We prove that any centered $1$-subgaussian random vector in $\mathbb{R}^{n}$ can be written as the sum of a universal number of standard Gaussian vectors. Following the work of the second-named author, this solves M. Talagrand's convexity problem, which in turn implies a combinatorial analogue of the problem.

We study the parafermionization of the Monster CFT with respect to its $\mathbb{Z}_{pA}$ subgroups, with $p$ an odd prime. Under certain assumptions, we show that the parafermionization is equal to a non-invertible gauging of $\mathcal{P}(p) \times \mathcal{P}(p)^\vee$, where $\mathcal{P}(p)$ is the theory of $\mathbb{Z}_p$-parafermions and $\mathcal{P}(p)^\vee$ is an appropriate dual theory, with global symmetry characterized by the centralizer of $\mathbb{Z}_{pA}$. By tracking the symmetries of $\mathcal{P}(p) \times \mathcal{P}(p)^\vee$ through the non-invertible gauging, we argue that the diagonal Monster CFT has $\mathrm{Rep}(\mathfrak{so}(3)_p) \boxtimes \mathrm{Rep}(\mathfrak{so}(3)_p)^\mathrm{op}$ symmetry, and hence that the holomorphic Monster theory has symmetry $\mathrm{Rep}(\mathfrak{so}(3)_p)$. We then compute the defect McKay-Thompson series associated to these symmetries, and prove that their invariance subgroups are $Γ_1(p+2)$.

Information abstraction reduces the computational cost of solving imperfect-information games by clustering information sets into a smaller number of \emph{buckets}. Existing methods either rely on domain-specific features such as rank or equity, which are inapplicable to games with non-standard payoff structures, or require expensive offline neural-network training on billions of samples. We propose \textbf{Warm-up Expected Value-based Abstraction (WEVA)}, a simple yet effective alternative: run a small number of Counterfactual Regret Minimization (CFR) iterations on the full game as a \emph{warm-up} phase, extract per-hand expected value features at every decision node, form a depth-weighted multi-node feature vector, and apply $k$-means++ clustering to obtain the abstraction mapping. WEVA requires no domain knowledge, no pre-training, and incurs only a small overhead on top of the abstract-game solve. Experiments on three structurally diverse games, with different bucket numbers and CFR variants, show that WEVA consistently outperforms equity-based and rank-based abstractions, reducing exploitability by up to over $80\%$. Surprisingly, as few as $W{=}10$ warm-up iterations already produce abstractions that outperform existing information abstraction methods in most settings. These results establish WEVA as an \emph{effective, efficient, and general} approach to information abstraction in imperfect-information extensive-form games.

The chromatic threshold of a graph $H$ is the minimum-degree density above which every $H$-free graph has bounded chromatic number. We study a two-color Ramsey analogue: for graphs $H_1$ and $H_2$, we ask for the minimum-degree density above which every graph that admits a red-blue edge-coloring with no red copy of $H_1$ and no blue copy of $H_2$ has bounded chromatic number. We give a complete answer when both $H_1$ and $H_2$ are 3-chromatic. The threshold takes exactly one of the five values \[ \frac23,\quad \frac57,\quad \frac34,\quad \frac79,\quad \frac45, \] and we characterize precisely which pairs $(H_1,H_2)$ give each value. The classification is determined by the ordinary chromatic thresholds of $H_1$ and $H_2$ and by their embeddability into a hierarchy of $C_5$-type Ramsey configurations.

In the classical Min-Sum Radii problem (MSR) we are given a set $X$ of $n$ points in a metric space and a positive integer $k\in [n]$. Our goal is to partition $X$ into $k$ subsets (the clusters) so as to minimize the sum of the radii of these clusters. The Min-Sum Diameters problem (MSD) is defined analogously, where instead of the radii of the clusters we consider their diameters. For both problems we present FPT approximation schemes for the natural parameter $k$. Specifically, given $ε>0$, we show how to compute $(1+ε)$-approximations for both MSD and MSR in time $(1/ε)^kn^{O(1)}$ and $(1/ε)^{O(k/ε\log 1/ε)}n^{poly(1/ε)}$ respectively. The previous best FPT approximation algorithms for these problems have approximation factors $4+ε$ and $2+ε$, respectively, and finding an FPT approximation scheme for both these problems had been outstanding open problems.

Shearing motion and magnetic flux cancellation around the polarity inversion line (PIL) play significant roles in the build-up of free magnetic energy and magnetic flux rope (MFR) in source region of major solar flares. Here we investigate the magnetic evolution of a highly-sheared PIL in active region (AR) 13842, hosting the largest X9.0 flare of Solar Cycle 25. Since 2024 September 29, a positive-polarity pore persistently drifted northward along the western side of the AR's main negative-polarity sunspot. The main sunspot remained stationary until negative-polarity patches successively emerged to its east and approached. Rear-ended by these same-polarity patches, the sunspot then began moving westward toward the opposite-polarity pore around October 1, forming a collisional PIL. Meanwhile, on the PIL's other side, the pore was also rear-ended by same-polarity patches sequentially emerging behind it, accelerating the shearing motion around the PIL, where frequent flux cancellations were also observed. Synchronous rapid accumulation of free magnetic energy and formation of MFR were then observed in the PIL, where multiple major flares successively occurred within two days. Before these large flares, the area and total free energy of the high-free-energy-density PIL region gradually decreased in the photosphere, which could be caused by the initial ascent of MFR before eruption and serve as a precursor of solar eruptions. These results suggest that persistent flux emergences with cross separation directions facilitates rapid formation of collisional shearing PIL and frequent flux cancellations, leading to repeated MFR formations and multiple large flares in a relatively short time.

Solar filament/prominence eruptions can significantly perturb geospace when originating from favorable source locations and directions. While stellar analogs have been recently reported, the disk locations and magnetic environments of their source regions remain spatially unresolved on other stars. To bridge this gap, we investigate the typical Sun-as-a-star H$α$ temporal spectral characteristics of solar filament/prominence eruptions with different source region locations (on-disk vs. limb, active region vs. quiet-Sun region). It is revealed that limb eruptions are characterized by blueshifted/redshifted emission caused by the bright off-limb erupting structures, whereas on-disk eruptions may show blueshifted absorptions due to the dark erupting filaments. Among the limb eruptions, front-side limb eruptions usually display line center emission before the blueshifted/redshifted emission, while far-side limb eruptions show the opposite sequence. Moreover, the magnetic environment at source also shapes the spectral characteristics. On-disk filament eruptions from active region exhibit much more intense flare-ribbon-dominated line center emission features compared with those from quiet-Sun region. Limb active region eruptions often show single-wing emissions, whereas large-scale quiet-Sun region (quiescent) prominence eruptions frequently display expansion-induced emission in both wings followed by line center absorption due to the disappearance of bright prominence. These distinct Sun-as-a-star H$α$ spectral characteristics, dependent on eruption location, provide a diagnostic basis for inferring source regions of stellar filament/prominence eruptions from spatially unresolved H$α$ spectra.

Recent progress in automated repair of performance bugs demands realistic, executable benchmarks. However, existing C++ performance benchmarks are largely built from competitive programming submissions, and recent real-world benchmarks predominantly target Python and .NET. To fill this gap, we present CppPerf-Mine, a configurable pipeline that mines execution-time-improving patches from open-source C++ repositories on GitHub by combining structural commit filtering, an LLM-based commit classifier, and a containerized build & test stage that produces fully reproducible Docker images for each patch. Using CppPerf-Mine, we build CppPerf-DB, a benchmark comprising 347 manually verified patches from 42 mature C++ repositories, 39% of which are multi-file, enabling the evaluation of repository-level repair tools. In our preliminary study, OpenHands correctly fixes only 13.5% of the patches in CppPerf-DB, confirming that real-world C++ performance repair remains an open challenge. CppPerf-Mine and CppPerf-DB are open-source and publicly available at: https://doi.org/10.5281/zenodo.20097425. In addition, a demonstration video is available at: https://www.youtube.com/watch?v=nixlupIgSdM.

We consider nonlinear functionals of discrete Gaussian free fields with ergodic random conductances on a class of random subgraphs of $\mathbb{Z}^{2}$, including i.i.d. supercritical percolation clusters, where the conductances are possibly unbounded but satisfy an integrability condition. As our main result, we show that, for almost every realisation of the environment, the nonlinear functionals of the rescaled field converge to their continuum counterparts in the Sobolev space $H^{-s}(D)$ for suitable $s > 0$. To obtain the latter, we establish pointwise bounds for the Green's function of the associated random walk among random conductances with Dirichlet boundary conditions, which are valid for all $d \geq 2$.

The strong decay $ψ(4040)\to D\bar D$ is anomalously suppressed despite ample phase space, whereas the $D\bar D^*$ and $D_s\bar D_s$ channels remain sizable. In this work, we study this suppression and the associated open-charm hierarchy in the framework of the instantaneous Bethe--Salpeter equation combined with the relativistic $^3P_0$ model, with the pair-creation strength fixed independently from $ψ(3770)\to D\bar D$. Within this framework, we show that the suppressed $D\bar D$ mode can be understood as a consequence of node-induced cancellations in the relativistic decay amplitude. The $D\bar D$ amplitude is strongly reduced because the corresponding overlap integral receives comparable positive and negative contributions from different momentum regions, whereas the $D\bar D^*$ and $D_s\bar D_s$ channels do not undergo the same strong cancellation. This interpretation is further supported by the pronounced sensitivity of the $D\bar D$ width to the initial mass, the charged-neutral $D$-meson mass splitting, and the dip structure in the mass dependence of the partial width. Our results provide a dynamical explanation of the suppressed $D\bar D$ mode and the core open-charm hierarchy of $ψ(4040)$ within a conventional $3\,{}^3S_1$ charmonium picture, while the precise value of the near-vanishing $D\bar D$ width remains model dependent.

Distance measurements are fundamental to radio pulsars' use as astrophysical probes of General Relativity and the interstellar medium. One of the primary methods for determining pulsar distances is HI kinematics, which leverages the radial velocities of HI absorption and emission features detected along pulsar lines-of-sight. This method necessarily assumes a model for Galactic rotation, our knowledge of which continues to evolve in both accuracy and precision. In this research note, we derive kinematic distances for 66 pulsars with archival HI radial velocity measurements using a state-of-the-art Galactic rotation curve. The results and software are provided in an online repository. Our kinematic distances differ by $<1σ$ from published parallaxes for nearly all pulsars in the sample that have both types of distance measurement available. Comparison to the NE2025 Galactic electron density model shows general consistency between measured and predicted distances.

We reformulate the definition of privacy in the private information retrieval (PIR) problem to accommodate flexible privacy requirements. We focus on graph-replicated PIR, with a generalized privacy requirement, instead of requiring all messages to be private from all servers, during retrieval. Towards this, we define a privacy requirement set for each server, which can be an arbitrary subset of all message indices, as long as the stored message indices are in their privacy requirement set. Since both the storage and privacy requirement sets have many possibilities, we focus on two specific storage settings, namely the path and cyclic graphs. We consider several privacy settings for each of them, which are not necessarily the same, to give different examples for privacy sets. Of particular interest are the privacy sets that comprise the indices of messages stored at servers within a neighborhood range. The neighborhood range parameter allows a transition from the recently introduced local PIR [1] to the standard graph-replicated PIR. In these cases, we derive bounds on the capacity or find the exact capacity.

We rethink the definition of privacy in multi-server, graph-replicated private information retrieval (PIR) systems, and introduce a novel setting where the user's privacy is governed by the servers' storage structure. In particular, while retrieving a message from a server, the user is concerned with hiding their desired message index from the server, only if the server stores the corresponding message. We coin this privacy requirement as local user privacy and the resulting PIR problem as local PIR on the graph. Our goal is to measure the gain in communication efficiency of local PIR, compared to that of canonical PIR, by establishing its capacity, i.e., the maximum number of message symbols retrieved, per downloaded symbol. To this end, we observe a remarkable gain in the local PIR capacity of graphs, that are disjoint union of distinct graphs, which is multiplicative, compared to the PIR capacity, when the individual graphs are identical. For connected graphs, we propose schemes to establish capacity lower bounds for edge-transitive and bipartite graphs, which are greater than the best-known PIR capacity bounds. Finally, we derive the exact local PIR capacity for the cyclic graph, and the path graph with an odd number of vertices.

A microscopic framework that combines the Hartree-Fock-Bogoliubov (HFB) approach with the fusion by diffusion (FBD) model is proposed to investigate the synthesis mechanism of superheavy nuclei (SHN). For the reaction $^{48}\text{Ca}+^{208}\text{Pb}$, the calculated evaporation-residue cross section (ERCS) reproduces the experimental data reasonably well. The method enables self-consistent extraction of the fusion injection point and inner barrier from HFB potential-energy surfaces (PES), thereby incorporating nuclear structure effects while eliminating phenomenological tuning at the fusion stage. For cold-fusion reactions, the PES features a hyperasymmetric valley driven by shell effects. This $^{208}$Pb anchored valley connects the entrance channel to compound nucleus formation and provides an exit channel for cluster decay. We further investigate the cold-fusion reactions $^{54}\text{Cr}+^{208}\text{Pb}$ and $^{58}\text{Fe}+^{208}\text{Pb}$, obtaining a near-exponential decrease of $P_{\text{CN}}$ with compound-nucleus charge $Z$, consistent with established systematics. This approach demonstrates a self-consistent framework that can reduce uncertainties in the fusion stage of SHN production.

The Law of Universal Gravitation is part of middle and high school's general physics and astronomy curricula. This topic is included in the most popular physics textbooks available as a fact whose origin remains in the detailed work of Sir Isaac Newton 300 years ago. Consequently, its mathematical form is presented as an equation without any deductive process. Nevertheless, deduction of the mathematical form of this law is an opportunity to discuss how a deductive process can be performed using the data available on the Internet from reliable sources.

We study tridimensional tensors on the complex field from the point of view of hypermatrices, taking into consideration the problem of determining whether they are degenerate or not, concise or not, what is their essential format if they are non-coincise, and, in some cases, their tensor rank. We use a geometrical approach to these problems which, in part, goes back to Schläfli and consists in studying certain determinantal schemes associated to the hypermatrix.

Chromatic choosability is a notion of fundamental importance in list coloring. A graph $G$ is chromatic-choosable when its chromatic number, $χ(G)$, is equal to its list chromatic number $χ_{\ell}(G)$. In 1990, Kostochka and Sidorenko introduced the list color function of a graph $G$, denoted $P_{\ell}(G,m)$, which is the list analogue of the chromatic polynomial of $G$, $P(G,m)$. A graph $G$ is said to be enumeratively chromatic-choosable when $P_{\ell}(G,m)=P(G,m)$ for every $m \in \mathbb{N}$. Theta graphs and their generalizations have played an important role in graph coloring problems over the years; for example, they appear in the characterization of chromatic-choosable graphs with chromatic number 2. In this paper we characterize the enumeratively chromatic-choosable theta graphs. Our proof utilizes ideas from DP-coloring (a.k.a. correspondence coloring), providing yet another example of how the more general setting of DP-coloring can be leveraged to attack a problem in list coloring.

Modified gravity theories such as Modified Newtonian Dynamics (MOND) and Scalar-Tensor-Vector Gravity (STVG) have been proposed as alternatives to dark matter, but decisive tests have been hindered by degeneracies between baryonic structure and gravitational laws. Here we break this degeneracy using independent, high-precision constraints: the Milky Way radial rotation curve, vertical phase-space spirals from Gaia, and a broken-exponential stellar disk. A joint reconstruction of the radial and vertical gravitational fields reveals a structural inconsistency in modified gravity -- no model can simultaneously reproduce both observations. Our results strongly disfavor MOND at $>13σ$ and STVG at $>4σ$. In contrast, dark matter halo models naturally explain the observations, providing a self-consistent test of gravity on galactic scales.

In discrete-variable black-box optimization, the number of candidate solutions grows combinatorially, while each evaluation is often expensive. Therefore, it is important to identify promising solutions efficiently within a limited number of trials. Bayesian Optimization of Combinatorial Structures (BOCS), an existing parametric method, works effectively when only a small amount of data is available. However, as the number of observations increases, BOCS tends to repeatedly propose points that have already been evaluated, which leads to search stagnation. A random-point addition strategy has been proposed to address this issue when an evaluated point is proposed, but it cannot sufficiently exploit information from promising data obtained so far. In this study, we propose a hybrid method that uses BOCS as the main search framework and generates alternative unevaluated points using a Gaussian process only when search stagnation is detected. In the Gaussian-process-based component, multiple Lower Confidence Bound (LCB) acquisition functions are adaptively selected to dynamically control the balance between exploitation and exploration. Numerical experiments using fully connected Quadratic Unconstrained Binary Optimization (QUBO) and Higher-order Unconstrained Binary Optimization (HUBO) as black-box functions show that the proposed method finds solutions with better objective values than the conventional random-point addition method in both settings. Additional analyses show that its effectiveness comes from selecting points that promote search progress within Hamming-distance neighborhoods, rather than simply adding low-energy points near promising solutions. Experiments with sparse surrogate models for quantum annealer applications further suggest the importance of retaining near-fully connected representational capacity.

This article presents an arithmetic, called superposition relaxation, for bracketing the graph of a multivariate factorable function on a compact domain between a pair of underestimating and overestimating functions that are both separable. Propagation rules are established for affine and nonlinear composition operations, with a focus on exploiting global monotonicity and convexity properties in the composition. The local convergence properties of this arithmetic are also analyzed in both the pointwise and Hausdorff sense, including conditions under which quadratic pointwise convergence propagates through composition. Parameterizations of the univariate summands in a superposition relaxation either as piecewise-constant or continuous piecewise-linear functions are discussed for a practical implementation. It is shown through numerical case studies that superposition relaxations can be consistently tighter than McCormick relaxations, including for the relaxation of artificial neural networks. But superposition relaxations also incur a higher computational cost than McCormick relaxations. Further investigations are thus warranted as applications in global optimization seek to balance a relaxation's tightness with its computational cost.

This paper resolves the open larger-alphabet quotient case in the accepting-state complexity theory of permutation automata. Rauch and Holzer showed that, in the unary setting, the attainable right-quotient accepting-state complexities are exactly $[1,mn]$. We prove that over arbitrary alphabets the exact spectrum is $g^{\operatorname{asc}}_{-1,\mathrm{PFA}}(m,n)=\{0\}$ if $m=0$ or $n=0$, and $g^{\operatorname{asc}}_{-1,\mathrm{PFA}}(m,n)=\mathbb{N}_{>0}$ if $m,n\ge 1$. Thus, once both input languages are nonempty, every positive accepting-state complexity is attainable for right quotient, and $0$ is the only unavoidable magic value. The proof has two parts. First, we show that if $m,n\ge 1$, then the quotient language $KL^{-1}$ cannot be empty when $K$ and $L$ are accepted by permutation automata with $\operatorname{asc}(K)=m$ and $\operatorname{asc}(L)=n$; this follows from the bijectivity of the transition action. Second, for every $m,n\ge 1$ and every $α\ge m$, we construct a ternary witness pair $(A^{\mathrm{q}}_{m,α},B^{\mathrm{q}}_{n,α})$ such that $\operatorname{asc}(L(A^{\mathrm{q}}_{m,α}))=m$, $\operatorname{asc}(L(B^{\mathrm{q}}_{n,α}))=n$, and $\operatorname{asc}(L(A^{\mathrm{q}}_{m,α})L(B^{\mathrm{q}}_{n,α})^{-1})=α$. The high-range construction is group-theoretic: the words accepted by $B^{\mathrm{q}}_{n,α}$ induce exactly a point stabilizer in a symmetric group, and the standard quotient construction then saturates the original final set of $A^{\mathrm{q}}_{m,α}$ to a full orbit, yielding a minimal quotient automaton with exactly $α$ final states. Combined with the known unary interval $[1,mn]$, this yields the complete spectrum and resolves the larger-alphabet right-quotient case for permutation automata.

We present exact solutions for the non-equilibrium steady states of a class of dissipative spinless fermionic systems with arbitrary Hamiltonian pairing terms, global charging energy interactions, and uniform single particle loss on every site. Our exact solution is found by generalizing the coherent quantum absorber technique to fermionic systems, and our result establishes the existence of hidden time-reversal symmetry in driven-dissipative fermionic models. The steady state exhibits a first order phase transition in the particle density, with the resulting jump discontinuity in density persisting even for finite dissipation rates. A mean-field description of the model exhibits a bistable regime that encompasses the first-order transition line yet which fails to accurately predict its precise location via a Maxwell construction. We also show that the model's hidden time-reversal symmetry results in an Onsager symmetry of certain two-time correlation functions.

Analog quantum computation offers a route to machine learning using controllable physical dynamics as a computational resource. However, many existing approaches rely on task-specific protocols or observables that are difficult to access experimentally, limiting generality and implementation. Here we introduce Qlustering, an unsupervised clustering framework based on steady-state quantum transport in quantum networks governed by the GKSL master equation, developed through algorithm-hardware co-design. Data are encoded as input states, and cluster assignments are inferred from steady-state output currents, avoiding full state tomography in favor of accessible transport observables. The method realizes a hybrid classical-quantum workflow in which data preparation and training are performed classically, while clustering is carried out by transport dynamics. We benchmark the method on synthetic datasets, localization, and QM9 and Iris, finding competitive performance and stability over a broad range of dephasing strengths. These results show that unlabeled data structure can be extracted directly from steady-state transport observables, identifying terminal-current readout as a native, tomography-free mechanism for unsupervised learning in open quantum networks.

We construct moment functions that are Neyman-orthogonal to a chosen order in parametric moment condition models. These moment functions reduce sensitivity to nuisance estimation error and, as such, offer a unified and tractable route to higher-order debiasing in a wide range of econometric models. The number of additional nuisance parameters required by our construction, beyond those already present in the original moment conditions, is independent of the order of orthogonalization and can be reduced to a single scalar if desired.

This paper studies algorithmic meta theorems for property testing with \emph{constant running time} in the bounded degree model. In (Adler, Harwath 2018) it was shown that on graph classes $\mathcal C^{w}_d$ consisting of all graphs with both degree at most $d$ and treewidth at most $w$, every problem expressible in monadic second-order logic with counting (CMSO) is testable with \emph{polylogarithmic} running time (where $d,w\in \mathbb N$ are fixed). It was left open whether this can be improved to \emph{constant} running time. In this paper we give a positive answer for testing CMSO on classes $\mathcal C^{c}_d$, where $d$ bounds the degree and $c$ bounds the component size. Our main result shows constant time testability of first-order logic with modulo counting (FOMOD) on $\mathcal C^{c}_d$. For our proof we tailor Hanf normal form of FOMOD to our setting, and we exhibit a number-theoretic `patchability' condition that allows to infer global information on the input graph from a local sample of constant size. We believe that our `patchability' might be of independent interest. The step from FOMOD to CMSO then follows from a result by (Eickmeyer, Elberfeld, Harwath, 2017) on the expressive power of order invariant monadic second-order logic on classes of bounded treedepth.

Topologically-ordered quantum states with Abelian excitations can host defects that obey effective non-Abelian statistics, in principle allowing for quantum information processing via defect braiding. These extrinsic defects (or twists) are typically studied as static features of the lattice. However, an alternative proposal considers how an underlying topologically ordered quantum substrate can be locally perturbed to create and manipulate synthetic defects \cite{you_synthetic_2013}. Unfortunately, while largely referenced, elements of this proposal were never systematically studied. Understanding the energy spectrum is particularly important in finite size and finitely perturbed systems, which are crucial for experimental realizations. In this work we announce a significant step in this direction by explicitly constructing, simplifying, and numerically studying the spectral properties of synthetic defects in a model system. First, we introduce two alternative representations of this problem in both spin and Majorana languages. In the former we describe emergent virtual symmetries which constrain and simplify the problem and in the latter we show a direct connection to Kitaev's well-known Majorana chain. We utilize these simplifications to perform numerical calculations to indicate the location of the quantum phase transition driving the emergence of the synthetic defects. We conclude by discussing key steps for future work to more clearly and completely study this phenomena.

Null points in synthetically tunable molecular aggregates are predicted to generate flat energy bands analogous to those known in strongly correlated condensed-matter physics. For chemistry, null points provide a powerful design principle for photovoltaic materials with selective charge filtering similar to photosynthesis. However, null points have never been experimentally verified because their defining prediction - state localization with selective electron or hole transfer - has remained unobserved. Here, using a donor-acceptor dyad as a minimal model, we provide the first experimental observation of a null point. Impulsive pump-probe measurements reveal charge separation through a near-instantaneously generated locally excited-charge transfer (LE-CT) intermediate that emerges upon solvent stabilization of CT states. Polarization anisotropy directly reveals state localization and selective charge-filtering, spanning balanced electron-hole transfer to selective hole filtering consistent with synthetic design. A generalized vibronic theory of null points explains these observations and identifies the ideal synthetic parameters for achieving null points which are protected from the vibrational bath.

For an $n$-vertex graph $G$, let $z(G;k)$ denote the number of zero forcing sets of size $k$. A conjecture of Boyer et al. asserts that the path $P_n$ maximizes these numbers coefficientwise among all $n$-vertex graphs; equivalently, the zero forcing polynomial of every $n$-vertex graph should be coefficientwise dominated by that of $P_n$. We prove this path-extremal conjecture for distance-hereditary graphs. This extends the previously known tree case to a much larger class that includes, in particular, all trees and all cographs. We then use canonical split decomposition to push the argument one step beyond the distance-hereditary setting. Specifically, we show that if a split-prime graph $H$ and all of its induced subgraphs are path-extremal, then every connected graph whose canonical split decomposition has a unique prime bag whose label graph is isomorphic to $H$ is also path-extremal. As a corollary, for each fixed $m$, if every induced subgraph of every split-prime graph on at most $m$ vertices is path-extremal, then so is every connected graph whose canonical split decomposition has a unique prime bag of size at most $m$. Thus, on these classes, the conjecture reduces to a finite verification problem on bounded-order prime cores. Our proofs combine two counting mechanisms for non-forcing sets -- fort obstructions arising from twin pairs and a leaf recurrence -- with the accessibility description of graph-labelled trees in the canonical split decomposition. This yields a new positive instance of the path-extremal conjecture and identifies a natural structural frontier for further progress.

We study the status of preservation theorems such as the Łoś-Tarski theorem and the homomorphism preservation theorem in the context of semiring semantics. Semiring semantics has its origins in the provenance analysis of database queries. Depending on the underlying semiring, it allows us to track which atomic facts are responsible for the truth of a statement or practical information about the evaluation such as costs or confidence. The systematic development of semiring semantics for first-order logic and other logical systems raises the question to what extent classical model-theoretic results can be generalised to this setting and how such results depend on the underlying semiring. The definitions of semantic properties such as preservation under extensions, substructures, or homomorphisms naturally generalise to the setting of semiring semantics. However, the status of the corresponding preservation theorem strongly depends on the algebraic properties of the particular semirings. We prove that these preservation theorems do indeed hold for all lattice semirings (a quite large class, encompassing practically relevant semirings and in particular all min-max semirings). The proofs combine adaptations of the classical compactness and amalgamation methods with specific reduction methods for logical entailment that have been developed in semiring semantics. On the other side, variants of the existential preservation theorem fail for many other semirings, including the tropical semiring, the Viterbi semiring, the Łukasiewicz semiring, and the natural semiring. Surprisingly, the existential preservation theorem does hold for finite interpretations in a number of semirings, including all lattice semirings. Thus, the situation for these semirings is in sharp contrast to the Boolean case, where the Łoś-Tarski theorem holds in general, but not in the finite.

At the center of our Galaxy lies the bright γ-ray point-like source HESS J1745-290, which is compatible in position with Sgr A star, although an association between the two remains uncertain. Using data obtained between 2004 and 2019 with the High Energy Stereoscopic System (H.E.S.S.) on the Galactic center region, we studied the variability of HESS J1745-290 over 353 hours of observations collected over 16 years, representing the largest dataset gathered yet on this region at TeV energies. We performed a 3D maximum-likelihood analysis of the central source and the diffuse γ-ray emission in the Galactic center region. This analysis allowed us to extract the spectral and morphological intrinsic behavior of the two components. By performing this analysis on an annual basis, we derived the light curve of HESS J1745-290 and the diffuse emission over the past 16 years. The 3D maximum-likelihood analysis method allowed us to separate the central source from the overlapping diffuse emission, enabling a recalibration of the former by the latter and alleviating some of the systematic effects. We find no long-term or yearly variability. We also provide an estimate of the sensitivity of H.E.S.S. to variation of this specific source over 16 years. We rule out any yearly gamma-ray flux variation of this source larger than 30 percent, as well as any linear flux variation exceeding 30% over this time period.