Tootfinder

Broncos LB Dre Greenlaw heads to injured reserve with lingering quad injury: Source https://www.nytimes.com/athletic/6647395/2025/09/20/broncos-dre-greenlaw-injured-reserve-quad/

Terry McLaurin injury update: Commanders WR exits vs. Raiders with quad ailment https://www.sportingnews.com/us/nfl/washington-commanders/news/terry-mclaurin-injury-update-commanders-raiders/3eae3feb8fba978ec68…

We demand that the University of Galway immediately end their contract with Technion. We cannot be complicit in genocide by having ties to the apartheid state of Israel.
Boycott. Divest. Sanction.

Titans activate L'Jarius Sneed from PUP after quad injury, knee clean-up procedure

https://www.cbssports.com/nfl/news/titans-activate-ljarius-sne…

Can LLMs Judge Debates? Evaluating Non-Linear Reasoning via Argumentation Theory Semantics
Reza Sanayei, Srdjan Vesic, Eduardo Blanco, Mihai Surdeanu
https://arxiv.org/abs/2509.15739

Can LLMs Judge Debates? Evaluating Non-Linear Reasoning via Argumentation Theory Semantics
Large Language Models (LLMs) excel at linear reasoning tasks but remain underexplored on non-linear structures such as those found in natural debates, which are best expressed as argument graphs. We evaluate whether LLMs can approximate structured reasoning from Computational Argumentation Theory (CAT). Specifically, we use Quantitative Argumentation Debate (QuAD) semantics, which assigns acceptability scores to arguments based on their attack and support relations. Given only dialogue-formatte…

PerchMobi^3: A Multi-Modal Robot with Power-Reuse Quad-Fan Mechanism for Air-Ground-Wall Locomotion
Yikai Chen, Zhi Zheng, Jin Wang, Bingye He, Xiangyu Xu, Jialu Zhang, Huan Yu, Guodong Lu
https://arxiv.org/abs/2509.12620

PerchMobi^3: A Multi-Modal Robot with Power-Reuse Quad-Fan Mechanism for Air-Ground-Wall Locomotion
Achieving seamless integration of aerial flight, ground driving, and wall climbing within a single robotic platform remains a major challenge, as existing designs often rely on additional adhesion actuators that increase complexity, reduce efficiency, and compromise reliability. To address these limitations, we present PerchMobi^3, a quad-fan, negative-pressure, air-ground-wall robot that implements a propulsion-adhesion power-reuse mechanism. By repurposing four ducted fans to simultaneously p…

Representation of solutions to continuous and discrete first-order linear matrix equations with pure delay involving nonpermutable matrices
Javad A. Asadzade, Nazim I. Mahmudov
https://arxiv.org/abs/2509.16845

Representation of solutions to continuous and discrete first-order linear matrix equations with pure delay involving nonpermutable matrices
This paper presents a unified treatment of first-order linear matrix equations (FLMEs) with pure delay in both continuous and discrete time, under the general setting of noncommutative coefficient matrices. In the continuous-time framework, we consider delayed matrix differential equations of the form \[ \dot{X}(\vartheta) = A_0 X(\vartheta-σ) + X(\vartheta-σ) A_1 + G(\vartheta), \quad \vartheta \geq 0, \] where \(A_0, A_1 \in \mathbb{R}^{d \times d}\) satisfy \(A_0 A_1 \neq A_1 A_0\)…

The new observations about the parameter-dependent Schr\"{o}dinger-Poisson system
Chen Huang, Sihua Liang, Lei Ma
https://arxiv.org/abs/2508.12732 https://

The new observations about the parameter-dependent Schrödinger-Poisson system
In this paper, we study the existence results of solutions for the following Schrödinger-Poisson system involving different potentials: \begin{equation*} \begin{cases} -Δu+V(x)u-λϕu=f(u)&\quad\text{in}~\mathbb R^3, -Δϕ=u^2&\quad\text{in}~\mathbb R^3. \end{cases} \end{equation*} We first consider the case that the potential $V$ is positive and radial so that the mountain pass theorem could be implied. The other case is that the potential $V$ is coercive and sign-changing, which…

Commanders WR McLaurin out 4th straight game https://www.espn.com/nfl/story/_/id/46626684/commanders-wr-mclaurin-quad-miss-fourth-consecutive-game

The ideal structure of the minimal Tensor Product of Ternary Rings of Operators
Vandana Rajpal, Arpit Kansal
https://arxiv.org/abs/2508.12374 https://arxiv…

The ideal structure of the minimal Tensor Product of Ternary Rings of Operators
Let \( V \) be a ternary ring of operator and \( B \) a \( C^* \)-algebra. We study the structure of the ideal space of the operator space injective tensor product \( V \otimes^{\mathrm{tmin}} B \) via two maps: \[ Φ(I, J) = \ker(q_I \otimes^{\mathrm{tmin}} q_J) \quad \text{and} \quad Δ(I, J) = I \otimes^{\mathrm{tmin}} B + V \otimes^{\mathrm{tmin}} J. \] We show that \( Φ\) is continuous with respect to the hull-kernel topology, and that its restriction to primitive and prime ideals defines…

Titans' Ridley, Key ruled out Sunday vs. Patriots https://www.espn.com/nfl/story/_/id/46627988/titans-calvin-ridley-hamstring-arden-key-quad-vs-pats

On the generalized Fermat equation $x^{13} y^{13} = z^n$
Alex J. Best, Sander R. Dahmen, Nuno Freitas
https://arxiv.org/abs/2510.12092 https://arxiv.org/…

On the generalized Fermat equation $x^{13} + y^{13} = z^n$
Let $n \in \mathbb{Z}_{\geq 2}$. We study the generalized Fermat equation \[x^{13}+y^{13}=z^n, \quad x,y,z \in \mathbb{Z}, \quad \gcd(x,y,z)=1.\] Using a combination of techniques, including the modular method, classical descent, unit sieves, and Chabauty and Mordell--Weil sieve methods over number fields, we show that for $n=5$ all its solutions $(a,b,c)$ are trivial, i.e. satisfy $abc=0$. Under the assumption of GRH, we also show that for $n=7$ there are only trivial solutions. Furthermore, w…

Terry McLaurin (quad) ruled out again ahead of Commanders-Cowboys game: 'It's just not ready yet' https://www.nfl.com/news/terry-mclaurin-quad-out-commanders-cowboys-deebo-samuel

Mit Quad abgestürzt - Nach intensiver Suche: Steirer tot aufgefunden #News #Nachrichten

Terry McLaurin injury: Commanders WR out vs. Cowboys as quad issue sidelines star for fourth game

https://www.cbssports.com/nfl/news/terry-mclaurin-injury-commanders-wr-out-v…

A Note on Eigenvalues of Perturbed Hermitian Matrices
Chi-Kwong Li, Ren-Cang Li
https://arxiv.org/abs/2508.08203 https://arxiv.org/pdf/2508.08203

A Note on Eigenvalues of Perturbed Hermitian Matrices
Let $$ A=\left(\begin{array}{cc} H_1 & E^*\\ E & H_2\end{array}\right) \quad \hbox{ and } \quad \wtd A=\left(\begin{array}{cc} H_1 & O\\ O & H_2\end{array}\right)$$ be two $N$-by-$N$ Hermitian matrices with eigenvalues $λ_1 \ge \cdots \ge λ_{N}$ and $\wtd λ_1 \ge \cdots \ge \wtd λ_N$, respectively. \iffalse There are two kinds of perturbation bounds on $|λ_i - \wtd λ_i|$: $|λ_i- \wtd λ_i| \le \|E\|$, where $\|E\|$ is the largest singular valu…

Log-Concavity and Log-Convexity of Restricted Infinite Products
Krystian Gajdzica, Bernhard Heim, Markus Neuhauser
https://arxiv.org/abs/2509.11246 https://

Log-Concavity and Log-Convexity of Restricted Infinite Products
In this paper we provide a classification on the sign distribution of $Δ_{E,\ell}(n):= p_{E,\ell }(n)^2 - p_{E,\ell }(n-1) \, p_{E,\ell }(n+1)$, where \begin{equation*} \sum_{n =0}^{\infty} p_{E,\ell }(n) \, q^n := \prod_{n \in S} \left(1 - q^n \right)^{-f_{\ell}(n)},\quad (\ell \in \mathbb{N}, f_1\equiv 1). \end{equation*} We take the product over $1\in S \subset \mathbb{N}$ and denote the complement by $E$, the set of exceptions. In the case of $\ell=1$ and $E$ the multiples of $k$, $p_{E,…

at public records in #brooklyn, dreamy/loopy chamber #jazz for a summer's eve in bitchin' quad sound by johnson/wilkes/uhlmann, which (i kept thinking) would probably get somewhat lost if played outdoors on a summer's eve. perhaps the best room for pure listening in

A porous medium equation with spatially inhomogeneous absorption. Part II: Large time behavior
Razvan Gabriel Iagar, Diana-Rodica Munteanu
https://arxiv.org/abs/2508.11046 https…

A porous medium equation with spatially inhomogeneous absorption. Part II: Large time behavior
We study the large time behavior of solutions to the Cauchy problem for the quasilinear absorption-diffusion equation $$ \partial_tu=Δu^m-|x|^σu^p, \quad (x,t)\in\real^N\times(0,\infty), $$ with exponents $p>m>1$ and $σ>0$ and with initial conditions either satisfying $$ u_0\in L^{\infty}(\real^N)\cap C(\real^N), \quad \lim\limits_{|x|\to\infty}|x|^θu_0(x)=A\in(0,\infty) $$ for some $θ\geq0$. A number of different asymptotic profiles are identified, and unifor…

Progress!
STM32H750 Trion T20 setup is now able to:
* Initialize QUADSPI-connected flash on MCU board and use it for microkvs. Currently in x1 mode, quad will come later once I debug a bit
* Bootloader attempts to boot an image in QUADSPI flash, but it fails because there's no code there
* Connect to FPGA via FMC-APB bridge
* Identify FPGA via hardcoded IDCODE in device info block (there doesn't seem to be any ICAP-equivalent in Trion that lets you determi…

A few people look tiny in the slot between Olin and Uris Libraries from the Arts Quad to Ho Plaza
#photo #photography #buildings

Design and Validation of Metasurface Transmitarrays for Time-Reversal Microwave Hyperthermia in Deep Brain Tumors
Alireza Rahmani, Mohammad Javad Hajiahmadi, Reza Faraji-Dana
https://arxiv.org/abs/2509.14429

Design and Validation of Metasurface Transmitarrays for Time-Reversal Microwave Hyperthermia in Deep Brain Tumors
This work presents a systematic design of a metasurface-based microwave hyperthermia system, leveraging the time-reversal technique to achieve precise focusing of electromagnetic energy at deep-seated brain tumors. The manuscript includes detailed numerical simulations on realistic and spherical head models, as well as the experimental realization of a quad-layer transmissive metasurface and an electrothermal head phantom. This represents the first experimental demonstration of metasurface-base…

Algebra bundles, projective flatness and rationally-deformed tori
Alexandru Chirvasitu
https://arxiv.org/abs/2509.10812 https://arxiv.org/pdf/2509.10812

Algebra bundles, projective flatness and rationally-deformed tori
We show that isomorphism classes $[\mathcal{A}]$ of flat $q\times q$ matrix bundles $\mathcal{A}$ (or projectively flat rank-$q$ complex vector bundles $\mathcal{E}$) on a pro-torus $\mathbb{T}$ are in bijective correspondence with the Čech cohomology group $H^2(\mathbb{T},μ_q:=\text{$q^{th}$ roots of unity})$ (respectively $H^2(\mathbb{Z})$) via the image of $[A]\in H^1(\mathbb{T},PGL(q,\mathcal{C}_{\mathbb{T}}))$ through $H^1(\mathbb{T},PGL(q,\mathcal{C}_{\mathbb{T}}))\xrightarrow{\quad}H^2…

Weak Existence and Uniqueness for Super-Brownian Motion with Irregular Drift
Leonid Mytnik, Johanna Weinberger
https://arxiv.org/abs/2509.10984 https://arx…

Weak Existence and Uniqueness for Super-Brownian Motion with Irregular Drift
We establish weak existence and uniqueness for random field solutions of the one-dimensional SPDE \[ d_tX_t = \frac{1}{2}ΔX_t +h(X_t)+ \sqrt{X_t}\dot{W}, \quad t\geq 0,\] where $\dot{W}$ is space-time white noise and $h$ is a bounded drift with $h(0)\geq 0$. The proof relies on an extension of the duality relation of the super-Brownian motion, which allows us to treat a broad class of admissible drifts, including functions that are non-Lipschitz or discontinuous at zero.

Diagonal Frobenius Number via Gomory's Relaxation and Discrepancy
Dmitry Gribanov, Dmitry Malyshev, Panos Pardalos
https://arxiv.org/abs/2509.05629 https://

Diagonal Frobenius Number via Gomory's Relaxation and Discrepancy
For a matrix $A \in Z^{k \times n}$ of rank $k$, the diagonal Frobenius number $F_{\text{diag}}(A)$ is defined as the minimum $t \in Z_{\geq 1}$, such that, for any $b \in \text{span}_{Z}(A)$, the condition \begin{equation*} \exists x \in R_{\geq 0}^n,\, x \geq t \cdot 1 \colon \quad b = A x \end{equation*} implies that \begin{equation*} \exists z \in Z_{\geq 0}^n \colon\quad b = A z. \end{equation*} In this work, we show that \begin{equation*} F_{\text{diag}}(A) = Δ+ O(\log k), \end{e…

Waves, structures, and the Riemann problem for a system of hyperbolic conservation laws
A. P. Chugainova, D. V. Treschev
https://arxiv.org/abs/2510.02070 https://

Waves, structures, and the Riemann problem for a system of hyperbolic conservation laws
A system of hyperbolic conservation laws $$ \partial_t u + \partial_x \partial_u Q = 0, \quad Q = u_1^3 / 3 + u_1 u_2^2, \qquad u = u(x,t) \in\mR^2, $$ as well as its viscous regularization $$ \partial_t u + \partial_x \partial_u Q = \calM \partial_x^2 u, \qquad \calM = \diag (μ_1,μ_2), \quad μ_1>0,\, μ_2>0, $$ are studied. It is assumed that admissible shocks are those that satisfy the requirement of existence of a structure (the traveling wave criterion). A solution of the Riemann…

In case you have a #FlipperZero, tune in to the livestream The Flipper Blackhat by @…
https://

The Flipper Blackhat WHY2025
I developed a 100% open-source (Quad core A7, 512MB RAM) Linux-enabled Flipper module for WiFi pentesting and ethical hacking!

Commanders' Wise carted off, out with quad injury https://www.espn.com/nfl/story/_/id/46229655/commanders-wise-carted-field-game-quad-injuary

Green\'s Mapping and Julia Sets
Ilia Binder, Paul F. X. M\"uller, Peter Yuditskii
https://arxiv.org/abs/2508.04207 https://arxiv.org/pdf/2508.0420…

Greenś Mapping and Julia Sets
In March 1999, the first named author (Binder) posed the problem of showing that a ``good direction'' $ψ\in [0,2]$ exists, for any Green's mapping $T:H\rightarrow\tilde Ω$, i.e., \begin{equation}\label{binder} \int\limits_0\limits^{1} |T''(re^{iπψ})|dr <\infty, \quad\text{ for at least one } \quad ψ\in [0,2]. \end{equation} Presently this problem is open even in the special case where $\partial Ω$ is a uniformly perfect subset of the real line. In this paper we obtain a posi…

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…

Photon spheres near black holes in a model with anisotropic fluid
V. D. Ivashchuk, S. V. Bolokhov, F. B. Belissarova, N. Kydyrbay, A. N. Malybayev, G. S. Nurbakova, B. Zheng
https://arxiv.org/abs/2509.08465

Photon spheres near black holes in a model with anisotropic fluid
The semi-review paper studies the null geodesics which appear for black hole solutions in the gravitational $4d$ model with anisotropic fluid. The equations of state for the fluid and solutions itselves depend upon integer parameter $q = 1, 2, ...$: $p_r = -ρc^2 (2q-1)^{-1}, \quad p_t = - p_r$, where $ρ$ is the mass density, $c$ is speed of light, $p_r$ and $p_t$ are pressures in radial and transverse to radial directions, respectively. The circular null geodesics are explored and the master …

A class of Truncated Freud polynomials
Juan Carlos Garc\'ia-Ardila, Francisco Marcell\'an, Misael E. Marriaga
https://arxiv.org/abs/2510.09214 https://

A class of Truncated Freud polynomials
Consider the following truncated Freud linear functional $\mathbf{u}_z$ depending on a parameter $z$, $$\langle\mathbf{u}_z,p\rangle=\int_0^\infty p(x)e^{-zx^4}dx,\quad z>0.$$ The aim of this work is to analyze the properties of the sequence of orthogonal polynomials $(P_n)_{n\geq 0}$ with respect to $\mathbf{u}_z$. Such a linear functional is semiclassical and, as a consequence, we get the system of nonlinear difference equations (Laguerre-Freud equations) that the coefficients of the three-te…

Congruences modulo powers of 2 and 3 for overpartition $k$-tuples
G. Kavya Keerthana, S. Ananya, Ranganatha D
https://arxiv.org/abs/2509.17705 https://arxi…

Congruences modulo powers of 2 and 3 for overpartition $k$-tuples
Let $\overline{p}_{k}(n)$ denote the number of overpartition $k$-tuples of $n$. In 2023, Saikia \cite{saikia} conjectured the following congruences: \begin{align*} \overline{p}_{q}(8n+2)& \equiv 0 \pmod{4},\quad \overline{p}_{q}(8n+3)\equiv 0 \pmod{8},\quad \overline{p}_{q}(8n+4) \equiv 0 \pmod{2},\\ \overline{p}_{q}(8n+5)& \equiv 0 \pmod{8},\quad \overline{p}_{q}(8n+6) \equiv 0 \pmod{8},\quad \overline{p}_{q}(8n+7)\equiv 0 \pmod{32}, \end{align*} where $n\geq0$ and $q$ is prime. Recently, Sell…

The functional Loomis-Whitney type inequality in the Heisenberg groups and Projection theorems over finite fields
Daewoong Cheong, Thang Pham, Dung The Tran
https://arxiv.org/abs/2510.05022

The functional Loomis-Whitney type inequality in the Heisenberg groups and Projection theorems over finite fields
We develop a functional Loomis--Whitney framework on the finite Heisenberg groups $\mathbb{H}^n(\mathbb{F}_q)$ and discover connections to the boundedness and orthogonal projection problems. For $n=1$ we determine the sharp region of exponents $(u_1,u_2)$ for which the associated bilinear projection form is bounded uniformly in $q$, namely \[ \frac{1}{u_1}+\frac{2}{u_2}\le 2 \quad\text{and}\quad \frac{2}{u_1}+\frac{1}{u_2}\le 2, \] which includes the endpoint $L^{3/2}\times L^{3/2}\to L^1$. For…

Korean Students demonstrate a giant video game on the side of a truck on the arts quad
#photo #photography #cornell

Orbits and attainable Hamiltonian diffeomorphisms of mechanical Liouville equations
Bettina Kazandjian, Eugenio Pozzoli, Mario Sigalotti
https://arxiv.org/abs/2509.24960 https:/…

Orbits and attainable Hamiltonian diffeomorphisms of mechanical Liouville equations
We study the approximate controllability problem for Liouville transport equations along a mechanical Hamiltonian vector field. Such PDEs evolve inside the orbit $$\mathcal{O}(ρ_0):=\left\{ρ_0\circ Φ\mid Φ\in {\rm DHam}(T^*M)\right\},\quad ρ_0\in L^r(T^*M,\R), \quad r\in[1,\infty),$$ where $ρ_0$ is the initial density and ${\rm DHam}(T^*M)$ is the group of Hamiltonian diffeomorphisms of the cotangent bundle manifold $T^*M$. The approximately reachable densities from $ρ_0$ are thus contai…

Isoperimetric Bounds for Weighted Steklov Eigenvalues with Radial Weights
Friedemann Brock, Francesco Chiacchio
https://arxiv.org/abs/2510.12631 https://ar…

Isoperimetric Bounds for Weighted Steklov Eigenvalues with Radial Weights
We study the following class of Steklov eigenvalue problems: \[ \nabla \cdot \bigl( w \nabla u \bigr) = 0 \quad \text{in } Ω, \qquad \frac{\partial u}{\partial ν} = γv u \quad \text{on } \partial Ω, \] where $w$ and $v$ are prescribed positive radial functions, $Ω$ is a Lipschitz domain in $\mathbb{R}^N$ with $N \geq 2$ and $ν$ denotes its outward unit normal. Extending classical results in the unweighted case due to Weinstock, the first author, and others, we establish isoperimetric ineq…

A Note on the failure of temporal regularity for stochastic PDEs
Antonio Agresti, Mark Veraar
https://arxiv.org/abs/2509.07803 https://arxiv.org/pdf/2509.0…

A Note on the failure of temporal regularity for stochastic PDEs
We consider solutions to linear parabolic SPDEs of the form \[ \mathrm{d} u(t) + A u(t)\, \mathrm{d} t = g(t)\, \mathrm{d} β, \qquad u(0)=0, \] where $A$ is a positive, invertible, and self-adjoint operator on a Hilbert space $X$, $β$ is a one-dimensional Brownian motion, and $g(t)\equiv x\in X$. We show that, for all $α\in [0,\frac{1}{2}),$ \[ u\in L^2(Ω;W^{α,2}(0,T;\mathsf{D}(A^{1/2}))) \quad \text{ if and only if }\quad x\in \mathsf{D}(A^α). \] In particular, there is a lack of persist…

Finite element analysis of a nonlinear heat Equation with damping and pumping effects
Rishabh Shukla, Wasim Akram, Manil T. Mohan
https://arxiv.org/abs/2510.10210 https://

Finite element analysis of a nonlinear heat Equation with damping and pumping effects
We study the following nonlinear heat equation with damping and pumping effects (a reaction-diffusion equation) posed on a bounded simply connected convex domain $Ω\subset \mathbb{R}^d$, $d \geq 1$ with Lipschitz boundary $\partialΩ$: $$ \frac{\partial u(t)}{\partial t} - νΔu(t) + α|u(t)|^{p-2}u(t) - \sum_{\ell=1}^M β_{\ell} |u(t)|^{q_{\ell}-2}u(t) = f(t), \quad t>0, $$ subject to homogeneous Dirichlet boundary conditions and the initial condition $u(0)=u_0$, where $2 …

On the Piecewise Linear Perturbations of the Doubling Map
Anubrato Bhattacharyya, Kuntal Banerjee
https://arxiv.org/abs/2510.11441 https://arxiv.org/pdf/25…

On the Piecewise Linear Perturbations of the Doubling Map
Inspired by the 2007 work by M.~Misiurewicz and A.~Rodrigues [Double Standard Maps, M. Misiurewicz, A. Rodrigues, Communications in Mathematical Physics], we consider a family of circle maps that are perturbations of the doubling map on the circle by a piecewise linear map. We call this the \textit{piecewise linear perturbation of the doubling map} (PLPDM) and it is given by the formula, $f_{a,b}(x)= \displaystyle \bigl(2x+a+\dfrac{b}{2} S(x) \bigr) // 1 \quad {\text {for }} x, a, b \in [0,1]…

Sharp non-existence threshold for a parabolic Hardy-H{\'e}non equation with quasilinear diffusion
Razvan Gabriel Iagar (LAMA), Philippe Lauren\c{c}ot (LAMA)
https://arxiv.org/abs/2508.06164

Sharp non-existence threshold for a parabolic Hardy-H{é}non equation with quasilinear diffusion
Optimal conditions for initial data leading to non-existence of non-negative solutions to the Cauchy problem for the parabolic Hardy-H{é}non equation $$ \partial\_tu=Δu^m+|x|^σu^p, \quad (t,x)\in(0,\infty)\times\mathbb{R}^N, $$ with $m>0$, $σ>0$ and $p>\max\{1,m\}$, are identified. Assuming that the initial condition satisfies $$ u\_0\in L^{\infty}(\mathbb{R}^N), \quad \lim\limits\_{|x|\to\infty}|x|^γu\_0(x)=L\in(0,\infty), \quad u\_0\geq0, $$ it is shown that…

RARO: Reliability-aware Conversion with Enhanced Read Performance for QLC SSDs
Yanyun Wang, Dingcui Yu, Yina Lv, Yunpeng Song, Yumiao Zhao, Liang Shi
https://arxiv.org/abs/2508.19530

RARO: Reliability-aware Conversion with Enhanced Read Performance for QLC SSDs
Quad-level cell (QLC) flash offers significant benefits in cost and capacity, but its limited reliability leads to frequent read retries, which severely degrade read performance. A common strategy in high-density flash storage is to program selected blocks in a low-density mode (SLC), sacrificing some capacity to achieve higher I/O performance. This hybrid storage architecture has been widely adopted in consumer-grade storage systems. However, existing hybrid storage schemes typically focus on …

A Survey on a Pair of Diophantine Equations
Hung Viet Chu, Rishabh Gulecha, Sicheng Guo, Nathanael Johnson, Steven J. Miller, Yeju Shin
https://arxiv.org/abs/2509.01781 https://…

A Survey on a Pair of Diophantine Equations
Given two relatively prime numbers $a$ and $b$, it is known that exactly one of the two Diophantine equations has a nonnegative integral solution $(x,y)$: $$ ax + by \ =\ \frac{(a-1)(b-1)}{2}\quad \mbox{ and }\quad 1 + ax + by \ =\ \frac{(a-1)(b-1)}{2}. $$ Furthermore, the solution is unique. This paper surveys recent results on finding the solution and determining which equation is used when $a$ and $b$ are taken from certain sequences. We contribute to the literature by finding $(x,y)$ when…

An analytical parameterization for all solutions of the two-dimensional moment problem under Carleman-type conditions
Sergey M. Zagorodnyuk
https://arxiv.org/abs/2508.10823 http…

An analytical parameterization for all solutions of the two-dimensional moment problem under Carleman-type conditions
The two-dimensional moment problem consists of finding a positive Borel measure $μ$ in $\mathbb{R}^2$ such that $\int_{\mathbb{R}^2} t_1^m t_2^n dμ= s_{m,n}$, $m,n=0,1,2,...$, where $s_{m,n}$ are prescribed real constants (moments). We study this moment problem in the case when the sequence $\{ s_{m,n} \}_{m,n=0}^\infty$ is positive semi-definite, and the following Carleman-type conditions hold: $$ \sum_{k=1}^\infty \frac{1}{ \sqrt[2k]{ s_{2m,2k} + s_{2m+2,2k} } } = \infty,\quad m=0,1,2,.... …

Terry McLaurin injury: Commanders WR out for Week 4 matchup vs. Falcons with quad issue

https://www.cbssports.com/nfl/news/terry-mclaurin-injury…

Trace and Extension Theorems for Besov Functions in Doubling Metric Measure Spaces
Iv\'an Caama\~no, Josh Kline
https://arxiv.org/abs/2510.01385 https://

Trace and Extension Theorems for Besov Functions in Doubling Metric Measure Spaces
In the setting of a non-complete doubling metric measure space $(Ω,d,μ)$, we construct various bounded linear trace and extension operators for homogeneous and inhomogeneous Besov spaces $B^α_{p,q}$. Equipping the boundary $\partialΩ:=\overlineΩ\setminusΩ$ with a measure which is codimension $θ$ Ahlfors regular with respect to $μ$, these operators take the form \[ T:B^α_{p,q}(Ω)\to B^{α-θ/p}_{p,q}(\partialΩ),\quad E:B^α_{p,q}(\partialΩ)\to B^{α+θ/p}_{p,q}(Ω). \] The trac…

Commanders DL Deatrich Wise Jr. carted off after suffering quad injury against Packers

https://www.cbssports.com/nfl/news/commanders-dl-dea…

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…

Tight bounds for judicious 3-partitions of graphs
Peiru Kuang, Yan Wang
https://arxiv.org/abs/2509.20994 https://arxiv.org/pdf/2509.20994

Tight bounds for judicious 3-partitions of graphs
In this paper, we show that every graph with $m$ edges admits a 3-partition such that \[ \max_{1 \leq i \leq 3} e(V_i) \leq \frac{m}{9} + \frac{1}{9}h(m) \quad \text{and} \quad e(V_1, V_2, V_3) \geq \frac{2}{3}m + \frac{1}{3}h(m), \] where $h(m) = \sqrt{2m + 1/4} - 1/2$. This answers a problem of Bollobás and Scott affirmatively. We also solve several related problems of Bollobás and Scott. All of our results are tight.

Primes of the form $ax by$ in certain intervals with small solutions
Yuchen Ding, Takao Komatsu, Honghu Liu
https://arxiv.org/abs/2510.01781 https://arxiv.…

Primes of the form $ax+by$ in certain intervals with small solutions
Let $1

Source: McLaurin seeks more opinions on injury https://www.espn.com/nfl/story/_/id/46356868/commanders-mclaurin-seeks-more-input-quad-injury

Growth rates for the H\"older coefficients of the linear stochastic fractional heat equation with rough dependence in space
Chang Liu, Bin Qian, Ran Wang
https://arxiv.org/abs/2507.22379

Growth rates for the Hölder coefficients of the linear stochastic fractional heat equation with rough dependence in space
We study the linear stochastic fractional heat equation $$ \frac{\partial}{\partial t}u(t,x)=-(-Δ)^{\fracα{2}}u(t,x)+\dot{W}(t,x), \quad t>0, \quad x\in\mathbb{R}, $$ where $-(-Δ)^{\fracα{2}}$ denotes the fractional Laplacian with power $α\in (1,2)$, and the driving noise $\dot{W}$ is a centered Gaussian field that is white in time and has the covariance of a fractional Brownian motion with Hurst parameter $H\in\left(\frac{2-α}{2},\frac{1}{2}\right)$. We establish exact asymptotics for …

Superior Highly Composite Numbers and the Explicit Upper Bound of Generalized Divisor Functions
Lee-Peng Teo
https://arxiv.org/abs/2508.06764 https://arxiv…

Superior Highly Composite Numbers and the Explicit Upper Bound of Generalized Divisor Functions
For $k\geq 2$, we give a detailed exposition of the superior $k$-highly composite numbers. We then consider the function \[f_k(n)=\frac{\log d_k(n)\log\log n}{\log k\log n},\quad n\geq 3\] which has a maximum value $λ(k)$ at a superior $k$-highly composite number. We develop an efficient algorithm to compute $λ(k)$ and the positive integer $N_{\max}(k)$ where $f_k$ achieves the value $λ(k)$. The results for $2\leq k\leq 100$ are tabled.

Comparison principle for Singular Fractional $ g- $Laplacian Problems
Abdelhamid Gouasmia, Kaushik Bal
https://arxiv.org/abs/2507.21185 https://arxiv.org/p…

Comparison principle for Singular Fractional $ g- $Laplacian Problems
In this paper, we establish a novel comparison principle of independent interest and prove the uniqueness of weak solutions within the local Orlicz--Sobolev space framework, for the following class of fractional elliptic problems: \begin{equation*} (-Δ)^{s}_{g} u = f(x) u^{-α} + k(x) u^β, \quad u > 0 \quad \text{in } Ω; \quad u = 0 \quad \text{in } \mathbb{R}^{N} \setminus Ω, \end{equation*} where \( Ω\subset \mathbb{R}^{N} \) is a smooth bounded domain, \( α> 0 \), and \( β> 0 …

Level sets of the Takagi function
Lai Jiang, Ting-Ting Ying, Yi-Yang Zhang
https://arxiv.org/abs/2508.21683 https://arxiv.org/pdf/2508.21683

Level sets of the Takagi function
In this paper, we investigate the Takagi function, $$ T_r(x) = \sum_{n=0}^{\infty} \frac{ϕ(r^n x)}{r^n} ,\quad x\in [0,1], \quad r \in \mathbb{Z}^+, $$ where $ϕ(x)={\rm dist}(x,\mathbb{Z})$ represents the distance from $x$ to the nearest integer. We generalize the concept of local level sets and find that the expected number of local level sets contained in a level set $L_r(y)$, with $y$ chosen at random, is $1 + 1/r$ for every even integer $r \geq 2$.

Terry McLaurin injury: Commanders WR battling quad injury; status for Week 4 uncertain

https://www.cbssports.com/nfl/news/terry-mclaurin-injury-c…

Linearly Stable KAM Tori for One Dimensional Forced Kirchhoff Equations under Periodic Boundary Conditions
Yin Chen, Jiansheng Geng, Guangzhao Zhou
https://arxiv.org/abs/2509.04861

Linearly Stable KAM Tori for One Dimensional Forced Kirchhoff Equations under Periodic Boundary Conditions
We prove an abstract infinite dimensional KAM theorem, which could be applied to prove the existence and linear stability of small-amplitude quasi-periodic solutions for one dimensional forced Kirchhoff equations with periodic boundary conditions \[ u_{tt}-(1+\int_{0}^{2π} |u_x|^2 dx)u_{xx}+ M_ξu+εg(\barωt,x) =0,\quad u(t,x+2π)=u(t,x),\] where $M_ξ$ is a real Fourier multiplier, $g(\barωt,x)$ is real analytic with forced Diophantine frequencies $\barω$, $ε$ is a small p…

On three classes of logarithmic integrals
Necdet Bat{\i}r, Nandan Sai Dasireddy
https://arxiv.org/abs/2510.00870 https://arxiv.org/pdf/2510.00870

On three classes of logarithmic integrals
In this paper, we evaluate the following families of definite integrals in closed form and we show that they are expressible only in terms of the dilogarithm function and the inverse tangent integral, and elementary functions. \begin{equation*} \int_{0}^{1}\frac{\log\big(x^m+1\big)}{x+1}\thinspace{\rm d}x \quad \mbox{and}\quad \int_{0}^{1}\frac{\log\big(x^m+1\big)}{x^2+1}\thinspace{\rm d}x, \end{equation*} where $m$ is a positive odd integer. When $m$ is a positive even integer, these integrals…

On a new region for the Lane-Emden conjecture in higher dimensions
Kui Li, Mingxiang Li, Juncheng Wei
https://arxiv.org/abs/2510.06613 https://arxiv.org/pd…

On a new region for the Lane-Emden conjecture in higher dimensions
We study the Lane-Emden conjecture, which asserts the non-existence of non-trivial, non-negative solutions to the Lane-Emden system \[ -Δu = v^p, \quad -Δv = u^q, \quad x \in \mathbb{R}^n\] in the subcritical regime. By employing an Obata-type integral inequality, Picone's identity, and exploiting the scaling invariance of the system, we prove that the conjecture holds for any dimension $n \geq 5$ and exponents satisfying $p\geq 1,q\geq 1$, and \[ \frac{1}{p+1} + \frac{1}{q+1} \geq …

Wide Angle shot outside Duffield Hall on Cornell's Engineering Quad
#photo #photography #cornell #buildings

The Boundary Principle of a Single Big Jump: Refined Asymptotics for Branching Processes with Immigration
Yunfan Zhao
https://arxiv.org/abs/2509.05650 https://

The Boundary Principle of a Single Big Jump: Refined Asymptotics for Branching Processes with Immigration
We analyze the stationary tail of a fixed-point equation arising in branching processes with state-independent immigration, when both immigration and offspring distributions have heavy tails with boundary index one. We prove that \[ P(X > x) \sim \frac{1}{(1-b)(1+x)}, \quad x \to \infty, \] and provide a refined asymptotic identifying negligible logarithmic corrections. Our approach develops a closure principle for subexponential summations and a cluster expansion representation, which disentan…

The Diophantine Frobenius Problem revisited
Yuchen Ding, Weijia Wang, Hao Zhang
https://arxiv.org/abs/2509.08599 https://arxiv.org/pdf/2509.08599

The Diophantine Frobenius Problem revisited
Let $k\ge 2$ and $a_1, a_2, \cdots, a_k$ be positive integers with \[ \gcd(a_1, a_2, \cdots, a_k)=1. \] It is proved that there exists a positive integer $G_{a_1, a_2, \cdots, a_k}$ such that every integer $n$ strictly greater than it can be represented as the form \[ n=a_1x_1+a_2x_2+\cdots+a_kx_k, \quad (x_1, x_2, \cdots, x_k\in\mathbb{Z}_{\ge 0},~\gcd(x_1, x_2, \cdots, x_k)=1). \] We then investigate the size of $G_{a_1, a_2}$ explicitly. Our result strengthens the primality requirement of $x…

Serrin's overdetermined theorem within Lipschitz domains
Hongjie Dong, Yi Ru-Ya Zhang
https://arxiv.org/abs/2509.05155 https://arxiv.org/pdf/2509.05155…

Serrin's overdetermined theorem within Lipschitz domains
Let $Ω\subset\mathbb R^n$ be a Lipschitz domain, $K$ be a (bounded) ellipsoid centered at the origin and $H$ be the associated Wulff potential. We prove that, % under a $L^2$-Dini-VMO condition and a non-degeneracy condition on $|Du|$ near the boundary, $Ω$ satisfies the following Serrin-type overdetermined system $$u \in W^{1,2}(\mathbb R^n), \quad u=0\ \text{ a.e. in }\R^n\setminus Ω,\quad Δ_H u=\mathbf{c}\mathscr{H}^{n-1}|_{\partial^*Ω} - \mathbf{1}_Ω\,dx,$$ in the w…

Homeomorphism theorem for sums of translates on the real axis
Tatiana M. Nikiforova
https://arxiv.org/abs/2509.07776 https://arxiv.org/pdf/2509.07776

Homeomorphism theorem for sums of translates on the real axis
In this paper, we study sums of translates on the real axis. These functions generalize logarithms of weighted algebraic polynomials. Namely, we are dealing with the following functions \[ F(\mathbf{y},t) := J(t) + \sum \limits_{j=1}^n K_j(t-y_j), \quad \mathbf{y} := (y_1,\ldots,y_n), \ y_1 \le \ldots \le y_n, \] where the field function $J$ is a function defined on $\mathbb{R}$, which is "admissible" for the kernels $K_1,\ldots,K_n$ concave on $(-\infty,0)$ and on $(0,\infty)$ and having a sin…

Terry McLaurin injury update: Commanders star WR could return for Week 7 vs. Cowboys, per report

https://www.cbssports.com/nfl/news/terry-mclaurin-injury-update-…

Monotonicity and local uniqueness for an isotropic nonlocal elliptic equation
Yi-Hsuan Lin
https://arxiv.org/abs/2510.10408 https://arxiv.org/pdf/2510.1040…

Monotonicity and local uniqueness for an isotropic nonlocal elliptic equation
We extend monotonicity-based inversion methods to an inverse coefficient problem for the isotropic nonlocal elliptic equation \[ (-\nabla \cdot σ\nabla)^s u = 0 \quad \text{in } Ω\subset \mathbb{R}^n, \] where $0 < s < 1$, $n \geq 3$, and $Ω$ is a bounded open set. We establish a monotonicity relation between the leading coefficient $σ$ and the (partial) exterior Dirichlet-to-Neumann (DN) map. Our main result shows that a monotonicity ordering of the coefficients implies a correspon…

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…

On the ternary Estermann problem with almost proportional summands
Firuz Rakhmonov
https://arxiv.org/abs/2510.05602 https://arxiv.org/pdf/2510.05602…

On the ternary Estermann problem with almost proportional summands
For $n \geq 3$, an asymptotic formula is derived for the number of representations of a sufficiently large natural number $N$ in the form $p_1+p_2+m^n=N$, where $p_1$, $p_2$ $-$ prime numbers, $m$ $-$ natural number satisfying the conditions $$ \left|p_k-μ_kN\right|\le H, \quad k=1,2,\qquad \left|m^n-μ_3N\right|\le H,\qquad H \ge N^{1-\frac1{n(n-1)}} {\mathscr{L}}^{\frac{2^{n+1}}{n-1}+n-1},$$ for $μ_1+μ_2+μ_3=1, \ \ μ_i >0, \mathscr{L} = \ln{N}. $ Keywords: Estermann…

A note on zeros of derivatives of ultraspherical Bessel functions
Tao Jiang
https://arxiv.org/abs/2508.16426 https://arxiv.org/pdf/2508.16426

A note on zeros of derivatives of ultraspherical Bessel functions
For any fixed $ν\ge 0, δ\in \mathbb R$ and $x>0$, we investigate the positive zeros of the derivatives $j'_{ν,δ}(x)$ and $y'_{ν,δ}(x)$, where \begin{equation*} j_{ν,δ}(x)=x^{-δ}J_ν(x)\quad\text{and} \quad y_{ν,δ}(x)=x^{-δ}Y_ν(x). \end{equation*} We derive asymptotic expansions for their $k$-th positive zeros as $k\rightarrow \infty$.

Normalized solutions for the nonlinear Schr\"{o}dinger equation with potentials
Matteo Rizzi, Xueqin Peng
https://arxiv.org/abs/2507.20961 https://arx…

Normalized solutions for the nonlinear Schrödinger equation with potentials
In this paper, we find normalized solutions to the following Schrödinger equation \begin{equation}\notag \begin{aligned} &-Δu-\fracμ{|x|^2}h(x)u+λu =f(u)\quad\text{in}\quad\mathbb{R}^{N},\\ & u>0,\quad \int_{\mathbb{R}^{N}}u^2dx=a^2, \end{aligned} \end{equation} where $N\geq3$, $a>0$ is fixed, $f$ satisfies mass-subcritical growth conditions and $h$ is a given bounded function with $||h||_\infty\le 1$. The $L^2(\mathbb{R}^N)$-norm of $u$ is fixed and $λ$ appea…

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…

Narayana numbers that are products of two Fibonacci numbers
Japhet Odjoumani
https://arxiv.org/abs/2508.02688 https://arxiv.org/pdf/2508.02688

Narayana numbers that are products of two Fibonacci numbers
Let $\{N_m\}_{m\ge0}$ be the Narayana's cows sequence given by $N_0=0$, $N_1=1=N_2=1$ and \[ N_{m+3}=N_{m+2}+N_m,\quad \text{ for }\; m\geq 0 \] and let $\{F_n\}_{n\ge0}$ be the Fibonacci sequence. In this paper we solve explicitely the Diophantine equation \[ N_m=F_nF_k, \] in positive unknowns $m,\,n$ and $k$. That is, we find the non-zero narayana numbers that are products of two Fibonacci numbers.

Comparison Principle, A.B.P.-type estimates for solutions of quasi-linear elliptic equations in non-divergence form and some implications
Junior da S. Bessa, Reshmi Biswas, Jo\~ao Vitor da Silva, Ginaldo S\'a, Makson Santos
https://arxiv.org/abs/2508.18495

Comparison Principle, A.B.P.-type estimates for solutions of quasi-linear elliptic equations in non-divergence form and some implications
In this work, we establish global gradient estimates to solutions of quasilinear elliptic models in non-divergence form with general degeneracy law and a Hamiltonian term, given by $$ -Ψ(x, |\nabla u|)Δ_p^{\mathrm{N}}u(x)+\mathscr{H}(x,\nabla u)=f(x) \quad \mathrm{in} \quad Ω, \quad \mathrm{for} \,\,\,1

Broncos' key free-agent addition Dre Greenlaw, DL Malcolm Roach out Week 1 vs. Titans https://www.nytimes.com/athletic/6602163/2025/09/05/broncos-dre-greenlaw-out-titans/

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 $…

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…

Terry McLaurin injury update: Commanders star WR ruled out for Week 5 game vs. Chargers

https://www.cbssports.com/nfl/news/terry-mclaurin-injury…

Symmetry and monotonicity of singular solutions for the Hartree equation
Ying Cai, Guangze Gu, Aleks Jevnikar
https://arxiv.org/abs/2508.07893 https://arxi…

Symmetry and monotonicity of singular solutions for the Hartree equation
In this paper we are concerned with positive singular solutions of the following nonlocal Hartree equation $$-Δu\!=\Big( \int_{\mathbb{R}^N\setminus Γ}\frac{F(u(y))}{|x-y|^μ}dy \Big)f (u(x)), \quad x\in \mathbb{R}^N\setminusΓ,$$ where $F$ is the primitive of $f$ and $Γ$ is the singular set. Under suitable assumptions, we prove that $u$ is symmetric and monotone with respect to the singular set by using moving plane methods. Furthermore, we complement this study by showing the exi…

Commanders WR McLaurin ruled out vs. Chargers https://www.espn.com/nfl/story/_/id/46469001/commanders-receiver-terry-mclaurin-ruled-vs-chargers

Fractional Morrey-Sobolev type embeddings and nonlocal subelliptic problems with oscillating nonlinearities on stratified Lie groups
Sekhar Ghosh, Tianxiang Gou, Vishvesh Kumar, Vicentiu D. Radulescu
https://arxiv.org/abs/2509.18587

A duality approach to the fractional Laplacian with measure data
Kenneth H. Karlsen, Francesco Petitta, Suleyman Ulusoy
https://arxiv.org/abs/2508.06390 https://

A duality approach to the fractional Laplacian with measure data
We describe a duality method to prove both existence and uniqueness of solutions to nonlocal problems like $$ (-Δ)^s v = μ\quad \text{in}\ \mathbb{R}^N, $$ with vanishing conditions at infinity. Here $μ$ is a bounded Radon measure whose support is compactly contained in $\mathbb{R}^N$, $N\geq2$, and $-(Δ)^s$ is the fractional Laplace operator of order $s\in (1/2,1)$.

Global weak solutions to a doubly degenerate nutrient taxis system on the whole real line
Federico Herrero-Herv\'as
https://arxiv.org/abs/2508.07503 https://

Global weak solutions to a doubly degenerate nutrient taxis system on the whole real line
This work addresses the one-dimensional Cauchy problem for the doubly degenerate nutrient taxis model \begin{equation*} \begin{cases} \displaystyle \frac{\partial u}{\partial t} = \frac{\partial}{\partial x}(u v u_x) - \frac{\partial}{\partial x}(u^2 v v_x) + u v, & x\in \mathbb{R}, ~t>0, \\ \displaystyle \frac{\partial v}{\partial t} = \frac{\partial^2 v}{\partial x^2} - u v, & x\in \mathbb{R}, ~t>0, \\ u(x,0) = u_0(x) \geq 0, \quad v(x,0) = v_0(x)>0, ~ & x\in \mathbb{R}, \end{cases}…

Infinite families of congruences for the second order mock theta function $\mathcal{B}(q)$
Hemjyoti Nath, Hirakjyoti Das
https://arxiv.org/abs/2509.20708 https://

Infinite families of congruences for the second order mock theta function $\mathcal{B}(q)$
The arithmetic properties of the second order mock theta function $\mathcal{B}(q)$, introduced by McIntosh, defined by \begin{equation*} \mathcal{B}(q) := \sum_{n \geq 0} \frac{q^n (-q;q^2)_n}{(q;q^2)_{n+1}} = \sum_{n \geq 0}b(n)q^n, \end{equation*} have been extensively studied. For instance, for all $n\ge0$, Kaur and Rana established congruences such as for all $n\ge0$, \begin{align*} b(12n+10) &\equiv 0 \pmod{36}, \quad b(18n+16) \equiv 0 \pmod{72}, \end{align*} Chen and Mao proved tha…

Diffuse measures and nonlinear parabolic equations
Francesco Petitta, Augusto C. Ponce, Alessio Porretta
https://arxiv.org/abs/2508.06384 https://arxiv.org…

Diffuse measures and nonlinear parabolic equations
Given a parabolic cylinder $Q =(0,T)\timesΩ$, where $Ω\subset \mathbb{R}^{N}$ is a bounded domain, we prove new properties of solutions of \[ u_t-Δ_p u = μ\quad \text{in $Q$} \] with Dirichlet boundary conditions, where $μ$ is a finite Radon measure in $Q$. We first prove a priori estimates on the $p$-parabolic capacity of level sets of $u$. We then show that diffuse measures (i.e.\@ measures which do not charge sets of zero parabolic $p$-capacity) can be strongly approximated by the measu…

Nonradial normalized solutions for a quasilinear Schr\"{o}dinger equation via dual approach
Lin Zhang
https://arxiv.org/abs/2509.05557 https://arxiv.o…

Nonradial normalized solutions for a quasilinear Schrödinger equation via dual approach
In this paper, we will utilize the dual method to construct multiple nonradial normalized solutions of the following quasilinear Schrödinger equation: \begin{equation*} -Δu-Δ(|u|^{2})u-μu=|u|^{p-2}u, \qquad in \quad \mathbb{R}^N, \end{equation*} subject to a mass-subcritical constraint. It should be emphasized that the nonradial result of this equation is new. Besides that, when considering the nonradial problem, it is necessary to construct a new workspace to ensure the…

On Brezis-Nirenberg problems: open questions and new results in dimension six
Fengliu Li, Giusi Vaira, Juncheng Wei, Yuanze Wu
https://arxiv.org/abs/2509.19863 https://

On Brezis-Nirenberg problems: open questions and new results in dimension six
In this paper, we consider the Brezis-Nirenberg problem \begin{equation*} \left\{\begin{aligned} &-Δu = λu+|u|^{2^*-2}u, \quad &\mbox{in}\,Ω,\\ &u=0,\quad &\mbox{on}\, \partialΩ, \end{aligned}\right. \end{equation*} where $Ω$ is a smoothly bounded domain of $\mathbb R^N$ with $N\geq 3$, $λ>0$ is a parameter and $2^*=\frac{2N}{N-2}$ is the critical Sobolev exponent. We first recall the history of the Brezis-Nirenberg problem and then provide new results of it in dimension six. Fi…

$L^p$ Hardy inequalities with homogeneous weights
Subhajit Roy
https://arxiv.org/abs/2509.05674 https://arxiv.org/pdf/2509.05674

$L^p$ Hardy inequalities with homogeneous weights
For $p\in (1,\infty)$ and $α\in\mathbb{R}$, we consider measurable functions $g$ on $\mathbb{S}^{N-1}$ that satisfy the following weighted Hardy inequality: \begin{equation}\label{abs} \int_{\mathbb{R}^N}\frac{ g (x/|x|)}{|x|^{p+α}}|u(x)|^p dx \leq C\int_{\mathbb{R}^N}\frac{|\nabla u(x)|^p}{|x|^α} dx, \quad\forall\,u\in \mathcal{C}_c^\infty(\mathbb{R}^N), \end{equation} for some constant $C>0$. Depending on $N$, $p$, and $α$, we identify s…

On a general class of free boundary Monge-Amp\`ere equations
Tristan C. Collins, Benjy Firester
https://arxiv.org/abs/2508.05551 https://arxiv.org/pdf/2508…

On a general class of free boundary Monge-Ampère equations
We solve a general class of free boundary Monge-Ampère equations given by \[ \det D^2u = λ\dfrac{f(-u)}{g(u^\star)h(\nabla u)}χ_{\{u<0\}} \; \text{ in } \mathbb{R}^n, \quad \nabla u (\mathbb{R}^n) = P \] where $P$ is a bounded convex set containing the origin, and $h>0$ on $P$. We consider applications to optimal transport with degenerate densities, Monge-Ampère eigenvalue problems, and geometric problems including a hemispherical Minkowski problem and free boundary Kähler-Ricci solitons…

Regularity theory for mixed local-nonlocal problem involving general stable operators
Pedro Fellype Pontes, Minbo Yang
https://arxiv.org/abs/2510.06569 https://

Regularity theory for mixed local-nonlocal problem involving general stable operators
In this paper, we study the regularity of solutions to a linear elliptic equation involving a mixed local-nonlocal operator of the form $$Lu - \operatorname{div}\big(a(x)\nabla u(x)\big)= f, \quad \text{in } Ω\subset \mathbb{R}^n,$$ where $L$ is a general stable Lévy type operator and $a(\cdot)$ is a positive Hölder continuous weight. By establishing a maximum principle and a Liouville-type result in the entire space, we are able to derive the interior regularity and the regularity up to the…

Liouville theorem for the inequality $\Delta_m u f(u)\leq 0$ on Riemannian manifolds
Biqiang Zhao
https://arxiv.org/abs/2509.16659 https://arxiv.org/pdf/25…

Liouville theorem for the inequality $Δ_m u+f(u)\leq 0$ on Riemannian manifolds
In this paper, we study the quasilinear inequality $ Δ_m u+f(u)\leq 0$ on a complete Riemannian manifold, where \begin{align*} m>1,α>m-1 \quad and \quad f(t)> 0,αf(t)-tf^{'}(t)\geq 0, \forall t>0. \end{align*} If for some point $x_0$ and large enough $r$, \begin{align*} vol B_r(x_0)\leq C r^p ln^q r, \end{align*} where $p=\frac{mα}{α-(m-1)},q=\frac{m-1}{α-(m-1)}$ and $B_r(x_0) $ is a geodesic ball of radius $r$ centered at $x_0$, then the inequality possesses no positive weak solution. Th…

Two alternative proofs of weak Harnack inequality for mixed local and nonlocal $p$-Laplace equations with a nonhomogeneity
Prashanta Garain
https://arxiv.org/abs/2510.04065 http…

Two alternative proofs of weak Harnack inequality for mixed local and nonlocal $p$-Laplace equations with a nonhomogeneity
We study a class of mixed local and nonlocal $p$-Laplace equations with prototype \[ -Δ_p u + (-Δ_p)^s u = f \quad \text{in } Ω, \] where $Ω\subset \mathbb{R}^n$ is bounded and open. We provide sufficient condition on $f$ to ensure weak Harnack inequality with a tail term for sign-changing supersolutions. Two different proofs are presented, avoiding the Krylov--Safonov covering lemma and expansion of positivity: one via the John--Nirenberg lemma, the other via the Bombieri--Giusti lemma. To…

Trace and Observability Inequalities for Laplace Eigenfunctions on the Torus
Nicolas Burq, Pierre Germain, Massimo Sorella, Hui Zhu
https://arxiv.org/abs/2507.16599

Trace and Observability Inequalities for Laplace Eigenfunctions on the Torus
We investigate trace and observability inequalities for Laplace eigenfunctions on the d-dimensional torus, with respect to arbitrary Borel measures $μ$. Specifically, we characterize the measures $μ$ for which the inequalities $$ \int |u|^2 d μ\lesssim \int |u|^2 d x \quad \text{(trace)}, \qquad \int |u|^2 d μ\gtrsim \int |u|^2 d x \quad \text{(observability)}$$ hold uniformly for all eigenfunctions $u$ of the Laplacian. Sufficient conditions are …

Global Dynamics of the Non-Radial Energy-Critical Inhomogeneous Biharmonic NLS
Carlos M. Guzm\'an, Sahbi Keraani, Chengbin Xu
https://arxiv.org/abs/2508.02796 https://

Global Dynamics of the Non-Radial Energy-Critical Inhomogeneous Biharmonic NLS
We investigate the focusing inhomogeneous nonlinear biharmonic Schrödinger equation \[ i\partial_t u + Δ^2 u - |x|^{-b}|u|^p u = 0 \quad \text{on } \mathbb{R} \times \mathbb{R}^N, \] in the energy-critical regime, $p = \frac{8 - 2b}{N - 4}$, and $5 \leq N < 12$. We focus on the challenging non-radial setting and establish global well-posedness and scattering under the subcritical assumption $ \sup_{t \in I} \|Δu(t)\|_{L^2} < \|ΔW\|_{L^2}, $ where $W$ …

Homogeneous steady states for the generalized surface quasi-geostrophic equations
Ken Abe, Javier G\'omez-Serrano, In-Jee Jeong
https://arxiv.org/abs/2510.03009 https://

Homogeneous steady states for the generalized surface quasi-geostrophic equations
We consider homogeneous (stationary self-similar) solutions to the generalized surface quasi-geostrophic (gSQG) equations parametrized by the constant $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…

On finite-energy solutions of Kazan-Warner equations on the lattice graph
Huyuan Chen, Bobo hua
https://arxiv.org/abs/2509.01155 https://arxiv.org/pdf/2509…

On finite-energy solutions of Kazan-Warner equations on the lattice graph
We investigate finite-energy solutions to Kazdan-Warner type equations in 2-dimensional integer lattice graph $$ - Δu= \varepsilon e^{κu} +βδ_0\quad {\rm in}\ \mathbb{Z}^2,$$ where $\varepsilon=\pm1$, $κ>0$ and $β\in\mathbb{R}$. When $\varepsilon=1$, we prove the existence of a continuous family of finite-energy solutions for some parameter $κ$. This provides a partial resolution of the open problem on the existence of finite-energy solutions to the Liouville equation. When $\varepsi…

Traveling wave solutions for the generalized Burgers-Fisher equation
Razvan Gabriel Iagar, Ariel S\'anchez
https://arxiv.org/abs/2509.24909 https://arx…

Traveling wave solutions for the generalized Burgers-Fisher equation
Traveling wave solutions, in the form $u(x,t)=f(x+ct)$, to the generalized Burgers-Fisher equation $$ \partial_tu=u_{xx}+k(u^n)_x+u^p-u^q, \quad (x,t)\in\mathbb{R}\times(0,\infty), $$ with $n\geq2$, $p>q\geq1$ and $k>0$, are classified with respect to their speed $c\in(-\infty,\infty)$ and the behavior at $\pm\infty$. The existence and uniqueness of traveling waves with any speed $c\in\mathbb{R}$ is established and their behavior as $x\to\pm\infty$ is described. In particular, it is shown that …

On the existence of normalized solutions to a class of fractional Choquard equation with potentials
Yongpeng Chen, Zhipeng Yang, Jianjun Zhang
https://arxiv.org/abs/2507.23363 h…

On the existence of normalized solutions to a class of fractional Choquard equation with potentials
This paper investigates the existence of normalized solutions to the nonlinear fractional Choquard equation: $$ (-Δ)^s u+V(x) u=λu+f(x)\left(I_α*\left(f|u|^q\right)\right)|u|^{q-2} u+g(x)\left(I_α*\left(g|u|^p\right)\right)|u|^{p-2} u, \quad x \in \mathbb{R}^N $$ subject to the mass constraint $$ \int_{\mathbb{R}^N}|u|^2 d x=a>0, $$ where $N>2 s, s \in(0,1), α\in(0, N)$, and $\frac{N+α}{N} \leq q

Necessary and Sufficient Conditions for the Maz'ya-Shaposhnikova Formula in (Fractional) Sobolev Spaces
Elisa Davoli, Giovanni Di Fratta, Rossella Giorgio, Andrea Pinamonti
https://arxiv.org/abs/2509.23226

Necessary and Sufficient Conditions for the Maz'ya-Shaposhnikova Formula in (Fractional) Sobolev Spaces
We investigate the asymptotic behavior, as $\varepsilon \to 0$, of nonlocal functionals $$ \mathcal{F}_{\varepsilon}(u) = \iint_{\mathbb{R}^N\times\mathbb{R}^N} ρ_{\varepsilon}(y-x)\,|u(x)-u(y)|^p\,dx\,dy,\qquad u\in L^p(\mathbb{R}^N),\quad 1\leqslant p<\infty, $$ associated with a general family of nonnegative measurable kernels $\{ρ_{\varepsilon}\}_{\varepsilon>0}$. Our primary aim is to single out the weakest moment-type assumptions on the family $\{ρ_{\varepsilon}\}_{\varepsi…

Liouville theorems for anisotropic $p$-Laplace equations with a semilinear term
Weizhao Liang, Tian Wu, Jin Yan
https://arxiv.org/abs/2507.20182 https://ar…

Liouville theorems for anisotropic $p$-Laplace equations with a semilinear term
In this paper, we investigate Liouville theorems for solutions to the anisotropic $p$-Laplace equation $$-Δ_p^H u=-\operatorname{div}(a(\nabla u))=f(u),\quad\text{in }\mathbb{R}^n,$$ where the semilinear term $f$ may be positive, negative, or sign-changing. When $f$ is positive (negative) and satisfies certain conditions, Serrin's technique is applied to show that every positive supersolution (subsolution) must be constant. For the subcritical case, we use the invariant tensor method to prove …