Tootfinder

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…

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…

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…

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.

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…

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…

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…

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

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.

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…

@… 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.

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…

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…

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…

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-…

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…

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.

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.

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,…

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…

Ramanujan, Landau and Casimir, divergent series: a physicist point of view
Gilles Montambaux
https://arxiv.org/abs/2506.08664 https://

Ramanujan, Landau and Casimir, divergent series: a physicist point of view
It is a popular paradoxical exercise to show that the infinite sum of positive integer numbers is equal to -1/12, sometimes called the Ramanujan sum. Here we propose a qualitative approach, much like that of a physicist, to show how the value -1/12 can make sense and, in fact, appears in certain physical quantities where this type of summation is involved. At the light of two physical examples, taken respectively from condensed matter -- the Landau diamagnetism -- and quantum electrodynamics --…

This https://arxiv.org/abs/2506.03770 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_ees…

Multiuser Beamformig for Pinching-Antenna Systems: An Element-wise Optimization Framework
The pinching-antenna system (PASS) reconstructs wireless channels through pinching beamforming, i.e., optimizing the activated locations of pinching antennas (PAs) along the waveguide. The aim of this article is to investigate the joint design of baseband beamforming and pinching beamforming. A low-complexity element-wise sequential optimization framework is proposed to address the sum-rate maximization problem in PASS-enabled downlink and uplink channels. i) For the downlink scenario, maximum …

This https://arxiv.org/abs/2409.17464 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_grqc_…

Computation of $\langle Φ^2\rangle$ and quantum fluxes at the polar interior of a spinning black hole
Renormalization of physical quantities for quantum field theories in curved spacetimes can be achieved via the consistent subtraction of counterterms within a regularization scheme such as a point-splitting method. Pragmatic mode-sum regularization (PMR) is a point-splitting method which is particularly suitable for rotating black hole spacetimes. We extend and tailor the t-splitting variant of PMR specifically for the interior of a Kerr black hole on the axis of rotation, focusing on a minimal…

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 Bi-Orthogonal Structure-Preserving eigensolver for large-scale linear response eigenvalue problem
Yu Li, Zijing Wang, Yong Zhang
https://arxiv.org/abs/2506.08355

A Bi-Orthogonal Structure-Preserving eigensolver for large-scale linear response eigenvalue problem
The linear response eigenvalue problem, which arises from many scientific and engineering fields, is quite challenging numerically for large-scale sparse/dense system, especially when it has zero eigenvalues. Based on a direct sum decomposition of biorthogonal invariant subspaces and the minimization principles in the biorthogonal complement, using the structure of generalized nullspace, we propose a Bi-Orthogonal Structure-Preserving subspace iterative solver, which is stable, efficient, and o…

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 …

This https://arxiv.org/abs/2503.21940 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Normalized vector solutions of nonlinear Schrödinger systems
Given $μ>0$ we look for solutions $ λ\in\mathbb{R}$ and $v_1,\dots,v_k\in H^1(\mathbb{R}^N)$ of the system \[ \begin{cases} \displaystyle -Δv_i+ λv_i+V_i(x)v_i = \sum_{\substack{j=1}}^kβ_{ij} v_iv_j^2 &\text{ in } \mathbb{R}^N, \text{ } i=1,\dots,k,\newline \displaystyle \int_{\mathbb{R}^N} \left(v_1^2+\dots+v_k^2 \right)\mathrm{d} x = μ, \end{cases}\] where $N=1,2,3$, $V_i:\mathbb R^N\to \mathbb R$ and $β_{ij}\in\mathbb{R}$ satisfy $β_{ij}=β_{ji}$ and $β_{ii}>0$. Under …

Telescoping Algorithms for $\Sigma^*$-Extensions via Complete Reductions
Shaoshi Chen, Yiman Gao, Hui Huang, Carsten Schneider
https://arxiv.org/abs/2506.08767

Telescoping Algorithms for $Σ^*$-Extensions via Complete Reductions
A complete reduction on a difference field is a linear operator that enables one to decompose an element of the field as the sum of a summable part and a remainder such that the given element is summable if and only if the remainder is equal to zero. In this paper, we present a complete reduction in a tower of $Σ^*$-extensions that turns to a new efficient framework for the parameterized telescoping problem. Special instances of such $Σ^*$-extensions cover iterative sums such as the harmonic …

Nonlinear input-output analysis of transitional shear flows using small-signal finite-gain $\mathcal{L}_p$ stability
Zhengyang Wei, Chang Liu
https://arxiv.org/abs/2506.08129

Nonlinear input-output analysis of transitional shear flows using small-signal finite-gain $\mathcal{L}_p$ stability
Input-output analysis has been widely used to predict the transition to turbulence in wall-bounded shear flows, but it typically does not capture the full nonlinear effects involved. This work employs the Small-Signal Finite-Gain (SSFG) Lp stability theorem to assess nonlinear input-output amplification and demonstrates its applicability using a nine-mode shear flow model with a random body force. Our analysis employs Linear Matrix Inequalities (LMI) and Sum-of-Squares (SOS) as the primary tool…

This https://arxiv.org/abs/2407.18152 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Kronecker coefficients and Harrison centres of the representation ring of the symmetric group
We present a computational approach to studying the structure of the representation ring of the symmetric group in dimension six. The Kronecker coefficients and all power formulae of irreducible representations of $S_6$ are computed using the character theory of finite groups. In addition, considering direct sum decomposition of tensor products of different irreducible representations of $S_6$, we characterise generators of the representation ring $\mathcal{R}(S_6)$, show that its unit group $U…

This https://arxiv.org/abs/2405.10178 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Infinite Divisibility of the Product of Two Correlated Normal Random Variables and Exact Distribution of the Sample Mean
We prove that the distribution of the product of two correlated normal random variables with arbitrary means and arbitrary variances is infinitely divisible. We also obtain exact formulas for the probability density function of the sum of independent copies of such random variables.

Sum Rules in Quantum Liquids
Riccardo Fantoni
https://arxiv.org/abs/2507.07950 https://arxiv.org/pdf/2507.07950

Sum Rules in Quantum Liquids
We review the linear response theory in the horizontal quantum liquids framework spanning from Coulomb liquids to Atomic gases. There are several well known references about this subject, like the onset of the Kramers-Kronig relations and the fluctuation-dissipation theorem. For the Coulomb systems we show the connection between the linear response function and the dielectric function which settles a parallelism between statistical mechanics and electrostatic properties. For (very degenerate, d…

The atoms of graph product von Neumann algebras
Ian Charlesworth, David Jekel
https://arxiv.org/abs/2506.09000 https://arxiv.org/pdf/…

The atoms of graph product von Neumann algebras
We completely classify the atomic summands in a graph product $(M,φ) = *_{v \in \mathcal{G}} (M_v,φ_v)$ of von Neumann algebras with faithful normal states. Each type I factor summand $(N,ψ)$ is a tensor product of type I factor summands $(N_v,ψ_v)$ in the individual algebras. The existence of such a summand and its weight in the direct sum can be determined from the $(N_v,ψ_v)$'s using explicit polynomials associated to the graph.

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…

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 $…

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…

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 $p$.

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…

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…

CNFs and DNFs with Exactly $k$ Solutions
L. Sunil Chandran, Rishikesh Gajjala, Kuldeep S. Meel
https://arxiv.org/abs/2506.07268 https://

CNFs and DNFs with Exactly $k$ Solutions
Model counting is a fundamental problem that consists of determining the number of satisfying assignments for a given Boolean formula. The weighted variant, which computes the weighted sum of satisfying assignments, has extensive applications in probabilistic reasoning, network reliability, statistical physics, and formal verification. A common approach for solving weighted model counting is to reduce it to unweighted model counting, which raises an important question: {\em What is the minimum …

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…

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.

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.

SDEs with critical general distributional drifts: sharp solvability and blow ups
D. Kinzebulatov, R. Vafadar
https://arxiv.org/abs/2506.09244 https://

SDEs with critical general distributional drifts: sharp solvability and blow ups
We establish weak well-posedness for SDEs having discontinuous diffusion coefficients and general distributional drifts that may introduce blow up effects. Our drifts satisfy minimal assumptions, i.e.\,we assume only that the Cauchy problem for the Kolmogorov backward equation is well-posed in the standard Hilbert triple $W^{1,2} \hookrightarrow L^2 \hookrightarrow W^{-1,2}$. By a result of Mazya and Verbitsky, these assumptions are precisely those drifts that can be represented as the sum of a…

On the statistical nature of Betti numbers and Euler characteristic of smooth random fields
Pravabati Chingangbam
https://arxiv.org/abs/2507.06255 https://…

On the statistical nature of Betti numbers and Euler characteristic of smooth random fields
We represent excursion sets of smooth random fields as unions of a topological basis consisting of a sequence of simply and multiply connected compact subsets of the underlying manifold. The associated coefficients, which are non-negative discrete random variables, reflect the randomness of the field. Betti numbers of the excursion sets can be expressed as summations over the coefficients, and the Euler characteristic and the sum of Betti numbers can also be expressed as their (alternating) sum…

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 …

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.

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…

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…

Square Functions and Variational Estimates for Ritt Operators on $L^1$
Jennifer Hults, Karin Reinhold-Larsson
https://arxiv.org/abs/2507.07256 https://

Square Functions and Variational Estimates for Ritt Operators on $L^1$
Let $T$ be a bounded operator. We say $T$ is a Ritt operator if $\sup_n n\lVert T^n-T^{n+1}\rVert<\infty$. It is know that when $T$ is a positive contraction and a Ritt operator in $L^p$, $1

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…

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…

On Sketching Trimmed Statistics
Honghao Lin, Hoai-An Nguyen, David P. Woodruff
https://arxiv.org/abs/2506.07342 https://arxiv.org/pdf…

On Sketching Trimmed Statistics
We present space-efficient linear sketches for estimating trimmed statistics of an $n$-dimensional frequency vector $x$, e.g., the sum of $p$-th powers of the largest $k$ frequencies (i.e., entries) in absolute value, or the $k$-trimmed vector, which excludes the top and bottom $k$ frequencies. This is called the $F_p$ moment of the trimmed vector. Trimmed measures are used in robust estimation, as seen in the R programming language's `trim.var' function and the `trim' parameter in the mean fun…

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.

Radiative decays of $P$-wave charmed baryons in the $SU(3)$ flavor $\bf6_F$ representation
Xuan Luo, Hua-Xing Chen, Er-Liang Cui, Hui-Min Yang, Dan Zhou, Zhi-Yong Zhou
https://arxiv.org/abs/2506.08335

Radiative decays of $P$-wave charmed baryons in the $SU(3)$ flavor $\bf6_F$ representation
We perform a comprehensive investigation of the radiative decays of $P$-wave charmed baryons in the $SU(3)$ flavor $\mathbf{6}_F$ representation, employing the light-cone QCD sum rule approach within the framework of heavy quark effective theory. We analyze their electromagnetic transitions into ground-state charmed baryons via photon emission. When combined with the mass spectra and strong decay properties previously studied in Ref.~\cite{Yang:2021lce}, our results constitute a systematic and …

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…

The Strong Law of Large Numbers for random semigroups on uniformly smooth Banach spaces
S. V. Dzhenzher, V. Zh. Sakbaev
https://arxiv.org/abs/2507.07658 ht…

The Strong Law of Large Numbers for random semigroups on uniformly smooth Banach spaces
We consider random linear continuous operators $Ω\to \mathcal{L}(X, X)$ on a Banach space $X$. For example, such random operators may be random quantum channels. The Law of Large Numbers is known when $X$ is a Hilbert space, in the form of the usual Law of Large Numbers for random operators, and in some other particular cases. Instead of the sum of i.i.d. variables, there may be considered the composition of random semigroups $e^{A_it/n}$. We obtain the Strong Law of Large Numbers in Strong Op…

Left-continuous pseudo-t-norms on modular lattices
Peng He, Xue-ping Wang
https://arxiv.org/abs/2506.07567 https://arxiv.org/pdf/2506…

Left-continuous pseudo-t-norms on modular lattices
This article focuses on the relationship between pseudo-t-norms and the structure of lattices. First, we establish a necessary and sufficient condition for the existence of a left-continuous t-norm on the ordinal sum of two disjoint complete lattices. Then, we define the $1$-distributivity of a lattice, which is applied for characterizing a complete atomistic lattice that has a left-continuous pseudo-t-norm. We also describe the forbidden structures of a finite modular lattice that is a $1$-dis…

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…

Kuznecov remainders and generic metrics
Vadim Kaloshin, Emmett L. Wyman, Yakun Xi
https://arxiv.org/abs/2507.06887 https://arxiv.org/pdf/2507.06887 https://arxiv.org/html/2507.06887
arXiv:2507.06887v1 Announce Type: new
Abstract: We obtain improved remainder estimates in the Kuznecov sum formula for period integrals of Laplace eigenfunctions on a Riemannian manifold $M$. Building upon the two-term asymptotic expansion established in [arXiv:2204.13525], we prove that for a Baire-generic class of metrics, the oscillatory second term in the Kuznecov formula can be eliminated, yielding an improved remainder estimate.
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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.

An interpretation of the fully-charmed scalar state $X(6200)$ as a molecular di-meson
\"O. E. Demircan, H. Da\u{g}, H. Sundu, J. Y. S\"ung\"u, E. V. Veliev
https://arxiv.org/abs/2506.08589

An interpretation of the fully-charmed scalar state $X(6200)$ as a molecular di-meson
The LHCb Collaboration recently observed new structures in the invariant mass spectrum of $J/ψJ/ψ$ meson pairs produced in proton-proton collisions, including a narrow peak around $6.2$ GeV. This study investigates the thermal behavior of this newly discovered fully-charm resonance, assuming it to be an $η_cη_c$ molecule with quantum numbers $J^{PC} = 0^{++}$. Employing thermal QCD sum rules up to dimension four, we analyze the temperature dependence of the mass and the decay constant. Our …

Fractional Programming for Stochastic Precoding over Generalized Fading Channels
Wenyu Wang, Kaiming Shen
https://arxiv.org/abs/2507.06944 https://

Fractional Programming for Stochastic Precoding over Generalized Fading Channels
This paper seeks an efficient algorithm for stochastic precoding to maximize the long-term average weighted sum rates throughout a multiple-input multiple-output (MIMO) network. Unlike many existing works that assume a particular probability distribution model for fading channels (which is typically Gaussian), our approach merely relies on the first and second moments of fading channels. For the stochastic precoding problem, a naive idea is to directly apply the fractional programming (FP) meth…

A study on preservation of parallel pairs and triangle equality attainment pairs
Jayanta Manna, Kalidas Mandal, Kallol Paul, Debmalya Sain
https://arxiv.org/abs/2507.07774

A study on preservation of parallel pairs and triangle equality attainment pairs
This paper investigates the structure and preservation of parallel pairs and triangle equality attainment (TEA) pairs in normed linear spaces. We begin by providing functional characterizations of these pairs in normed linear spaces and provide a characterization of finite-dimensional Banach spaces with numerical index one. We then study parallel pairs and TEA pairs in the $p$-direct sum of normed linear spaces. Next, we show that in finite-dimensional Banach spaces, a bijective bounded linear …

This https://arxiv.org/abs/2504.11666 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

A note on the second supplementary law of rational power residue symbols
As a natural generalization of the Legendre symbol, the $q$-th power residue symbol $(a/p)_q$ is defined for primes $p$ and $q$ with $p\equiv 1 \bmod q$. In this paper, we generalize the second supplementary law by providing an explicit condition for $(q/p)_q = 1$, when $p$ has a special form $p = \sum_{i=0}^{q-1} m^i n^{q-1-i}$. This condition is expressed in terms of the polylogarithm $\mathrm{Li}_{1-q}(x)$ of negative index. Our proof relies on an argument similar to Lemmermeyer's proof of E…

Hybrid Quantum Convolutional Neural Network-Aided Pilot Assignment in Cell-Free Massive MIMO Systems
Doan Hieu Nguyen, Xuan Tung Nguyen, Seon-Geun Jeong, Trinh Van Chien, Lajos Hanzo, Won Joo Hwang
https://arxiv.org/abs/2507.06585

Hybrid Quantum Convolutional Neural Network-Aided Pilot Assignment in Cell-Free Massive MIMO Systems
A sophisticated hybrid quantum convolutional neural network (HQCNN) is conceived for handling the pilot assignment task in cell-free massive MIMO systems, while maximizing the total ergodic sum throughput. The existing model-based solutions found in the literature are inefficient and/or computationally demanding. Similarly, conventional deep neural networks may struggle in the face of high-dimensional inputs, require complex architectures, and their convergence is slow due to training numerous …

Langlands branching rule for type B snake modules
Jingmin Guo, Jian-Rong Li, Keyu Wang
https://arxiv.org/abs/2507.06570 https://arxiv…

Langlands branching rule for type B snake modules
We prove that each snake module of the quantum Kac-Moody algebra of type $B_n^{(1)}$ admits a Langlands dual representation, as conjectured by Frenkel and Hernandez (\textit{Lett. Math. Phys.} (2011) 96:217-261). Furthermore, we establish an explicit formula, called the Langlands branching rule, which gives the multiplicities in the decomposition of the character of a snake module of the quantum Kac-Moody algebra of type $B_n^{(1)}$ into a sum of characters of irreducible representations of its…

Average Nikolskii factors for random diffusion polynomials on closed Riemannian manifolds
Yun Ling, Heping Wang
https://arxiv.org/abs/2507.06505 https://…

Average Nikolskii factors for random diffusion polynomials on closed Riemannian manifolds
For $1\le p,q\le \infty$, the Nikolskii factor for a diffusion polynomial $P_{\bf a}$ of degree at most $n$ is defined by $$N_{p,q}(P_{\bf a})=\frac{\|P_{\bf a}\|_{q}}{\|P_{\bf a}\|_{p}},\ \ P_{\bf a}({\bf x})=\sum_{k:λ_{k}\leq n}a_{k}ϕ_{k}({\bf x}),$$ where ${\bf a}=\{a_k\}_{λ_k\le n}$, and $\{(ϕ_k,-λ_k^2)\}_{k=0}^\infty$ are the eigenpairs of the Laplace-Beltrami operator $Δ_{\mathbb M}$ on a closed smooth Riemannian manifold $\mathbb M$ with normalized Riemannian me…

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…

Qualitative properties of solutions to parabolic anisotropic equations: Part I -- Expansion of positivity
Simone Ciani, Eurica Henriques, Mariia Savchenko, Igor I. Skrypnik
https://arxiv.org/abs/2507.06714 https://arxiv.org/pdf/2507.06714 https://arxiv.org/html/2507.06714
arXiv:2507.06714v1 Announce Type: new
Abstract: We prove expansion of positivity and reduction of the oscillation results to the local weak solutions to a doubly nonlinear anisotropic class of parabolic differential equations with bounded and measurable coefficients, whose prototype is \begin{equation*} u_t-\sum\limits_{i=1}^N \left( u^{(m_i-1)(p_i-1)} \ |u_{x_i}|^{p_i-2} \ u_{x_i} \right)_{x_i}=0 , \end{equation*} for a restricted range of $p_i$s and $m_i$s, that reflects their competition for the diffusion. The positivity expansion relies on an exponential shift and is presented separately for singular and degenerate cases. Finally we present a study of the local oscillation of the solution for some specific ranges of exponents, within the singular and degenerate cases.
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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.

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…

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.

Robust signal decompositions on the circle
Aral Kose, Daniel Liberzon
https://arxiv.org/abs/2507.07007 https://arxiv.org/pdf/2507.070…

Robust signal decompositions on the circle
We consider the problem of decomposing a piecewise constant function on the circle into a sum of indicator functions of closed circular disks in the plane, whose number and location are not a priori known. This represents a situation where an agent moving on the circle is able to sense its proximity to some landmarks, and the goal is to estimate the number of these landmarks and their possible locations -- which can in turn enable control tasks such as motion planning and obstacle avoidance. Mo…

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 …

Refuting Perfect Matchings in Spectral Expanders is Hard
Ari Biswas, Rajko Nenadov
https://arxiv.org/abs/2506.07700 https://arxiv.org…

Refuting Perfect Matchings in Spectral Expanders is Hard
This work studies the complexity of refuting the existence of a perfect matching in spectral expanders with an odd number of vertices, in the Polynomial Calculus (PC) and Sum of Squares (SoS) proof system. Austrin and Risse [SODA, 2021] showed that refuting perfect matchings in sparse $d$-regular \emph{random} graphs, in the above proof systems, with high probability requires proofs with degree $Ω(n/\log n)$. We extend their result by showing the same lower bound holds for \emph{all} $d$-regul…

Decentralized Optimization on Compact Submanifolds by Quantized Riemannian Gradient Tracking
Jun Chen, Lina Liu, Tianyi Zhu, Yong Liu, Guang Dai, Yunliang Jiang, Ivor W. Tsang
https://arxiv.org/abs/2506.07351

Decentralized Optimization on Compact Submanifolds by Quantized Riemannian Gradient Tracking
This paper considers the problem of decentralized optimization on compact submanifolds, where a finite sum of smooth (possibly non-convex) local functions is minimized by $n$ agents forming an undirected and connected graph. However, the efficiency of distributed optimization is often hindered by communication bottlenecks. To mitigate this, we propose the Quantized Riemannian Gradient Tracking (Q-RGT) algorithm, where agents update their local variables using quantized gradients. The introducti…

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.

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 $D_{1,1,1}$.

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…

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…

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…

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…

An inexact inertial projective splitting algorithm with strong convergence
M. Marques Alves, J. E. Navarro Caballero, R. T. Marcavillaca
https://arxiv.org/abs/2507.05382

An inexact inertial projective splitting algorithm with strong convergence
We propose and study a strongly convergent inexact inertial projective splitting (PS) algorithm for finding zeros of composite monotone inclusion problems involving the sum of finitely many maximal monotone operators. Strong convergence of the iterates is ensured by projections onto the intersection of appropriately defined half-spaces, even in the absence of inertial effects. We also establish iteration-complexity results for the proposed PS method, which likewise hold without requiring inerti…

Combinatorics of descent algebras and graph coverings
Philippe Biane
https://arxiv.org/abs/2506.05528 https://arxiv.org/pdf/2506.0552…

Combinatorics of descent algebras and graph coverings
We give a direct combinatorial proof that the product of two descent classes in a symmetric group is a sum of descent classes. The proof is based on the fact that the group product gives a covering map when descent classes are endowed with the graph structure coming from the weak order. The main geometric argument is valid for any Coxeter group, even infinite ones for which the descent algebra does not exist.

$q$-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*…

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…

On exotic matrix exponential sums and Bessel-Speh functions
Elad Zelingher
https://arxiv.org/abs/2507.06394 https://arxiv.org/pdf/250…

On exotic matrix exponential sums and Bessel-Speh functions
We define the notions of non-abelian exotic Gauss sums and of exotic matrix Kloosterman sums, the latter one generalizing the notions of Katz's exotic Kloosterman sums and of twisted matrix Kloosterman sums. Using Kondo's Gauss sum and results about Shintani lifts, we reduce non-abelian exotic Gauss sums to products of classical (exotic) Gauss sums. Using Macdonald's characteristic maps, we generalize our previous result and show that exotic matrix Kloosterman sums can be expressed as products …

A Proximal Variable Smoothing for Minimization of Nonlinearly Composite Nonsmooth Function -- Maxmin Dispersion and MIMO Applications
Keita Kume, Isao Yamada
https://arxiv.org/abs/2506.05974

A Proximal Variable Smoothing for Minimization of Nonlinearly Composite Nonsmooth Function -- Maxmin Dispersion and MIMO Applications
We propose a proximal variable smoothing algorithm for a nonsmooth optimization problem whose cost function is the sum of three functions including a weakly convex composite function. The proposed algorithm has a single-loop structure inspired by a proximal gradient-type method. More precisely, the proposed algorithm consists of two steps: (i) a gradient descent of a time-varying smoothed surrogate function designed partially with the Moreau envelope of the weakly convex function; (ii) an appli…

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…

Divided difference operators for Hessenberg representations
Mathieu Guay-Paquet
https://arxiv.org/abs/2507.05614 https://arxiv.org/pd…

Divided difference operators for Hessenberg representations
The equivariant cohomology ring of a regular semisimple Hessenberg variety in type A is a free module over the equivariant cohomology ring of a point. When equipped with Tymoczko's dot action, it becomes a twisted representation of the symmetric group, and the character of this representation is given by the chromatic quasisymmetric function of an indifference graph. In this note, we use divided difference operators to decompose this representation as a direct sum of sub-representations in a wa…

Improving elliptic curve rank classification using multi-value and learned Mestre-Nagao sums
Zvonimir Bujanovi\'c, Matija Kazalicki, Domagoj Vlah
https://arxiv.org/abs/2506.07967

Improving elliptic curve rank classification using multi-value and learned Mestre-Nagao sums
Determining the rank of an elliptic curve E/Q remains a central challenge in number theory. Heuristics such as Mestre--Nagao sums are widely used to estimate ranks, but there is considerable room for improving their predictive power. This paper introduces two novel methods for enhancing rank classification using Mestre--Nagao sums. First, we propose a ``multi-value'' approach that simultaneously uses two distinct sums, S_0 and S_5, evaluated over multiple ranges. This multi-sum perspective sign…

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…

Unraveling $K(1690)$ as a pseudoscalar $ud\bar{d}\bar{s}$ tetraquark state
Jin-Peng Zhang, Xu-Liang Chen, Zi-Xi Ou-Yang, Xiang Yu, Wei Chen, Jia-Jun Wu
https://arxiv.org/abs/2507.05726

Unraveling $K(1690)$ as a pseudoscalar $ud\bar{d}\bar{s}$ tetraquark state
The recent observed $K (1690)$ has been identified as a supernumerary pseudoscalar resonance signal in the strange-meson spectrum predicted by quark model calculations. It is the best candidate of a strange crypto-exotic state. In this work, we systematically study the hadron masses of $ud\bar{d}\bar{s}$ tetraquark states with $J^P = 0^-$ in the method of QCD sum rules (QCDSR). For ten interpolating currents, we calculate the correlation functions up to dimension-8 nonperturbative condensates. …

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.

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.

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/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…

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 \[ \int _0^1\left |\sum _{n\leq x}d(n)e(nα)\right |dα\asymp \sqrt x.\]