Tootfinder

Proof of the exact diffusion constant via first passage time in quasi-periodic potentials
Ming Gong
https://arxiv.org/abs/2510.10435 https://arxiv.org/pdf/…

Proof of the exact diffusion constant via first passage time in quasi-periodic potentials
Brownian motion in terms of Lifson and Jackson (LJ) formula has been widely explored in periodic systems and it has been believed for a long time that the LJ formula only applies to periodic potentials. Recently we show that for the following Brownian motion $γ\dot{x} = -U'(x) + ξ$, where $U(x)$ is the quasi-periodic potential, the effective diffusion constant can still be described by the LJ formula $D^* = D/(\langle \exp(βU)\rangle \langle \exp(-βU)\rangle)$, where the average is redefine…

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…

Revisiting Cases 2 and 11 of the Map Color Theorem
Timothy Sun
https://arxiv.org/abs/2509.06407 https://arxiv.org/pdf/2509.06407

Revisiting Cases 2 and 11 of the Map Color Theorem
In 1968, Ringel and Youngs solved the remaining cases of the orientable Map Color Theorem by finding genus embeddings of the complete graphs $K_n$, for sufficiently large $n \equiv 2, 8, 11 \pmod{12}$. Following the approach previously explored by the author for $n \equiv 8 \pmod{12}$, we aim to streamline their constructions for $n \equiv 2, 11 \pmod{12}$ by finding families of current graphs with simpler patterns for the arc labelings.

A bilinear form for spin$^c$ manifolds
Huijun Yang
https://arxiv.org/abs/2509.01979 https://arxiv.org/pdf/2509.01979

A bilinear form for spin$^c$ manifolds
For any $(8n {+} 2)$-dimensional closed oriented spin$^{c}$ manifold, denote by $[M]$ the fundamental class of $M$, $\langle \cdot ~, ~\cdot \rangle$ the Kronecker product, and $ρ_{2} \colon H^{4n}(M; \mathbb{Z}) \rightarrow H^{4n}(M; \mathbb{Z}/2)$ the $\bmod ~ 2$ reduction homomorphism. In this paper, we will prove that the following identity \[ \langle ρ_2(t) \cdot Sq^2 ρ_2 (t), [M] \rangle = \langle ρ_2 (t) \cdot Sq^2 v_{4n}(M), [M]\rangle, \] holds for any torsion class $t \in H^{4n}(M…

Rescaling of unconditional Schauder frames in Hilbert spaces and completely bounded maps
Anton Tselishchev
https://arxiv.org/abs/2508.02802 https://arxiv.o…

Rescaling of unconditional Schauder frames in Hilbert spaces and completely bounded maps
We prove that if every element $u$ in a Hilbert space $H$ admits a representation as unconditionally convergent series $$u=\sum_{k=1}^\infty \langle u, y_k\rangle x_k,$$ then there exist nonzero scalars $\{α_k\}_{k=1}^\infty$ such that both sequences $\{α_k x_k\}_{k=1}^\infty$ and $\{\overlineα_k^{-1}y_k\}_{k=1}^\infty$ are frames. Our result has the following equivalent reformulation: if $Φ:\ell^\infty\to B(H)$ is a bounded linear map such that for every element of the unit vector basis $e…

Study on the certain type of nonlinear algebraic partial differential equation in $\mathbb{C}^m$
Sujoy Majumder, Debabrata Pramanik, Nabadwip Sarkar
https://arxiv.org/abs/2508.15857

Study on the certain type of nonlinear algebraic partial differential equation in $\mathbb{C}^m$
In the paper, using Nevanlinna's value distribution theory of meromorphic functions in $\mathbb{C}^m$, we study for the existence of entire solutions $f$ in $\mathbb{C}^m$ of the following algebraic partial differential equation \[f^n(z)+P_d(f(z))=p(z)e^{\langle α,\ol z\rangle},\] where $P_d(f)$ is an algebraic differential polynomial in $f$ of degree $d \leq n-2$, $n \geq 3$ is an integer, $p$ is a non-zero polynomial, $α=(α_{1},\ldots,α_{m})\neq (0,\ldots,0)$ and $\langle α,\ol z\rangle=…