Tootfinder

A New Proof of Meissner's Optimal Bound on the Degree of a Poincar\'e Multiplier and an Improved Optimal Degree Multiplier
Brittany Riggs
https://arxiv.org/abs/2509.09008

A New Proof of Meissner's Optimal Bound on the Degree of a Poincaré Multiplier and an Improved Optimal Degree Multiplier
Let $f$ be a monic univariate polynomial. We say that $f$ is positive if $f(x)$ is positive over all $x > 0$. If all the coefficients of $f$ are non-negative, then $f$ is trivially positive. In 1883, Poincaré proved that $f$ is positive if and only if there exists a monic polynomial $g$ such that all the coefficients of $gf$ are non-negative. Such polynomial $g$ is called a Poincaré multiplier for the positive polynomial $f$. Of course one hopes to find a multiplier with smallest degree. In 1…

Johnson homomorphisms and the second rational cohomology of handlebody Torelli groups
Annie Holden
https://arxiv.org/abs/2509.03742 https://arxiv.org/pdf/2…

Johnson homomorphisms and the second rational cohomology of handlebody Torelli groups
We introduce two Torelli subgroups of the handlebody group. The group $HI_{g,p}^b$ is the subgroup of the handlebody group acting trivially on the first homology of the boundary surface, and $H_B I_{g,p}^b$ is the subgroup of the handlebody group acting trivially on the first homology of the handlebody. Using the symplectic representation and the Johnson homomorphisms for the Torelli subgroups of the mapping class group and of $\operatorname{Aut}(F_g)$, we define abelian quotients of these hand…

Few Bilinear Operators on Spaces of Continuous Functions
Leandro Candido
https://arxiv.org/abs/2510.05322 https://arxiv.org/pdf/2510.05322

Few Bilinear Operators on Spaces of Continuous Functions
Motivated by recent work exhibiting a locally compact scattered space $L$ constructed under Ostaszewski's $\clubsuit$-principle, which yielded a complete classification of linear operators on $C_0(L\times L)$, we extend the analysis to the bilinear setting. We show that, for this space $L$, every bilinear operator $G:C_0(L)\times C_0(L)\to C_0(L)$ admits a unique decomposition into the sum of trivially predictable components. This establishes a bilinear analogue of the few operators phenomenon.

Triacontagonal proofs of the Bell-Kochen-Specker theorem
P. K. Aravind, Justin Y. J. Burton, Guillermo Nunez-Ponasso, D. Richter
https://arxiv.org/abs/2510.01063 https://…

Triacontagonal proofs of the Bell-Kochen-Specker theorem
Coxeter pointed out that a number of polytopes can be projected orthogonally into two dimensions in such a way that their vertices lie on a number of concentric regular triacontagons (or 30-gons). Among them are the 600-cell and 120-cell in four dimensions and Gosset's polytope in eight dimensions. We show how these projections can be modified into Kochen-Secker diagrams from which parity proofs of the Bell-Kochen-Specker theorem are easily extracted. Our construction trivially yields parity pr…

Effective height bounds via Mordell Weil lattice symmetries
Madhavi Prakash
https://arxiv.org/abs/2509.22672 https://arxiv.org/pdf/2509.22672

Effective height bounds via Mordell Weil lattice symmetries
We obtain explicit, computable upper bounds for the Neron-Tate height of rational points on curves of genus at least two over number fields. The bounds use automorphisms acting on the Mordell-Weil lattice of the Jacobian. We prove an averaged spectral-gap criterion that replaces the 'large enough automorphism group' requirement of prior work. As a special case, when some automorphism acts trivially on V the method gives the sharper bound. In rank 2 we provide a practical Bravais-lattice test, u…

The bispectral problem, the Darboux process, monodromy and the Hermite operator
M. M. Castro, F. A. Gr\"unbaum
https://arxiv.org/abs/2509.04158 https://

The bispectral problem, the Darboux process, monodromy and the Hermite operator
The complete solution of the bispectral problem for the Schrödinger operator $L=-\tfrac{d^2}{dx^2}+V(x)$ in [DG] (J. J. Duistermaat and F. A. Grünbaum, Differential equations in the spectral parameter, Comm. Math. Phys. 103 (1986), 177-240) is obtained by the application of the Darboux process to the cases of $V=0$ and $V(x)=-\tfrac{1}{4x^2}$. Both of these cases are trivially bispectral and after repeated applications of the Darboux process one gets either a pair of rank one bundles of bispe…

LSM and CPT
Nathan Seiberg, Shu-Heng Shao, Wucheng Zhang
https://arxiv.org/abs/2508.17115 https://arxiv.org/pdf/2508.17115…

LSM and CPT
We study a number of 1+1d lattice models with anti-unitary symmetries that simultaneously reflect space and reverse time. Some of these symmetries are anomalous, leading to Lieb-Schultz-Mattis-type constraints, thus excluding a trivially gapped phase. Examples include a mod 8 anomaly in the Majorana chain and various mod 2 anomalies in the spin chain. In some cases, there is an exact, non-anomalous lattice symmetry that flows in the continuum to CPT. In some other cases, the CPT symmetry of the…

Finite generation of abelianizations of the genus 3 Johnson kernel and the commutator subgroup of the Torelli group for $\mathrm{Out}(F_3)$
Alexander A. Gaifullin
https://arxiv.org/abs/2507.20710

Finite generation of abelianizations of the genus 3 Johnson kernel and the commutator subgroup of the Torelli group for $\mathrm{Out}(F_3)$
Let $Σ_g^b$ be a compact oriented surface of genus $g$ with $b$ boundary components, where $b\in\{0,1\}$. The Johnson kernel $\mathcal{K}_g^b$ is the subgroup of the mapping class group $\mathrm{Mod}(Σ_g^b)$ generated by Dehn twists about separating simple closed curves. Let $F_n$ be a free group with $n$ generators. The Torelli group for $\mathrm{Out}(F_n)$ is the subgroup $\mathrm{IO}_n\subset\mathrm{Out}(F_n)$ consisting of all outer automorphisms that act trivially on the abelianization o…

Bootstrap Policy Iteration for Stochastic LQ Tracking with Multiplicative Noise
Jiayu Chen, Zhenhui Xu, Xinghu Wang
https://arxiv.org/abs/2508.20394 https://

Bootstrap Policy Iteration for Stochastic LQ Tracking with Multiplicative Noise
This paper studies the optimal tracking control problem for continuous-time stochastic linear systems with multiplicative noise. The solution framework involves solving a stochastic algebraic Riccati equation for the feedback gain and a Sylvester equation for the feedforward gain. To enable model-free optimal tracking, we first develop a two-phase bootstrap policy iteration (B-PI) algorithm, which bootstraps a stabilizing control gain from the trivially initialized zero-value start and proceeds…

On the Landis Conjecture for Positive Quasi-linear Operators on Graphs
Ujjal Das, Matthias Keller, Yehuda Pinchover
https://arxiv.org/abs/2509.20559 https://

On the Landis Conjecture for Positive Quasi-linear Operators on Graphs
We prove a Landis type unique continuation result for positive quasi-linear operators on graphs. Specifically, we give decay criteria that ensures when a harmonic function for a positive quasilinear Schrödinger operator with potential less than 1 is trivially zero. The assumption of positivity of the operator allows the application of criticality theory such as the Liouville comparison theorem. Furthermore, our results fundamentally build on the so called simplified energy. As an application w…

Basis pursuit by inconsistent alternating projections
Roger Behling, Yunier Bello-Cruz, Luiz-Rafael Santos, Paulo J. S. Silva
https://arxiv.org/abs/2508.15026 https://

Basis pursuit by inconsistent alternating projections
Basis pursuit is the problem of finding a vector with smallest $\ell_1$-norm among the solutions of a given linear system of equations. It is a famous convex relaxation of what the literature refers to as sparse affine feasibility problem, in which sparse solutions to underdetermined systems are sought. In addition to enjoying convexity, the basis pursuit can be trivially rewritten as a linear program, and thus standard tools in linear programming apply for solving it. In turn, we tackle the …

Robustness and resilience of complex networks
Oriol Artime, Marco Grassia, Manlio De Domenico, James P. Gleeson, Hernan A. Makse, Giuseppe Mangioni, Matjaz Perc, Filippo Radicchi
https://arxiv.org/abs/2509.19867

Robustness and resilience of complex networks
Complex networks are ubiquitous: a cell, the human brain, a group of people and the Internet are all examples of interconnected many-body systems characterized by macroscopic properties that cannot be trivially deduced from those of their microscopic constituents. Such systems are exposed to both internal, localized, failures and external disturbances or perturbations. Owing to their interconnected structure, complex systems might be severely degraded, to the point of disintegration or systemic…

The Safety Gap Toolkit: Evaluating Hidden Dangers of Open-Source Models
Ann-Kathrin Dombrowski, Dillon Bowen, Adam Gleave, Chris Cundy
https://arxiv.org/abs/2507.11544

The Safety Gap Toolkit: Evaluating Hidden Dangers of Open-Source Models
Open-weight large language models (LLMs) unlock huge benefits in innovation, personalization, privacy, and democratization. However, their core advantage - modifiability - opens the door to systemic risks: bad actors can trivially subvert current safeguards, turning beneficial models into tools for harm. This leads to a 'safety gap': the difference in dangerous capabilities between a model with intact safeguards and one that has been stripped of those safeguards. We open-source a toolkit to est…

The gauge theory dual of the bilayer XY model with second order Josephson coupling
Pye Ton How, Sungkit Yip
https://arxiv.org/abs/2507.19401 https://arxiv.…

The gauge theory dual of the bilayer XY model with second order Josephson coupling
We formulate a duality transformation for a bilayer XY model where the layers are coupled by second order Josephson effect, which favors inter-layer phase difference of either $0$ or $π$. The model may represent a bilayer superconductor or a spin-1 ferromagnetic Bose gas in the easy-plane limit. The second order Josephson term is mapped to a U(1) gauge field, known to be trivially confining in two dimensions, and we argue that a Coulomb-gas analysis is not applicable to the dual theory. Instea…

Supersymmetry and integrability of the elliptic $\mathrm{AdS}_3 \times \mathrm{S}^3 \times \mathrm{T}^4$ superstring
Ben Hoare, Fiona K. Seibold
https://arxiv.org/abs/2507.18548

Supersymmetry and integrability of the elliptic $\mathrm{AdS}_3 \times \mathrm{S}^3 \times \mathrm{T}^4$ superstring
We construct a 1-parameter family of Ramond-Ramond fluxes supporting the elliptic $\mathrm{AdS}_3 \times \mathrm{S}^3 \times \mathrm{T}^4$ metric with constant dilaton and preserving 8 of the 16 supercharges of the undeformed background. On the supersymmetric locus, we compute the tree-level worldsheet S-matrix in uniform light-cone gauge up to quadratic order in fermions and find that it non-trivially satisfies the classical Yang-Baxter equation. Moreover, imposing classical integrability and …

Notes on symplectic action on $(2,1)$-cycles on $K3$ surfaces
Ken Sato
https://arxiv.org/abs/2509.13491 https://arxiv.org/pdf/2509.13491

Notes on symplectic action on $(2,1)$-cycles on $K3$ surfaces
In this paper, we propose and study a conjecture that symplectic automorphisms of a $K3$ surface $X$ act trivially on the indecomposable part $\mathrm{CH}^2(X,1)_{\mathrm{ind}}\otimes \mathbb{Q}$ of Bloch's higher Chow group. This is a higher Chow analogue of Huybrechts' conjecture on the symplectic action on $0$-cycles. We give several partial results verifying our conjecture, some conditional and some unconditional. Our unconditional results include the full proof for Kummer surfaces of produ…

On the structure and skeletals of principal ideal graphs of inverse semigroups
C S Preenu, R S Indu, Sijo George
https://arxiv.org/abs/2509.15656 https://a…

On the structure and skeletals of principal ideal graphs of inverse semigroups
The principal left ideal graph of a semigroup is a simple graph whose vertices are the non-zero elements of the semigroup, and two vertices are adjacent if their principal left ideals intersect non-trivially. In this paper, we study the structure of the principal ideal graphs of inverse semigroups, particularly symmetric inverse semigroups. We also introduce the concept of skeletal of a graph and show that the principal ideal graph of an inverse semigroup has a skeletal, which is a simple graph…

Depth one part of Tannakian groups of filtrations
Payman Eskandari
https://arxiv.org/abs/2507.11717 https://arxiv.org/pdf/2507.11717

Depth one part of Tannakian groups of filtrations
Let $(F_r M)_{r\in\mathbb{Z}}$ be a finite filtration on an object $M$ of a neutral Tannakian category $\mathbf{T}$ in characteristic zero. Let $u(M)=u^F(M)$ be the Lie algebra of the subgroup of the Tannakian fundamental group of $M$ that acts trivially on the associated graded $Gr^FM$. The filtration $F_\bullet M$ induces a filtration on the internal Hom $\underline{Hom}(M,M)$, which in turn induces a filtration $F_\bullet u(M)$ on $u(M)$. This filtration on $u(M)$ is concentrated in negative…