Tootfinder

On the sum of a prime and a square-free number with divisibility conditions
Ethan S. Lee, Rowan O'Clarey
https://arxiv.org/abs/2506.11814 https://

On the sum of a prime and a square-free number with divisibility conditions
We prove that every natural number greater than $2$ can be written as the sum of a prime and a square-free number that is co-prime to any prime $q > 3$. We also demonstrate that this result can be extended in a natural way, by assuming the Generalised Riemann Hypothesis is true.

🤼 To understand America today, study the zero-sum mindset, writes Stefanie Stantcheva
Young people and city-dwellers are among those most likely to see one group’s gain as another’s loss
(great insight into political psychology!)
https://www.

To understand America today, study the zero-sum mindset, writes Stefanie Stantcheva
Young people and city-dwellers are among those most likely to see one group’s gain as another’s loss

Linear-quadratic stochastic nonzero-sum differential games between graphon teams
De-xuan Xu, Zhun Gou, Nan-jing Huang
https://arxiv.org/abs/2506.11468 http…

Linear-quadratic stochastic nonzero-sum differential games between graphon teams
We study a class of nonzero-sum stochastic differential games between two teams with agents in each team interacting through graphon aggregates. On the one hand, in each large population group, agents act together to optimize a common social cost function. On the other hand, these two groups compete with each other, forming a Nash game between two graphon teams. We note that the original problem can be equivalently formulated as an infinite-dimensional two-agent Nash game. Applying the dynamic …

A Parallelizable Approach for Characterizing NE in Zero-Sum Games After a Linear Number of Iterations of Gradient Descent
Taemin Kim, James P. Bailey
https://arxiv.org/abs/2507.11366

A Parallelizable Approach for Characterizing NE in Zero-Sum Games After a Linear Number of Iterations of Gradient Descent
We study online optimization methods for zero-sum games, a fundamental problem in adversarial learning in machine learning, economics, and many other domains. Traditional methods approximate Nash equilibria (NE) using either regret-based methods (time-average convergence) or contraction-map-based methods (last-iterate convergence). We propose a new method based on Hamiltonian dynamics in physics and prove that it can characterize the set of NE in a finite (linear) number of iterations of altern…

Lattice paths and the Geode
Ira M. Gessel
https://arxiv.org/abs/2507.09405 https://arxiv.org/pdf/2507.09405

Lattice paths and the Geode
Let $t_1,t_2,\dots$ be variables, and let $S$ be the formal power series in the variables $t_1, t_2,\dots$ satisfying $S=1+\sum_{i=1}^\infty t_n S^n.$ Let $S_1 =\sum_{n=1}^\infty t_n$. Wildberger and Rubine recently showed that there is a formal power series $G$ in the $t_i$, which they called the Geode, satisfying $S=1+GS_1$. In this paper we discuss some of the properties of the Geode and of the related series $H=G/S$, which satisfies $S=1/(1-HS_1)$. We show that \begin{equation*} G=\biggl(1-…

Let me explain what I did today… no, there is too much, let me sum up:

ADHO (adho.org) updates at the opening of #DH2025 from Diane and Michael - the Italian Digital Humanities association officially joins ADHO, and announces awards including the Zampoli prize to Stylo software, the conference bursaries winners and Fortier prize nominees. Also note the Code of Conduct!

If I had to sum up my expectations for what is and isn't #news it would have to be:
I don't wanna hear about what the rich are doin'
I don't wanna go to where, where the rich are goin'

Witnessing PPT entanglement via rank analysis of (sub)matrices
Aabhas Gulati
https://arxiv.org/abs/2506.11346 https://arxiv.org/pdf/2…

Witnessing PPT entanglement via rank analysis of (sub)matrices
We develop a new method for entanglement detection in bipartite quantum states, using the violation of the rank-1 generated property of matrices. The positive-semidefinite matrices form a convex cone that has extremal elements of only rank-1. But convex conic subsets resulting from presence of linear constraints allow extremal elements of rank > 1. The problem of deciding when a matrix is rank-1 generated, i.e a sum of rank-1 PSD matrices, has been studied extensively in optimization theory. Th…

ILCL: Inverse Logic-Constraint Learning from Temporally Constrained Demonstrations
Minwoo Cho, Jaehwi Jang, Daehyung Park
https://arxiv.org/abs/2507.11000 …

ILCL: Inverse Logic-Constraint Learning from Temporally Constrained Demonstrations
We aim to solve the problem of temporal-constraint learning from demonstrations to reproduce demonstration-like logic-constrained behaviors. Learning logic constraints is challenging due to the combinatorially large space of possible specifications and the ill-posed nature of non-Markovian constraints. To figure it out, we introduce a novel temporal-constraint learning method, which we call inverse logic-constraint learning (ILCL). Our method frames ICL as a two-player zero-sum game between 1) …

Convolutive sequences, I: Through the lens of integer partition functions
Shane Chern, Dennis Eichhorn, Shishuo Fu, James A. Sellers
https://arxiv.org/abs/2507.10965

Convolutive sequences, I: Through the lens of integer partition functions
Motivated by the convolutive behavior of the counting function for partitions with designated summands in which all parts are odd, we consider coefficient sequences $(a_n)_{n\ge 0}$ of primitive eta-products that satisfy the generic convolutive property \begin{align*} \sum_{n\ge 0} a_{mn} q^n = \left(\sum_{n\ge 0} a_n q^n\right)^m \end{align*} for a specific positive integer $m$. Given the results of an exhaustive search of the Online Encyclopedia of Integer Sequences for such sequences…

Sum Rate Maximization for Pinching Antennas Assisted RSMA System With Multiple Waveguides
Peiyu Wang, Hong Wang, Rongfang Song
https://arxiv.org/abs/2506.10596

Sum Rate Maximization for Pinching Antennas Assisted RSMA System With Multiple Waveguides
In this letter, a pinching antennas (PAs) assisted rate splitting multiple access (RSMA) system with multiple waveguides is investigated to maximize sum rate. A two-step algorithm is proposed to determine PA activation scheme and optimize the waveguide beamforming. Specifically, a low complexity spatial correlation and distance based method is proposed for PA activation selection. After determining the PA activation status, a semi-definite programming (SDP) based successive convex approximation…

On the Tschirnhausen module of coverings of curves on decomposable ruled surfaces and applications
Youngook Choi, Hristo Iliev, Seonja Kim
https://arxiv.org/abs/2507.11304

On the Tschirnhausen module of coverings of curves on decomposable ruled surfaces and applications
We show that for two classes of $m$-secant curves $X \subset S$, with $m \geq 2$, where $f : S = \mathbb{P} (\mathcal{O}_Y \oplus \mathcal{O}_Y (E)) \to Y$ and $E$ is a non-special divisor on a smooth curve $Y$, the Tschirnhausen module $\mathcal{E}^{\vee}$ of the covering $φ= f_{|_X} : X \to Y$ decomposes completely as a direct sum of line bundles. Specifically, we prove that: for $X \in |\mathcal{O}_S (mH)|$, where $H$ denotes the tautological divisor on $S$, one has $ \mathca…

Linear representations of the mapping class group of dimension at most $3g-3$
Julian Kaufmann, Nick Salter, Zhong Zhang, Xiyan Zhong
https://arxiv.org/abs/2507.11365

Linear representations of the mapping class group of dimension at most $3g-3$
We classify representations of the mapping class group of a surface of genus $g$ (with at most one puncture or boundary component) up to dimension $3g-3$. Any such representation is the direct sum of a representation in dimension $2g$ or $2g+1$ (given as the action on the (co)homology of the surface or its unit tangent bundle) with a trivial representation. As a corollary, any linear system on the moduli space of Riemann surfaces of genus $g$ in this range is of algebro-geometric origin.

Bicriteria Polygon Aggregation with Arbitrary Shapes
Lotte Blank, David Eppstein, Jan-Henrik Haunert, Herman Haverkort, Benedikt Kolbe, Philip Mayer, Petra Mutzel, Alexander Naumann, Jonas Sauer
https://arxiv.org/abs/2507.11212

Bicriteria Polygon Aggregation with Arbitrary Shapes
We study the problem of aggregating polygons by covering them with disjoint representative regions, thereby inducing a clustering of the polygons. Our objective is to minimize a weighted sum of the total area and the total perimeter of the regions. This problem has applications in cartographic map generalization and urban analytics. Here, the polygons represent building footprints and the clusters may represent urban areas. Previous approaches forced the boundaries of the regions to come from a…

Sharp Stability of Global Compactness on the Heisenberg Group
Hua Chen, Yun-lu Fan, Xin Liao
https://arxiv.org/abs/2506.11404 https://

Sharp Stability of Global Compactness on the Heisenberg Group
In this paper, we establish a sharp stability inequality on the Heisenberg group for functions that are close to the sum of m weakly interacting Jerison-Lee bubbles. As a consequence, we obtain a sharp quantitative stability of global compactness for non-negative functions on Heisenberg group.

OffsetCrust: Variable-Radius Offset Approximation with Power Diagrams
Zihan Zhao, Pengfei Wang, Minfeng Xu, Shuangmin Chen, Shiqing Xin, Changhe Tu, Wenping Wang
https://arxiv.org/abs/2507.10924

OffsetCrust: Variable-Radius Offset Approximation with Power Diagrams
Offset surfaces, defined as the Minkowski sum of a base surface and a rolling ball, play a crucial role in geometry processing, with applications ranging from coverage motion planning to brush modeling. While considerable progress has been made in computing constant-radius offset surfaces, computing variable-radius offset surfaces remains a challenging problem. In this paper, we present OffsetCrust, a novel framework that efficiently addresses the variable-radius offsetting problem by computing…

Semi-empirical Pseudopotential Method for Monolayer Transitional-Metal Dichalcogenides
Raj Kumar Paudel, Chung-Yuan Ren, Yia-Chung Chang
https://arxiv.org/abs/2506.11360

Semi-empirical Pseudopotential Method for Monolayer Transitional-Metal Dichalcogenides
We present a Semiempirical Pseudopotential Method for accurately computing the band structures and Bloch states of monolayer transition-metal dichalcogenides (TMDCs), including MoS2, MoSe2, WS2, and WSe2. Our approach combines local and non-local pseudopotentials, carefully fitted to replicate fully self-consistent density-functional theory results, while using only a minimal set of empirical parameters. By expressing the total potential as a sum of a few separable components, we achieve both a…

Solving Distance-Based Optimization Problems Using Optical Hardware
Guangyao Li, Richard Zhipeng Wang, Natalia G. Berloff
https://arxiv.org/abs/2507.11378 …

Solving Distance-Based Optimization Problems Using Optical Hardware
We present a practical approach to solving distance-based optimization problems using optical computing hardware. The objective is to minimize an energy function defined as the weighted sum of squared differences between measured distances and the squared Euclidean distances of point coordinates. By representing coordinates as complex numbers, we map the optimization problem onto optical fields, enabling its solution through either the canonical transformation (CT) method or the gain-based bifu…

Probability-One Optimization of Generalized Rayleigh Quotient Sum For Multi-Source Generalized Total Least-Squares
Dominik Friml, Pavel V\'aclavek
https://arxiv.org/abs/2506.09573

Probability-One Optimization of Generalized Rayleigh Quotient Sum For Multi-Source Generalized Total Least-Squares
This paper addresses the global optimization of the sum of the Rayleigh quotient and the generalized Rayleigh quotient on the unit sphere. While various methods have been proposed for this problem, they do not guarantee convergence to the global maximizer. To overcome this limitation, we introduce a probability-one homotopy optimization method that, under certain conditions, guarantees convergence to the global maximizer. The proposed method is analyzed alongside state-of-the-art approaches thr…

Samsung agrees to acquire Seattle-based Xealth, which helps healthcare providers manage digital health tools and has raised over $50M, for an undisclosed sum (Taylor Soper/GeekWire)
https://www.geekwire.com/2025/samsung-acquiring…

CDC urges summer camps to check for measles immunity, as U.S. nears record (Alexander Tin/CBS News)
https://www.cbsnews.com/news/cdc-urges-summer-camps-to-check-for-measles-immunity-as-u-s-nears-record/
http://www.memeorandum.com/250613/p135#a250613p135

Pricing energy spread options with variance gamma-driven Ornstein-Uhlenbeck dynamics
Tim Leung, Kevin Lu
https://arxiv.org/abs/2507.11480 https://

Pricing energy spread options with variance gamma-driven Ornstein-Uhlenbeck dynamics
We consider the pricing of energy spread options for spot prices following an exponential Ornstein-Uhlenbeck process driven by a sum of independent multivariate variance gamma processes. Within this class of mean-reverting, infinite activity price processes, the Esscher transform is used to obtain an equivalent martingale measure. We focus on the weak variance alpha-gamma process and show that it is not closed under the Esscher transform. By deriving an analytic expression for the cumulant gene…

Way More Than the Sum of Their Parts: From Statistical to Structural Mixtures
James P. Crutchfield
https://arxiv.org/abs/2507.07343 https://

Way More Than the Sum of Their Parts: From Statistical to Structural Mixtures
We show that mixtures comprised of multicomponent systems typically are much more structurally complex than the sum of their parts; sometimes, infinitely more complex. We contrast this with the more familiar notion of statistical mixtures, demonstrating how statistical mixtures miss key aspects of emergent hierarchical organization. This leads us to identify a new kind of structural complexity inherent in multicomponent systems and to draw out broad consequences for system ergodicity.

Invariants for sum-rank metric codes
Paolo Santonastaso, Ferdinando Zullo
https://arxiv.org/abs/2507.04158 https://arxiv.org/pdf/2507…

Invariants for sum-rank metric codes
The code equivalence problem is central in coding theory and cryptography. While classical invariants are effective for Hamming and rank metrics, the sum-rank metric, which unifies both, introduces new challenges. This paper introduces new invariants for sum-rank metric codes: generalised idealisers, the centraliser, the center, and a refined notion of linearity. These lead to the definition of nuclear parameters, inspired by those used in division algebra theory, where they are crucial for pro…

On the correlations between character sums of division polynomials under shifts
Subham Bhakta, Igor E. Shparlinski
https://arxiv.org/abs/2507.09271 https:/…

On the correlations between character sums of division polynomials under shifts
Let $E$ be an elliptic curve over the finite field $\mathbb{F}_p$, and $P \in E(\mathbb{F}_p)$ be an $\mathbb{F}_p$-rational point. We study the sums \[ S_{χ,P}(N,h) = \sum_{n=1}^N χ(ψ_n(P)) χ(ψ_{n+h}(P)), \] where $ψ_n(P)$ denotes the $n$-th division polynomial evaluated at $P$, and $χ$ is a multiplicative character of $\mathbb{F}_p^{*}$. We estimate $S_{χ,P}(N,h)$ on average over $h$ over a rather short interval $h \in [1, H]$. We also obtain a multidimensional generalisation of this …

Nash equilibria in semidefinite games and Lemke-Howson paths
Constantin Ickstadt, Thorsten Theobald, Elias Tsigaridas, Antonios Varvitsiotis
https://arxiv.org/abs/2506.11940

Nash equilibria in semidefinite games and Lemke-Howson paths
We consider an algorithmic framework for two-player non-zero-sum semidefinite games, where each player's strategy is a positive semidefinite matrix with trace one. We formulate the computation of Nash equilibria in such games as semidefinite complementarity problems and develop symbolic-numeric techniques to trace generalized Lemke-Howson paths. These paths generalize the piecewise affine-linear trajectories of the classical Lemke-Howson algorithm for bimatrix games, replacing them with nonline…

In 73/74 my first job was in a car accessory shop and paint factors, for the magnificent sum of £10.50 a week.
One of its few attractions was that it also sold 8 track music and for weeks I'd play "Holland" by The Beach Boys.
It may not be the strongest album but it kept me warm.
#Music

This https://arxiv.org/abs/2503.13527 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_hepp…

Isospin sum rules for bottom-baryon weak decays
Isospin symmetry, as the most precise flavor symmetry, can be used to extract information about hadronic dynamics. The effective Hamiltonian operators of bottom quark weak decays are zero under a series of isospin lowering operators $I_-^n$, which permits us to generate isospin sum rules without the Wigner-Eckart invariants. In this work, we derive hundreds of isospin sum rules for the two- and three-body non-leptonic decays of bottom baryons. They provide hints for new decay modes and the isos…

Energy Efficiency Optimization of Finite Block Length STAR-RIS-aided MU-MIMO Broadcast Channels
Mohammad Soleymani, Ignacio Santamaria, Eduard Jorswieck, Robert Schober, Lajos Hanzo
https://arxiv.org/abs/2506.11594

Energy Efficiency Optimization of Finite Block Length STAR-RIS-aided MU-MIMO Broadcast Channels
Energy-efficient designs are proposed for multi-user (MU) multiple-input multiple-output (MIMO) broadcast channels (BC), assisted by simultaneously transmitting and reflecting (STAR) reconfigurable intelligent surfaces (RIS) operating at finite block length (FBL). In particular, we maximize the sum energy efficiency (EE), showing that STAR-RIS can substantially enhance it. Our findings demonstrate that the gains of employing STAR-RIS increase when the codeword length and the maximum tolerable b…

This https://arxiv.org/abs/2505.17282 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_csLG_…

Attention with Trained Embeddings Provably Selects Important Tokens
Token embeddings play a crucial role in language modeling but, despite this practical relevance, their theoretical understanding remains limited. Our paper addresses the gap by characterizing the structure of embeddings obtained via gradient descent. Specifically, we consider a one-layer softmax attention model with a linear head for binary classification, i.e., $\texttt{Softmax}( p^\top E_X^\top ) E_X v = \frac{ \sum_{i=1}^T \exp(p^\top E_{x_i}) E_{x_i}^\top v}{\sum_{j=1}^T \exp(p^\top E_{x_{j…

Everything’s a pound
https://www.bbc.co.uk/news/articles/c36594lr29ko

Poundland: Struggling bargain chain sold for £1
The firm with 825 UK stores was sold for a "nominal" sum to a US investment firm.

A note on instantaneous gelation for coagulation kernels vanishing on the diagonal
Iulia Cristian (LJLL), Barbara Niethammer, Juan J. L. Vel\'azquez
https://arxiv.org/abs/2506.11573

A note on instantaneous gelation for coagulation kernels vanishing on the diagonal
We prove that instantaneous gelation (i.e., instantaneous loss of mass) occurs for coagulation equations with sum-type kernels of homogeneity greater than one which vanish on the diagonal. Our proof includes solutions that are Radon measures if we exclude the case of initial data that are a single Dirac delta.

Self-reverse labelings of distance magic graphs
Petr Kov\'a\v{r}, Ksenija Rozman, Primo\v{z} \v{S}parl
https://arxiv.org/abs/2507.11226 https://…

Self-reverse labelings of distance magic graphs
A graph is distance magic if it admits a bijective labeling of its vertices by integers from $1$ up to the order of the graph in such a way that the sum of the labels of all the neighbors of a vertex is independent of a given vertex. We introduce the concept of a self-reverse distance magic labeling of a regular graph which allows for a more compact description of the graph and the labeling in terms of the corresponding quotient graph. We show that the members of several known infinite families…

Movable-Antenna Array Enhanced Downlink NOMA
Nianzu Li, Peiran Wu, Lipeng Zhu, Derrick Wing Kwan Ng
https://arxiv.org/abs/2506.11438 https://

Movable-Antenna Array Enhanced Downlink NOMA
Movable antenna (MA) has gained increasing attention in the field of wireless communications due to its exceptional capability to proactively reconfigure wireless channels via localized antenna movements. In this paper, we investigate the resource allocation design for an MA array-enabled base station serving multiple single-antenna users in a downlink non-orthogonal multiple access (NOMA) system. We aim to maximize the sum rate of all users by jointly optimizing the transmit beamforming and th…

Bayesian variable selection in a Cox proportional hazards model with the "Sum of Single Effects" prior
Yunqi Yang, Karl Tayeb, Peter Carbonetto, Xiaoyuan Zhong, Carole Ober, Matthew Stephens
https://arxiv.org/abs/2506.06233

Bayesian variable selection in a Cox proportional hazards model with the "Sum of Single Effects" prior
Motivated by genetic fine-mapping applications, we introduce a new approach to Bayesian variable selection regression (BVSR) for time-to-event (TTE) outcomes. This new approach is designed to deal with the specific challenges that arise in genetic fine-mapping, including: the presence of very strong correlations among the covariates, often exceeding 0.99; very large data sets containing potentially thousands of covariates and hundreds of thousands of samples. We accomplish this by extending the…

Counting sums of two powers
Anand Patel
https://arxiv.org/abs/2507.08337 https://arxiv.org/pdf/2507.08337

Counting sums of two powers
We generalize two well-known enumerative facts. The first, due to Clebsch, says that a general binary sextic form is expressible as the sum of a cube and a square in 40 different ways. The second, due to Zariski and later Vakil, states that a general pencil of binary 12-ics contains exactly 3762 cubes plus squares, ignoring scaling.

On Polynomial Stochastic Barrier Functions: Bernstein Versus Sum-of-Squares
Peter Amorese, Morteza Lahijanian
https://arxiv.org/abs/2506.09164 https://

On Polynomial Stochastic Barrier Functions: Bernstein Versus Sum-of-Squares
Stochastic Barrier Functions (SBFs) certify the safety of stochastic systems by formulating a functional optimization problem, which state-of-the-art methods solve using Sum-of-Squares (SoS) polynomials. This work focuses on polynomial SBFs and introduces a new formulation based on Bernstein polynomials and provides a comparative analysis of its theoretical and empirical performance against SoS methods. We show that the Bernstein formulation leads to a linear program (LP), in contrast to the se…

Generalization of Ramanujan's formula for the sum of half-integer powers of consecutive integers via formal Bernoulli series
Max A. Alekseyev, Rafael Gonzalez, Keryn Loor, Aviad Susman, Cesar Valverde
https://arxiv.org/abs/2506.05961

Generalization of Ramanujan's formula for the sum of half-integer powers of consecutive integers via formal Bernoulli series
Faulhaber's formula expresses the sum of the first $n$ positive integers, each raised to an integer power $p\geq 0$ as a polynomial in $n$ of degree $p+1$. Ramanujan expressed this sum for $p\in\{\frac12,\frac32,\frac52,\frac72\}$ as the sum of a polynomial in $\sqrt{n}$ and a certain infinite series. In the present work, we explore the connection to Bernoulli polynomials, and by generalizing those to formal series, we extend the Ramanujan result to all positive half-integers $p$.

Spanning subgraphs and spectral radius in graphs
Sizhong Zhou
https://arxiv.org/abs/2507.11078 https://arxiv.org/pdf/2507.11078

Spanning subgraphs and spectral radius in graphs
A spanning tree $T$ of a connected graph $G$ is a subgraph of $G$ that is a tree covers all vertices of $G$. The leaf distance of $T$ is defined as the minimum of distances between any two leaves of $T$. A fractional matching of a graph $G$ is a function $h$ assigning every edge a real number in $[0,1]$ so that $\sum\limits_{e\in E_G(v)}{h(e)}\leq1$ for any $v\in V(G)$, where $E_G(v)$ denotes the set of edges incident with $v$ in $G$. A fractional matching of $G$ is called a fractional perfect …

Bootstrapping Gravity with Crossing Symmetric Dispersion Relations
Celina Pasiecznik
https://arxiv.org/abs/2506.09884 https://arxiv.o…

Bootstrapping Gravity with Crossing Symmetric Dispersion Relations
We derive bounds on Wilson coefficients in gravitational effective field theories using fully crossing symmetric dispersion relations. These sum rules naturally isolate finite subsets of low-energy couplings without relying on the forward limit or specific high-energy completions. We validate our method by matching bounds computed previously for scalar scattering with gravity as well as for supergraviton scattering. For graviton scattering we use crossing symmetric functions that combine variou…

Nonrelativistic limit of ground states to $L^2$-supercritical nonlinear Dirac equations
Pan Chen, Yanheng Ding, Qi Guo
https://arxiv.org/abs/2507.11203 htt…

Nonrelativistic limit of ground states to $L^2$-supercritical nonlinear Dirac equations
In this paper, we study the existence and nonrelativistic limit of normalized ground states for the following nonlinear Dirac equation with power-type potentials \begin{equation*} \begin{cases} &-i c\sum\limits_{k=1}^3α_k\partial_k u +mc^2 β{u}- |{u}|^{p-2}{u}=ω{u}, \\ &\displaystyle\int_{\mathbb{R}^3}\vert u \vert^2 dx =1. \end{cases} \end{equation*} We demonstrate the existence of ground states for large $c$ and establish the uniqueness of the associated Lagrange multiplier f…

This https://arxiv.org/abs/2307.08780 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_csLO_…

Discounted-Sum Automata with Multiple Discount Factors
Discounting the influence of future events is a key paradigm in economics and it is widely used in computer-science models, such as games, Markov decision processes (MDPs), reinforcement learning, and automata. While a single game or MDP may allow for several different discount factors, nondeterministic discounted-sum automata (NDAs) were only studied with respect to a single discount factor. It is known that every class of NDAs with an integer as the discount factor has good computational prop…

Best approximation of a three-variable function by sum of one-variable coordinate functions
Rashid A. Aliev, Vugar A. Guliyev, Amil F. Jabiyev
https://arxiv.org/abs/2507.04747

Best approximation of a three-variable function by sum of one-variable coordinate functions
Consider the following approximation problem of a continuous function of three variables by the sum of three continuous functions of one variable: \[ E(f,Ω)=\inf||f(x,y,z)-ϕ(x)-ψ(y)-ω(z)||_{\infty}=? \] where $f(x,y,z)$ is a given continuous function defined on $Ω=[0,1]^3$ and the infimum runs over all triplets of continuous functions $ϕ(x),ψ(y),ω(z)$ defined on the unit interval $[0,1]$. In this paper, we will prove a formula to calculate the error $E(f,Ω)$ und…

An Interpretable Two-Stage Feature Decomposition Method for Deep Learning-based SAR ATR
Chenwei Wang, Renjie Xu, Congwen Wu, Cunyi Yin, Ziyun Liao, Deqing Mao, Sitong Zhang, Hong Yan
https://arxiv.org/abs/2506.09377

An Interpretable Two-Stage Feature Decomposition Method for Deep Learning-based SAR ATR
Synthetic aperture radar automatic target recognition (SAR ATR) has seen significant performance improvements with deep learning. However, the black-box nature of deep SAR ATR introduces low confidence and high risks in decision-critical SAR applications, hindering practical deployment. To address this issue, deep SAR ATR should provide an interpretable reasoning basis $r_b$ and logic $λ_w$, forming the reasoning logic $\sum_{i} {{r_b^i} \times {λ_w^i}} =pred$ behind the decisions. Therefore,…

@… Hi Sean. I’ve been doing science all night, and I think I’ve uncovered a new layer to Autechre (the “tapestry behind the cotton wool”, as Virgina Woolf might put it.) Please bear with me as I explain:
A = 1
U = 21
T = 20
E = 5
C = 3
H = 8
R = 18
E = 5
− [1, 21, 20, 5, 3, 8, 18, 5]
Sum = 81
Your name, Se…

This https://arxiv.org/abs/2412.00040 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

A Short Proof of Knuth's Old Sum
We give a short proof of the well-known Knuth's old sum and provide some generalizations. Our approach utilizes the binomial theorem and integration formulas derived using the Beta function. Several new polynomial identities and combinatorial identities are derived.

NFL summer trade proposals: Alvin Kamara to the Broncos and other big-name deals for 2025

https://www.cbssports.com/nfl/news/nfl-summer-trade…

Sum Rate Maximization for Wireless Powered Pinching-Antenna Systems (PASS)
Yixuan Li, Ji Wang, Ming Zeng, Yuanwei Liu
https://arxiv.org/abs/2506.00355 http…

Sum Rate Maximization for Wireless Powered Pinching-Antenna Systems (PASS)
In this letter, we investigate a novel wireless powered communication network (WPCN) enabled by a pinching-antenna system (PASS), in which multiple pinching antennas (PAs) are activated on a waveguide to establish strong line-of-sight (LoS) links with multiple devices. In this system, time division multiple access (TDMA) and non-orthogonal multiple access (NOMA) protocols are adopted to fully explore the potential of the wireless information transmission. To maximize the sum rate of the PASS-WP…

This https://arxiv.org/abs/2505.19184 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_csCL_…

When Two LLMs Debate, Both Think They'll Win
Can LLMs accurately adjust their confidence when facing opposition? Building on previous studies measuring calibration on static fact-based question-answering tasks, we evaluate Large Language Models (LLMs) in a dynamic, adversarial debate setting, uniquely combining two realistic factors: (a) a multi-turn format requiring models to update beliefs as new information emerges, and (b) a zero-sum structure to control for task-related uncertainty, since mutual high-confidence claims imply systemati…

Renormalization of bosonic quadratic Hamiltonians involving rank-one perturbations
Thomas Gamet (UMPA-ENSL)
https://arxiv.org/abs/2506.08590 https://

Renormalization of bosonic quadratic Hamiltonians involving rank-one perturbations
In this article, we study the renormalization of a bosonic quadratic Hamiltonian with an ultraviolet divergence. The Hamiltonian is composed of the sum of a free energy and the square of the smeared field operator. We explicitly diagonalize the Hamiltonian via Bogoliubov transformations, and the diagonalized Hamiltonian is easier to study. We show a renormalization of both the energy and the coupling constant depending on the form factor of the interaction.

Whereas Leibniz’s task was to find the sum of infinitely many known terms, Hounsfield wondered:
Could the process be run in reverse?
With enough total dimmings, from enough directions, could we work backward to deduce the unknown absorption at each point along the many beams
— and use that to see inside the brain?
Most radiologists thought the idea was crazy. -- But one doctor was willing to listen.
He handed Hounsfield a jar containing a human brain with a t…

How a Puzzle About Fractions Got Brain Scans Rolling
A story of bowling pins, patterns and medical miracles.

Improving Online Bin Covering with Little Advice
Andrej Brodnik, Bengt J. Nilsson, Gordana Vujovi\'c
https://arxiv.org/abs/2506.09004 https://

Improving Online Bin Covering with Little Advice
The online bin covering problem is: given an input sequence of items find a placement of the items in the maximum number of bins such that the sum of the items' sizes in each bin is at least~1. Boyar~{\em et~al}.\@~\cite{boyar2021} present a strategy that with $O(\log \log n)$ bits of advice, where $n$ is the length of the input sequence, achieves a competitive ratio of $8/15\approx0.5333\ldots$. We show that with a strengthened analysis and some minor improvements, the same strategy achieves t…

Existence and Uniqueness Property On a Generalized Ledin-Brousseau Sum
Ivan Hadinata
https://arxiv.org/abs/2507.03813 https://arxiv.o…

Existence and Uniqueness Property On a Generalized Ledin-Brousseau Sum
In this paper, we present the existence and uniqueness property on a finite sum involving a polynomial and a homogeneous linear recurrence sequence. This finite sum is of the form $\sum_{k=1}^n P(k)s_{hk+r}$ where $n$ is a positive integer, $P(x)$ is a polynomial in $\mathbb C[x]$, $h$ and $r$ are some integers, and $(s_k)_{k\in\mathbb Z}$ is a homogeneous linear recurrence sequence of degree $m\geq 2$ with some constraints.

This https://arxiv.org/abs/2502.11351 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_hepp…

Review of the QCD sum rules for exotic states
We review the exotic states, such as the $X$, $Y$, $Z$, $T$ and $P$ states, and present their possible assignments based on the QCD sum rules. We present many predictions which can be confronted to the experimental data in the future to diagnose the exotic states. Furthermore, we also mention other theoretical methods.

GlobalFoundries agrees to acquire MIPS, a developer of RISC-V-based solutions and IP, with the deal expected to close in H2 2025 (Anton Shilov/Tom's Hardware)
https://www.tomshardware.com/pc-components…

GlobalFoundries to make RISC-V CPUs — fab acquires MIPS, will integrate RISC-V and AI IP into its portfolio
For an unknown sum.

A c-theorem for the effective central charge in the R=1 replica limit, and applications to systems with measurement-induced randomness
Rushikesh A. Patil, Andreas W. W. Ludwig
https://arxiv.org/abs/2507.07959

A c-theorem for the effective central charge in the R=1 replica limit, and applications to systems with measurement-induced randomness
We present a general theorem demonstrating non-perturbatively the decrease of the "effective central charge" $c_{\text{eff}}=(d c/dR)|_{R=1}$ under renormalization group (RG) flow in the $R\rightarrow1$ replica limit of a $R$-copy $2D$ conformal field theory (CFT) action $S_{*}$ perturbed by a replica interaction of the form $$-\mathbb{S}=-\sum_{a=1}^{R}S_{*}^{(a)}+Δ\int d^2 x \sum_{\substack{a,b=1\\ a\neq b}}^{R}φ^{(a)}(x)φ^{(b)}(x).$$ Here $φ$ is a scaling field belonging to the CFT …

Regular sets in Cayley sum graphs on generalized dicyclic groups
Meiqi Peng, Yuefeng Yang, Wenying Zhu
https://arxiv.org/abs/2507.07736 https://

Regular sets in Cayley sum graphs on generalized dicyclic groups
For a graph $Γ=(V(Γ),E(Γ))$, a subset $C$ of $V(Γ)$ is called an $(α,β)$-regular set in $Γ$, if every vertex of $C$ is adjacent to exactly $α$ vertices of $C$ and every vertex of $V(Γ)\setminus C$ is adjacent to exactly $β$ vertices of $C$. In particular, if $C$ is an $(α,β)$-regular set in some Cayley sum graph of a finite group $G$ with connection set $S$, then $C$ is called an $(α,β)$-regular set of $G$. In this paper, we consider a generalized dicyclic group $G$ and for each s…

This https://arxiv.org/abs/2405.08485 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Doubly relaxed forward-Douglas--Rachford splitting for the sum of two nonconvex and a DC function
In this paper, we consider a class of structured nonconvex nonsmooth optimization problems whose objective function is the sum of three nonconvex functions, one of which is expressed in a difference-of-convex (DC) form. This problem class covers several important structures in the literature including the sum of three functions and the general DC program. We propose a splitting algorithm and prove the subsequential convergence to a stationary point of the problem. The full sequential convergenc…

Explicit Constructions of Sum-Rank Metric Codes from Quadratic Kummer Extensions
Zhu Yunlong, Zhao Chang-An
https://arxiv.org/abs/2506.18653 https://

Explicit Constructions of Sum-Rank Metric Codes from Quadratic Kummer Extensions
This paper presents new constructions of sum-rank metric codes derived from algebraic function fields, while the existing results about such codes remain limited. A central challenge in this field involves the determination of parameters. We address this challenge through quadratic Kummer extensions, and propose two general constructions of $2\times2$ sum-rank codes. The novel sum-rank metric code constructions derived from algebraic function fields demonstrate superior code lengths compared to…

Large value estimates for Dirichlet polynomials, and the density of zeros of Dirichlet's $L$-functions
Bin Chen
https://arxiv.org/abs/2507.08296 https:…

Large value estimates for Dirichlet polynomials, and the density of zeros of Dirichlet's $L$-functions
It is proved that \[ \sum_{χ\bmod q}N(σ, T, χ) \lesssim_ε (qT)^{7(1-σ)/3+ε}, \] where $N(σ, T, χ)$ denote the number of zeros $ρ= β+ it$ of $L(s, χ)$ in the rectangle $σ\leq β\leq 1$, $|t| \leq T$. The exponent $7/3$ improves upon Huxley's earlier exponent of $12/5$. The key innovation lies in deriving a sharp upper bound for sums involving affine transformations with GCD twists, which emerges from our application of the Guth-Maynard method. As corollaries, we obtain two …

A characterization of compact operators on $\ell^p$-spaces
Mortaza Abtahi
https://arxiv.org/abs/2506.06726 https://arxiv.org/pdf/2506…

A characterization of compact operators on $\ell^p$-spaces
Let $A$ be a Banach space, $p>1$, and $1/p+1/q=1$. If a sequence $a=(a_i)$ in $A$ has a finite $p$-sum, then the operator $Λ_a:\ell^q\to A$, defined by $Λ_a(β)=\sum_{i=1}^\infty β_i a_i, β=(β_i)\in \ell^q$, is compact. We present a characterization of compact operators $Λ:\ell^q\to A$, and prove that $Λ$ is compact if and only if $Λ=Λ_a$, for some sequence $a=(a_i)$ in $A$ with $\{(ϕ(a_i)): ϕ\in A^*, \|ϕ\|\leq 1\}$ being a totally bounded set in $\ell^p$. For a sequence $…

Correlators from Amplitubes
Ross Glew
https://arxiv.org/abs/2507.07199 https://arxiv.org/pdf/2507.07199

Correlators from Amplitubes
Recently, the wavefunction coefficients for conformally coupled scalars in an FRW cosmology have been presented as a sum over amplitude-like functions known as {\it amplitubes}. In this work we extend this analysis to full {\it correlation functions}. Remarkably, the amplitube expansion of the correlator exhibits many vanishing contributions that are otherwise present in the expansion of the wavefunction. Moreover, while the wavefunction coefficients suffer from relative minus signs between ter…

QCD sum rule analysis of $J^{PC}=1^{--}$ light hybrid mesons
Shuang-Hong Li, Zhuo-Ran Huang, Wei Chen, Hong-Ying Jin
https://arxiv.org/abs/2506.22412 https…

QCD sum rule analysis of $J^{PC}=1^{--}$ light hybrid mesons
We performed a comprehensive next-to-leading order (NLO) QCD sum rule analysis of the light hybrid meson with quantum numbers $J^{PC}=1^{--}$. Utilizing both Laplace sum rules (LSR) and Gaussian sum rules (GSR), we investigate the two-point correlation function of the light hybrid interpolating current, including condensates up to dimension-8. A conservative range $2-2.5\text{GeV}$ is obtained. Notably, the GSR yield mass predictions $2.27\text{GeV}$ for the $s\bar{s}g$ and $2.19\text{GeV}$ for…

Finding a solution to the Erd\H{o}s-Ginzburg-Ziv theorem in $O(n\log\log\log n)$ time
Yui Hin Arvin Leung
https://arxiv.org/abs/2507.08139 https://

Finding a solution to the Erdős-Ginzburg-Ziv theorem in $O(n\log\log\log n)$ time
The Erdős-Ginzburg-Ziv theorem states that for any sequence of $2n-1$ integers, there exists a subsequence of $n$ elements whose sum is divisible by $n$. In this article, we provide a simple, practical $O(n\log\log n)$ algorithm and a theoretical $O(n\log\log\log n)$ algorithm, both of which improve upon the best previously known $O(n\log n)$ approach. This shows that a specific variant of boolean convolution can be implemented in time faster than the usual $O(n\log n)$ expected from FFT-based…

Spectral Efficiency Maximization for DMA-enabled Multiuser MISO with Statistical CSI
Hao Xu, Boyu Ning, Chongjun Ouyang, Hongwen Yang
https://arxiv.org/abs/2506.09570

Spectral Efficiency Maximization for DMA-enabled Multiuser MISO with Statistical CSI
Dynamic metasurface antennas (DMAs) offer the potential to achieve large-scale antenna arrays with low power consumption and reduced hardware costs, making them a promising technology for future communication systems. This paper investigates the spectral efficiency (SE) of DMA-enabled multiuser multiple-input single-output (MISO) systems in both uplink and downlink transmissions, using only statistical channel state information (CSI) to maximize the ergodic sum rate of multiple users. For the u…

Every real number is a sum of two real numbers with diverging partial quotients
Dmitry Gayfulin, Erez Nesharim
https://arxiv.org/abs/2507.04521 https://

Every real number is a sum of two real numbers with diverging partial quotients
We show that every irrational number is a sum of two real numbers with diverging partial quotients. The proof is constructive. The key towards these results is an algorithm which was recently developed by Nikita Shulga, and our study of this algorithm is of independent interest.

Topped baryons from QCD sum rules
Shu-Wei Zhang, Wei-Han Tan, Xuan Luo, Hua-Xing Chen
https://arxiv.org/abs/2507.05895 https://arxiv.…

Topped baryons from QCD sum rules
The recent CMS observation of a near-threshold enhancement in top quark pair production provides the first experimental indication of a short-lived pseudoscalar $t\bar{t}$ bound state, commonly referred to as toponium. While most existing studies focus on toponium using perturbative QCD near threshold, baryonic configurations containing a single top quark -- singly topped baryons -- offer a complementary nonperturbative perspective. Notably, singly topped baryons are expected to exhibit a longe…

ASMOP: Additional sampling stochastic trust region method for multi-objective problems
Nata\v{s}a Krklec Jerinki\'c, Luka Rute\v{s}i\'c
https://arxiv.org/abs/2506.10976

ASMOP: Additional sampling stochastic trust region method for multi-objective problems
We consider an unconstrained multi-criteria optimization problem with finite sum objective functions. The proposed algorithm belongs to a non-monotone trust-region framework where additional sampling approach is used to govern the sample size and the acceptance of a candidate point. Depending on the problem, the method can result in a mini-batch or an increasing sample size approach. Therefore, this work can be viewed as an extension of additional sampling trust region method for scalar finite …

A weighted divisor problem and exponential sum
Kritika Aggarwal, Debika Banerjee
https://arxiv.org/abs/2507.01891 https://arxiv.org/p…

A weighted divisor problem and exponential sum
In this paper, we investigate a weighted divisor problem involving the exponential sum of $D_{(1)}(n)$, the $n$th coefficient in the Dirichlet series expansion of $ζ'(s)^2$. We establish a truncated Voronoï type formula for the error term of $\sum_{n\leq x}D_{(1)}(n)e(nh/k)$, analogous to the results obtained by Jutila. Utilizing this truncated formula, we derive a mean square estimate of the error term. In addition, we study the associated Riesz sum and the corresponding error term, along wi…

Degree Sequence of Albertson and $\sigma$-Indices on Trees of Order $n\geqslant 3$
Jasem Hamoud, Artem Kornosov
https://arxiv.org/abs/2506.08213 https://…

Degree Sequence of Albertson and $σ$-Indices on Trees of Order $n\geqslant 3$
In this paper, we presented a study of topological indices on trees, where we show a relationship with irregularity of Albertson index and minimum, maximum degrees $δ,Δ$ of graph $G$, where contribute vital roles in determining connection, shading, component incorporation, and realisability where well-known Albertson index as: $\operatorname{irr}(G)=\sum_{uv\in E(G)}\lvert d_u(G)-d_v(G) \rvert$. The sigma index on trees that we introduced as $σ(T)=(d_1-1)^3+\sum_{i=1}^{3}(d_i-1)(d_i-2)+(d_3-…

Kurdyka-\L ojasiewicz exponent via sqaure parametrization
Wenqing Ouyang
https://arxiv.org/abs/2506.10110 https://arxiv.org/pdf/2506.…

Kurdyka-Łojasiewicz exponent via sqaure parametrization
We consider one of the most common reparameterization techniques, the square transformation. Assuming the original objective function is the sum of a smooth function and a polyhedral function, we study the variational properties of the objective function after reparameterization. In particular, we first study the minimal norm of the subdifferential of the reparameterized objective function. Second, we compute the second subderivative of the reparameterized objective function on a linear subspac…

This https://arxiv.org/abs/2410.10884 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

On evaluation of the lattice sum $\frac{1}{|x||y||x+y|}$
We study lattice sums $\sum 1/|x||y||x+y|$ and $\sum 1/|x|^2|y|^2|x+y|^2$ taken over the pairs $(x,y)$ of primitive lattice vectors in $\ZZ^2$ that span a parallelogram of area one. We prove convergence of these and similar sums, propose a new method to evaluate $\sum 1/|x|^2|y|^2|x+y|^2$, which, in turn, gives a new and simple proof of Zagier's formula for $D_{1,1,1}$.

The Neighbour Sum Problem on Trees
Sayan Dutta, Sohom Gupta
https://arxiv.org/abs/2506.23965 https://arxiv.org/pdf/2506.23965

The Neighbour Sum Problem on Trees
A graph $\mathcal G = (\mathcal V, \mathcal E)$ is said to satisfy the Neighbour Sum Property if there exists some $f:\mathcal V\to\mathbb R$ such that $f\not\equiv 0$ and it maps every vertex to the sum of the values taken by its neighbours. In this article, we provide an algorithm to check whether a given finite tree satisfies the neighbour sum property. We also find a large class of trees on $n$ vertices that satisfy the property.

QCD sum rule study on excited light meson operators
Wei-Han Tan, Wen-Ying Liu, Hong-Zhou Xi, Hua-Xing Chen
https://arxiv.org/abs/2505.24376 https://…

QCD sum rule study on excited light meson operators
We apply the QCD sum rule method to systematically study excited light meson operators and calculate their decay constants. These operators are constructed by explicitly adding one covariant derivative to the quark-antiquark pair. In total, twelve such operators are constructed, among which ten are subjected to detailed numerical analyses. The considered quark contents include $\bar{q}q$, $\bar{q}s$, and $\bar{s}s$ ($q = u/d$), allowing the formation of various $SU(3)$ flavor nonets. For instan…

Asymptotic properties of zeros of Riemann zeta function
Juan Arias de Reyna, Yves Meyer
https://arxiv.org/abs/2507.07253 https://arxi…

Asymptotic properties of zeros of Riemann zeta function
We try to define the sequence of zeros of the Riemann zeta function by an intrinsic property. Let $(z_k)_{k\in \mathbb{N}}$ be the sequence of nontrivial zeros of $ζ(s)$ with positive imaginary part. We write $z_k= 1/2+iτ_k$ (RH says that these $τ_k$ are all real). Then the sequence $(τ_k)_{k\in \mathbb{N}},$ satisfies the following asymptotic relation \[\sum_{k\in\mathbb{N}}\frac{2x}{x^2+τ_k^2}\simeq \frac12\log\frac{x}{2π}+\sum_{n=1}^\infty \frac{a_n}{x^n},\,\,x\to +\infty\] where $a_{2…

Refinement of a conjecture on positive square energy of graphs
Saieed Akbari, Hitesh Kumar, Bojan Mohar, Shivaramakrishna Pragada, Shengtong Zhang
https://arxiv.org/abs/2506.07264

Refinement of a conjecture on positive square energy of graphs
Let $G$ be a simple graph of order $n$ with eigenvalues $λ_1(G)\geq \cdots \geq λ_n(G)$. Define \[s^+(G)=\sum_{λ_i >0} λ_i^2(G), \quad s^-(G)=\sum_{λ_i<0} λ_i^2(G).\] It was conjectured by Elphick, Farber, Goldberg and Wocjan that for every connected graph $G$ of order $n$, $s^+(G) \ge n-1.$ We verify this conjecture for graphs with domination number at most 2. We then strengthen the conjecture as follows: if $G$ is a connected graph of order $n$ and size $m \geq n+1$, then $s^+(G) \geq n…

This https://arxiv.org/abs/2506.01870 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Various irrational series involving binomial coefficients
Motivated by Galois theory, we propose some new irrational series of Ramanujan's type or Zeilberger's type. For example, we conjecture that \begin{equation*}\sum_{k=1}^\infty\frac{\binom{2k}k^2\binom{4k}{2k}}{(84-32\sqrt6)^{4k}} \left(280k+5\sqrt6+26\right)=\frac3π(2\sqrt3+33\sqrt2) \end{equation*} and \begin{align*}&\sum_{k=1}^\infty\frac{(27(37102-15147\sqrt6))^k}{k^3\binom{2k}k^2\binom{3k}k} \left(9k(11\sqrt6+51)-2(19\sqrt6+54)\right) \\&\qquad=\frac{5625}2\left(\sqrt6 L_{-24}(2)-3G\r…

A matheuristic for solving the single row facility layout problem
Thomas Pammer, Markus Sinnl
https://arxiv.org/abs/2506.09793 https://

A matheuristic for solving the single row facility layout problem
The single row facility layout problem (SRFLP) is a well-studied NP-hard combinatorial optimization problem with applications in manufacturing and logistics systems. In the SRFLP, a set of facilities with lengths is given, as well as weights between each pair of facilities. The facilities must be arranged on a line, such that the sum of the weighted center-to-center distances is minimized. In this work, we introduce a novel matheuristic approach that integrates exact optimization into a m…

This https://arxiv.org/abs/2505.01329 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_hepp…

Semileptonic Decays of $D \to ρl^+ ν$ and $D_{(s)} \to K^\ast l^+ ν$ from Light-Cone Sum Rules
We investigate the semileptonic decays of charmed mesons to light vector mesons within the framework of light-cone sum rules. Our calculation is performed at leading order in QCD coupling, incorporating contributions up to twist-five accuracy from both two-particle and three-particle light-cone distribution amplitudes. Transition form factors are predicted twist by twist to assess the convergence property of the operator product expansion. It is verified that the twist-four and twist-five contr…

This https://arxiv.org/abs/2505.03447 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Asymptotic formula for the sum of a prime and a square-full number in intervals shorter than $X^{1/2}$
Let $R(N)$ be the number of representations of $N$ as a sum of a prime and a square-full number weighted with logarithmic function. In $2024$, the author and Y. Suzuki obtained an asymptotic formula for the sum of $R(N)$ over positive integers $N$ in a short interval ($X$, $X+H$] for $X^{\frac{1}{2}+\varepsilon} \le H < X^{1-\varepsilon}$. In this article, we improve the range of $H$, that is, we prove the same asymptotic formula for $X^{\frac{32-4\sqrt{15}}{49}+\varepsilon} \le H \le X^{1- \va…

Polarized Deep Inelastic Scattering as $x \to 1$ using Soft Collinear Effective Theory
Jaipratap Singh Grewal, Aneesh V. Manohar, Jyotirmoy Roy
https://arxiv.org/abs/2507.07175

Polarized Deep Inelastic Scattering as $x \to 1$ using Soft Collinear Effective Theory
We use Soft Collinear Effective Theory (SCET) to factorize the polarized Deep Inelastic Scattering (DIS) structure functions $g_1(x)$ and $g_2(x)$, and to sum Sudakov double logarithms of $1-x$. The analysis is done both in terms of lightcone parton distributions and their moments. Computing $g_2$ requires subleading SCET operators which contain gluons. We calculate the one-loop matching coefficients from QCD onto these subleading SCET operators, and the one-loop matching from SCET onto the par…

Ramanujan's partition generating functions modulo $\ell$
Kathrin Bringmann, William Craig, Ken Ono
https://arxiv.org/abs/2506.06101 https://

Ramanujan's partition generating functions modulo $\ell$
For the partition function $p(n)$, Ramanujan proved the striking identities $$ P_5(q):=\sum_{n\geq 0} p(5n+4)q^n =5\prod_{n\geq 1} \frac{\left(q^5;q^5\right)_{\infty}^5}{(q;q)_{\infty}^6}, $$ $$ P_7(q):=\sum_{n\geq 0} p(7n+5)q^n =7\prod_{n\geq 1}\frac{\left(q^7;q^7\right)_{\infty}^3}{(q;q)_{\infty}^4}+49q \prod_{n\geq 1}\frac{\left(q^7;q^7\right)_{\infty}^7}{(q;q)_{\infty}^8}, $$ where $(q;q)_{\infty}:=\prod_{n\geq 1}(1-q^n).$ As these identities imply his celebrated congruences modulo 5 …

Dissipativity-based time domain decomposition for optimal control of hyperbolic PDEs
B\'alint Farkas, Birgit Jacob, Manuel Schaller, Merlin Schmitz
https://arxiv.org/abs/2507.07812

Dissipativity-based time domain decomposition for optimal control of hyperbolic PDEs
We propose a time domain decomposition approach to optimal control of partial differential equations (PDEs) based on semigroup theoretic methods. We formulate the optimality system consisting of two coupled forward-backward PDEs, the state and adjoint equation, as a sum of dissipative operators, which enables a Peaceman-Rachford-type fixed-point iteration. The iteration steps may be understood and implemented as solutions of many decoupled, and therefore highly parallelizable, time-distributed …

Extremal Zagreb indices of bicyclic hypergraphs
Hong Zhou, Changjiang Bu
https://arxiv.org/abs/2506.08875 https://arxiv.org/pdf/2506.…

Extremal Zagreb indices of bicyclic hypergraphs
The Zagreb index of a hypergraph is defined as the sum of the squares of the degrees of its vertices. A connected $k$-uniform hypergraph with $n$ vertices and $m$ edges is called bicyclic if $n=m(k-1)-1$. In this paper, we determine the hypergraphs with the maximum and minimum Zagreb indices among all linear bicyclic uniform hypergraphs.

This https://arxiv.org/abs/2505.03447 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Asymptotic formula for the sum of a prime and a square-full number in intervals shorter than $X^{1/2}$
Let $R(N)$ be the number of representations of $N$ as a sum of a prime and a square-full number weighted with logarithmic function. In $2024$, the author and Y. Suzuki obtained an asymptotic formula for the sum of $R(N)$ over positive integers $N$ in a short interval ($X$, $X+H$] for $X^{\frac{1}{2}+\varepsilon} \le H < X^{1-\varepsilon}$. In this article, we improve the range of $H$, that is, we prove the same asymptotic formula for $X^{\frac{32-4\sqrt{15}}{49}+\varepsilon} \le H \le X^{1- \va…

Lower bounds for high moments of zeta sums
Zikang Dong, Weijia Wang, Hao Zhang
https://arxiv.org/abs/2506.09334 https://arxiv.org/pdf…

Lower bounds for high moments of zeta sums
In this article, we investigate high moments of zeta sums $\sum_{n\le x}n^{i t}$. We show unconditional lower bounds for them.

Composite Optimization with Indicator Functions: Stationary Duality and a Semismooth Newton Method
Penghe Zhang, Naihua Xiu, Houduo Qi
https://arxiv.org/abs/2506.08374

Composite Optimization with Indicator Functions: Stationary Duality and a Semismooth Newton Method
Indicator functions of taking values of zero or one are essential to numerous applications in machine learning and statistics. The corresponding primal optimization model has been researched in several recent works. However, its dual problem is a more challenging topic that has not been well addressed. One possible reason is that the Fenchel conjugate of any indicator function is finite only at the origin. This work aims to explore the dual optimization for the sum of a strongly convex function…

An Equivalence Between Erd\H{o}s's Square Packing Conjecture and the Convergence of an Infinite Series
Anshul Raj Singh
https://arxiv.org/abs/2506.23284

An Equivalence Between Erdős's Square Packing Conjecture and the Convergence of an Infinite Series
Let $f(n)$ denote the maximum sum of the side lengths of $n$ non-overlapping squares packed inside a unit square. We prove that $f(n^2+1) = n$ for all positive integers $n$ if and only if the sum $\sum_{k\geq 1}(f(k^2+1)-k)$ converges. We also show that if $f(k^2+1) = k$, for infinitely many positive integers then $f(k^2+1) = k$ for all positive integers.

Notes on sum-free sets in abelian groups
Nathana\"el Hassler, Andrew Treglown
https://arxiv.org/abs/2506.17401 https://arxiv.org…

Notes on sum-free sets in abelian groups
In this paper we highlight a few open problems concerning maximal sum-free sets in abelian groups. In addition, for most even order abelian groups $G$ we asymptotically determine the number of maximal distinct sum-free subsets in $G$. Our proof makes use of the container method.

$q$-Congruences for Z.-W. Sun's generalized polynomials $w^{(\alpha)}_k(x)$
Lin-Yue Li, Rong-Hua Wang
https://arxiv.org/abs/2507.04653 https://

$q$-Congruences for Z.-W. Sun's generalized polynomials $w^{(α)}_k(x)$
In 2022, Z.-W. Sun defined \begin{equation*} w_k^{(α)}{(x)}=\sum_{j=1}^{k}w(k,j)^αx^{j-1}, \end{equation*} where $k,α$ are positive integers and $w(k,j)=\frac{1}{j}\binom{k-1}{j-1}\binom{k+j}{j-1}$. Let $(x)_{0}=1$ and $(x)_{n}=x(x+1)\cdots(x+n-1)$ for all $n\geq 1$. In this paper, it is proved by $q$-congruences that for any positive integers ${α,β, m,n,r}$, we have \begin{equation*} \frac{(2,n)}{n(n+1)(n+2)}\sum_{k=1}^{n}k^r(k+1)^r(2k+1)w_{k}^{(α)}(x)^{m}\in\mathbb{Z}[x], \end{equation*…

On zero-sum subsequences in a finite abelian group of length not exceeding a given number
Kevin Zhao
https://arxiv.org/abs/2506.21383 https://

On zero-sum subsequences in a finite abelian group of length not exceeding a given number
Let $G$ be an additive finite abelian group and let $k\in [\exp(G),\mathsf{D}(G)-1]$ be a positive integer. Denote by $\mathsf{s}_{\leq k}(G)$ the smallest positive integer $l\in \mathbb{N}\cup \{+\infty\}$ such that each sequence of length $l$ over $G$ has a non-empty zero-sum subsequence of length at most $k$. Let $k_G\in [\exp(G),\mathsf{D}(G)-1]$ be the smallest positive integer such that $\mathsf{s}_{\leq \mathsf{D}(G)-d}(G)\leq \mathsf{D}(G)+d$ for $\mathsf{D}(G)-d\geq k_G$. We conjecture…

On the sum of a prime and two Fibonacci numbers
Ji-Zhen Xu, Yong-Gao Chen
https://arxiv.org/abs/2506.03631 https://arxiv.org/pdf/2506…

On the sum of a prime and two Fibonacci numbers
Let $\{f_n\}$ be the Fibonacci sequence. For any positive integer $n$, let $r(n)$ be the number of solutions of $n=p+f_{k_1^{2}} +f_{k_{2}^{2}}$, where $p$ is a prime and $k_1, k_2$ are nonnegative integers with $k_1\le k_2$. In this paper, it is proved that $\{ n : r(n)=0\} $ contains an infinite arithmetic progression, and both sets $\{ n : r(n)=1\} $ and $\{ n : r(n)\ge 2\}$ have positive asymptotic densities.

Semisimple algebras related to immaculate tableaux
John M. Campbell
https://arxiv.org/abs/2507.02539 https://arxiv.org/pdf/2507.02539…

Semisimple algebras related to immaculate tableaux
Given a direct sum $A$ of full matrix algebras, if there is a combinatorial interpretation associated with both the dimension of $A$ and the dimensions of the irreducible $A$-modules, then this can be thought of as providing an analogue of the famous Frobenius-Young identity $n! = \sum_{λ\vdash n} ( f^λ )^{2}$ derived from the semisimple structure of the symmetric group algebra $\mathbb{C}S_{n}$, letting $f^λ$ denote the number of Young tableaux of partition shape $λ\vdash n$. By letting $g…

This https://arxiv.org/abs/2505.10090 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Sombor index of clean graphs
Let $G = (V, E)$ be a graph with the vertex set $V (G)$ and edge set $E(G)$. The Sombor index of $G$, $SO(G)$, is defined as $\sum_{uv\in E(G)} \sqrt{deg(u)^2 + deg(v)^2}$, where $deg(u)$ is the degree of vertex $u$ in $V (G)$. The clean graph of a ring R, denoted by $Cl(R)$, is a graph with vertex set $\{(e, u) : e \in Id(R), u \in U(R)\}$ and two distinct vertices $(e, u)$ and$(f, v)$ are adjacent if and only if $ef = 0$ or $uv = 1$ ($Id(R)$ and $U(R)$ are the sets of idempotents and unit ele…

An Equivalent Representation of Generalized Differentials
Valentin Suder
https://arxiv.org/abs/2507.07337 https://arxiv.org/pdf/2507.…

An Equivalent Representation of Generalized Differentials
We propose an equivalent formula for the higher-order derivatives used in the study of Generalized Almost Perfect Nonlinear functions over an arbitrary finite field of characteristic $p$. The result is obtained by counting the number of subsets of the prime field with a fixed cardinality for which the sum of their elements is constant. We then ask related questions regarding the diversity of higher-order derivatives.

On the Estrada Index and Spectral Properties of $k$-Uniform Hypergraphs
Liya Jess Kurian, Chithra A. V
https://arxiv.org/abs/2507.06322 https://

On the Estrada Index and Spectral Properties of $k$-Uniform Hypergraphs
Various properties of a hypergraph can be explored through its spectrum. In this paper,we estimate the bound for the sum of $t,\,t\in[1,n]$, largest eigenvalues of a $k$-uniform hypergraph of order $n$. Also, we characterize the $k$-uniform hypergraph with two distinct eigenvalues. We establish bounds for the Estrada index of $k$-uniform hypergraphs based on their number of edges, order and energy. In addition, we obtain the second largest Estrada index among the unicyclic hypergraphs.

This https://arxiv.org/abs/2503.18789 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

On the Sylvester program and Cayley algorithm for vector partition reduction
A vector partition problem asks for a number of nonnegative integer solutions to a system of several linear Diophantine equations with integer nonnegative coefficients. J.J. Sylvester put forward an idea of reduction of vector partition to a sum of scalar partitions. In the simplest case of two equations with positive coefficients A. Cayley performed a reduction of the corresponding double partition to a sum of scalar partitions using an algorithm subject to a set of conditions on the coefficie…

This https://arxiv.org/abs/2404.04747 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

The $L^1$ mean of the exponential sum of the divisor function
We give a relatively simple proof that \[ \int _0^1\left |\sum _{n\leq x}d(n)e(nα)\right |dα\asymp \sqrt x.\]

This https://arxiv.org/abs/2506.01870 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Various irrational series involving binomial coefficients
Motivated by Galois theory, we propose some new irrational series of Ramanujan's type or Zeilberger's type. For example, we conjecture that \begin{equation*}\sum_{k=1}^\infty\frac{\binom{2k}k^2\binom{4k}{2k}}{(84-32\sqrt6)^{4k}} \left(280k+5\sqrt6+26\right)=\frac3π(2\sqrt3+33\sqrt2) \end{equation*} and \begin{align*}&\sum_{k=1}^\infty\frac{(27(37102-15147\sqrt6))^k}{k^3\binom{2k}k^2\binom{3k}k} \left(9k(11\sqrt6+51)-2(19\sqrt6+54)\right) \\&\qquad=\frac{5625}2\left(\sqrt6 L_{-24}(2)-3G\r…