f$ is a function which is strictly positive on every compactly contained subs…