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1. Off-shell minimal form factors

   https://doi.org/10.1007/JHEP07(2025)231  

   Belitsky, A V; Bork, L V (July 2025, Journal of High Energy Physics)

   A<sc>bstract</sc> We study off-shelln-particle form factors of half-BPS operators built fromncomplex scalar fields at the two-loop order in the planar maximally supersymmetric Yang-Mills theory (sYM). These are known as minimal form factors. We construct their representation as a sum of independent scalar Feynman integrals relying on two complementary techniques. First, by going to the Coulomb branch of the theory by employing the spontaneous symmetry breaking which induces masses, but only for external particles while retaining masslessness for virtual states propagating in quantum loops. For a low number of external legs, this entails an uplift of massless integrands to their massive counterparts. Second, utilizing the$$ \mathcal{N} $$ $N$= 1 superspace formulation of$$ \mathcal{N} $$ $N$= 4 sYM and performing algebra of covariant derivatives off-shell. Both techniques provide identical results. These form factors are then studied in the near-mass-shell limit with the off-shellness regularizing emerging infrared divergences. We observe their exponentiation and confirm the octagon anomalous dimension, not the cusp, as the coefficient of the Sudakov double logarithmic behavior. By subtracting these singularities and defining a finite remainder, we verified that its symbol is identical to the one found a decade ago in the conformal case. Beyond-the-symbol contributions are different in the two cases, however. 

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2. Algorithms for covering multiple submodular constraints and applications

   https://doi.org/10.1007/s10878-022-00874-x  

   Chekuri, Chandra; Inamdar, Tanmay; Quanrud, Kent; Varadarajan, Kasturi; Zhang, Zhao (June 2022, Journal of Combinatorial Optimization)

   Abstract We consider the problem of covering multiple submodular constraints. Given a finite ground setN, a weight function$$w: N \rightarrow \mathbb {R}_+$$ $w:N\to R_{+}$,rmonotone submodular functions$$f_1,f_2,\ldots ,f_r$$ $f_1,f_2,\dots ,f_r$overNand requirements$$k_1,k_2,\ldots ,k_r$$ $k_1,k_2,\dots ,k_r$the goal is to find a minimum weight subset$$S \subseteq N$$ $S\subseteq N$such that$$f_i(S) \ge k_i$$ $f_i\left(S\right)\ge k_i$for$$1 \le i \le r$$ $1\le i\le r$. We refer to this problem asMulti-Submod-Coverand it was recently considered by Har-Peled and Jones (Few cuts meet many point sets. CoRR.arxiv:abs1808.03260Har-Peled and Jones 2018) who were motivated by an application in geometry. Even with$$r=1$$ $r=1$Multi-Submod-Covergeneralizes the well-known Submodular Set Cover problem (Submod-SC), and it can also be easily reduced toSubmod-SC. A simple greedy algorithm gives an$$O(\log (kr))$$ $O(log(kr\left)\right)$approximation where$$k = \sum _i k_i$$ $k={\sum }_i{k}_i$and this ratio cannot be improved in the general case. In this paper, motivated by several concrete applications, we consider two ways to improve upon the approximation given by the greedy algorithm. First, we give a bicriteria approximation algorithm forMulti-Submod-Coverthat covers each constraint to within a factor of$$(1-1/e-\varepsilon )$$ $(1-1/e-\epsilon )$while incurring an approximation of$$O(\frac{1}{\epsilon }\log r)$$ $O(\frac{1}{ϵ}logr)$in the cost. Second, we consider the special case when each$$f_i$$ $f_i$is a obtained from a truncated coverage function and obtain an algorithm that generalizes previous work on partial set cover (Partial-SC), covering integer programs (CIPs) and multiple vertex cover constraints Bera et al. (Theoret Comput Sci 555:2–8 Bera et al. 2014). Both these algorithms are based on mathematical programming relaxations that avoid the limitations of the greedy algorithm. We demonstrate the implications of our algorithms and related ideas to several applications ranging from geometric covering problems to clustering with outliers. Our work highlights the utility of the high-level model and the lens of submodularity in addressing this class of covering problems. 

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3. Large charge ’t Hooft limit of $$ \mathcal{N} $$ = 4 super-Yang-Mills

   https://doi.org/10.1007/JHEP02(2024)047  

   Caetano, João; Komatsu, Shota; Wang, Yifan (February 2024, Journal of High Energy Physics)

   A<sc>bstract</sc> The planar integrability of$$ \mathcal{N} $$ $N$= 4 super-Yang-Mills (SYM) is the cornerstone for numerous exact observables. We show that the large charge sector of the SU(2)$$ \mathcal{N} $$ $N$= 4 SYM provides another interesting solvable corner which exhibits striking similarities despite being far from the planar limit. We study non-BPS operators obtained by small deformations of half-BPS operators withR-chargeJin the limitJ→ ∞ with$$ {\lambda}_J\equiv {g}_{\textrm{YM}}^2J/2 $$ ${\lambda }_J\equiv g_{\mathrm{YM}}^{2}J/2$fixed. The dynamics in thislarge charge ’t Hooft limitis constrained by a centrally-extended$$ \mathfrak{psu} $$ $\mathrm{psu}$(2|2)²symmetry that played a crucial role for the planar integrability. To the leading order in 1/J, the spectrum is fully fixed by this symmetry, manifesting the magnon dispersion relation familiar from the planar limit, while it is constrained up to a few constants at the next order. We also determine the structure constant of two large charge operators and the Konishi operator, revealing a rich structure interpolating between the perturbative series at weak coupling and the worldline instantons at strong coupling. In addition we compute heavy-heavy-light-light (HHLL) four-point functions of half-BPS operators in terms of resummed conformal integrals and recast them into an integral form reminiscent of the hexagon formalism in the planar limit. For general SU(N) gauge groups, we study integrated HHLL correlators by supersymmetric localization and identify a dual matrix model of sizeJ/2 that reproduces our large charge result atN= 2. Finally we discuss a relation to the physics on the Coulomb branch and explain how the dilaton Ward identity emerges from a limit of the conformal block expansion. We comment on generalizations including the large spin ’t Hooft limit, the combined largeN-largeJlimits, and applications to general$$ \mathcal{N} $$ $N$= 2 superconformal field theories. 

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4. Collinear anatomy

   https://doi.org/10.1007/JHEP05(2025)117  

   Belitsky, AV (May 2025, Journal of High Energy Physics)

   A<sc>bstract</sc> We study the collinear factorization of off-shell scattering amplitudes in maximally supersymmetric Yang-Mills (sYM) theory. These are constructed starting from six-dimensional$$ \mathcal{N} $$ $N$= (1) sYM, taking advantage of an available unconstrained spinor-helicity formalism combined with a unitarity-cut sewing procedure. After generalized dimensional reduction, their collinear behavior is dissected with assistance from the Method of Regions. We then construct off-shell splitting amplitudes directly using the same techniques, establishing equivalence to the amplitude analysis. The calculations are performed at one-loop order. 

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5. NeuralSAT: A High-Performance Verification Tool for Deep Neural Networks

   https://doi.org/10.1007/978-3-031-98679-6_19  

   Duong, Hai; Nguyen, ThanhVu; Dwyer, Matthew B (July 2025, Springer Nature Switzerland)

   Abstract Deep Neural Networks (DNNs) are increasingly deployed in critical applications, where ensuring their safety and robustness is paramount. We present$$_\text {CAV25}$$ ${}_{\mathrm{CAV}25}$, a high-performance DNN verification tool that uses the DPLL(T) framework and supports a wide-range of network architectures and activation functions. Since its debut in VNN-COMP’23, in which it achieved the New Participant Award and ranked 4th overall,$$_\text {CAV25}$$ ${}_{\mathrm{CAV}25}$has advanced significantly, achieving second place in VNN-COMP’24. This paper presents and evaluates the latest development of$$_\text {CAV25}$$ ${}_{\mathrm{CAV}25}$, focusing on the versatility, ease of use, and competitive performance of the tool.$$_\text {CAV25}$$ ${}_{\mathrm{CAV}25}$is available at:https://github.com/dynaroars/neuralsat. 

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