Tootfinder

1/ C has a wealthy grandmother who has offered us a large sum of money towards a house. She’s even agreed to make up the difference from her estate if she dies within 7 years, to cover off inheritance tax. She has however insisted that that sum be ringfenced so that it ‘stays within the family’ - meaning C and her 2 biological kids. I don’t really know how to feel about this. I that since I earn more than double what C does, we call the sum ‘ours’ if we’re together 10 or more years.

The inverse of the (alternating) infinite sum of the reciprocal of the weighted sum for generalized Fibonacci sub-sequences
Yongkang Wan, Zhonghao Liang, Qunying Liao
https://arxiv.org/abs/2509.14569

The inverse of the (alternating) infinite sum of the reciprocal of the weighted sum for generalized Fibonacci sub-sequences
In this paper, for the generalized Fibonacci sequence $\left\{W_n\left(a,b,p,q\right)\right\}$, by using elementary methods and techniques, we give the asymptotic estimation values of $\left(\sum\limits_{k=n}^{\infty}\frac{1}{\sum\limits_{i=0}^{t}s_{i}W_{mk+l_i}}\right)^{-1}$ and $\left(\sum\limits_{k=n}^{\infty}\frac{\left(-1\right)^k}{\sum\limits_{i=0}^{t}s_{i}W_{mk+l_i}}\right)^{-1}$, respectively. In particular, for some special $a,b,p,q,m,t,s_i$ and $l_i\left(0\leq i\leq t \right)$, Theore…

🇺🇦 #NowPlaying on KEXP's #DriveTime
ROSALÍA:
🎵 Porcelana
#ROSALÍA
https://beatsbyalpaca.bandcamp.com/track/rosal-a-porcelana-alpacas-ego-sum-mix
https://open.spotify.com/track/3hETtcSLmNncBjPMLHtVZs

Minimum Sum Coloring with Bundles in Trees and Bipartite Graphs
Takehiro Ito, Naonori Kakimura, Naoyuki Kamiyama, Yusuke Kobayashi, Yoshio Okamoto
https://arxiv.org/abs/2509.15080

Minimum Sum Coloring with Bundles in Trees and Bipartite Graphs
The minimum sum coloring problem with bundles was introduced by Darbouy and Friggstad (SWAT 2024) as a common generalization of the minimum coloring problem and the minimum sum coloring problem. During their presentation, the following open problem was raised: whether the minimum sum coloring problem with bundles could be solved in polynomial time for trees. We answer their question in the negative by proving that the minimum sum coloring problem with bundles is NP-hard even for paths. We compl…

The whole day so far, after doing some chores around the city, is depressing a few layers high ∑ and a few layers deep ↓
I am somewhere in the middle ∝ and miraculously alright, but I don't like it.

The good thing about AI is that if you have a fragment of a stupid idea, it can easily turn it into a whole stupid thing
The Whole of the Bum
I pictured a cheek,
You showed me the pair.
I saw just a glimpse,
You bared it with flair.
I looked at a crack,
You gave me the run.
I saw a shadow,
You showed me the bum.
Chorus
Oh, I saw a dimple,
You saw the sum.
I saw a wrinkle,
You saw the whole of the bum.
I saw a half…

The Sum Leaks More Than Its Parts: Compositional Privacy Risks and Mitigations in Multi-Agent Collaboration
Vaidehi Patil, Elias Stengel-Eskin, Mohit Bansal
https://arxiv.org/abs/2509.14284

The Sum Leaks More Than Its Parts: Compositional Privacy Risks and Mitigations in Multi-Agent Collaboration
As large language models (LLMs) become integral to multi-agent systems, new privacy risks emerge that extend beyond memorization, direct inference, or single-turn evaluations. In particular, seemingly innocuous responses, when composed across interactions, can cumulatively enable adversaries to recover sensitive information, a phenomenon we term compositional privacy leakage. We present the first systematic study of such compositional privacy leaks and possible mitigation methods in multi-agent…

I love little programming katas like this where the theoretical solution is kinda obvious but the implementation needs many things to wrap your head around.
Here: sum to numbers as strings (that will possibly exceed the max value of your numeric data type).
https://www.codewars.com/kata/5324945e2ece

Created two small linters improving an ergonomics of type switches in Go by adding exhaustive match check. With those tools adding a new case become a lint error.
https://github.com/gomoni/sumlint
1. oneoflint does an exhaustive check for code implementing protobuf's oneof

GitHub - gomoni/sumlint: Exhaustiveness checks for your interface "sum types" using a lightweight naming convention + `go vet`.
Exhaustiveness checks for your interface "sum types" using a lightweight naming convention + `go vet`. - gomoni/sumlint

Proof of a Generalized Ryu-Takayanagi Conjecture
Artem Averin
https://arxiv.org/abs/2508.14877 https://arxiv.org/pdf/2508.14877

Proof of a Generalized Ryu-Takayanagi Conjecture
We derive a generalized version of the Ryu-Takayanagi formula for the entanglement entropy in arbitrary diffeomorphism invariant field theories. We use a recent framework which expresses the measurable quantities of a quantum theory as a weighted sum over paths in the theory's phase space. If this framework is applied to a field theory on a spacetime foliated by a hypersurface $Σ,$ the choice of a codimension-2 surface $B$ without boundary contained in $Σ$ specifies a submanifold in the phase…

Zero-sum turn games using Q-learning: finite computation with security guarantees
Sean Anderson, Chris Darken, Jo\~ao Hespanha
https://arxiv.org/abs/2509.13585 https://

Zero-sum turn games using Q-learning: finite computation with security guarantees
This paper addresses zero-sum ``turn'' games, in which only one player can make decisions at each state. We show that pure saddle-point state-feedback policies for turn games can be constructed from dynamic programming fixed-point equations for a single value function or Q-function. These fixed-points can be constructed using a suitable form of Q-learning. For discounted costs, convergence of this form of Q-learning can be established using classical techniques. For undiscounted costs, we provi…

Every sufficiently large odd integer is a sum of two positive perfect squares and a prime
Ricardo Adonis Caraccioli Abrego
https://arxiv.org/abs/2509.14246 https://

Every sufficiently large odd integer is a sum of two positive perfect squares and a prime
We prove that every sufficiently large odd integer is a sum of two positive squares and a prime. Let R(n) be the number of representations n = x^2 + y^2 + p with x, y >= 1 and p prime. We show that R(n) > 0 for all odd n >= n0 and obtain the asymptotic formula R(n) = S(n) C n / log n, where S(n) > 0 is the singular series and C > 0 is the singular integral. The proof uses the Hardy-Littlewood circle method: on the major arcs we extract the main term, and on the minor arcs we combine an L2 bound…

If $\sum_n n! c_n z^n$ is entire and $c_n$ does not terminate, then $\sum_n c_n z^n$ has infinitely many zeros
Alann Rosas
https://arxiv.org/abs/2510.10054 https://

If $\sum_n n! c_n z^n$ is entire and $c_n$ does not terminate, then $\sum_n c_n z^n$ has infinitely many zeros
We prove that if $\sum_n n! c_n z^n$ is entire and $c_n$ does not terminate, then $\sum_n c_n z^n$ has infinitely many zeros. We then use this result to give alternative proofs that the Le Roy functions $f_r(z)=\sum_{n=0}^\infty \frac{z^n}{(n!)^r}$ for $r>1$ and Bessel functions $J_α(z)=\sum_{m=0}^\infty \frac{(-1)^m}{m!Γ(m+α+1)}\left(\frac{z}{2}\right)^{2m+α}$ for $α\in\mathbb R$ have infinitely many zeros.

A sum rule for derangements
Jean-Christophe Pain
https://arxiv.org/abs/2509.14672 https://arxiv.org/pdf/2509.14672

A sum rule for derangements
We propose a sum rule for derangements. Three different proofs are provided. The first one involves integral representations and the second one relies on the Hermite identity for the integer part of the product of an integer by a real number. The third one, which, unlike the previous ones, is not based on induction, proceeds directly from a recurrence relation for the number of derangements.

In a short blog post for the MacArthur Foundation, my colleague Veronica Thamaini and I discuss the Abstract Wikipedia proposal, one of five finalists for the ongoing 100&Change program.
A Path to a World Where Everyone Can Share in the Sum of All Knowledge
https://www.

A Path to a World Where Everyone Can Share in the Sum of All Knowledge

Existence, asymptotic behaviors, and high-dimensional uniqueness of topological solutions to the skew-symmetric Chern-Simons system on lattice graphs
Honggang Liu
https://arxiv.org/abs/2509.14538

Existence, asymptotic behaviors, and high-dimensional uniqueness of topological solutions to the skew-symmetric Chern-Simons system on lattice graphs
In this paper, we consider the topological solutions to the skew-symmetric Chern-Simons system on lattice graphs: $$\left\{\begin{aligned} Δu &=λ\mathrm{e}^{\upsilon}(\mathrm{e}^{u}-1)+4π\sum\limits_{j=1}^{k_1}m_jδ_{p_j}, Δ\upsilon&=λ\mathrm{e}^{u}(\mathrm{e}^{\upsilon}-1)+4π\sum\limits_{j=1}^{k_2}n_jδ_{q_j}, \end{aligned} \right. $$ here, $λ\in\mathbb{R}_+$, $k_1$ and $k_2$ are two positive integers, $m_j\in\mathbb{N}\, (j=1,2,\cdot\cdot\cdot,k_1)$, $n_j\in\mathbb{N}\,(j=1,2,\cdot\cdo…

Strongly clean ring elements that are one-sided inverses
George M. Bergman
https://arxiv.org/abs/2508.14396 https://arxiv.org/pdf/2508.14396

Strongly clean ring elements that are one-sided inverses
A longstanding open question is whether every strongly clean ring (ring in which every element is strongly clean, i.e., is the sum of an idempotent and a unit which commute with each other) is Dedekind-finite (has the property that every element with a one-sided inverse is invertible). We give an example of a ring with two strongly clean elements that are one-sided, but not two-sided, inverses of one another, suggesting that the answer to that question may be negative. We then discuss possibl…

Gaussian Multiplier Bootstrap Procedure for the $\kappa$th Largest Coordinate of High-Dimensional Statistics
Yixi Ding, Qizhai Li, Yuke Shi, Liuquan Sun
https://arxiv.org/abs/2508.14400

Gaussian Multiplier Bootstrap Procedure for the $κ$th Largest Coordinate of High-Dimensional Statistics
We consider the problem of Gaussian and bootstrap approximations for the distribution of the $κ$th-largest statistic in high dimensions. This statistic, defined as the $κ$th-largest component of the sum of independent random vectors, is critical in numerous high-dimensional estimation and testing problems. Such a problem has been studied previously for $κ=1$ (i.e., maxima). However, in many applications, a general $κ\geq1$ is of great interest, which is addressed in this paper. By invoking …

Tight Regret Upper and Lower Bounds for Optimistic Hedge in Two-Player Zero-Sum Games
Taira Tsuchiya
https://arxiv.org/abs/2510.11691 https://arxiv.org/pdf…

Tight Regret Upper and Lower Bounds for Optimistic Hedge in Two-Player Zero-Sum Games
In two-player zero-sum games, the learning dynamic based on optimistic Hedge achieves one of the best-known regret upper bounds among strongly-uncoupled learning dynamics. With an appropriately chosen learning rate, the social and individual regrets can be bounded by $O(\log(mn))$ in terms of the numbers of actions $m$ and $n$ of the two players. This study investigates the optimality of the dependence on $m$ and $n$ in the regret of optimistic Hedge. To this end, we begin by refining existing …

Tencent and Baidu sold $3.3B combined in offshore bonds this year, the most ever for Chinese tech companies, as companies secure cheap financing for AI buildout (Janice Huang/Bloomberg)
https://www.bloomberg.com/news/articles/2025-…

Larry Summers "stepping back" from public roles after Epstein files release (Rebecca Falconer/Axios)
https://www.axios.com/2025/11/18/larry-summers-harvard-epstein
http://www.memeorandum.com/251117/p166#a251117p166

Efficiently learning depth-3 circuits via quantum agnostic boosting
Srinivasan Arunachalam, Arkopal Dutt, Alexandru Gheorghiu, Michael de Oliveira
https://arxiv.org/abs/2509.14461

Efficiently learning depth-3 circuits via quantum agnostic boosting
We initiate the study of quantum agnostic learning of phase states with respect to a function class $\mathsf{C}\subseteq \{c:\{0,1\}^n\rightarrow \{0,1\}\}$: given copies of an unknown $n$-qubit state $|ψ\rangle$ which has fidelity $\textsf{opt}$ with a phase state $|ϕ_c\rangle=\frac{1}{\sqrt{2^n}}\sum_{x\in \{0,1\}^n}(-1)^{c(x)}|x\rangle$ for some $c\in \mathsf{C}$, output $|ϕ\rangle$ which has fidelity $|\langle ϕ| ψ\rangle|^2 \geq \textsf{opt}-\varepsi…

Geometric representation of CP phases $\delta_{\rm PDG}, \delta_{\rm KM}$ in flavor mixing matrix and its sum rule by alternative unitarity triangle and quadrangle
Masaki J. S. Yang
https://arxiv.org/abs/2509.11596

Geometric representation of CP phases $δ_{\rm PDG}, δ_{\rm KM}$ in flavor mixing matrix and its sum rule by alternative unitarity triangle and quadrangle
In this letter, we present a geometric representation of the CP phases $δ_{\rm PDG}$ and $δ_{\rm KM}$ in the PDG and Kobayashi--Maskawa parameterizations of the mixing matrix as angles on the complex plane. The sum rule with the unitarity triangle $δ_{\rm PDG} + δ_{\rm KM} = π- α+ γ$ is geometrically expressed as a quadrangle on the complex plane, combination of a unitarity triangle and an alternative triangle. Furthermore, a new set of inverse unitarity triangles is defined from the inv…

Learning Rate Should Scale Inversely with High-Order Data Moments in High-Dimensional Online Independent Component Analysis
M. Oguzhan Gultekin, Samet Demir, Zafer Dogan
https://arxiv.org/abs/2509.15127

Learning Rate Should Scale Inversely with High-Order Data Moments in High-Dimensional Online Independent Component Analysis
We investigate the impact of high-order moments on the learning dynamics of an online Independent Component Analysis (ICA) algorithm under a high-dimensional data model composed of a weighted sum of two non-Gaussian random variables. This model allows precise control of the input moment structure via a weighting parameter. Building on an existing ordinary differential equation (ODE)-based analysis in the high-dimensional limit, we demonstrate that as the high-order moments increase, the algorit…

On Laplacian and Signless Laplacian Permanental Polynomials of Some Well-known Graphs
Sarbari Mitra, Soumya Bhoumik
https://arxiv.org/abs/2509.14389 https://

On Laplacian and Signless Laplacian Permanental Polynomials of Some Well-known Graphs
The permanent of an $n \times n$ matrix $M = (m_{ij})$ is defined as $\mathrm{per}(M) = \sum_{σ\in S_n} \prod_{i=1}^n m_{i,σ(i)}$, where $S_n$ denotes the symmetric group on $\{1,2,\ldots,n\}$. The permanental polynomial of $M$, is defined by $ψ(M;x) = \mathrm{per}(xI_n - M)$. We study two fundamental variants: the Laplacian permanental polynomial $ψ(L(G);x)$ and signless Laplacian permanental polynomial $ψ(Q(G);x)$ of a graph $G$. A graph is said to be {determined} by its (signless) Lapla…

Re$^3$MCN: Cubic Newton Variance Reduction Momentum Quadratic Regularization for Finite-sum Non-convex Problems
Dmitry Pasechnyuk-Vilensky, Dmitry Kamzolov, Martin Tak\'a\v{c}
https://arxiv.org/abs/2510.08714

Re$^3$MCN: Cubic Newton + Variance Reduction + Momentum + Quadratic Regularization for Finite-sum Non-convex Problems
We analyze a stochastic cubic regularized Newton method for finite sum optimization $\textstyle\min_{x\in\mathbb{R}^d} F(x) \;=\; \frac{1}{n}\sum_{i=1}^n f_i(x)$, that uses SARAH-type recursive variance reduction with mini-batches of size $b\sim n^{1/2}$ and exponential moving averages (EMA) for gradient and Hessian estimators. We show that the method achieves a $(\varepsilon,\sqrt{L_2\varepsilon})$-second-order stationary point (SOSP) with total stochastic oracle calls $n + \widetilde{\mathcal…

Improvement of Drmota-Verwee's effective Erdos-Wintner theorem for Zeckendorf expansions
Johann Verwee
https://arxiv.org/abs/2509.14974 https://arxiv.o…

Improvement of Drmota-Verwee's effective Erdos-Wintner theorem for Zeckendorf expansions
We revisit the effective Erdos-Wintner theorem for Zeckendorf expansions. Drmota and the author obtained a uniform Kolmogorov bound whose error involves $T\sum_{j>L-2h}|f(F_j)|$, which assumes absolute convergence of the linear tail $\sum_j f(F_j)$. We remove this assumption. Grouping the transfer matrices in pairs and working to second order on the logarithm of the product, after extracting the common linear phase along the dominant direction, yields a quadratic tail $T^2\sum_{j>L-2h…

Efficient Computation of Time-Index Powered Weighted Sums Using Cascaded Accumulators
Deijany Rodriguez Linares, Oksana Moryakova, H{\aa}kan Johansson
https://arxiv.org/abs/2509.15069

Efficient Computation of Time-Index Powered Weighted Sums Using Cascaded Accumulators
This letter presents a novel approach for \mbox{efficiently} computing time-index powered weighted sums of the form $\sum_{n=0}^{N-1} n^{K} v[n]$ using cascaded accumulators. Traditional direct computation requires $K{\times}N$ general multiplications, which become prohibitive for large $N$, while alternative strategies based on lookup tables or signal reversal require storing entire data blocks. By exploiting accumulator properties, the proposed method eliminates the need for such storage and …

Nearly Tight Bounds for the Online Sorting Problem
Yossi Azar, Debmalya Panigrahi, Or Vardi
https://arxiv.org/abs/2508.14287 https://arxiv.org/pdf/2508.142…

Nearly Tight Bounds for the Online Sorting Problem
In the online sorting problem, a sequence of $n$ numbers in $[0, 1]$ (including $\{0,1\}$) have to be inserted in an array of size $m \ge n$ so as to minimize the sum of absolute differences between pairs of numbers occupying consecutive non-empty cells. Previously, Aamand {\em et al.} (SODA 2023) gave a deterministic $2^{\sqrt{\log n} \sqrt{\log \log n + \log (1/\varepsilon)}}$-competitive algorithm when $m = (1+\varepsilon) n$ for any $\varepsilon \ge Ω(\log n/n)$. They also showed a lower b…

On the extension of a class of Hermite bivariate interpolation problems
Hakop Hakopian, Anush Khachatryan
https://arxiv.org/abs/2509.14359 https://arxiv.or…

On the extension of a class of Hermite bivariate interpolation problems
We characterize the sets of solvability for Hermite bivariate interpolation problems when the sum of multiplicities is at most $2n + 2$, with $n$ the degree of the polynomial space. This result extends an earlier theorem (2000) by one of the authors concerning the case $2n+1$. The latter theorem, in turn, can be regarded as a natural generalization of a classical theorem of Severi (1921).

Did you choose your major poorly in college, and are so obsessed with being in London that you are willing to accept a sub-living wage to reside there? Or is your partner well enough paid to maintain you, but you don't want to be bored at home? Are you independently wealthy, perhaps? Please consider subsidizing the British State with your labor in exchange for a token sum of £38k per year.

k-SUM Hardness Implies Treewidth-SETH
Michael Lampis
https://arxiv.org/abs/2510.08185 https://arxiv.org/pdf/2510.08185

k-SUM Hardness Implies Treewidth-SETH
We show that if k-SUM is hard, in the sense that the standard algorithm is essentially optimal, then a variant of the SETH called the Primal Treewidth SETH is true. Formally: if there is an $\varepsilon>0$ and an algorithm which solves SAT in time $(2-\varepsilon)^{tw}|ϕ|^{O(1)}$, where $tw$ is the width of a given tree decomposition of the primal graph of the input, then there exists a randomized algorithm which solves k-SUM in time $n^{(1-δ)\frac{k}{2}}$ for some $δ>0$ and all su…

Functorial equivalence classes of $2$-blocks of tame representation type
Robert Boltje, Serge Bouc, Deniz Y{\i}lmaz
https://arxiv.org/abs/2509.14682 https://

Functorial equivalence classes of $2$-blocks of tame representation type
For any block of a finite group over an algebraically closed field of characteristic $2$ which has dihedral, semidihedral, or generalized quaternion defect groups, we determine explicitly the decomposition of the associated diagonal $p$-permutation functor over an algebraically closed field $\mathbb{F}$ of characteristic $0$ into a direct sum of simple functors. As a consequence we see that two blocks with dihedral, semidihedral, or generalized quaternion defect groups are functorially equivale…

🇺🇦 #NowPlaying on KEXP's #MiddayShow
Rochelle Jordan:
🎵 Sum
#RochelleJordan
https://rochellejordan.bandcamp.com/track/sum
https://open.spotify.com/track/3MIVJlKGcLfYNLso4pp6dn

Sideband Spectroscopy in the Strong Driving Regime: Volcano Transparency and Sideband Anomaly
Luka Antonic, Sergey Hazanov, Sergei Masis, Daniel Podolsky, Eyal Buks
https://arxiv.org/abs/2508.14781

Sideband Spectroscopy in the Strong Driving Regime: Volcano Transparency and Sideband Anomaly
We study the response of a spin to two crossed magnetic fields: a strong and fast transverse field, and a weak and slow longitudinal field. We characterize the sideband response at the sum and the difference of driving frequencies over a broad range of parameters. In the strong transverse driving regime, the emission spectrum has a characteristic volcano lineshape with a narrow central transparency region surrounded by asymmetric peaks. Next, we couple the spin to a nonlinear cavity that both d…

Detection of ferroic octupole ordering in $d$-wave altermagnetic rutile-type compounds
Masaichiro Mizumaki, Norimasa Sasabe, Takayuki Uozumi, Rikuto Oiwa, Hiroaki Kusunose
https://arxiv.org/abs/2509.14716

Detection of ferroic octupole ordering in $d$-wave altermagnetic rutile-type compounds
We propose that X-ray absorption and emission magnetic circular dichroism (XAS-MCD and XES-MCD) are promising measurements to directly detect ferroic higher-rank multipoles as candidate order parameters in altermagnets. Using the sum rules for XES-MCD and connecting them to multipole language, we demonstrate that the expectation value of the magnetic octupole moment is finite in the $d$-wave altermagnetic candidate rutile-type compounds TF$_2$ (T=transition metal). We also perform spectral calc…

🇺🇦 #NowPlaying on #BBC6Music's #CraigCharles
ROSALÍA:
🎵 Porcelana
#ROSALÍA
https://beatsbyalpaca.bandcamp.com/track/rosal-a-porcelana-alpacas-ego-sum-mix
https://open.spotify.com/track/3hETtcSLmNncBjPMLHtVZs

Generalised Reachability Games Revisited
Sougata Bose (UMONS - Universit\'e de Mons), Daniel Hausmann (University of Liverpool), Soumyajit Paul (University of Liverpool), Sven Schewe (University of Liverpool), Tansholpan Zhanabekova (University of Liverpool)
https://arxiv.org/abs/2509.14091

Generalised Reachability Games Revisited
Classic reachability games on graphs are zero-sum games, where the goal of one player, Eve, is to visit a vertex from a given target set, and that of other player, Adam, is to prevent this. Generalised reachability games, studied by Fijalkow and Horn, are a generalisation of reachability objectives, where instead of a single target set, there is a family of target sets and Eve must visit all of them in any order. In this work, we further study the complexity of solving two-player games on graph…

Discrete weakly stable distributions
Matthew Aldridge
https://arxiv.org/abs/2509.12070 https://arxiv.org/pdf/2509.12070

Discrete weakly stable distributions
A random variable X is strictly stable if a sum of independent copies of X has the same distribution as X up to scaling, and is stable (or weakly stable) if the sum has the same distribution as X up to both scaling and shifting. Steutel and Van Harn studied 'discrete strict stability' for random variables taking values in the non-negative integers, replacing scaling by the thinning operation. We extend this to 'discrete (weak) stability', where shifting is replaced by the addition of an indepen…

Incremental-Decremental Maximization
Yann Disser, Max Klimm, Annette Lutz, Lea Strubberg
https://arxiv.org/abs/2508.14516 https://arxiv.org/pdf/2508.14516

Incremental-Decremental Maximization
We introduce a framework for incremental-decremental maximization that captures the gradual transformation or renewal of infrastructures. In our model, an initial solution is transformed one element at a time and the utility of an intermediate solution is given by the sum of the utilities of the transformed and untransformed parts. We propose a simple randomized and a deterministic algorithm that both find an order in which to transform the elements while maintaining a large utility during all …

Diagonalization of the Toda flow on conjugacy classes of matrices with simple spectrum
David Mart\'inez Torres, Carlos Tomei
https://arxiv.org/abs/2509.13452 https://…

Diagonalization of the Toda flow on conjugacy classes of matrices with simple spectrum
We construct coordinates on conjugacy classes of traceless complex matrices with simple spectrum that diagonalize the non-periodic Toda vector field. By this we mean that the coordinates, defined on an open and dense neighborhood of any diagonal matrix in the conjugacy class, decouple the Toda vector field into a sum of multiples of the Euler and rotational vector field in $\mathbb{C}$. Using Lie theoretic methods, we extend this construction from $\mathfrak{sl}_\mathbb{C}$ to arbitrary complex…

Diffraction-Unlimited Tip-Enhanced Sum-Frequency Vibrational Nanoscopy
Shota Takahashi, Koichi Kumagai, Atsunori Sakurai, Tatsuto Mochizuki, Tomonori Hirano, Akihiro Morita, Toshiki Sugimoto
https://arxiv.org/abs/2509.09179

Diffraction-Unlimited Tip-Enhanced Sum-Frequency Vibrational Nanoscopy
Sum-frequency generation (SFG) is a powerful second-order nonlinear spectroscopic technique that provides detailed insights into molecular structures and absolute orientations at surfaces and interfaces. However, conventional SFG based on far-field schemes suffers from the diffraction limit of light, which inherently averages spectroscopic information over micrometer-scale regions and obscures nanoscale structural inhomogeneity. Here, we overcome this fundamental limitation by leveraging a high…

Quantum State Recovery via Direct Sum Formalism Without Measurement Outcomes
Taiga Suzuki, Masayuki Ohzeki
https://arxiv.org/abs/2509.11066 https://arxiv.o…

Quantum State Recovery via Direct Sum Formalism Without Measurement Outcomes
This study proposes a new approach to quantum state recovery following measurement. Specifically, we introduce a special operation that transfers the probability amplitude of the quantum state into its orthogonal complement. This operation is followed by a measurement performed on this orthogonal subspace, enabling the undisturbed original quantum state to be regained. Remarkably, this recovery is achieved without dependence of the post-measurement operation on the measurement outcome, thus all…

🇺🇦 Auf radioeins läuft...
Rosalía:
🎵 Porcelana
#NowPlaying #Rosalía
#newRelease 🆕 album
https://beatsbyalpaca.bandcamp.com/track/rosal-a-porcelana-alpacas-ego-sum-mix
https://open.spotify.com/track/3hETtcSLmNncBjPMLHtVZs

Resolving the Body-Order Paradox of Machine Learning Interatomic Potentials
Sanggyu Chong, Tong Jiang, Michelangelo Domina, Filippo Bigi, Federico Grasselli, Joonho Lee, Michele Ceriotti
https://arxiv.org/abs/2509.14146

Resolving the Body-Order Paradox of Machine Learning Interatomic Potentials
In many cases, the predictions of machine learning interatomic potentials (MLIPs) can be interpreted as a sum of body-ordered contributions, which is explicit when the model is directly built on neighbor density correlation descriptors, and implicit when the model captures the correlations through non-linear functions of low body-order terms. In both cases, the "effective body-orderedness" of MLIPs remains largely unexplained: how do the models decompose the total energy into body-ordered contr…

Improved Online Sorting
Jubayer Nirjhor, Nicole Wein
https://arxiv.org/abs/2508.14361 https://arxiv.org/pdf/2508.14361

Improved Online Sorting
We study the online sorting problem, where $n$ real numbers arrive in an online fashion, and the algorithm must immediately place each number into an array of size $(1+\varepsilon) n$ before seeing the next number. After all $n$ numbers are placed into the array, the cost is defined as the sum over the absolute differences of all $n-1$ pairs of adjacent numbers in the array, ignoring empty array cells. Aamand, Abrahamsen, Beretta, and Kleist introduced the problem and obtained a deterministic a…

Imma be rich! Thanks Elon!

On a Restriction Problem of Hickman and Wright for the Parabola in $\mathbb{Z}/N\mathbb{Z}$ for Squarefree $N$
Nathaniel Kingsbury-Neuschotz
https://arxiv.org/abs/2509.09885 htt…

On a Restriction Problem of Hickman and Wright for the Parabola in $\mathbb{Z}/N\mathbb{Z}$ for Squarefree $N$
Hickman and Wright proved an $L^2$ restriction estimate for the parabola $Σ$ in $\mathbb{Z}/N\mathbb{Z}$ of the form $$\left(\frac{1}{|Σ|}\sum\limits_{m\inΣ}|\widehat{f}(m)|^2 \right)^{\frac{1}{2}}\leq C_εN^ε\cdot N^{-1}\left(\sum\limits_{x\in (\mathbb{Z}/N\mathbb{Z})^2}|f(x)|^\frac{6}{5}\right)^\frac{5}{6}$$ for all functions $f:(\mathbb{Z}/N\mathbb{Z})^2\rightarrow \mathbb{C}$ and any $ε>0$, and that this bound is sharp when $N$ has a large square facto…

Two-Timescale Sum-Rate Maximization for Movable Antenna Enhanced Systems
Xintai Chen, Biqian Feng, Yongpeng Wu, Derrick Wing Kwan Ng, Robert Schober
https://arxiv.org/abs/2509.04062

Two-Timescale Sum-Rate Maximization for Movable Antenna Enhanced Systems
This paper studies a novel movable antenna (MA)-enhanced multiuser multiple-input multiple-output downlink system designed to improve wireless communication performance. We aim to maximize the average achievable sum rate through two-timescale optimization exploiting instantaneous channel state information at the receiver (I-CSIR) for receive antenna position vector (APV) design and statistical channel state information at the transmitter (S-CSIT) for transmit APV and covariance matrix design. W…

Improved Bounds for the Index Conjecture in Zero-Sum Theory
Andrew Pendleton
https://arxiv.org/abs/2510.11976 https://arxiv.org/pdf/2510.11976

Improved Bounds for the Index Conjecture in Zero-Sum Theory
The Index Conjecture in zero-sum theory states that when $n$ is coprime to $6$ and $k$ equals $4$, every minimal zero-sum sequence of length $k$ modulo $n$ has index $1$. While other values of $(k,n)$ have been studied thoroughly in the last 30 years, it is only recently that the conjecture has been proven for $n>10^{20}$. In this paper, we prove that said upper bound can be reduced to $4.6\cdot10^{13}$, and lower under certain coprimality conditions. Further, we verify the conjecture for $n<1.…

Turnpike properties for zero-sum stochastic linear quadratic differential games of Markovian regime switching system
Xun Li, Fan Wu, Xin Zhang
https://arxiv.org/abs/2509.09358 h…

Turnpike properties for zero-sum stochastic linear quadratic differential games of Markovian regime switching system
This paper investigates the long-time behavior of zero-sum stochastic linear-quadratic (SLQ) differential games within Markov regime-switching diffusion systems and establishes the turnpike property of the optimal triple. By verifying the convergence of the associated coupled differential Riccati equations (CDREs) along with their convergence rate, we show that, for a sufficiently large time horizon, the equilibrium strategy in the finite-horizon problem can be closely approximated by that of t…

On the Exact Sum PDF and CDF of {\alpha}-{\mu} Variates
Fernando Dar\'io Almeida Garc\'ia, Francisco Raimundo Albuquerque Parente, Michel Daoud Yacoub, Jose C\^andido Silveira Santos Filho
https://arxiv.org/abs/2510.03850

On the Exact Sum PDF and CDF of α-μ Variates
The sum of random variables (RVs) appears extensively in wireless communications, at large, both conventional and advanced, and has been subject of longstanding research. The statistical characterization of the referred sum is crucial to determine the performance of such communications systems. Although efforts have been undertaken to unveil these sum statistics, e.g., probability density function (PDF) and cumulative distribution function (CDF), no general efficient nor manageable solutions ca…

Frozen-corner enumeration of Alternating Sign Matrices
Filippo Colomo, Andrei G. Pronko
https://arxiv.org/abs/2509.14006 https://arxiv.org/pdf/2509.14006…

Frozen-corner enumeration of Alternating Sign Matrices
An Alternating Sign Matrix (ASM) is a square matrix with entries in $\{0,1,-1\}$, and such that: $i)$ in each row and columns, nonzero entries alternate in sign; $ii)$ for any given row or column, entries sum up to 1. We define the frozen-square enumeration as the enumeration of $n\times n$ ASMs under the refinement of having, located in a corner, an $s\times s$ square of entries that are all zeroes. We state a conjectural formula for such enumeration, in terms of a Fredholm type determinant of…

On the universal Casimir spectrum
R. L. Mkrtchyan
https://arxiv.org/abs/2509.13707 https://arxiv.org/pdf/2509.13707…

On the universal Casimir spectrum
We conjecture the connection between $su$ and $so$ members of universal, in Vogel's sense, multiplets. The key element is the notion of the {\it vertical componentwise sum} $\oplus_v$ of Young diagrams. Representations in the decomposition of the power of the adjoint representation of $su(N)$ algebra can be parameterized by a couple of $N$-independent Young diagrams $λ$ and $τ$, with equal area. We assume that the $so(N)$ member of the universal (Casimir) multiplet of a given $su(N)$ repres…

The Modular Energy Function: Exact Identities and a Primality Equivalence
Es-said En-naoui
https://arxiv.org/abs/2509.10592 https://arxiv.org/pdf/2509.1059…

The Modular Energy Function: Exact Identities and a Primality Equivalence
We introduce and study the arithmetic function E_m(n), defined as the sum of the remainders of n when divided by the first m positive integers. Although the definition is elementary, the function encodes rich arithmetic structure. In this first part we develop exact algebraic and combinatorial identities, including floor-sum and divisor-sum reformulations, quotient-residue decompositions, sharp bounds and extremal classes, and symmetry relations. We also establish a diagonal identity linking E_…

Consistent causal discovery with equal error variances: a least-squares perspective
Anamitra Chaudhuri, Yang Ni, Anirban Bhattacharya
https://arxiv.org/abs/2509.15197 https://…

Consistent causal discovery with equal error variances: a least-squares perspective
We consider the problem of recovering the true causal structure among a set of variables, generated by a linear acyclic structural equation model (SEM) with the error terms being independent and having equal variances. It is well-known that the true underlying directed acyclic graph (DAG) encoding the causal structure is uniquely identifiable under this assumption. In this work, we establish that the sum of minimum expected squared errors for every variable, while predicted by the best linear c…

$\Xi_c \to \Xi$ Semileptonic Decays: An LCSR View on the Experiment-Lattice Tension
T. M. Aliev, S. Bilmis, M. Savci
https://arxiv.org/abs/2509.10181 https://

$Ξ_c \to Ξ$ Semileptonic Decays: An LCSR View on the Experiment-Lattice Tension
We present a light-cone QCD sum rule analysis of the semileptonic decays of $Ξ_c$ baryons, focusing on the channels $Ξ_c^0 \to Ξ^- \ell^+ ν_\ell$, and $Ξ_c^+ \to Ξ^0 \ell^+ ν_\ell$. The transition form factors are calculated within the light-cone QCD sum rules framework, using the distribution amplitudes of the heavy $Ξ_c$ baryons. The obtained form factors are then used to compute the differential and total decay widths, as well as the branching fractions. Our numerical results for the…

N-Parties Private Structure and Parameter Learning for Sum-Product Networks
Xenia Heilmann, Ernst Althaus, Mattia Cerrato, Nick Johannes Peter Rassau, Mohammad Sadeq Dousti, Stefan Kramer
https://arxiv.org/abs/2510.05946

N-Parties Private Structure and Parameter Learning for Sum-Product Networks
A sum-product network (SPN) is a graphical model that allows several types of probabilistic inference to be performed efficiently. In this paper, we propose a privacy-preserving protocol which tackles structure generation and parameter learning of SPNs. Additionally, we provide a protocol for private inference on SPNs, subsequent to training. To preserve the privacy of the participants, we derive our protocol based on secret sharing, which guarantees privacy in the honest-but-curious setting ev…

Most incompatible measurements and sum-of-squares optimisation
S\'ebastien Designolle
https://arxiv.org/abs/2509.10381 https://arxiv.org/pdf/2509.10381…

Most incompatible measurements and sum-of-squares optimisation
Measurement incompatibility, or joint measurability, is a cornerstone of quantum theory and a useful resource. For finite-dimensional systems, quantifying this resource and establishing universal bounds valid for all measurements is a long-standing problem. In this work, we exhibit analytical universal parent measurements giving access to bounds that beat the state of the art. In particular, we can show that, for relevant robustnesses, sets of anticommuting observables give rise to the most inc…

A Poisson representation of the positive sojourn time of L\'evy processes
Helmut H. Pitters
https://arxiv.org/abs/2509.11776 https://arxiv.org/pdf/2509…

A Poisson representation of the positive sojourn time of Lévy processes
We study the distribution of the positive sojourn time $$ A_t:= \int_0^t \mathbf 1\{ X_s>0 \}ds $$ of an arbitrary Lévy process $X:= (X_t)_{t\geq 0}$. For an exponential random variable $E^{(q)}$ of rate $q>0$ independent of $X$ we show the representation in law \begin{align*} A_{E^{(q)}} =_d \sum_{T\in Π} T \end{align*} as the sum of points of a Poisson process $Π$ with intensity given explicitely in terms of the positivity $t\mapsto \mathbb P\{X_t>0\}$. This representation raises some fu…

On the Bit Size of Sum-of-Squares Proofs for Symmetric Formulations
Alex Bortolotti, Monaldo Mastrolilli, Marilena Palomba, Luis Felipe Vargas
https://arxiv.org/abs/2509.06928 h…

On the Bit Size of Sum-of-Squares Proofs for Symmetric Formulations
The Sum-of-Squares (SoS) hierarchy is a powerful framework for polynomial optimization and proof complexity, offering tight semidefinite relaxations that capture many classical algorithms. Despite its broad applicability, several works have revealed fundamental limitations to SoS automatability. (i) While low-degree SoS proofs are often desirable for tractability, recent works have revealed they may require coefficients of prohibitively large bit size, rendering them computationally infeasible.…

Affine-Orthogonal Manifolds and Deformation to Levi-Civita Connections
Mihail Cocos
https://arxiv.org/abs/2510.09648 https://arxiv.org/pdf/2510.09648

Affine-Orthogonal Manifolds and Deformation to Levi-Civita Connections
We study a class of affine manifolds equipped with a flat affine connection $\nabla$ and a global Riemannian metric $g$ that is diagonal in local affine coordinates. These structures are closely related to \emph{Hessian manifolds}, where the metric locally arises as the Hessian of a smooth potential. For example, the Hopf manifold $(\mathbb{R}^{n+1}\setminus \{0\}) / \langle x \mapsto 2x \rangle$ with metric $g = (\sum_i x_i^2)^{-1} \sum_i dx_i^2$ admits a proper deformation of $\nabla$ into it…

On some conjectural supercongruences involving the sequence $t_n(x)$
Hui-Li Han, Chen Wang
https://arxiv.org/abs/2510.11338 https://arxiv.org/pdf/2510.1133…

On some conjectural supercongruences involving the sequence $t_n(x)$
In this paper, we study some supercongruences involving the sequence $$ t_n(x)=\sum_{k=0}^n\binom{n}{k}\binom{x}{k}\binom{x+k}{k}2^k $$ and solve some open problems. For any odd prime $p$ and $p$-adic integer $x$, we determine $\sum_{n=0}^{p-1}t_n(x)^2$ and $\sum_{n=0}^{p-1}(n+1)t_n(x)^2$ modulo $p^2$; for example, we establish that \begin{align*} \sum_{n=0}^{p-1}t_n(x)^2\equiv\begin{cases} \left(\dfrac{-1}{p}\right)\pmod{p^2},&\text{if }2x\equiv-1\pmod{p},\\[8pt] (-1)^{\langle x\rangle…

A Particle-Flow Algorithm for Free-Support Wasserstein Barycenters
Kisung You
https://arxiv.org/abs/2509.11435 https://arxiv.org/pdf/2509.11435

A Particle-Flow Algorithm for Free-Support Wasserstein Barycenters
The Wasserstein barycenter extends the Euclidean mean to the space of probability measures by minimizing the weighted sum of squared 2-Wasserstein distances. We develop a free-support algorithm for computing Wasserstein barycenters that avoids entropic regularization and instead follows the formal Riemannian geometry of Wasserstein space. In our approach, barycenter atoms evolve as particles advected by averaged optimal-transport displacements, with barycentric projections of optimal transport …

To cover a permutohedron
Bochao Kong, Ji Zeng
https://arxiv.org/abs/2509.13877 https://arxiv.org/pdf/2509.13877

To cover a permutohedron
The permutohedron $P_n$ of order $n$ is a polytope embedded in $\mathbb{R}^n$ whose vertex coordinates are permutations of the first $n$ natural numbers. It is obvious that $P_n$ lies on the hyperplane $H_n$ consisting of points whose coordinates sum up to $n(n+1)/2$. We prove that if the vertices of $P_n$ are contained in the union of $m$ affine hyperplanes different from $H_n$, then $m\geq n$ when $n \geq 3$ is odd, and $m \geq n-1$ when $n \geq 4$ is even. This confirms a recent conjecture o…

SoftBank-owned PayPay takes a 40% stake in Binance Japan for an undisclosed sum, to capitalize on Japan's crypto boom; PayPay filed for a US IPO in August (Mayumi Negishi/Bloomberg)
https://www.bloomberg.com/news/articles/20

Scale-Invariant Regret Matching and Online Learning with Optimal Convergence: Bridging Theory and Practice in Zero-Sum Games
Brian Hu Zhang, Ioannis Anagnostides, Tuomas Sandholm
https://arxiv.org/abs/2510.04407

Scale-Invariant Regret Matching and Online Learning with Optimal Convergence: Bridging Theory and Practice in Zero-Sum Games
A considerable chasm has been looming for decades between theory and practice in zero-sum game solving through first-order methods. Although a convergence rate of $T^{-1}$ has long been established since Nemirovski's mirror-prox algorithm and Nesterov's excessive gap technique in the early 2000s, the most effective paradigm in practice is *counterfactual regret minimization*, which is based on *regret matching* and its modern variants. In particular, the state of the art across most benchmarks …

Enhancing Sum Capacity via Quantum and No-Signaling Cooperation Between Transmitters
Seung-Hyun Nam, Hyun-Young Park, Jiyoung Yun, Ashutosh Rai, Si-Hyeon Lee, Joonwoo Bae
https://arxiv.org/abs/2509.08219

Enhancing Sum Capacity via Quantum and No-Signaling Cooperation Between Transmitters
We consider a communication scenario over a discrete memoryless interference channel or multiple access channel without feedback, where transmitters exploit classical, quantum, or no-signaling cooperation. In this scenario, several previous works have shown that the sum capacities of channels involving pseudo-telepathy games can be enhanced by quantum or no-signaling cooperation. However, a full characterization of which channels admit such an improvement remains open. By focusing on the common…

Sum of short exponential sums with prime numbers
Firuz Rakhmonov
https://arxiv.org/abs/2510.09102 https://arxiv.org/pdf/2510.09102

Sum of short exponential sums with prime numbers
For sufficiently large integers $K$, $x$, $y$, and $q$ satisfying $K \le y < x$, where $f(u) = αu^n + α_{n-1}u^{n-1} + \ldots + α_1 u$ is a polynomial of degree $n$ with real coefficients, $n$ is a fixed positive integer, $α$ is a real number such that $\left|α- \frac{a}{q}\right| \le \frac{1}{q^2}$, $(a, q) = 1$, $q \ge 1$ and $\mathscr{L} = \ln x$, an estimate of the form $$ \sum_{k=1}^K \left| \sum_{x - y < p \le x} e(kf(p)) \right| \ll K y \left( \frac{1}{q} + \frac…

Bounded Littlewood identities with fixed number of odd rows or odd columns
JiSun Huh, Jang Soo Kim, Christian Krattenthaler, Soichi Okada
https://arxiv.org/abs/2510.11054 https:…

Bounded Littlewood identities with fixed number of odd rows or odd columns
A Littlewood identity is an identity equating a sum of Schur functions with an infinite product. A bounded Littlewood identity is one where the sum is taken over the partitions with a bounded number of rows or columns. The price to pay is that the infinite product has to be replaced by a determinant. The focus of this article is on refinements of such bounded Littlewood identities where one also prescribes the number of odd-length rows or columns of the partitions. Goulden [{\it Discrete Math.}…

Analysis of the hidden-charm pentaquark candidates in the $J/\psi p$ mass spectrum with QCD sum rules
Zhi-Gang Wang
https://arxiv.org/abs/2509.05648 https://

Analysis of the hidden-charm pentaquark candidates in the $J/ψp$ mass spectrum with QCD sum rules
In this work, we distinguish the isospin unambiguously to construct the diquark-diquark-antiquark type five-quark currents with the isospin $I=\frac{1}{2}$, and study the $uudc\bar{c}$ pentaquark states with the QCD sum rules systematically for the first time. Then we obtain the mass spectrum of the diquark-diquark-antiquark type $uudc\bar{c}$ pentaquark states with the isospin-spin-parity $IJ^P=\frac{1}{2}{\frac{1}{2}}^-$, $\frac{1}{2}{\frac{3}{2}}^-$ and $\frac{1}{2}{\frac{5}{2}}^-$, and make…

Single-trace current correlators for 2d models of 4d gluon scattering
Sean Seet
https://arxiv.org/abs/2509.12200 https://arxiv.org/pdf/2509.12200

Single-trace current correlators for 2d models of 4d gluon scattering
Correlators of affine Kac-Moody currents evaluate to a sum over multi-color-trace contributions. We present a `single-trace current system' defined on a genus 0 Riemann surface, whose correlators evaluate to precisely the single-trace term in the evaluation of a standard affine Kac-Moody at level 1. We demonstrate that using the single-trace current system in genus 0 Berkovits-Witten twistor string correlators gives pure $\mathcal{N} = 4$ super Yang-Mills tree amplitudes, and explain how this w…

Variable Selection for Additive Global Fr\'echet Regression
Haoyi Yang, Satarupa Bhattacharjee, Lingzhou Xue, Bing Li
https://arxiv.org/abs/2509.13685 https://

Variable Selection for Additive Global Fréchet Regression
We present a novel framework for variable selection in Fréchet regression with responses in general metric spaces, a setting increasingly relevant for analyzing non-Euclidean data such as probability distributions and covariance matrices. Building on the concept of (weak) Fréchet conditional means, we develop an additive regression model that represents the metric-based discrepancy of the response as a sum of covariate-specific nonlinear functions in reproducing kernel Hilbert spaces (RKHS). …

Solving Zero-Sum Games with Fewer Matrix-Vector Products
Ishani Karmarkar, Liam O'Carroll, Aaron Sidford
https://arxiv.org/abs/2509.04426 https://arxiv…

Solving Zero-Sum Games with Fewer Matrix-Vector Products
In this paper we consider the problem of computing an $ε$-approximate Nash Equilibrium of a zero-sum game in a payoff matrix $A \in \mathbb{R}^{m \times n}$ with $O(1)$-bounded entries given access to a matrix-vector product oracle for $A$ and its transpose $A^\top$. We provide a deterministic algorithm that solves the problem using $\tilde{O}(ε^{-8/9})$-oracle queries, where $\tilde{O}(\cdot)$ hides factors polylogarithmic in $m$, $n$, and $ε^{-1}$. Our result improves upon the state-of-the…

On integers with many representations as the sum of $k$th powers of primes
Anay Aggarwal
https://arxiv.org/abs/2509.11558 https://arxiv.org/pdf/2509.11558

On integers with many representations as the sum of $k$th powers of primes
For a natural number $k>1$, let $f_k(n)$ denote the number of distinct representations of a natural number $n$ of the form $p^k+q^k$ for primes $p,q$. We prove that, for all $k>1$, $$\limsup_{n\to\infty}f_k(n)=\infty.$$ This positively answers a conjecture of Erdos, which asks if there are natural numbers $n$ with arbitrarily many distinct representations of the form $p_1^k+p_2^k+\dots+p_k^k$ for primes $p_1,p_2,\dots,p_k$.

Verizon agrees to buy Starry, which provides home internet via millimeter wave tech to ~100K customers and filed for bankruptcy in 2023, for an undisclosed sum (Masha Abarinova/Fierce Network)
https://www.fierce-network.com/broadband/verizon-buys-starry-what-y…

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 $\sumelements$.

Constructing new heavy-quark sum rule for $b \to c$ meson and baryon decays
Motoi Endo, Syuhei Iguro, Satoshi Mishima, Ryoutaro Watanabe
https://arxiv.org/abs/2509.02006 https:/…

Constructing new heavy-quark sum rule for $b \to c$ meson and baryon decays
We study heavy-hadron semileptonic decays proceeding via $b \to c$ transition, such as $B \to D^{(*)}τ\barν_τ$ and $Λ_b \to Λ_cτ\barν_τ$. In the heavy-quark limit, where the heavy-quark symmetry holds, we construct a general heavy-quark sum rule for these decays based on the spin decomposition picture. The relation holds directly for the squared amplitudes without requiring phase-space integration. We then apply this relation to reproduce the sum rule among $B \to D^{(*)}τ\barν_τ$ an…

Qualcomm acquires Italian open-source electronics platform Arduino for an undisclosed sum and will retain the Arduino brand; Arduino releases a Pi-like board (Emma Roth/The Verge)
https://www.theverge.com/news/794452/qualcomm-arduino-acquisition-uno-q
…

Qualcomm is acquiring DIY electronics platform Arduino
Qualcomm is acquiring the DIY electronics platform Arduino, and launching a new Uno Q that connects to a new App Lab platform.

Improved Rates for Stochastic Variance-Reduced Difference-of-Convex Algorithms
Anh Duc Nguyen, Alp Yurtsever, Suvrit Sra, Kim-Chuan Toh
https://arxiv.org/abs/2509.11657 https://…

Improved Rates for Stochastic Variance-Reduced Difference-of-Convex Algorithms
In this work, we propose and analyze DCA-PAGE, a novel algorithm that integrates the difference-of-convex algorithm (DCA) with the ProbAbilistic Gradient Estimator (PAGE) to solve structured nonsmooth difference-of-convex programs. In the finite-sum setting, our method achieves a gradient computation complexity of $O(N + N^{1/2}\varepsilon^{-2})$ with sample size $N$, surpassing the previous best-known complexity of $O(N + N^{2/3}\varepsilon^{-2})$ for stochastic variance-reduced (SVR) DCA meth…

Sublinear-Time Algorithms for Diagonally Dominant Systems and Applications to the Friedkin-Johnsen Model
Weiming Feng, Zelin Li, Pan Peng
https://arxiv.org/abs/2509.13112 https:…

Sublinear-Time Algorithms for Diagonally Dominant Systems and Applications to the Friedkin-Johnsen Model
We study sublinear-time algorithms for solving linear systems $Sz = b$, where $S$ is a diagonally dominant matrix, i.e., $|S_{ii}| \geq δ+ \sum_{j \ne i} |S_{ij}|$ for all $i \in [n]$, for some $δ\geq 0$. We present randomized algorithms that, for any $u \in [n]$, return an estimate $z_u$ of $z^*_u$ with additive error $\varepsilon$ or $\varepsilon \lVert z^*\rVert_\infty$, where $z^*$ is some solution to $Sz^* = b$, and the algorithm only needs to read a small portion of …

On the Diminished Sombor Index of Fixed-Order Molecular Graphs With Cyclomatic Number at Least 3
Abdulaziz Mutlaq Alotaibi, Abdulaziz M. Alanazi, Taher S. Hassan, Akbar Ali
https://arxiv.org/abs/2509.12294

On the Diminished Sombor Index of Fixed-Order Molecular Graphs With Cyclomatic Number at Least 3
For a graph $G$ with edge set $E$, let $d(u)$ denote the degree of a vertex $u$ in $G$. The diminished Sombor (DSO) index of $G$ is defined as $DSO(G)=\sum_{uv\in E}\sqrt{(d(u))^2+(d(v))^2}(d(u)+d(v))^{-1}$. The cyclomatic number of a graph is the smallest number of edges whose removal makes the graph acyclic. A connected graph of maximum degree at most $4$ is known as a molecular graph. The primary motivation of the present study comes from a conjecture concerning the minimum DSO index of fixe…

Periods detecting Eisenstein series and sums of $L$-values I
Weixiao Lu, Guodong Xi
https://arxiv.org/abs/2510.12446 https://arxiv.org/pdf/2510.12446

Periods detecting Eisenstein series and sums of $L$-values I
We study the automorphic period associated to a $G$-Hamiltonian variety $M$ whose dual is $\check{M} = T^*(\check{G}/\check{L})$, where $\check{G}$ is a general linear group and $\check{L}$ is a Levi subgroup. For certain cuspidal Eisenstein series, we prove that their period is equal to a finite sum of special values of $L$-functions. This sum is indexed by the fixed points of the associated extended $L$-parameter on $\check{M}$, confirming a conjecture by Ben-Zvi-Sakellaridis-Venkatesh in thi…

On the (Im)possibility of Electrically Charged Planck Relics
Stefano Profumo
https://arxiv.org/abs/2509.12520 https://arxiv.org/pdf/2509.12520

On the (Im)possibility of Electrically Charged Planck Relics
I revisit whether black-hole remnants, from sub-Planckian compact objects to Planck relics and up to (super)massive black holes, can preserve Standard-Model (SM) electric charge. Two exterior-field mechanisms -- Coulomb-focused capture from ambient media and QED Schwinger pair production -- robustly neutralize such objects across cosmic history. I first derive the general capture rate including both Coulomb and gravitational focusing, and sum the stepwise discharge time in closed form via the t…

An Exclusive-Sum-of-Products Pipeline for QAOA
Matthew Brunet, Shilpi Shah, Mostafa Atallah, Anthony Wilkie, Rebekah Herrman
https://arxiv.org/abs/2508.21686 https://

An Exclusive-Sum-of-Products Pipeline for QAOA
The quantum approximate optimization algorithm is commonly used to solve combinatorial optimization problems. While unconstrained problems map naturally into the algorithm, incorporating constraints typically requires penalizing constraint violations in the objective function. In this work, we propose an alternative approach that encodes constraints as Boolean expressions in exclusive-sum-of-products (ESOP) form before penalization. We test this method on the maximum independent set problem usi…

Subgroup perfect codes of $S_n$ in Cayley sum graphs
Ankan Shaw, Biswajit Mondal, Satya Bagchi
https://arxiv.org/abs/2509.05069 https://arxiv.org/pdf/2509.…

Subgroup perfect codes of $S_n$ in Cayley sum graphs
A perfect code in a graph $Γ= (V, E)$ is a subset $C$ of $V$ such that no two vertices in $C$ are adjacent, and every vertex in $V \setminus C$ is adjacent to exactly one vertex in $C$. Let $ G $ be a finite group, and let $ S $ be a square-free normal subset of $ G $. The Cayley sum graph of $ G $ with respect to $ S $ is a simple graph with vertex set $ G $ and two vertices $ x $ and $ y $ are adjacent if $ xy\in S .$ A subset $ C $ of $ G $ is called perfect code of $ G $ if there exists a …

The Online Submodular Cover Problem
Anupam Gupta, Roie Levin
https://arxiv.org/abs/2510.08883 https://arxiv.org/pdf/2510.08883

The Online Submodular Cover Problem
In the submodular cover problem, we are given a monotone submodular function $f$, and we want to pick the min-cost set $S$ such that $f(S) = f(N)$. Motivated by problems in network monitoring and resource allocation, we consider the submodular cover problem in an online setting. As a concrete example, suppose at each time $t$, a nonnegative monotone submodular function $g_t$ is given to us. We define $f^{(t)} = \sum_{s \leq t} g_s$ as the sum of all functions seen so far. We need to maintain a …

On the Ratat-Goormaghtigh equation and integer points close to the graph of a smooth function
Tomohiro Yamada
https://arxiv.org/abs/2510.11252 https://arxi…

On the Ratat-Goormaghtigh equation and integer points close to the graph of a smooth function
We prove that the sum of reciprocals $1/x$ of integer solutions of $(x^m-1)/(x-1)=N$ with $x, m\geq 2$ for a given integer $N$ except the smallest $x$ is smaller than $5.9037$. If we limit $x$ to be prime, then the sum is smaller than $0.73194$.

Divergent Infinite Series -- Ramanujan's Initial Intuition
Mario M. Attard
https://arxiv.org/abs/2510.08652 https://arxiv.org/pdf/2510.08652

Divergent Infinite Series -- Ramanujan's Initial Intuition
This paper investigates Srinivasa Ramanujan's initial intuitive methodology for assigning the finite value -1/12 to the sum of the divergent infinite series of all positive integers. We systematically examine Ramanujan's initial method, originally sketched in his notebooks, and set the methodology into an algebraic framework. The methodology has limited applicability to other classes of divergent series. The methodology is extended to assign a Ramanujan smoothed sum to the infinite sequences of…

Resonance sum rules: an application to the square well potential
Zi-Xi Ou-Yang, Philipp Gubler, Makoto Oka, Guang-Juan Wang, Jia-Jun Wu
https://arxiv.org/abs/2509.24336 https://…

Resonance sum rules: an application to the square well potential
We propose an extension of the Quantum Chromodynamics (QCD) sum rules, termed the Resonance sum rules (RSR), to access resonance poles in the complex energy plane. By strategically introducing a contour in the complex plane and conformal mapping, the method intends to reach resonance poles on the second Riemann sheet. To validate this approach, we apply RSR to the square-well potential model, for which the pole locations are known. The analysis demonstrates a successful extraction of the pole p…

Special restricted partition functions for the stable sheaf cohomology on flag varieties
Mircea Cimpoeas
https://arxiv.org/abs/2509.09418 https://arxiv.org…

Special restricted partition functions for the stable sheaf cohomology on flag varieties
Let $\mathbf a:=(a_1,\ldots,a_r)$ be a sequence of positive integers, $d\geq 2$ and $j\geq 1$, some integers. We study the functions $p_{\mathbf a,d}(n):=$ the number of integer solutions $(x_1,\dots,x_r)$ of $\sum_{i=1}^r a_ix_i=n$, with $x_i\geq 0$ and $x_i \equiv 0,1(\bmod\;d)$, for all $1\leq i\leq r$, and $p_{\mathbf a,d}(n;j):=$ the number of $(x_1,\ldots,x_r)$ as above which satisfy also the condition $\sum_{i=1}^r \left(x_i-(d-2)\left\lfloor \frac{x_i}{d} \right\rfloor\right) =j$. We …

Triple convolution sums of the generalised divisor functions
Bikram Misra, Biswajyoti Saha
https://arxiv.org/abs/2509.04610 https://arxiv.org/pdf/2509.0461…

Triple convolution sums of the generalised divisor functions
We study the triple convolution sum of the generalised divisor functions given by $$\sum_{n\leq x} d_k(n+h)d_l(n)d_m(n-h),$$ where $h \le x^{1-ε}$ for any $ε>0$ and $d_k(n)$ denotes the generalised divisor function which counts the number of ways $n$ can be written as product of $k$ many positive integers. The purpose of this paper is two-fold. Firstly, we note a predicted asymptotic estimate for the above sum, where the constant appearing in the estimate can be obtained from the theory of Di…

On the number of triangles in $K_4$-free graphs
Jialin He, Jie Ma, Yan Wang, Chunlei Zu
https://arxiv.org/abs/2509.12100 https://arxiv.org/pdf/2509.12100…

On the number of triangles in $K_4$-free graphs
Erdős asked whether for any $n$-vertex graph $G$, the parameter $p^*(G)=\min \sum_{i\ge 1} (|V(G_i)|-1)$ is at most $\lfloor n^2/4\rfloor$, where the minimum is taken over all edge decompositions of $G$ into edge-disjoint cliques $G_i$. In a restricted case (also conjectured independently by Erdős), Győri and Keszegh [Combinatorica, 37(6) (2017), 1113--1124] proved that $p^*(G)\leq \lfloor n^2/4\rfloor$ for all $K_4$-free graphs $G$. Motivated by their proof approach, they conjectu…

A degree sum condition for the existence of a quasi 5-contractible edge in a quasi 5-connected graph
Shuai Kou, Chengfu Qin, Weihua Yang, Mingzu Zhang, Shuang Zhao
https://arxiv.org/abs/2509.25809

A degree sum condition for the existence of a quasi 5-contractible edge in a quasi 5-connected graph
An edge of a quasi $k$-connected graph is said to be quasi $k$-contractible if the contraction of the edge results in a quasi $k$-connected graph. We show that every 5-connected graph contains a quasi 5-contractible edge. Furthermore, we prove that a quasi 5-connected graph possesses a quasi 5-contractible edge, if the degree sum of any two vertices with distance at most two is at least 9. This result strengthens a theorem proved by Kriesell when $k=4$ (M. Kriesell, A degree sum condition for t…

Burgess-type character sum estimates over generalized arithmetic progressions of rank $2$
Ali Alsetri, Xuancheng Shao
https://arxiv.org/abs/2509.07765 https://

Burgess-type character sum estimates over generalized arithmetic progressions of rank $2$
We extend the classical Burgess estimates to character sums over proper generalized arithmetic progressions (GAPs) of rank $2$ in prime fields $\mathbb{F}_p$. The core of our proof is a sharp upper bound for the multiplicative energy of these sets, established by adapting an argument of Konyagin and leveraging tools from the geometry of numbers. A key step in our argument involves establishing new upper bounds for the sizes of Bohr sets, which may be of independent interest.

Restricted Jacobi permutations
Alyssa G. Henke, Kyle R. Hoffman, Derek H. Stephens, Yongwei Yuan, Yan Zhuang
https://arxiv.org/abs/2509.11494 https://arxiv…

Restricted Jacobi permutations
Jacobi permutations, introduced by Viennot in the context of Jacobi elliptic functions, are counted by the Euler numbers $E_{n}$ appearing in the series expansion $\sec x+\tan x=\sum_{n=0}^{\infty}E_{n}x^{n}/n!$. We conduct a systematic study of pattern avoidance in Jacobi permutations, achieving a complete enumeration of Jacobi permutations avoiding a prescribed set of length 3 patterns. In the case of a single pattern restriction, we obtain refined enumerations with respect to several permuta…

On the $p$-adic transcendence of $\sum_{k=1}^\infty p^{-1/p^k}$
Shanwen Wang, Yijun Yuan
https://arxiv.org/abs/2509.24609 https://arxiv.org/pdf/2509.24609

On the $p$-adic transcendence of $\sum_{k=1}^\infty p^{-1/p^k}$
Let $p$ be a prime number. In this article, we prove that the $p$-adic Hahn series $\sum_{k=1}^\infty p^{-1/p^k}$, which is the mixed-characteristic analogue of Abhyankar's solution $\sum_{k=1}^\infty t^{-1/p^k}$ to the Artin-Schreier equation $X^p-X-t^{-1}=0$ over $\mathbf{F}_p\left(\!\left(t\right)\!\right)$, is a $p$-adic complex number, but not a $p$-adic algebraic number. Based on this result, we formulate a conjecture about the possible order type of the support of an algebraic $p$-adic H…

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,…

On the verification of a Nicolas inequality
Orlando Galdames-Bravo
https://arxiv.org/abs/2509.11182 https://arxiv.org/pdf/2509.11182

On the verification of a Nicolas inequality
Nicolas inequality we deal can be written as \begin{equation}\label{Nicineq} e^γ\log\log N_x < \dfrac{N_x}{φ(N_x)}\,, \end{equation} where $x\ge 2$, $N_x$ denotes the product of the primes less or equal than $x$, $γ$ is the Euler constant and $φ$ is the Euler totient function. We show that there is a large $x_0>0$ such that this inequality fails infinitely often for integers $x\ge x_0$. To this aim we analyze the sign of the Big-o function in the Mertens estimate for the sum of reciprocals …

On the Squared Distance Matrix of a Starlike Block Graph
Joyentanuj Das, Sumit Mohanty
https://arxiv.org/abs/2509.10854 https://arxiv.org/pdf/2509.10854

On the Squared Distance Matrix of a Starlike Block Graph
Let $D(G)$ be the distance matrix of a simple connected graph $G$. The Hadamard product $D(G)~\circ~ D(G)$ is called the squared distance matrix of $G$, and is denoted by $Δ(G)$. A simple connected graph is called a starlike block graph if it has a central cut vertex, and each of its blocks is a complete graph. Let $ \mathcal{S}(n_1, n_2, \ldots, n_b)$ be the starlike block graph with blocks $K_{n_1+1}, K_{n_2+1}, \ldots, K_{n_b+1} $ on $n=1 + \sum_{i=1}^b n_i$ vertices. In this article, we co…