Tootfinder

Quasi-Whittaker modules
Cunguang Cheng, Wenting Gao, Shiyuan Liu, Kaiming Zhao, Yueqiang Zhao
https://arxiv.org/abs/2508.05917 https://arxiv.org/pdf/2508.0…

Quasi-Whittaker modules
In this paper, a general setting is proposed to define a class of modules over nonsemisimple Lie algebras $\mathfrak{g}$ induced by a nonperfect ideal $\mathfrak{p}$. This class of Lie algebras includes many well-known Lie algebras, and some of this class of modules are Whittaker modules and others are not. We call these modules quasi-Whittaker modules. By introducing a new concept: the Whittaker annihilator for universal quasi-Whittaker modules, we are able to determine the necessary and suffi…

The convex structure of the Parisi formula for multi-species spin glasses
Hong-Bin Chen, Victor Issa, Jean-Christophe Mourrat
https://arxiv.org/abs/2508.06397 https://

The convex structure of the Parisi formula for multi-species spin glasses
We study the free energy of mean-field multi-species spin glasses with convex covariance function. For such models with $D$ species, the Parisi formula is known to be valid, and expresses the limit free energy as a supremum over monotone probability measures on $\mathbb{R}_+^D$. We show here that one can transform this representation into a supremum over all probability measures on $\mathbb{R}_+^D$ of a concave functional. We then deduce that the Parisi formula admits a unique maximizer. Using …

Enhancing Data Processing Efficiency in Blockchain Enabled Metaverse over Wireless Communications
Liangxin Qian, Jun Zhao
https://arxiv.org/abs/2507.04657 …

Enhancing Data Processing Efficiency in Blockchain Enabled Metaverse over Wireless Communications
In the rapidly evolving landscape of the Metaverse, enhanced by blockchain technology, the efficient processing of data has emerged as a critical challenge, especially in wireless communication systems. Addressing this challenge, our paper introduces the innovative concept of data processing efficiency (DPE), aiming to maximize processed bits per unit of resource consumption in blockchain-empowered Metaverse environments. To achieve this, we propose the DPE-Aware User Association and Resource A…

In Ursula K. Le Guin's "A Man of the People" (part of "Four Ways to Forgiveness") there's a scene where the Hainish protagonist begins studying history. It's excellent in many respects, but what stood out the most to me was the softly incomprehensible idea of a people with multiple millions of years of recorded history. As one's mind starts to try to trace out the implications of that, it dawns on you that you can't actually comprehend the concept. Like, you read the sentence & understood all the words, and at first you were able to assemble them into what seemed like a conceptual understanding, but as you started to try to fill out that understating, it began to slip away, until you realized you didn't in fact have the mental capacity to build a full understanding and would have you paper things over with a shallow placeholder instead.
I absolutely love that feeling, as one of the ways in which reading science fiction can stretch the brain, and I connected it to a similar moment in Tsutomu Nihei's BLAME, where the android protagonists need to ride an elevator through the civilization/galaxy-spanning megastructure, and turn themselves off for *millions of years* to wait out the ride.
I'm not sure why exactly these scenes feel more beautifully incomprehensible than your run-of-the-mill "then they traveled at lightspeed for a millennia, leaving all their family behind" scene, other than perhaps the authors approach them without trying to use much metaphor to make them more comprehensible (or they use metaphor to emphasize their incomprehensibility).
Do you have a favorite mind=expanded scene of this nature?
#AmReading

Ground state energies of multipartite $p$-spin models -- partially lifted RDT view
Mihailo Stojnic
https://arxiv.org/abs/2509.05916 https://arxiv.org/pdf/2…

Ground state energies of multipartite $p$-spin models -- partially lifted RDT view
We consider ground state energies (GSE) of multipartite $p$-spin models. Relying on partially lifted random duality theory (pl RDT) concepts we introduce an analytical mechanism that produces easy to compute lower and upper GSE bounds for \emph{any} spin sets. We uncover that these bounds actually match in case of fully spherical sets thereby providing optimal GSE values for spherical multipartite pure $p$-spin models. Numerical evidence further suggests that our upper and lower bounds may matc…

The Taylor Measure and its Applications
Athanasios Micheas
https://arxiv.org/abs/2508.04760 https://arxiv.org/pdf/2508.04760…

The Taylor Measure and its Applications
We propose and study a novel collection of signed measures, which will be apply called Taylor measures. Stochastic versions of the new measures are also defined and studied. We illustrate, through examples, how the deterministic and stochastic versions of the proposed Taylor measures emerge as a unifying framework that includes many concepts from mathematics and probability theory as special cases.

Factorizations in Geometric Lattices
Alex Aguila, Elvis Cabrera, Jyrko Correa-Morris
https://arxiv.org/abs/2506.14892 https://arxiv.o…

Factorizations in Geometric Lattices
This article investigates atomic decompositions in geometric lattices isomorphic to the partition lattice $Π(X)$ of a finite set $X$, a fundamental structure in lattice theory and combinatorics. We explore the role of atomicity in these lattices, building on concepts introduced by D.D. Anderson, D.F. Anderson, and M. Zafrullah within the context of factorization theory in commutative algebra. As part of the study, we first examine the main characteristics of the function $\mathfrak{N}\colon Π…

Differential Shannon and R\'enyi entropies revisited
Yuliya Mishura, Kostiantyn Ralchenko
https://arxiv.org/abs/2508.04373 https://arxiv.org/pdf/2508.0…

Differential Shannon and Rényi entropies revisited
Shannon entropy for discrete distributions is a fundamental and widely used concept, but its continuous analogue, known as differential entropy, lacks essential properties such as positivity and compatibility with the discrete case. In this paper, we analyze this incompatibility in detail and illustrate it through examples. To overcome these limitations, we propose modified versions of Shannon and Rényi entropy that retain key properties, including positivity, while remaining close to the clas…

Density convergence of spatial average of solution to a one dimensional stochastic wave equation
Chengbo Sun, Yaozhong Hu
https://arxiv.org/abs/2508.01872 https://

Density convergence of spatial average of solution to a one dimensional stochastic wave equation
In this paper we study the spatial averages of the solution of a one-dimensional stochastic wave equation driven by a Gaussian multiplicative noise, which is white in time and has a homogeneous spatial covariance described by the Riesz kernel. We establish the rate of convergence for the uniform distance between the density of spatial averages and the standard normal density. The proof combines Malliavin calculus with Stein's method for normal approximations. The key technical challenges lie in…

On the Smallest Singular Value of Log-Concave Random Matrices
Manuel Fernandez V, Galyna V. Livshyts, Stephanie Mui
https://arxiv.org/abs/2508.17745 https://

On the Smallest Singular Value of Log-Concave Random Matrices
Let $A$ be an $N\times n$ random matrix whose entries are coordinates of an isotropic log-concave random vector in $\mathbb{R}^{Nn}$. We prove sharp lower tail estimates for the smallest singular value of $A$ in the following cases: (1) when $N=n$ and $A$ is drawn from an unconditional distribution, with no independence assumption; (2) when the columns of $A$ are independent and $N\geq n$; (3) when $A$ is sufficiently tall, that is $N\geq (1+λ)n$ for any positive constant $λ$.