Tootfinder

Applications of the L-space satellite formula
Daren Chen, Ian Zemke, Hugo Zhou
https://arxiv.org/abs/2509.20288 https://arxiv.org/pdf/2509.20288

Applications of the L-space satellite formula
We give a formula for the $τ$-invariant of a satellite knot $P(K,n)$ when $P$ is an L-space satellite operator. Our formula holds for general L-space satellite operators $P$ when the companion $K$ satisfies $ε(K)=1$. When $ε(K)$ is $0$ or $-1$, we state a formula which requires some additional assumptions on $P$ or $n$. Our main tool is our algorithm which computes the knot Floer complex of satellite knots constructed using L-space satellite operators, which we developed in a previous paper.…

The strange story of an almost unknown prime number counter: The Rafael Barrett formula
Eduardo Mizraji
https://arxiv.org/abs/2509.19324 https://arxiv.org/…

The strange story of an almost unknown prime number counter: The Rafael Barrett formula
In this brief article, we present the formula created by Rafael Barrett in 1903 in a note to Henri Poincaré, which remained unknown for decades. Once discovered in the 1930s by an Uruguayan mathematician, this formula was disseminated and analyzed in a journal published in Montevideo in 1935. In this study, we present Barrett's formula and discuss some of its challenges it may pose.

Bats are one of the few animals that I genuinely don't enjoy being around.... but I still want them to thrive!
This is really great news that a Canadian researcher might have found a treatment for white-nose syndrome!
" B.C.-based scientist Cori Lausen, nicknamed Bat Lady by her peers, believes a probiotic formula may be capable of saving them. This cocktail is a blend of four bacterial strains that suppress the fungus. These bacteria are naturally found in some bats, but not all. ”
#Bats #health #nature
https://www.cbc.ca/news/canada/british-columbia/bats-white-nose-syndrome-probiotic-9.6948885

Properties of quasi-projective dimension over abelian categories
Hongxing Chen, Xiaohu Chen, Mengge Liu
https://arxiv.org/abs/2509.20137 https://arxiv.org/…

Properties of quasi-projective dimension over abelian categories
Quasi-projective dimension was introduced by Gheibi, Jorgensen and Takahashi to generalize the Auslander-Buchsbaum formula and the depth formula in commutative algebra. In this paper, we establish some basic properties of quasi-projective dimensions of objects in abelian categories. Analogous to global dimension of rings, we also introduce the concept of quasi-global dimension for left Noetherian rings, and then compare quasi-global dimension with global dimension for a class of Nakayama algebr…

Higher order weighted Dirichlet type spaces with poly-superharmonic weights and Dirichlet type operators of finite order
Ashish Kujur, Md. Ramiz Reza
https://arxiv.org/abs/2509.19714

Higher order weighted Dirichlet type spaces with poly-superharmonic weights and Dirichlet type operators of finite order
We study higher-order weighted Dirichlet integral and weighted Dirichlet type spaces $\mathscr D_{w,k}$ where the associated admissible weights arise from a class of positive poly-superharmonic functions on the open unit disc. We establish a generalized higher order version of the Local Douglas formula for computing the higher order weighted Dirichlet integral induced by Dirac measures supported on the open unit disc. Building on this formula, we analyze the properties of the multiplication ope…

New research from UC San Diego and Princeton reveals the winning formula for climate action: carrot AND stick.
Their study shows that combining clean energy subsidies with pollution taxes drives faster adoption of green tech than either approach alone. Economic models prove this dual strategy can deliver rapid emissions cuts in the near term.

YouTube says it will no longer give data to US Billboard charts after January 16, saying Billboard's ranking formula doesn't count ad-supported streams fairly (Sarah Perez/TechCrunch)
https://techcrunch.com/2025/12/17/yout

YouTube to pull music data from Billboard's charts because it doesn't like its ranking formula | TechCrunch
YouTube takes its ball and goes home by pulling Billboard's access to its music data.

US TV rights for #F1 will go to Apple TV
It seems like they are keeping #F1TV but details are sparse on pricing

Linear relations between $L$-values of newforms and moments of elliptic $K$ integral
Kam Cheong Au
https://arxiv.org/abs/2509.19960 https://arxiv.org/pdf/2…

Linear relations between $L$-values of newforms and moments of elliptic $K$ integral
By systematically translating certain integrals involving moments of the elliptic integral into $L$-values of modular forms on $Γ_1(4), Γ(4)$ and $Γ_1(8)$, and then utilizing relations among the critical $L$-values of cuspidal newforms, we obtain a dimension formula for the space spanned by these integrals. We also report a hypergeometric motive associated to a weight $6$ newform.

Computing the Topological Degree of Maps Between 2-Spheres
Daniil Kucher
https://arxiv.org/abs/2509.20167 https://arxiv.org/pdf/2509.20167

Computing the Topological Degree of Maps Between 2-Spheres
We describe an effective method for computing the topological degree of continuous functions $R:S^2 \to S^2$, where $S^2$ is the Riemann sphere. Our approach generalizes the degree formula for rational functions of complex polynomials, $\frac{f}{g}$, without common zeros. To apply our method, it is necessary to represent the function $R$ as the ratio of two continuous complex-valued functions $f$ and $g$ without common zeros. By utilizing Hopf fibration, this method reduces the problem to compu…

Eagles make a change to get winning formula back on track: Will philosophical shift produce wins?

https://www.cbssports.com/nfl/news/eagles-make-a-change-to-get-winning-for…

FDA warns major retailers over recalled baby formula sales (Max Rego/The Hill)
https://thehill.com/policy/healthcare/5651675-infant-formula-recall-fda-compliance/
http://www.memeorandum.com/251216/p135#a251216p135

Can anybody please calculate for me the time #dilation numbers related to how much faster someone at the equator is moving compared to someone at, say, either of the poles? Or anywhere in between, really. Ideally, a simple formula that correlates degrees of latitude with speed of revolution would be great. I’m at 40°30’36”N, and 4960ft elevation. I presume elevation also factors into it.

“Items that I have bought regularly have gone up in price steadily
From hair dye to baby formula,
our grocery list has gotten smaller while our budget has had to increase.
Meats like steak are a no-go for our household.”
https://www.theguardian.com/us-news…

Colleges are visiting my kid's school today, and each one does their best to make their name stand out.

Snapchat launches Topic Chats, which enables public conversations on subjects like F1, and plans to use AI and other safety tech to moderate the chats (Aisha Malik/TechCrunch)
https://techcrunch.com/2025/11/18/snapchat-rolls-out-topic-chats-for-publ…

Snapchat rolls out 'Topic Chats' for public conversations | TechCrunch
The social network says the new feature will let users chat about what’s happening in the world and across various communities, from Formula 1 to the reality TV show "Below Deck."

The Motivic Picard--Lefschetz Formula
Ran Azouri, Emil Jacobsen
https://arxiv.org/abs/2510.12762 https://arxiv.org/pdf/2510.12762

The Motivic Picard--Lefschetz Formula
We prove a motivic enhancement of the classical Picard--Lefschetz formula. Our proof is completely motivic, and yields a description of the motivic nearby cycles at a quasi-homogeneous singularity, as well as its monodromy, in terms of an embedding of projective hypersurfaces.

A (very) simple proof of the gravitational energy formula of polytropic spheres
Luca Ciotti (Dept. of Physics,Astronomy, Bologna University, Italy)
https://arxiv.org/abs/2510.11882

A (very) simple proof of the gravitational energy formula of polytropic spheres
It is shown how the well-known formula for the gravitational energy of self-gravitating regular polytropes of finite mass can be obtained in an elementary way by using Gauss's divergence theorem and the Chandrasekhar virial tensor, without resorting to lengthy algebra, to specific properties of Lane-Emden functions, and to thermodynamics arguments, as is instead commonly found in standard treatises and in astrophysical literature. The present approach, due to its simplicity, can be particularly…

Our collective intellectual culture seems to have calcified around a cohort of thinkers who achieved prominence roughly ten years ago and have been coasting ever since. 
This isn't about Malcolm Gladwell specifically, though he'll appear as a recurring character. 
The Gladwell formula, if you haven't encountered it, goes something like this:
take a subject that seems simple,
complicate it with research that seems to undermine common sense,
then re…

Comfort Food for the Thinking Class: The Great Intellectual Stagnation
Wander into any bookstore (I dare you.)  The non-fiction table will be all but dominated by the usual suspects: Malcolm Gladwell's latest exploration of how some counterintuitive thing is actually the opposite of what you'd expect, a David Brooks meditation on character and virtue, something by Michael Lewis about how

Feynman Formula for Discrete-time Quantum Walks
Jean-Pierre Fouque, Tomoyuki Ichiba, Ka Lok Lam
https://arxiv.org/abs/2510.12038 https://arxiv.org/pdf/2510…

Feynman Formula for Discrete-time Quantum Walks
We explicitly connect (discrete-time) quantum walks on Z with a four-state Markov additive process via a Feynman-type formula (2.5). Using this representation, we derive a relation between the spectral decomposition of the Markov additive process and the limiting density of the homogeneous quantum walk. In addition, we consider a space-time rescaling of quantum walks, which leads to a system of quantum transport PDEs in continuous time and space with a phase interaction term. Our probabilistic …

ngl the Formula E gen 4 car looks DELICIOUS 😍

Kronecker second limit formula for real quadratic fields
YoungJu Choie, Rahul Kumar
https://arxiv.org/abs/2510.10554 https://arxiv.org/pdf/2510.10554

Kronecker second limit formula for real quadratic fields
In this paper, the second Kronecker ``limit" formula for a real quadratic field is established for the first time. More precisely, we obtain the second Kronecker limit formula of Zagier's zeta function. Using the reduction theory of Zagier, which connects Zagier's zeta function to the zeta function of real quadratic fields, we express the values of the zeta function of narrow ideal classes in real quadratic fields at natural arguments in terms of an analytic function which we call the \emph{hig…

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…

Compounding formula approach to chromatin and active polymer dynamics
Takahiro Sakaue, Enrico Carlon
https://arxiv.org/abs/2510.07949 https://arxiv.org/pdf…

Compounding formula approach to chromatin and active polymer dynamics
Active polymers are ubiquitous in nature, and often kicked by persistent noises that break detailed balance. In order to capture the out-of-equilibrium dynamics of such active polymers, we propose a simple yet reliable analytical framework based on a compounding formula. Connecting polymeric dynamics to the isolated monomeric behavior via the notion of tension propagation, the formula allows us to clarify rich scaling scenarios alongside corresponding intuitive physical pictures. We demonstrate…

Great work, 'DOGE.' Very well done.
https://abcnews.go.com/GMA/Family/cdc-2-infants-hospitalized-botulism-outbreak-byheart-expands/story?id=127417947

CDC says 2 more infants hospitalized in botulism outbreak as ByHeart expands voluntary formula recall
At least 15 babies have been hospitalized with infant botulism.

Reverse Fa\`a di Bruno's Formula for Cartesian Reverse Differential Categories
Aaron Biggin (Macquarie University), Jean-Simon Pacaud Lemay (Macquarie University)
https://arxiv.org/abs/2509.20931

Reverse Faà di Bruno's Formula for Cartesian Reverse Differential Categories
Reverse differentiation is an essential operation for automatic differentiation. Cartesian reverse differential categories axiomatize reverse differentiation in a categorical framework, where one of the primary axioms is the reverse chain rule, which is the formula that expresses the reverse derivative of a composition. Here, we present the reverse differential analogue of Faa di Bruno's Formula, which gives a higher-order reverse chain rule in a Cartesian reverse differential category. To prop…

These are three arguments for web dev serv. APIs, even if you have to take a critical look at them in detail:
»Speed Comparison: Benchmarking programming languages using the Leibniz formula for calculating π«
— 2025-12-12
📊 https://niklas-heer.github.io/speed-comparison/…

"Before we go into what is missing, let us take a moment to understand why this partial story is so popular. Many software engineers do not engage with the broad “software engineering literature” very much: through the act of reading this magazine you are placing yourself at the pinnacle of software engineering curiosity!"
https…

The Other Side Of The Card
If you ask many developers what a "user story" is, you will be told that it is a software requirement expressed using a prescribed formula: "As a category of person, I want to do a thing so that some goal is attained."

Apple and F1 announce a five-year US media rights deal that will bring all races to Apple TV beginning in 2026; sources: Apple is paying about $140M per year (Alex Sherman/CNBC)
https://www.cnbc.com/2025/10/17/apple-f1-media-deal-streaming.html

Apple and F1 reach 5-year media deal, bringing all races to Apple TV streaming in the U.S.
Apple and Formula 1 announced their long-awaited five-year U.S. media rights deal that will bring all U.S. televised races to Apple TV

Been doing these demos for a month now & even the most jaded & cynical audience members all ask the same question:
"Um, so when is this coming out & how do I get it?" 😁
#Excel #Copilot #Microsoft

Shawn Brink (@Shawn_Brink@techhub.social)
Attached: 1 image Microsoft #Excel Power User Updates | Agent Mode, Copilot Function and Formula AI https://www.elevenforum.com/t/microsoft-excel-power-user-updates-agent-mode-copilot-function-and-formula-ai.41677/

A combinatorial formula for Interpolation Macdonald polynomials
Houcine Ben Dali, Lauren Williams
https://arxiv.org/abs/2510.02587 https://arxiv.org/pdf/25…

A combinatorial formula for Interpolation Macdonald polynomials
In 1996, Knop and Sahi introduced a remarkable family of inhomogeneous symmetric polynomials, defined via vanishing conditions, whose top homogeneous parts are exactly the Macdonald polynomials. Like the Macdonald polynomials, these interpolation Macdonald polynomials are closely connected to the Hecke algebra, and admit nonsymmetric versions, which generalize the nonsymmetric Macdonald polynomials. In this paper we give a combinatorial formula for interpolation Macdonald polynomials in terms o…

Approximating quantum states by states of low rank
Nathaniel Johnston, Chi-Kwong Li
https://arxiv.org/abs/2510.08463 https://arxiv.org/pdf/2510.08463

Approximating quantum states by states of low rank
Given a positive integer k, it is natural to ask for a formula for the distance between a given density matrix (i.e., mixed quantum state) and the set of density matrices of rank at most k. This problem has already been solved when "distance" is measured in the trace or Frobenius norm. We solve it for all other unitary similarity invariant norms. We also present some consequences of our formula. For example, in the trace and Frobenius norms, the density matrix that is farthest from the set of l…

Homebrew removing support for `--no-quarantine` for casks
#homebrew

Apple and Formula One announce a five-year US media rights deal for all races starting in 2026; sources: Apple is paying $140M/year, up from ESPN's ~$85M/year (Alex Sherman/CNBC)
https://www.cnbc.com/2025/10/17/apple-f1-media-deal-streaming.html

The Democrats Have a New Winning Formula (Derek Thompson)
https://www.derekthompson.org/p/the-democrats-new-formula-the-affordability
http://www.memeorandum.com/251106/p9#a251106p9

Instability of the Sherman-Morrison formula and stabilization by iterative refinement
Behnam Hashemi, Yuji Nakatsukasa
https://arxiv.org/abs/2510.01696 https://

Instability of the Sherman-Morrison formula and stabilization by iterative refinement
Owing to its simplicity and efficiency, the Sherman-Morrison (SM) formula has seen widespread use across various scientific and engineering applications for solving rank-one perturbed linear systems of the form $(A+uv^T)x = b$. Although the formula dates back at least to 1944, its numerical stability properties have remained an open question and continue to be a topic of current research. We analyze the backward stability of the SM, demonstrate its instability in a scenario increasingly common …

The Chevalley--Weil formula for finite group actions on higher dimensional compact complex manifolds
Wenfei Liu, Renjie Lyu
https://arxiv.org/abs/2510.10430 https://

The Chevalley--Weil formula for finite group actions on higher dimensional compact complex manifolds
Building on the Atiyah--Singer holomorphic Lefschetz fixed-point theorem, we define ramification modules associated to the fixed loci of a finite group acting on a compact complex manifold. This allows us to generalize the Chevalley--Weil formula for compact Riemann surfaces to higher dimensions. More precisely, let $G$ be a finite group acting on a compact complex manifold $X$, and let $\mathcal{E}$ be a $G$-equivariant locally free sheaf on $X$. Then, in the representation ring $R(G)_\mathbb{…

An alternative proof of the asymptotic formula for the Fourier coefficients of the elliptic modular $j$-function
Karin Ikeda
https://arxiv.org/abs/2510.10598 https://

An alternative proof of the asymptotic formula for the Fourier coefficients of the elliptic modular $j$-function
In 1997, Báez-Duarte gave a probabilistic proof of the asymptotic formula for the partition function, which had originally been proved by Hardy-Ramanujan. Based on the probabilistic approach, this paper proves an asymptotic formula for the coefficients of the elliptic modular $j$-function using various expressions in terms of modular functions having simple infinite products.

Not in Sync: Unveiling Temporal Bias in Audio Chat Models
Jiayu Yao, Shenghua Liu, Yiwei Wang, Rundong Cheng, Lingrui Mei, Baolong Bi, Zhen Xiong, Xueqi Cheng
https://arxiv.org/abs/2510.12185

Not in Sync: Unveiling Temporal Bias in Audio Chat Models
Large Audio Language Models (LALMs) are increasingly applied to audio understanding and multimodal reasoning, yet their ability to locate when events occur remains underexplored. We present the first systematic study of temporal bias in LALMs, revealing a key limitation in their timestamp prediction. For example, when asked "At which second does the lecturer introduce the key formula?", models often predict timestamps that are consistently earlier or later than the ground truth. Through control…

Robust magnetic field estimates in star-forming galaxies with the equipartition formula in the absence of equipartition
H. -H. Sandy Chiu, Mateusz Ruszkowski, Maria Werhahn, Christoph Pfrommer, Timon Thomas
https://arxiv.org/abs/2510.03229

Robust magnetic field estimates in star-forming galaxies with the equipartition formula in the absence of equipartition
The equipartition model is widely used to estimate magnetic field strength from synchrotron intensity in radio galaxies, yet the validity of its underlying assumptions remains uncertain. Using an Arepo simulation which incorporates a two-moment cosmic ray (CR) transport scheme and a multiphase interstellar medium, we compare magnetic fields inferred from synthetic synchrotron emission maps with the true fields in the simulation. Starting from the derivation of the equipartition formula, we find…

New nickle release today. This adds complex numbers, adds more math functions and fixes some existing ones. I'm using all of this to generate test vectors for picolibc math functions. More news about that when I manage to get the tests through CI.
Here's the nickle bits:
https://github.com/keith-…

Quaternionic Pole Placement via Companion Forms and the Ackermann Formula
Michael Sebek
https://arxiv.org/abs/2509.21425 https://arxiv.org/pdf/2509.21425…

Quaternionic Pole Placement via Companion Forms and the Ackermann Formula
We present an extension of state-feedback pole placement for quaternionic systems, based on companion forms and the Ackermann formula. For controllable single-input quaternionic LTI models, we define a companion polynomial that right-annihilates its companion matrix, characterize spectra via right-eigenvalue similarity classes, and prove coefficient-matching design in controllable coordinates. We then derive a coordinate-free Ackermann gain expression valid for real target polynomials, and stat…

Application of a new iterative formula for computing $\pi$ and nested radicals with roots of $2$
Sanjar M. Abrarov, Rehan Siddiqui, Rajinder Kumar Jagpal, Brendan M. Quine
https://arxiv.org/abs/2509.25221

Application of a new iterative formula for computing $π$ and nested radicals with roots of $2$
In this work, we obtain an iterative formula that can be used for computing digits of $π$ and nested radicals of kind $c_n/\sqrt{2 - c_{n - 1}}$, where $c_0 = 0$ and $c_n = \sqrt{2 + c_{n - 1}}$. We also show how with help of this iterative formula the two-term Machin-like formulas for $π$ can be generated and approximated. Some examples with Mathematica codes are presented.

The propaganda is so wild. When I was younger, they'd lie about things 20-30 years prior, with the assumption that people will have forgotten. Now they are constantly attempting to rewrite history that happened a mere 5 years ago, thinking that they're slick.
To quote Orwell: "The Party told you to reject the evidence of your eyes and ears. It was their final, most essential command."

Supergroup Invariants and the Brane/Negative Brane Expansion
Kasia Budzik
https://arxiv.org/abs/2509.20451 https://arxiv.org/pdf/2509.20451

Supergroup Invariants and the Brane/Negative Brane Expansion
We propose a Molien--Weyl-type formula computing generating functions of invariants of supergroups $U(N|M)$, i.e. polynomials in supertraces, which arise as gauge groups of brane/negative brane systems in string theory. We either prove or numerically verify the formula in various examples. The formula further leads to a new expansion relating finite-$N$ and infinite-$N$ indices of $U(N)$ gauge theories. We comment on its relation to Murthy's Giant Graviton expansion, for which we suggest a…

General real-valued theories with the Schr\"oder-Bernstein property are stable
Alexander Berenstein, Nicol\'as Cuervo Ovalle, Isaac Goldbring
https://arxiv.org/abs/2510.11858

General real-valued theories with the Schröder-Bernstein property are stable
We show that every general theory à la Keisler with the Schröder-Bernstein property is stable. This generalizes the corresponding result from classical logic due to John Goodrick. Our proof uses the classical result (generalized to the case that the instability is witnessed by an infinitary formula) together with a discretization technique introduced by Keisler and the third-named author. We speculate on how our techniques could be adapted to show that every continuous theory with the Schröd…

Probably not among the juiciest topics in #NeoLatin and #Jesuit studies, but in the unlikely event that you feel like learning more about the legal basis of the Jesuits’ congregationes procuratorum, I’ve got you covered: “The Formula of the Congregation of Procurators: Introduction, Latin Text, …

The g-2 in the neutral Electroweak model with cutoff: convergent expansion, RG and the Jackiw-Weinberg formula
Vieri Mastropietro, Michele Bianchessi
https://arxiv.org/abs/2510.02092

The g-2 in the neutral Electroweak model with cutoff: convergent expansion, RG and the Jackiw-Weinberg formula
The prediction of the anomalous gyromagnetic factor of the electron, started with the evaluation of the electromagnetic contribution by Schwinger (1948) and of the weak contribution by Jackiw and Weinberg (1972), is one of the major successes of Quantum Field Theory and the Standard Model. The results obtained truncating the series are in spectacular agreement with experiments. Yet, a mathematical justification and an estimate of the truncation error are problematic, being such series diverging…

Variational formula in the Schottky model of Riemann surfaces
Andrei Bogatyrev
https://arxiv.org/abs/2509.22702 https://arxiv.org/pdf/2509.22702

Variational formula in the Schottky model of Riemann surfaces
A simple and efficient formula is proposed for varying Abelian integrals (including their periods) under variation of the generators of the classical Schottky group representing a Riemann surface.

Magnetic-Field-Induced Geometric Response of Mean-Field Projectors: Streda Formula and Orbital Magnetization
Jihang Zhu, Chunli Huang
https://arxiv.org/abs/2510.07001 https://…

Magnetic-Field-Induced Geometric Response of Mean-Field Projectors: Streda Formula and Orbital Magnetization
We study the magnetic-field response of interacting electron systems within mean-field theory using perturbation theory. We show that the linear response of the mean-field density-matrix to a weak magnetic field is purely geometric: it depends only on wavefunction derivatives, the Berry connections linking the occupied and unoccupied subspaces, and is independent of the interaction potential and the quasiparticle dispersion. This leads to compact, gauge-invariant projector expressions for both …

Feit's conjecture, the canonical Brauer induction formula, and Adams operations
Robert Boltje, Gabriel Navarro
https://arxiv.org/abs/2510.03179 https://

Feit's conjecture, the canonical Brauer induction formula, and Adams operations
This paper is motivated by a strong version of Feit's conjecture, first formulated by the authors in joint work with A. Kleshchev and P. H. Tiep in 2025, concerning the conductor $c(χ)$ of an irreducible character $χ$ of a finite group $G$. We connect the conjecture with the following construction: For any positive integer $n$ dividing the exponent of $G$ and for any character $χ$ of $G$, we introduce an integer-valued invariant $S(G,χ,n)$ which can be defined as the sum of certain coeffici…

Sources: Apple is set to announce an exclusive US media rights deal with Formula One, paying $140M per year; F1 races average 1.4M viewers on ESPN (Alex Sherman/CNBC)
https://www.cnbc.com/2025/10/16/apple-wants-to-reform-sports-streaming…

Generalizations of Langbein's Formula under Non-Stationarity, Mixed Populations, and Over- or Under-Dispersion in the Number of Exceedances
Francesco Dell'Aira, Antonino Cancelliere, Claudio I. Meier
https://arxiv.org/abs/2509.23546

Generalizations of Langbein's Formula under Non-Stationarity, Mixed Populations, and Over- or Under-Dispersion in the Number of Exceedances
Since its publication in 1949, Langbein's formula has been applied ubiquitously in both research documents and national guidelines concerning frequency analyses (FAs) of hydrologic extremes. Given a time series of independent peak-over-threshold (POT) events and the corresponding annual maxima (AM) series-defined as the subset of extremes representing the largest event in each year-the formula provides a theoretical relationship between the return period T derived from the AM series and the ave…

Verifying Memoryless Sequential Decision-making of Large Language Models
Dennis Gross, Helge Spieker, Arnaud Gotlieb
https://arxiv.org/abs/2510.06756 https://

Verifying Memoryless Sequential Decision-making of Large Language Models
We introduce a tool for rigorous and automated verification of large language model (LLM)- based policies in memoryless sequential decision-making tasks. Given a Markov decision process (MDP) representing the sequential decision-making task, an LLM policy, and a safety requirement expressed as a PCTL formula, our approach incrementally constructs only the reachable portion of the MDP guided by the LLM's chosen actions. Each state is encoded as a natural language prompt, the LLM's response is pa…

Im using case_when() quite a lot, case_match() is new to me: #rstats

A general vectorised switch() — case_match
This function allows you to vectorise multiple switch() statements. Each case is evaluated sequentially and the first match for each element determines the corresponding value in the output vector. If no cases match, the .default is used. case_match() is an R equivalent of the SQL "simple" CASE WHEN statement. Connection to case_when() While case_when() uses logical expressions on the left-hand side of the formula, case_match() uses values to match against .x with. The following two statement…

Monotonicity of Causal Killing Vectors and Geometry of ADM Mass Minimizers
Sven Hirsch, Lan-Hsuan Huang
https://arxiv.org/abs/2510.10306 https://arxiv.org/…

Monotonicity of Causal Killing Vectors and Geometry of ADM Mass Minimizers
We address two problems concerning the ADM mass-minimizing initial data sets. First, we show that the equality case of the positive mass theorem embeds into a pp-wave spacetime. Second, we show that positive Bartnik mass minimizers embed into strongly stationary vacuum spacetimes, thereby confirming the Bartnik stationary vacuum conjecture. A key ingredient is a new monotonicity formula for the Lorentzian length of a causal Killing vector field, which, among other applications, yields a strong …

Giving young players reps among 3 trends Cowboys must continue in Week 6 vs Panthers https://cowboyswire.usatoday.com/story/sports/nfl/cowboys/2025/10/10/3-trends-cowboys-must-carryover-to-panthers-game-in…

from my link log —
Ferrari’s Formula One pit-stops and handovers from surgery to intensive care.
https://gwern.net/doc/technology/2008-sower.pdf
saved 2025-11-26

Exact heat flux formula and its spectral decomposition in molecular dynamics for arbitrary many-body potentials
Markos Poulos, Donatas Surblys, Konstantinos Termentzidis
https://arxiv.org/abs/2509.26044

Exact heat flux formula and its spectral decomposition in molecular dynamics for arbitrary many-body potentials
In this study we have derived an exact framework for the calculation of the heat flux and its spec- tral decomposition in Molecular Dynamics (MD) for arbitrary many-body potentials. This work addresses several lacks and limitations of previous approaches and allows for the accurate computa- tional study of thermal properties in a wide variety of many-body systems with MD. We have tested our modifications with Green-Kubo (GK) and Non-Equilibrium MD (NEMD) simulations for vari- ous 2D and 3D mate…

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]…

Computationally and statistically efficient estimation of time-smoothed counterfactual curves
Herbert P. Susmann, Nicholas T. Williams, Richard Liu, Jessica G. Young, Iv\'an D\'iaz
https://arxiv.org/abs/2509.26554

Computationally and statistically efficient estimation of time-smoothed counterfactual curves
Longitudinal causal inference is concerned with defining, identifying, and estimating the effect of a time-varying intervention on a time-varying outcome that is indexed by a follow-up time. In an observational study, Robins's generalized g-formula can identify causal effects induced by a broad class of time-varying interventions. Various methods for estimating the generalized g-formula have been posed for different outcome types, such as a failure event indicator by a specified time (e.g. mort…

Fast and Accurate Intersections on a Sphere
Hongyu Chen, Paul A. Ullrich, Julian Panetta
https://arxiv.org/abs/2510.09892 https://arxiv.org/pdf/2510.09892

Fast and Accurate Intersections on a Sphere
We introduce a fast, high-precision algorithm for calculating intersections between great circle arcs and lines of constant latitude on the unit sphere. We first propose a simplified intersection point formula with improved speed and numerical robustness over the ones traditionally implemented in geoscience software. We then show how algorithms based on the concept of error-free transformations (EFT) can be applied to evaluate this formula within a relative error bound that is on the order of m…

A Poisson-Alekseev-Gr\"obner formula through Malliavin calculus for Poisson random integrals
Paul Maurer, J\'er\'emy Zurcher
https://arxiv.org/abs/2510.05300 https:…

A Poisson-Alekseev-Gröbner formula through Malliavin calculus for Poisson random integrals
In this paper, we establish an Alekseev--Gröbner formula for stochastic differential equations (SDEs) driven by a Poisson random measure, which express the global error between a functional of two processes solution of SDEs started at the same initial condition, in terms of the infinitesimal error (i.e, the difference between the SDEs coefficients). In particular, we consider the situation where the flow process is only assumed to be jointly stochastically continuous with respect to space and …

Sparse Regularization by Smooth Non-separable Non-convex Penalty Function Based on Ultra-discretization Formula
Natsuki Akaishi, Koki Yamada, Kohei Yatabe
https://arxiv.org/abs/2509.19886

Sparse Regularization by Smooth Non-separable Non-convex Penalty Function Based on Ultra-discretization Formula
In sparse optimization, the $\ell_{1}$ norm is widely adopted for its convexity, yet it often yields solutions with smaller magnitudes than expected. To mitigate this drawback, various non-convex sparse penalties have been proposed. Some employ non-separability, with ordered weighting as an effective example, to retain large components while suppressing small ones. Motivated by these approaches, we propose ULPENS, a non-convex, non-separable sparsity-inducing penalty function that enables contr…

Denominators of R-matrices, higher Dorey's rules and a generalization of T-systems for quantum affine algebras
Se-jin Oh, Travis Scrimshaw
https://arxiv.org/abs/2510.10874 h…

Denominators of R-matrices, higher Dorey's rules and a generalization of T-systems for quantum affine algebras
We construct a higher level analogue of Dorey's rule, which describe certain surjective morphisms between Kirillov--Reshetikhin (KR) modules over quantum affine algebras. Building on this, we establish a generalized T-system of short exact sequences and prove the denominator formula between KR modules in all nonexceptional types, except with only mild ambiguities persisting in type $C_n^{(1)}$. As a consequence, we can completely classify when a tensor product of KR modules is simple. These res…

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…

Equivalence criteria for the two--term functional equations for Herglotz--Zagier functions
Sumukha Sathyanarayana, N. Guru Sharan
https://arxiv.org/abs/2510.10088 https://

Equivalence criteria for the two--term functional equations for Herglotz--Zagier functions
We establish Kronecker limit type formula for the generalized Mordell-Tornheim zeta function $Θ(r,r,t,x)$ as a function of the third argument around $t=1-r$. We then show that the above Kronecker limit type formula is equivalent to the two-term functional equation for the higher Herglotz function obtained by Vlasenko and Zagier. We also show the equivalence between a previously known Kronecker limit type formula for $Θ(1,1,t,x)$ around $t=0$ and the two-term functional equation for the Herglo…

The Smith normal form of distance matrices of high dimensional trees
Carlos A. Alfaro, Jes\'us Uriel Medrano, Iv\'an T\'ellez T\'ellez
https://arxiv.org/abs/2510.04471

The Smith normal form of distance matrices of high dimensional trees
Graham-Lovász-Pollak \cite{GL,GP} obtained the celebrated formula $$\det({\sf D}(T_{n+1}))=(-1)^nn2^{n-1},$$ for the determinant of the distance matrix ${\sf D}(T_{n+1})$ for any tree $T_{n+1}$ with $n+1$ vertices. Later, Hou and Woo \cite{HW} extended this formula to the Smith normal form (SNF) obtaining that $\SNF({\sf D}(T_{n+1}))={\sf I}_2\oplus 2{\sf I}_{n-2}\oplus [2n]$, for any tree $T_{n+1}$ with $n+1$ vertices. A $k$-{\it tree} is either a complete graph on $k$ vertices or a graph o…

Correlated double ramification cycle formula
Thomas Blomme, Francesca Carocci
https://arxiv.org/abs/2509.24721 https://arxiv.org/pdf/2509.24721

Correlated double ramification cycle formula
We prove a refinement of Pixton's formula for the double ramification cycle with target variety which takes into account the correlator of a rubber map previously introduced by the authors. To do so, we need to: reinterpret the correlator in terms of (logarithmic) roots of the trivial line bundle; refine the natural stratification of the boundary of moduli of maps to keep track of how torsion line bundles on the target variety pull-back. We apply the refined DR cycle formula to compute 0-correl…

Exploiting higher-order correlation functions for photon-statistics-based characterization and reconstruction of arbitrary Gaussian states
Philip Heinzel, Ren\'e Sondenheimer
https://arxiv.org/abs/2510.09083

Exploiting higher-order correlation functions for photon-statistics-based characterization and reconstruction of arbitrary Gaussian states
Gaussian states are an essential building block for various applications in quantum optics and quantum information science, yet the precise relation between their second- and third-order correlation functions remains not fully explored. We discuss connections between these correlation functions by constructing an explicit decomposition formula for arbitrary sixth-order moments of ladder operators for general Gaussian states and demonstrate how the derived relations enable state classification f…

A new elementary proof of the formula $\sum\limits_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}$
Jia Li
https://arxiv.org/abs/2510.03233 https://arxiv.org/p…

A new elementary proof of the formula $\sum\limits_{n=1}^{\infty}\frac{1}{n^2}=\frac{π^2}{6}$
In this article, we provide a new elementary proof of the Basel problem.

Spin glasses and the Parisi formula
Jean-Christophe Mourrat
https://arxiv.org/abs/2510.01054 https://arxiv.org/pdf/2510.01054

Spin glasses and the Parisi formula
Spin glasses are models of statistical mechanics in which a large number of simple elements interact with one another in a disordered fashion. One of the fundamental results of the theory is the Parisi formula, which identifies the limit of the free energy of a large class of such models. Yet many interesting models remain out of reach of the classical theory, and direct generalizations of the Parisi formula yield invalid predictions. I will report here on some partial progress towards the reso…

Integral Matrices of Fixed Rank over Number Fields
Nihar Gargava, Vlad Serban, Maryna Viazovska, Ilaria Viglino
https://arxiv.org/abs/2510.11673 https://ar…

Integral Matrices of Fixed Rank over Number Fields
We prove an asymptotic formula for the number of fixed rank matrices with integer coefficients over a number field K/Q and bounded norm. As an application, we derive an approximate Rogers integral formula for discrete sets of module lattices obtained from lifts of algebraic codes. This in turn implies that the moment estimates of random lattices with a number field structure also carry through for large enough discrete sets of module lattices.

Hyperfunctions in $A$-model Localization
Emil Hakan Leeb-Lundberg
https://arxiv.org/abs/2509.25976 https://arxiv.org/pdf/2509.25976

Hyperfunctions in $A$-model Localization
We apply localization techniques to topologically $A$-twisted $\mathcal{N}=(2,2)$ supersymmetric theories of vector and chiral multiplets on $S^{2}$ and derive a novel exact formula for abelian observables, described by a distribution integrated along the real line. The distributional integral formula is verified by evaluating the correlator of the $A$-twisted $\mathbb{CP}^{N-1}$ gauged linear sigma model and confirming the standard selection rule. Finally, we use hyperfunctions to demonstrate …

Zeros of Holomorphic Functions in Commuting and Non-commuting Variables as Spectral Data
Poornendu Kumar, Jeet Sampat
https://arxiv.org/abs/2510.12718 https://

Zeros of Holomorphic Functions in Commuting and Non-commuting Variables as Spectral Data
We characterize the zero sets of functions in the Schur--Agler class over the unit polydisk as well as functions in the unit ball of the multiplier algebra of the Drury--Arveson space via operators associated with a unitary realization formula for these functions. To this end, new notions of `eigenvalues' for tuples of operators are introduced, where the eigenvalues depend on the operator space structure of the ambient domain. Several examples showcasing the properties of these eigenvalues and …

A recursive definition for the polymatroid Tutte polynomial
Xiaxia Guan, Xian'an Jin, Weiling Yang
https://arxiv.org/abs/2510.11046 https://arxiv.org/p…

A recursive definition for the polymatroid Tutte polynomial
The Tutte polynomial is a significant invariant of graphs and matroids. It is well-known that it has three equivalent definitions: bases expansion, rank generating function, and deletion-contraction formula. The polymatroid Tutte polynomial $\mathscr{T}_{P}$ generalizes the Tutte polynomial from matroids to polymatroids $P$. In \emph{[Adv. Math. 402 (2022) 108355.]} and \emph{[J. Combin. Theory Ser. A 188 (2022) 105584]}, the authors provided bases expansion and rank generating function constru…

Photometric stellar masses for galaxies in DESI Legacy Imaging Surveys
Ivana Ebrov\'a, Michal B\'ilek, Ji\v{r}\'i Eli\'a\v{s}ek
https://arxiv.org/abs/2510.02257 …

Photometric stellar masses for galaxies in DESI Legacy Imaging Surveys
In many areas of extragalactic astrophysics, we need to convert the luminosity of a galaxy into its stellar mass. In this work, we aim to find a simple and effective formula to estimate the stellar mass from the images of galaxies delivered by the currently popular DESI Legacy Imaging Surveys. This survey provides an unsurpassed combination of a deep imaging with an extensive sky coverage in up to four photometric bands. We calibrated the sought formula against a sample of local galaxies observ…

DE-Sinc approximation for unilateral rapidly decreasing functions and its computational error bound
Tomoaki Okayama
https://arxiv.org/abs/2510.11411 https://

DE-Sinc approximation for unilateral rapidly decreasing functions and its computational error bound
The Sinc approximation is known to be a highly efficient approximation formula for rapidly decreasing functions. For unilateral rapidly decreasing functions, which rapidly decrease as $x\to\infty$ but does not as $x\to-\infty$, an appropriate variable transformation makes the functions rapidly decreasing. As such a variable transformation, Stenger proposed $t = \sinh(\log(\operatorname{arsinh}(\exp x)))$, which enables the Sinc approximation to achieve root-exponential convergence. Recently, an…

Diophantine analysis and Arthur's trace formula
Yuchan Lee
https://arxiv.org/abs/2509.22314 https://arxiv.org/pdf/2509.22314

Diophantine analysis and Arthur's trace formula
Let $X$ be a $G$-homogeneous space over a number field $k$ such that $X\cong G_γ\backslash G$. Here, $G$ is a simply connected semisimple group over $k$ and $γ\in G(k)$ whose centralizer $G_γ$ is a maximal torus in $G$ which is anisotropic over $k$. We formulate the asymptotic for the number of integral points on $X$ bounded by a fixed norm $T>0$ as $T\rightarrow \infty$ in terms of $κ$-orbital integrals, which play a role in the stabilization of Arthur's trace formula. This formula coincid…

Small-$b$ expansion of the DOZZ formula for light operators
Franco Ferrari, Marcin R. Piatek, Artur R. Pietrykowski
https://arxiv.org/abs/2509.21182 https://

Small-$b$ expansion of the DOZZ formula for light operators
We present a systematic small-$b$ expansion of the Liouville DOZZ three-point structure constant in the light-operator regime $α_i = bσ_i$ as $b\to 0$. In this limit, the exact DOZZ function factorizes into a prefactor $\mathcal{P}(b;σ_1,σ_2,σ_3)$ and a power series in $b^2$, \[ C(bσ_1,bσ_2,bσ_3) =\mathcal{P}(b;σ_i)\Bigl[1+\sum_{n=1}^\infty b^{2n}\,Ω_n(σ_1,σ_2,σ_3)\Bigr]. \] Using Thorn's asymptotic expansion of the $Υ_b$-function, we derive closed-form expressions for the leading…

On the monodromy conjecture for determinantal varieties
Yifan Chen, Huaiqing Zuo
https://arxiv.org/abs/2510.11425 https://arxiv.org/pdf/2510.11425

On the monodromy conjecture for determinantal varieties
This paper presents a proof of the monodromy conjecture for determinantal varieties. Our strategy centers on an in-depth analysis of monodromy zeta functions, leveraging a generalized A'Campo formula, an examination of multiple contact loci, and the exploitation of the intrinsic symmetric structures inherent to these varieties. Keywords. monodromy conjecture, determinantal varieties, monodromy zeta function.

Inverse Kazhdan-Lusztig polynomials of matroids under deletion
Tom Braden, Luis Ferroni, Jacob P. Matherne, Nutan Nepal
https://arxiv.org/abs/2510.01086 https://

Inverse Kazhdan-Lusztig polynomials of matroids under deletion
We provide a deletion formula for the inverse Kazhdan--Lusztig polynomial and the inverse $Z$-polynomial of a matroid. Our formulas provide analogues to the deletion formulas of Braden--Vysogorets for Kazhdan--Lusztig and $Z$-polynomials. We discuss several consequences, which include closed formulas and recursions for these invariants on uniform matroids, projective geometries, glued cycles, and arbitrary matroids of corank $2$. As a relevant application of our deletion formula, we show the ex…

Summation Formulas for Harmonic Maass Forms
Nikolaos Diamantis, Joshua Pimm
https://arxiv.org/abs/2509.22607 https://arxiv.org/pdf/2509.22607

Summation Formulas for Harmonic Maass Forms
In a previous work (arXiv:2505.05574), a summation formula for harmonic Maass forms of polynomial growth was established. In this note, we use the theory of $L$-series of harmonic Maass forms to state and prove a summation formula for such forms without any restrictions on their growth. We deduce a summation formula for the partition function. We further employ the same theory to derive a result on classical modular forms, namely, a summation formula and the asymptotics of a Riesz sum attached …

New representations of the Hu-Meyer formulas and series expansion of iterated Stratonovich stochastic integrals with respect to components of a multidimensional Wiener process
Dmitriy F. Kuznetsov
https://arxiv.org/abs/2510.06981

New representations of the Hu-Meyer formulas and series expansion of iterated Stratonovich stochastic integrals with respect to components of a multidimensional Wiener process
The article is devoted to the systematic derivation of new representations of the Hu-Meyer formulas. The formula expressing a multiple Wiener stochastic integral through the sum of multiple Stratonovich stochastic integrals and the formula expressing a multiple Stratonovich stochastic integral through the sum of multiple Wiener stochastic integrals are derived for the case of a multidimensional Wiener process. The proof method proposed by the author in 2006 is applied. At that a different defin…

Relativistic Particle on Light-Front
Junmou Chen
https://arxiv.org/abs/2510.08983 https://arxiv.org/pdf/2510.08983

Relativistic Particle on Light-Front
We study Wigner's classification of massive one particle state in a general moving frame in the front form of Lorentz group, characterized and labeled by a special null vector -- reference vector. The little group transformation turns out to be equivalent to a change of reference vector, which also gives a concrete formula to compute the Wigner D-matrices with a general momentum. The little group transformation and Wigner D-matrices have both continuous zero momentum limit and infinite boost (m…

Weighted implicit-explicit discontinuous Galerkin methods for two-dimensional Ginzburg-Landau equations on general meshes
Zhen Guan, Xianxian Cao
https://arxiv.org/abs/2510.10283

Weighted implicit-explicit discontinuous Galerkin methods for two-dimensional Ginzburg-Landau equations on general meshes
In this paper, a second-order linearized discontinuous Galerkin method on general meshes, which treats the backward differentiation formula of order two (BDF2) and Crank-Nicolson schemes as special cases, is proposed for solving the two-dimensional Ginzburg-Landau equations with cubic nonlinearity. By utilizing the discontinuous Galerkin inverse inequality and the mathematical induction method, the unconditionally optimal error estimate in $L^2$-norm is obtained. The core of the analysis in thi…

Finite isoresidual covers in strata of $k$-differentials
Dawei Chen, Quentin Gendron, Miguel Prado, Guillaume Tahar
https://arxiv.org/abs/2510.01630 https://

Finite isoresidual covers in strata of $k$-differentials
Consider the strata of primitive $k$-differentials on the Riemann sphere whose singularities, except for two, are poles of order divisible by $k$. The map that assigns to each $k$-differential the $k$-residues at these poles is a ramified cover of its image. Generalizing results known in the case of abelian differentials, we describe the ramification locus of this cover and provide a formula, involving the $k$-factorial function, for the cardinality of each fiber. We prove this formula using in…

Rademacher-type exact formula and higher order Tur\'{a}n inequalities for cubic overpartitions
Archit Agarwal, Meghali Garg, Bibekananda Maji
https://arxiv.org/abs/2509.23151

Rademacher-type exact formula and higher order Turán inequalities for cubic overpartitions
In 1918, Hardy and Ramanujan made a breakthrough by developing the circle method to deduce an asymptotic formula for the partition function $p(n)$, which was later refined by Rademacher in 1937 to produce an absolutely convergent series representation for $p(n)$. Since then, Rademacher-type exact formulas for various partition functions have been investigated by many mathematicians. The concept of overpartitions was introduced by Lovejoy and Corteel in 2004. Kim, in 2010, studied an overpartiti…

Tripartite entanglement in the HaPPY code is not holographic
Sriram Akella
https://arxiv.org/abs/2510.08520 https://arxiv.org/pdf/2510.08520

Tripartite entanglement in the HaPPY code is not holographic
Holographic states satisfy several entropic inequalities owing to the Ryu-Takayangi formula. A drawback of these inequalities is that they only use bipartite entanglement in their formulation. We investigate a recently proposed "GHZ-forbidding" inequality, built out of the reflected entropy and the tripartite multi-entropy, that holds for holographic states. We show that the inequality is either violated or saturated, but never strictly satisfied, by stabilizer states, thereby showing that stab…

Symmetry of the refined $q,t$-Catalan polynomials for $\vec{k}$-Dyck paths
Menghao Qu, Yingrui Zhang
https://arxiv.org/abs/2510.08196 https://arxiv.org/pdf…

Symmetry of the refined $q,t$-Catalan polynomials for $\vec{k}$-Dyck paths
Pappe, Paul, and Schilling introduced two combinatorial statistics, depth and ddinv, associated with classical Dyck paths, and proved that the distributions of (area, depth) and (dinv, ddinv) are $q,t$-symmetric by constructing an involution on plane trees. They also provided a new formula for the original $q,t$-Catalan polynomials $C_{n}(q,t)$. We observe that depth is a slight modification of bounce, which was defined by the filling algorithm and ranking algorithm of Xin and the second author…

Degree 4 cohomological invariants of algebraic tori
Cyril Demarche, Hanqing Long
https://arxiv.org/abs/2510.09552 https://arxiv.org/pdf/2510.09552

Degree 4 cohomological invariants of algebraic tori
In this paper, we determine the motive of the classifying torsor of an algebraic torus. As a result, we give an exact sequence describing the degree 4 cohomological invariants of algebraic tori. Using results by Blinstein and Merkurjev, this provides a formula for the degree 4 unramified cohomology group of an algebraic torus, via a flasque resolution.

Boundary Conditions of Warped AdS$_3$ in New Massive Gravity
S. N. Sajadi, Supakchai Ponglertsakul
https://arxiv.org/abs/2510.06572 https://arxiv.org/pdf/2…

Boundary Conditions of Warped AdS$_3$ in New Massive Gravity
To satisfy the Cardy formula for Warped AdS$_3$ (WAdS$_{3}$) solutions in a quadratic ensemble, a specific set of boundary conditions has been proposed in \cite{Aggarwal:2020igb}. In this paper, these boundary conditions have been investigated in the three-dimensional new massive gravity (NMG) framework. The associated solution space, asymptotic symmetries, and charge algebra have been extracted. It has been shown that the surface charges are finite, but not integrable, and the integrability of…

Arithmetic Hirzebruch-Zagier divisors and central derivative values of Rankin-Selberg $L$-functions
Jeanine Van Order
https://arxiv.org/abs/2510.10303 https://

Arithmetic Hirzebruch-Zagier divisors and central derivative values of Rankin-Selberg $L$-functions
We give two distinct proofs of the Gross-Zagier formula in terms of sums of automorphic Green's functions realized as regularized theta lifts, including one involving arithmetic Hirzebruch-Zagier divisors on the Hilbert modular surface $X_0(N) \times X_0(N)$. We then describe applications to the refined conjecture of Birch and Swinnerton-Dyer. Through these calculations, we also describe known and conjectural relations of the central derivative values of Rankin-Selberg $L$-functions that appear…

Sum of elements preceding records in set partitions
Walaa Asakly, Noor Kezil
https://arxiv.org/abs/2510.06521 https://arxiv.org/pdf/2510.06521

Sum of elements preceding records in set partitions
In this paper, we aim to derive an explicit formula for the total number of elements preceding records over all set partitions of $[n]$ with exactly $k$ blocks, as well as an asymptotic estimate for the total sum of elements preceding records in all set partitions of $[n]$, expressed in terms of Bell numbers. To achieve this, we analyze the generating function that enumerates set partitions of $[n]$ according to this statistic, which we denote by $\sumelements$.

On the Birch and Swinnerton-Dyer formula modulo squares for certain quadratic twists of elliptic curves
Alexander J. Barrios, Chung Pang Mok
https://arxiv.org/abs/2510.00926 htt…

On the Birch and Swinnerton-Dyer formula modulo squares for certain quadratic twists of elliptic curves
Let $E/\mathbb{Q}$ be an elliptic curve with conductor $N=N_+N_-$, where $N_+$ and $N_-$ are coprime and $N_-$ is squarefree. Let $D$ be a positive fundamental discriminant satisfying the modified Heegner hypothesis with respect to $(N_+,N_-)$: primes dividing $N_+$ (resp. $N_-$) split (resp. are inert) in $\mathbb{Q}(\sqrt{D})$; we denote by $E^D/\mathbb{Q}$ the quadratic twist of $E/\mathbb{Q}$ by $D$. In the first half of the paper we consider the situation where $N_-$ is a squarefree produc…

A note on Poisson summation for GL(2)
Tian An Wong
https://arxiv.org/abs/2510.02273 https://arxiv.org/pdf/2510.02273

A note on Poisson summation for GL(2)
Using analytic number theory techniques, Altuğ showed that the contribution of the trivial representation to the Arthur-Selberg trace formula for GL(2) over $\Q$ could be cancelled by applying a modified Poisson summation formula to the regular elliptic contribution. Drawing on recent works, we re-examine these methods from an adelic perspective.

Mordell--Tornheim zeta function: Kronecker limit type formulas and Special values
Sumukha Sathyanarayana, N. Guru Sharan
https://arxiv.org/abs/2510.10093 https://

Mordell--Tornheim zeta function: Kronecker limit type formulas and Special values
In this paper, we establish Kronecker limit type formulas for the generalized Mordell--Tornheim zeta function $Θ(r,s,t,x)$ as a function of the third variable, in terms of Riemann-zeta and Gamma values. We also give series evaluations of $Θ(r,s,t,x)$ in terms of Herglotz-Zagier type functions, and their derivatives. As applications of this, we derive Kronecker limit type formula in the second variable and a new infinite family of modular relations called mixed functional equations. We also st…

Counting integral points in homogeneous spaces over function fields
Sheng Chen, Jing Liu
https://arxiv.org/abs/2510.06983 https://arxiv.org/pdf/2510.06983

Counting integral points in homogeneous spaces over function fields
We establish the asymptotic formula for the number of integral points in non-compact symmetric homogeneous spaces of semi-simple simply connected algebraic groups over global function fields, given by the sum of the products of local densities twisted by suitable Brauer elements.

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…